the bonded electrical resistance strain gage an introduction 019507209x

The Bonded Electrica l Resistanc eStrain Gag e

This page intentionally left blank

The Bonde dElectrical Resistanc e

Strain Gag eAn Introduction

WILLIAM M . MURRA YProfessor Emeritus

Massachusetts Institute of Technology

WILLIAM R . MILLE RProfessor Emeritus

The University of Toledo

New Yor k Oxfor dOXFORD UNIVERSIT Y PRES S

1992

Oxford Universit y Pres s

Oxford Ne w Yor k Toront oDelhi Bomba y Calcutt a Madra s Karach i

Kuala Lumpu r Singapor e Hon g Kon g Toky oNairobi Da r e s Salaam Cap e Tow n

Melbourne Aucklan d

and associate d companie s i nBerlin Ibada n

Copyright ; 199 2 b y Oxfor d Universit y Press , Inc .

Published b y Oxfor d Universit y Press , Inc. ,200 Madiso n Avenue . New Yor k 1001 6

Oxford i s a registere d trademar k o f Oxfor d Universit y Press

All right s reserved . N o par t o f this publicatio n ma y b e reproduced ,stored i n a retrieva l system , o r transmitted , in an y for m o r b y an y means ,

electronic, mechanical , photocopying, recording , o r otherwise .without th e prio r permissio n o f Oxford Universit y Press.

Library o f Congres s Cataloging-in-Publicatio n Dat aMurray , Willia m M .

The bonde d electrica l resistanc e strai n gag e :an introductio n / b y Willia m M. Murra y an d Willia m R . Miller.

p. cm . Include s bibliographica l reference s an d index .ISBN 0-19-507209- X

1. Strai n gages . 2 , Electri c resistanc e Measurement .I. Miller . Willia m R . (Willia m Ralph) , 1917 - . II . Title .

TA413.5.M87 199 2 624.1'76'028 7 dc2 0 91-4136 9

2 4 6 8 9 7 5 3 1

Printed i n th e Unite d State s o f Americ aon acid-fre e pape r

PREFACE

Experimental stres s analysi s i s an importan t too l i n th e overal l desig n an ddevelopment o f machinery an d structures . While analytica l technique s an dcomputer solution s ar e available during th e design stage, the results are stil ldependent o n many assumption s tha t mus t be made i n order t o adap t the mto th e problem s a t hand . Onl y whe n th e desig n i s fixed, the prototype s ar econstructed, and testin g is underway, can th e proble m area s b e realisticallydetermined, and thi s must b e done throug h experimenta l means .

One metho d o f findin g th e weaknesses , an d a metho d whic h i s use dextensively, i s through th e us e o f the electrica l resistanc e strai n gage . Strai ngages ar e relativel y lo w i n cost , easil y applie d b y a reasonabl y skille dtechnician, d o no t requir e extensiv e investment i n instrumentatio n (fo r th egeneral user) , and ye t they yield a wealth o f information in a relatively shorttime. The information and it s validity is, of course, dependent o n the trainin gand knowledg e o f th e enginee r wh o plan s th e test s an d reduce s th e data .The latter statemen t become s painfull y apparen t whe n one finds a user tryingto interpre t dat a fro m a singl e strai n gag e applie d i n a n unknow n biaxia lstress field.

In 1988 , th e author s decide d t o edi t Dr . Murray' s notes , whic h wer edeveloped ove r hi s extensiv e career , an d t o writ e a n introductor y tex t o nelectrical resistanc e strai n gages. Th e tex t is directed a t senio r an d first-yea rgraduate student s i n th e engineerin g disciplines , althoug h student s fro mother field s (geology , engineering physics , etc. ) wil l als o benefit .

The prerequisite s fo r a strai n gag e cours e ar e th e following : (1 ) Th ebasic courses in resistance o f materials. (2) An elementary course in electricalcircuits. (3) At least one course in mechanical or structural design is desirable.It follow s tha t the more experienc e student s have in analysis and design , themore the y wil l benefi t fro m a n experimenta l course . I t i s in th e laborator yand i n experimental courses tha t student s reall y develop a sens e o f securityin, an d a bette r understandin g of , the theor y the y have bee n expose d t o i ntheir analytica l studies .

The development o f stress an d strai n transformatio n equations an d th ecorresponding Mohr' s circles , a s wel l a s the stress-strai n relationships , ar ecovered in Chapter 2. Depending o n the student's preparation , th e instructo rmay us e this chapter fo r a rapid revie w or eliminate i t entirely. The authors ,however, hav e foun d i t beneficia l t o spen d a t leas t severa l period s o n th ematerial.

Basic electrica l circuit s ar e examine d i n Chapter s 3 throug h 5 . A nelementary circui t consisting of a single strain gage and its response t o strai nis first considered, followe d b y the potentiometric circui t and the Wheatston e

vi PREFAC E

bridge. In the development of the expressions for output voltage, as the straingage's resistanc e change s wit h increasin g loading , i s th e effec t o f circui tnonlinearity. Th e equation s ar e develope d s o tha t th e studen t ca n easil yhandle the intervening algebra between steps and thereb y see the nonlinearityterms unfold . I t i s importan t tha t student s recogniz e thi s an d understand ,when recordin g larg e strains , how t o correc t th e indicated strain s to obtai nthe actua l strains . Th e effec t o f resistanc e i n bot h th e powe r suppl y an dindicating mete r i s also accounte d for .

Lead-line resistance is considered i n the Wheatstone bridg e circuits. Thecircuits ar e th e ful l bridge , th e hal f bridg e wit h fou r wires , th e hal f bridg ewith three wires, the quarter bridg e wit h three wires , and th e quarte r bridg ewith tw o wires . The equation s ar e developed s o tha t th e nonlinearit y effect sare apparent .

Sensitivity variation in order to obtain a desired output is next discussedin Chapter 6 . Equations ar e developed , including nonlinearity effects, fo r th edesensitization o f single gages , half-bridg e circuits, and full-bridg e circuits.

Chapter 7 is devoted t o th e lateral , or transverse , effect o n strai n gages ,along wit h a discussio n o f th e method s use d t o determin e th e gag e facto rand th e transvers e sensitivit y factor o f strai n gages . Thi s i s followe d b yChapters 8 and 9 o n strai n gag e rosette s an d dat a reduction . I t i s shownhow t o reduc e rosett e dat a b y bot h analytica l method s an d graphica lmethods. This is followed b y considering transverse effects, usin g informationfrom Chapte r 7 , in rosett e dat a reduction.

Chapter 1 0 discusses ho w strai n gage s ma y b e use d t o measur e bot hnormal stresse s an d shearin g stresse s directly , while Chapte r 1 1 consider sthe effec t o f temperatur e o n strai n gag e readings . Temperature-induce dstrains ar e discussed , followe d b y a n examinatio n o f self-temperature -compensated gage s an d thei r therma l outpu t curve s whe n th e gage s ar ebonded t o severa l differen t materials . On e ca n se e ho w t o correc t th eindicated strain not onl y for the temperature-induced strain , but als o fo r thegage facto r variatio n resultin g from temperatur e change .

Several type s o f strain-gag e transducer s ar e covere d i n Chapte r 12 .Among them ar e th e axial-force load cell , the torque meter , the shear meter ,and th e pressur e transducer . Th e purpos e i s t o introduc e th e studen t t oseveral type s o f transducer s tha t coul d b e mad e an d calibrate d fo r hi s us ein th e laboratory .

At the time of Dr. Murray' s death on August 14, 1990, the major portio nof th e manuscrip t ha d bee n completed . I f there are error s o r discrepancies ,the faul t i s not hi s bu t mine . In completin g th e text , I gathered togethe r al lof th e sourc e materia l i n orde r t o giv e proper credit ; I sincerel y hope non ehas bee n overlooked .

A textboo k i s not th e wor k o f one o r severa l people alone . Al l of us ar einfluenced no t onl y by our contemporarie s bu t b y those wh o hav e precede dus (one has only to thin k of Professor Ott o Moh r t o realiz e this). Therefore,I want to acknowledg e our deb t t o al l of these people, no t th e least o f whom

PREFACE vi i

were our students . I wan t especially t o than k Marth a Watso n Spaldin g ofMeasurements Group, Inc. fo r her cooperation in furnishing a considerableamount o f material . I als o wan t t o acknowledg e th e assistanc e o f th efollowing companies: BLH Electronics, Inc.; Eaton Corporation, TransducerProducts; Electri x Industries , Inc. ; Hartru n Corporation ; Measurement sGroup, Inc. ; Stein Engineering Services, Inc.; and Texa s Measurements, Inc.

W. R . Miller

CONTENTS

1. Fundamenta l Concept s fo r Strai n Gages , 31.1 Introduction, 31.2 Characteristics Desired in a Strain Gage, 41.3 General Considerations, 51.4 Analysis of Strain Sensitivity in Metals, 141.5 Wire Strain Gages, 241.6 Foil Strain Gages, 291.7 Semiconductor Gages, 321.8 Some Other Types of Gages, 331.9 Brittle Lacquer Coatings, 36

2. Stress-Strai n Analysi s and Stress-Strai n Relations , 422.1 Introduction, 422.2 Basic Concepts of Stress, 432.3 Biaxial Stresses, 452.4 Mohr's Circle for Stress, 542.5 Basic Concepts of Strain, 612.6 Plane Strain, 622.7 Mohr's Circle for Strain, 682.8 Stress-Strain Relationships, 722.9 Application of Equations, 772.10 Stress and Strain Invariants, 81

3. Elementar y Circuits , 903.1 Introduction, 903.2 Constant- Voltage Circuit, 913.3 Constant-Current Circuit, 943.4 Advantages of the Constant-Current Circuit, 963.5 Fundamental Laws of Measurement, 97

x CONTENT S

4. Th e Potentiometri c Circuit, 1004.1 Introduction, 1004.2 Circuit Equations, 1014.3 Analysis of the Circuit. 1064.4 Linearity Considerations, 1194.5 Temperature Effects, 1294.6 Calibration, 141

5. Wheatston e Bridge , 1465.1 Introduction, 1465.2 Elementary Bridge Equations, 1495.3 Derivation of Elementary Bridge Equations, 1575.4 General Bridge Equations, 1725.5 Effect o f Lead-Line Resistance, 18 05.6 Circuit Calibration, 1935.7 Comments, 195

6. Sensitivit y Variation , 2056.1 Introduction, 2056.2 Analysis of Single Gage Desensitization, 2076.3 Analysis of Half-Bridge Desensitization, 2186.4 Analysis of Full-Bridge Sensitivity Variation, 227

1. Latera l Effect s i n Strai n Gages , 23 47.1 Significance of Strain Sensitivity and Gage Factor, 2347.2 Basic Equations for Unit Change in Resistance, 2367.3 Determination of Gage Factor and Transverse

Sensitivity Factor, 2427.4 Use of Strain Gages Under Conditions Differing from those

Corresponding to Calibration, 2467.5 Indication from a Pair of Like Strain Gages Crossed at

Right Angles, 248

8. Strai n Gage Rosette s and Dat a Analysis , 2538.1 Reason for Rosette Analysis, 2538.2 Stress Fields, 2538.3 Rosette Geometry, 2568.4 Analytical Solution for the Rectangular Rosette, 258

CONTENTS

8.5 Analytical Solution for the Equiangular or Delta Rosette, 2678.6 Rosettes with Four Strain Observations, 2758.7 Graphical Solutions, 281

9. Strai n Gag e Rosette s an d Transvers e Sensitivit y Effect , 29 19.1 Introduction, 2919.2 Two Identical Orthogonal Gages, 2919.3 Two Different Orthogonal Gages, 2949.4 Three-Element Rectangular Rosette, 2969.5 The Equiangular or Delta Rosette, 301

10. Stres s Gages , 31070.7 Introduction, 31 010.2 The Normal Stress Gage, 31010.3 The SR-4 Stress-Strain Gage, 31610.4 Electrical Circuit for Two Ordinary Gages to Indicate

Normal Stress, 32010.5 The V-Type Stress Gage, 32110.6 Application of a Single Strain Gage to Indicate

Principal Stress, 32610.7 Determination of Plane Shearing Stress, 327

11. Temperatur e Effect s o n Strai n Gages , 33711.1 Introduction, 33711.2 Basic Considerations of Temperature-Induced Strain, 33711.3 Self-Temperature-Compensated Strain Gages, 34311.4 Strain Gage-Test Material Mismatch, 34911.5 Compensating Gage, 353

12. Transducers , 36 072.7 Introduction, 36 012.2 Axial-Force Transducers, 36312.3 Simple Cantilever Beam, 36812.4 Bending Beam Load Cells, 37212.5 Shear Beam Load Cell, 37512.6 The Torque Meter, 37812.7 The Strain Gage Torque Wrench, 38012.8 Pressure Measurement, 382

xi

xii CONTENT S

13. Strai n Gag e Selectio n and Application , 39013.1 General Considerations, 39013.2 Strain Gage Alloys. 39113.3 Grid Backing Materials, 39313.4 Gage Length, Geometry, and Resistance, 39413.5 Adhesives , 39 613.6 Bonding a Strain Gage to a Specimen, 398

Answers t o Selecte d Problems, 402

Index, 405

The Bonded Electrical ResistanceStrain Gage

1FUNDAMENTAL CONCEPTS FOR STRAIN GAGES

1.1. Introduction

The constan t deman d fo r improvemen t i n th e desig n o f machin e an dstructural part s ha s le d t o th e developmen t o f various experimental techni-ques fo r determinin g stres s distributions . These experimenta l method s ar eemployed for both the checking of theoretical predictions, and the evaluationof stresse s i n situation s wher e mathematica l approache s ar e unavailabl e o runsuited.

However, sinc e stres s canno t b e measure d directly , th e experimenta lprocedures, o f necessity, make thei r approach throug h som e typ e o f strainmeasurement. The measured strain s are then converted into their equivalentvalues i n term s o f stress . I n orde r t o achiev e thi s ultimat e objective , som etype o f strain-indicating device o r measurin g device i s required.

In additio n t o thei r use s fo r stres s analysis , strai n gage s als o fin dwide applicatio n i n sensin g device s an d contro l devices . I n thes e applica -tions, th e strai n i n som e mechanica l par t i s use d a s a n indicatio n o f force ,bending, torque , pressure , acceleration , o r som e othe r quantit y relate d t ostrain.

Even th e mos t casua l surve y of the literatur e relatin g t o th e measure -ment of mechanical strain wil l yield information on a wide variety of deviceswhich have been developed fo r this purpose . I n addition to photoelasticity ,brittle lacquer (1 , 2, 3),1 and X-rays , one finds all sorts o f mechanical, optical ,and electrica l strai n gage s an d extensometers , an d variou s combination sthereof, whic h have bee n develope d fo r on e purpos e o r another , frequentl ywith regar d t o som e ver y specifi c application . I t i s ver y obviou s tha t th edevelopment o f a single instrument possessin g al l the optimu m characteris -tics, fo r al l applications , i s unlikely . However , a goo d approac h t o th eultimate i s stil l possible .

The brittl e lacque r markete d a s Tens-La c (1 , 2 ) i s n o longe r avail -able, althoug h Stresscoa t (3 ) ca n b e obtained . Thes e references , however,give a goo d descriptio n o f the us e o f brittle lacquer s i n experimenta l stres sanalysis.

1 Numbers in parentheses refe r t o Reference s a t th e en d o f a chapter.

4 TH E BONDE D ELECTRICA L RESISTANCE STRAIN GAG E

1.2. Characteristics desired in a strain gage

If w e se t ou t t o devis e a general-purpos e strai n gage , w e woul d probabl ymake a lis t o f all possible desire d characteristics . Some o f these include , no tnecessarily i n thei r order o f importance , th e following :

1. Abilit y t o measur e strain s precisel y unde r stati c an d dynami cconditions.

2. Smal l siz e and weight . The smal l size permits mounting th e instrumentin confine d locations , o r t o obtai n reasonabl y precis e indication sin region s o f hig h stres s gradient . Smal l weigh t i s require d s o tha tthe inerti a effect s i n th e gag e wil l b e negligibl e unde r dynami cconditions.

3. Th e possibility of remote observation and recording . This is very mucha relativ e requirement, sinc e remote migh t mean anythin g fro m a fewfeet i n the laborator y t o thousand s o f miles, as in th e cas e of a rocke tor missil e with radi o transmissio n (telemetering ) of th e signa l t o th elocation o f the observer .

4. Independenc e o f th e influenc e o f temperature . Thi s i s probabl y th emost difficul t requiremen t o f all . Ver y satisfactor y result s ca n b eachieved ove r smal l temperatur e excursions , bu t whe n th e tempera -ture ma y fluctuat e u p o r dow n i n th e rang e fro m abou t — 400°Fto +1500° F (-24 0 t o 815°C) , th e proble m become s exceedingl ydifficult.

5. Eas y installation . In order to b e commercially attractive, a strain gag eshould b e sufficientl y eas y t o instal l so that relatively unskilled peoplecan b e trained , i n a shor t spac e o f time , t o perfor m thi s operatio nsatisfactorily an d reliably .

6. Stabilit y o f calibration . I t i s extremely desirable tha t th e calibratio nshould b e stable ove r th e entir e range o f operating conditions .

7. Linea r respons e t o strain . Althoug h no t absolutel y essential , thi s i svery desirable . Smal l deviation s fro m linearit y ca n frequentl y b ebrought withi n tolerable limit s b y combinatio n (opposition ) wit h theinherent nonlinearit y of the electrica l circui t of which the gag e formsa part . Fo r large r departure s fro m linearity , the electrica l circui t canbe specially designed t o provid e automatic compensation (4 , 5). Whenlarge-scale computer s ar e employe d t o conditio n an d proces s th estrain gag e indications , provide d tha t th e relatio n betwee n strai n an dgage indicatio n i s known, thi s functio n ca n b e directl y programme dinto th e machine .

8. Lo w cost . Thi s i s anothe r relativ e consideratio n tha t depend s upo nthe work a t hand. Generally speaking, the cost of modern strain gagesis relatively insignificant in comparison wit h the other cost s associate dwith a n importan t project .

9. Dependability . Unles s th e strai n gag e indication s ca n b e depende dupon, it s us e become s ver y limited . Fortunately , th e strai n gage s

FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 5

available toda y ar e ver y dependabl e whe n used unde r th e conditionsfor whic h they were intended.

10. Th e possibility of operation as an individual strain gage , or in multiplearrangements, t o determin e quantitie s tha t ar e indicate d b y th esimultaneous observatio n o f strains a t mor e tha n on e location . Thi smeans that , fo r certain applications , w e should b e abl e t o us e strai ngages in multiple arrangements to perfor m automatic computation ofsome quantit y tha t i s related t o strain s a t severa l locations .

No on e ha s ye t developed a strai n gag e possessin g al l of these desire dcharacteristics. However , on e ca n generall y sa y tha t bonde d electrica lresistance strain gages (wire, foil, o r semiconductor ) come much nearer thanany othe r devic e to satisfyin g al l these requirements.

1.3. General considerations

Basic principle

In commo n wit h photoelasticit y an d stresscoat , th e basi c principl e under -lying th e operatio n o f electrica l resistanc e strai n gage s ha s bee n know nfor a long time. However, the application o f the principl e to strai n measure-ment (on a commercial scale ) is much more recent . In 185 6 Lord Kelvi n (6)reported hi s observation s tha t certai n electrica l conductor s h e ha d bee nstudying exhibite d a chang e i n electrica l resistanc e wit h chang e i nstrain.

The chang e o f electrica l resistanc e resultin g fro m mechanica l strai nrepresents th e basi c principl e upo n whic h electrical resistanc e strai n gage soperate. Fo r semiconducto r gages , the detai l o f the mean s b y whic h strai nchanges th e resistanc e seem s t o b e wel l understood , bu t fo r metalli cconductors (wir e or foil), we are still a long way from a complete understand-ing o f what takes place within the material .

Definition of strain sensitivity

When a conducto r i s trained i n th e axia l direction , it s lengt h wil l change ,and, i f unrestrained laterally , it s cross-sectiona l are a wil l als o chang e (th ePoisson effect) . Th e increas e in length, shown in Fig . 1.1 , is accompanied bya decrease i n the cross-sectional area , and vic e versa. In addition, the specifi cresistivity o f the materia l ma y change . These thre e influences, the chang e i nlength, th e chang e i n cross-sectiona l area , an d th e chang e i n specifi cresistivity, combin e t o produc e a chang e i n th e overal l electrica l resistanc eof th e conductor . Th e amoun t o f the resistanc e change , i n relatio n t o th echange i n lengt h o f th e conductor , i s an inde x o f wha t i s calle d th e strai nsensitivity of the materia l o f the conductor . This relationship is expressed a sa dimensionles s rati o calle d th e strain sensitivity factor. Fo r a straigh t

THE BONDE D ELECTRICA L RESISTANC E STRAIN GAG E

FIG. 1.1 . Schemati c diagra m o f strained conducto r (tensil e effec t shown) .

conductor o f uniform cros s section , thi s is expressed a s

unit change in resistanc eStrain sensitivity factor =

unit change i n length

unit change i n resistanc estrain

In symbols , thi s can b e written a s

where S , = strai n sensitivit y (factor) of the conducto r an d i sdimensionless; thi s is a physica l property o f the materia l

R = resistanc e i n ohm s

L = length i n inches

R, L = corresponding changes i n resistance and length, respectively,in ohm s an d inche s

E = L/ L = strai n alon g th e conducto r (dimensionless )

Examination o f Eq. (1 .1 ) and th e definition s of the symbol s wil l rais e aquestion regardin g th e values that should b e used fo r R an d L i n calculatin gthe strai n sensitivity . Do thes e symbol s represen t th e following?

6


1. Th e initial resistance, R0, and the initial length, L0, when the conductoris stress free? In whic h case the denominator , E, corresponds to nomina lstrain based o n L 0.

1. An y corresponding value s o f resistance an d lengt h which may prevai lafter a certain amoun t o f initial load ha s bee n applied?

3. Th e instantaneous values of resistance and lengt h which prevail duringinfinitely smal l change s o f lengt h an d resistance . I n whic h case , a s

L 0 , in the limit,

In Eq . (1.2) the denominator, e = dL/L, i s what is sometimes called th etrue strain (a s contrasted wit h th e nomina l strain) , and th e valu e of S,obtained i n thi s manne r i s sometime s calle d th e instantaneou s sen -sitivity factor, since it refers to the resistance and length in the stretchedcondition fo r which both R an d L ar e variabl e (7).

Except fo r th e specia l cas e in whic h R happen s t o b e directly proportionalto L , theoretically , these thre e mode s o f interpretation wil l yiel d differen tresults for the value of S,, the strain sensitivity factor. This means that we areconfronted wit h th e proble m o f havin g t o decid e upo n whic h particula rprocedure w e should follow . Fo r th e specia l case in which the resistanc e isdirectly proportiona l t o th e length , R = KL, wher e K i s a constant . Thus ,

R = K ( L) , and hence

Since R = pL/A, therefor e K = p/A, whic h means tha t t o fulfil l thi s condi-tion, the specific resistivity , p, will have to be proportional t o the area o f thecross section .

Elastic strains in metals

For smal l strain s with correspondingl y smal l changes i n resistance , such asmight b e expected i n metal s whe n strained withi n the elasti c limit , there isno problem. Here L0 an d L wil l be nearly equal and, likewise , R0 an d R wil lbe s o nearl y alik e i t wil l mak e n o noticeabl e differenc e i n th e valu e of S t,whether i t i s computed o n th e basi s o f L0 an d R 0, o r fro m th e value s of Land R whic h correspond t o th e elasti c limit. This i s a great convenience forthe followin g reasons :

1. Th e initia l resistance , R 0, an d th e initia l length , L 0, provid e goo dreferences from whic h the changes R and L ca n be readily determined.

THE BONDE D ELECTRICA L RESISTANC E STRAIN GAG E

2. Th e strai n sensitivity, S,, can b e determined fro m th e slop e o f the curvewhich i s established by plotting R/R 0 agains t L/L 0.

3. Th e analyse s o f the basi c electrica l circuits which are use d wit h strai ngages, develope d i n followin g chapters , sho w tha t th e output , o rindication, i s given in term s of R/R 0.

Plastic strains in metals

When a meta l conducto r i s strained beyon d th e elasti c limi t int o th e plasti crange, th e change s i n resistanc e an d lengt h (fro m th e initia l values ) wil lultimately becom e s o larg e tha t ther e wil l b e a considerabl e differenc ebetween R an d R (), an d als o betwee n L an d L 0.

When this happens, the previous approximate metho d o f determining Stfrom th e value s o f R 0 an d L 0 wil l n o longe r b e satisfactory . I t wil l b enecessary t o comput e th e instantaneou s valu e of S , from th e instantaneou svalues o f R an d L , accordin g t o Eq . (1.2) .

At first glance, this might appear to be a formidable task, but fortunatelythis i s not so . W e determin e a serie s of corresponding value s of R an d L a sthe conductor i s being stretched (o r compressed), an d then plot the logarithmof the dimensionless ratio, R/R 0, agains t th e logarith m o f the dimensionlessratio, L/L 0. Th e slop e o f the lin e thu s draw n represent s th e instantaneou svalue of the strain sensitivit y factor, St. Furthe r discussion wil l be found late rin th e chapter .

Semiconductor materials

The relativel y hig h strai n sensitivit y o f silico n an d germaniu m ha s mad ethese semiconducto r material s attractiv e fo r strai n gag e sensin g elements .For silicon , whic h is the preferre d material , the theoretica l valu e of St lies inthe rang e betwee n —15 0 and abou t +175 . Furthermore , b y suitabl eprocessing (doping) , silico n can b e produce d wit h an y arbitraril y specifiedvalue of S, within this range. For commercia l strain gages, in order t o achievea suitable compromise betwee n respons e t o strai n and respons e t o tempera -ture, i t i s usual t o proces s th e materia l fo r strai n sensitivities in th e rang e ofabout -10 0 t o abou t + 120.

The resistance-strai n relatio n fo r silico n i s somewha t mor e elaborat ethan tha t fo r metalli c conductors . I t i s nonlinear , an d ver y noticeabl yinfluenced b y temperature . Dorse y (8 , 9) give s the followin g expressio n fo runit chang e i n term s o f strain:

8


where R = chang e in resistanc e fro m R O(TO> (ohms )

R0(To} = resistance (ohms) of the unstressed material (prior to beingmounted a s a strai n gage ) a t temperatur e T 0, in Kelvin

T0 = temperatur e a t whic h R O(TO) wa s determined (Kelvin)

T = temperatur e (Kelvin)

e = strai n (dimensionless )

GF', C'2 = constant s fo r the particula r piec e o f material(dimensionless)

Equation (1.4 ) indicate s th e followin g characteristic s regardin g th erelation betwee n uni t chang e i n resistanc e an d strai n fo r silicon:

1. Th e strain sensitivity factor , which corresponds to the slope of the curveof R/R 0(:ro) vs . e, will be a variable whose value will depend upo n bot hthe strai n leve l and th e temperature.

2. Sinc e th e relationshi p expresse d i n Eq . (1.4 ) represent s a parabola ,one ca n expec t th e degre e o f nonlinearit y t o var y wit h strai n an dtemperature.

3. A t constan t temperature , T 0, Eq. (1.4 ) reduces t o

Hence, for this special condition show n in Fig. 1.2 , GF' correspond s t othe slop e o f th e curve , o r th e sensitivit y factor , fo r e = 0 , an d C' 2represents th e nonlinearit y constan t whic h determine s th e degre e o fdeparture o f the curv e fro m th e slop e a t th e poin t R = 0 , e = 0 , forwhich th e resistanc e equal s Ro(r 0i- Bake r (10 ) als o expresse s Eq . (1.5 )in essentiall y th e sam e form .

Over a limite d rang e o f strain , fo r exampl e abou t 60 0 microstrai n ( 1microstrain = 1 uin/in), an d particularl y at strai n level s wher e th e slop e ofthe curv e change s mor e gradually , th e variabl e strai n sensitivit y ca n b eapproximated b y a constant tha t corresponds t o the average value, and goodresults ma y b e expected fro m this . For large r range s o f strain , o r fo r mor eprecise indications , mor e elaborat e method s mus t be employed .

When th e temperatur e varies , the whol e problem o f relating resistancechanges t o strai n become s mor e complicated . Thi s i s du e t o th e fac t tha tchanges i n temperature , a s indicate d i n Eq . (1.4) , produc e change s i n th esensitivity. I n addition , th e valu e of R 0(To-> wil l als o chang e wit h variation sin th e referenc e temperature, T 0.

10 THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

FIG. 1.2. Schemati c diagra m fo r R/R n(TaR/R0(Tat whe n R = e = 0.)

al constan t temperature , T 0. (Resistance =

Desired properties of strain-sensitive materials

1. Linea r relation between unit chang e i n resistance an d chang e i n strain(i.e., constant sensitivity) .

2. Negligibl e effec t fro m temperature .3. Hig h strai n sensitivit y factor .4. Moderatel y hig h resistance.5. Abilit y t o b e connected t o lea d wire s easily.6. Lo w cost .7. Availability.8. Absenc e of creep and hysteresis .

One canno t expec t t o fin d al l th e desirabl e characteristic s i n an yparticular materia l withou t som e advers e properties , too . I n general , th eselection o f a materia l fo r th e sensin g element o f a strai n gage wil l resul t ina compromis e dependin g upo n th e intende d use o f the gage .

Properties of some metals

In vie w o f th e previou s discussio n o f strai n sensitivity , and th e propertie sdesired in strain sensing materials, let us look a t som e typica l characteristicsas represented b y a few metals. These are indicated i n Figs. 1. 3 and 1.4 , takenfrom th e wor k o f Jones an d Masle n (11) . In eac h case , th e percen t chang ein resistance , base d o n R 0, ha s bee n plotte d agains t percen t strain , o n th ebasis of L/L 0. Th e slope s of the line s represent S, . and th e differen t genera lrelationships ar e indicate d as follows :

vs

FUNDAMENTAL CONCEPT S FO R STRAI N GAGE S 1 1

1. Th e same linear relatio n betwee n R/R 0 and L/L 0 i n both th e elasticand plasti c ranges . Thi s conditio n i s represented b y anneale d copper ,as well as annealed copper-nicke l alloys like Ferry. This means that thestrain sensitivit y factor wil l b e th e sam e i n th e plasti c range a s i t i s inthe elasti c range . Thi s characteristi c i s highl y desirabl e because i teliminates al l concern abou t th e possibilit y o f a change in gag e factorin th e even t th e sensin g elemen t o f a strai n gag e migh t b e straine dbeyond it s elasti c limit . In consequence , thi s typ e o f materia l i s wel lsuited fo r gages whic h will be required t o measur e high elastic strains ,or bot h elasti c and plasti c strains .

2. Nonlinea r relationshi p such a s exhibited by nickel.3. Relationshi p approximated b y two straigh t lines indicating a change of

strain sensitivit y with the transitio n from elasti c to plasti c conditions .Some materials , suc h a s minalpha , manganin , an d har d silver -palladium, sho w a lowe r strai n sensitivit y at lo w strain s tha n a t hig hstrains.

4. Th e sam e genera l relationshi p a s indicate d i n Ite m (3) , bu t wit h th edifference tha t th e highe r strai n sensitivit y corresponds t o th e lowe rstrains, a s shown by rhodium-platinum .

For th e relation s indicate d i n Items (3 ) and (4) , the chang e i n slope a syielding set s i n i s no t abrupt , a s suggeste d b y th e graphs , bu t follow s asmooth transitio n fro m th e elasti c t o th e pasti c range .

Numerical values of the strain sensitivity factor

Table 1. 1 presents typica l strain sensitivit y values for a number o f metals a tlow strain , togethe r wit h correspondin g informatio n wit h respec t t o th eeffects o f temperature change s (12).

A mor e elaborat e tabulation , whic h includes some o f the pur e metal sand a numbe r o f alloy s (wit h approximat e compositions) , i s give n i n th eAppendix o f thi s chapter . Wher e possible , informatio n fo r sensitivitie s i nboth the elastic and plastic strain ranges, and for material in the cold workedand anneale d conditions , has bee n included .

Approximate composition s o f some o f the alloy s in Tabl e 1. 1 are givenin Tabl e 1.2 .

A stud y of the literatur e an d o f the tabulate d dat a i n th e Appendi x a tthe en d o f the chapte r yield s the followin g observations regardin g materia lproperties:

1. Differen t value s o f strain sensitivit y for har d an d anneale d condition sof the same materia l suggest s that th e degree o f cold working , and th eheat treatment , hav e a n influence . This i s o f particular importanc e i nrelation t o th e effect s o f temperature an d temperatur e compensation .

THE BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

FIG. 1.3. Typica l example s o f resistanc e chang e vs . strai n (Fro m ref. 11 wit h permissio n o fHMSO.)

2. Difference s i n sensitivit y fo r differen t lot s o f nominall y th e sam ematerial sugges t tha t difference s i n impurities , and i n trac e elements ,exert an influenc e o n th e physica l properties. This i s also of importancewith respec t t o temperatur e effects .

3. Fo r nearl y al l th e metal s investigated , th e strai n sensitivit y facto rappears t o approac h a valu e of 2.0 in th e plasti c range .

For larg e strain s (u p t o 3 0 percent), Weibul l (13 ) has reporte d som e ver yinteresting detailed experimenta l results on th e relatio n betwee n changes i nlength an d resistanc e for 0.45-mm diamete r Cope l wire . This i s a 5 5 percentcopper, 4 5 percent nicke l alloy.

From the data in Table 1.3 , the values of R/R0, L/L 0, R/R0, an d L/L 0,have been computed . Plot s of \n(R/R0) vs . ln(L/L0) an d R/R0 \sAL/L 0 ar eshown i n Fig . 1. 5 fo r comparativ e purposes . Fro m th e slop e o f th elogarithmic plot , whic h i s represente d b y a straigh t line , th e valu e o f th e

12

FIG. 1.4. Resistanc e chang e vs . strai n fo r anneale d Ferr y wir e (60/4 0 cupronickel) . (Fro mref. 12. )

Table 1.1. Typica l strain sensitivit y factor s

MaterialStrain sensitivity factor

(for small strains)

Stress in Ib/in equivalent to influenceof temperature change of 1°C for

installation on steel material"

ManganinNickelNichromePhosphor bronz e5% Iridium-Platinu mAdvanceCopelMonelIsoelastic

0.47— 12.1 (nonlinear)

2.11.95.12.1 (selected material )2.41.93.6

-400-13500

21007800

11600±30

-20080005000

Source: reference 12 ." One shoul d not e tha t thes e figures can onl y be considered a s semiquantitative indications because they willvary wit h hea t treatmen t an d col d workin g of the materia l an d als o wit h temperature level.

Table 1.2. Compositio n o f alloys

Material Composition

Advance an d Cope l5% Indium-platinu mIsoelastic

ManganinNichrome V

45% Ni; 55 % Cu5% Ir ; 95 % P t36% Ni; 8 % Cr; 52 % Fe;0.5% Mo; + (Mn, Si, Cu, V) = 3.5%4% Ni ; 12 % Mn; 84 % Cu80% Ni; 20% Cr

14 TH E BONDE D ELECTRICA L RESISTANC E STRAI N GAGE

Table 1.3. Weibull' s observation s fro m stati c tes ton Cope l wir e

Initial diameter = 0.45 mm; initial length = 125 mm\L (mm) R (ohms)

0.006.25

12.5018.7525.0031.2537.50

0.3760.4140.4550.4970.5420.5880.635

Source: referenc e 13 . Reprinte d b y permission , r 194 8 Mac -millan Magazine s Ltd .

strain sensitivit y facto r i s found t o b e

Weibull does not stat e the metallurgical condition o f the wire , but fro mthe magnitud e (6 0 percent ) o f th e elongatio n reporte d fo r on e o f hi sspecimens, i t is assumed tha t th e material wa s in the annealed condition . H ealso report s essentiall y comparabl e result s for a dynami c tes t o n 0.45-m mdiameter wir e wit h a lengt h o f 10 1 mm. Th e maximu m strai n reache d 3 4percent wit h a velocity of 6.2 m/sec for the moving head of the testing device.

The 0.45m m (0.017 7 in) wir e diamete r whic h Weibul l investigate d i ssomewhat large r tha n the 1-mi l (0.001-in) size normally employed for bonde dstrain gages . Wit h th e smalle r diameter , smalle r ultimat e elongatio n i sexpected becaus e mino r variation s i n diamete r wil l have , relatively , muchgreater influence . Shou b (14 ) report s elongation s u p t o 2 2 percen t fo rspecially anneale d constanta n wir e of 0.001 i n diameter . His result s indicat ea straight-lin e relationship , wit h a slop e o f 2.02 , fo r th e plo t o f log (R/R 0)vs. log (L/L 0). Thi s confirm s Weibull's observations .

1.4. Analysis of strain sensitivity in metals

The general case

Figure 1. 6 shows a metal conductor o f uniform cross sectio n (no t necessaril yrectangular, althoug h thi s i s shown) referre d t o th e axe s X , Y , and Z . W ewant t o establis h a n expressio n fo r the rati o o f unit chang e i n resistanc e i nthe X directio n t o th e uni t chang e i n length , in term s o f strains e x, e y, an de. (in th e direction s o f the X , Y , and Z axes , respectively ) and th e materia lproperty o f the conductor .


FIG. 1.5. Weibull' s experimenta l result s from 0.45-m m diameter Copel wire. (From ref . 13. )

The expressio n fo r th e resistanc e in th e X directio n ca n b e written as

where R = resistance i n length L (ohms )p = specific resistivity of the materia l (ohms-in )L = length (in)A = area o f the cross section (in2)

16 THE BONDE D ELECTRICA L RESISTANC E STRAIN GAG E

Fie. 1.6. Meta l conductor referre d t o X , Y , and Z axes .

By multiplying the numerato r an d denominato r o f the right-han d ter mby th e lengt h L , Eq . (1.7) can b e rewritte n as

where V — LA = volum e (in3). By takin g th e logarith m o f bot h sides , Eq .(1.8a) become s

Differentiation o f Eq . (1.8b ) results in

Equation (1.9 ) expresses th e uni t chang e i n resistanc e i n term s o f th e uni tchanges i n resistivity , length, an d volume .

We no w postulat e tha t th e uni t change i n resistivit y ca n b e relate d t othe uni t chang e i n volum e a s follows :

where m = a functio n o f th e materia l propertie s an d th e tw o ratio s o f th etransverse t o the longitudinal strain. Fo r th e elastic strains , an d fixed valuesof th e tw o strai n ratios , som e material s exhibi t a constan t valu e o f th efunction m . This relatio n i s stated b y Biermas z e t al . (15) , who give s credi tfor i t t o Bridgeman . Meie r (16 ) uses th e sam e relatio n i n a slightl y differen tform.

FUNDAMENTAL CONCEPT S FO R STRAI N GAGES 1 7

By substituting th e valu e of dp/p give n by Eq . (1.10 ) into Eq . (1.9) , wemay write

or

Dividing al l terms o f Eq. (1.11 ) by dL/L, w e obtai n

Equation (1.12 ) indicates that , for plastic deformation (which takes place a tconstant volume , s o tha t d V = 0), th e valu e o f th e instantaneou s strai nsensitivity ca n b e expected t o b e 2 for an y strai n condition.

Since dL/L = ex, an d because dV/V = (sx + sy + ez), Eq. (1.12) can beexpressed i n term s o f the strain s a s follows :

Special case of a uniform straight wire

For th e specia l cas e o f a straigh t wir e o f any unifor m cross section , whichis free t o contrac t or expan d laterall y due t o th e Poisso n effect , th e ratio s oflateral t o axia l strain ar e give n by the expressio n

where v = Poisson' s ratio o f the material .When th e value s o f the strai n ratios , give n for thi s specia l cas e b y Eq .

(1.14), ar e substitute d into Eq . (1.13 ) for strain sensitivity , we arrive a t

For smal l changes , suc h a s encountere d withi n th e elasti c range s o fmetals, Eq . (1.15 ) can b e modified to rea d


Equations (1.15 ) and (1.16 ) indicat e tw o interestin g characteristic s i nregard t o th e strai n sensitivit y of a wire .

1. I f the materia l property i s such that m = 1 , then, regardless o f the valu eof Poisson' s ratio , th e strai n sensitivit y factor o f th e meta l wil l b e 2 .This mean s th e strai n sensitivit y will b e th e sam e i n th e elasti c an dplastic ranges , eve n thoug h ther e wil l be a chang e i n v as on e proceed sfrom elasti c to plastic conditions . Conversely, this also tells us that onl ythose materials whose strain sensitivity is 2 can hav e the same sensitivit yin bot h th e elasti c an d plasti c ranges .

2. Fo r perfectl y plasti c deformation, which takes place at constant volume,dV - 0 and v = 0.5 . Therefore , n o matte r wha t th e valu e of m is, thestrain sensitivit y factor fo r plasti c deformation wil l b e 2 , as previousl yindicated b y Eq . (1.12) . Thi s mean s that , fo r plasti c deformation , al lmetals shoul d exhibi t a strai n sensitivit y factor o f 2 . Thi s i s substan -tiated b y th e result s o f tests, a s indicate d i n th e tabulatio n presente din the Appendi x of this chapter, for which, in almost al l cases, th e strai nsensitivities i n th e hig h strai n range s approximat e a valu e o f 2.

The sligh t deviation o f some o f the value s from 2 i s probably du eto th e effec t o f a certai n amoun t o f elastic strain whic h wil l b e presen tduring th e plasti c deformation . The fe w cases involvin g larger devia -tions fro m 2 likel y correspon d t o rathe r incomplet e o r gradua l plasti cdeformation, and possibl y the influence o f some typ e of work hardening.

Equations (1.15 ) an d (1.16 ) can no w b e converted int o a mor e familia rform customaril y foun d i n th e literature . Expansio n o f the secon d ter m o nthe right-han d sid e o f these equation s result s i n th e expressio n

In order to write Eq . (1.17) in a differen t form , the change i n the volum eof th e wir e a s i t i s straine d axiall y can b e considered . Th e unstraine d wir evolume i s

Taking th e logarith m o f both side s an d the n differentiatin g yield s

As th e wir e i s strained , it s lengt h increase s b y dL , bu t du e t o th e Poisso neffect it s diamete r decrease s b y ( — v dL/L)D, wher e D i s th e wir e diameter .


The fina l wir e diameter i s

The chang e i n area ca n no w b e written as

If th e higher-orde r ter m i n Eq . (d ) i s neglected, the n w e can writ e

Substituting the valu e o f dA/A give n b y Eq . (e ) into Eq. (b ) give s

Thus, Eq . (f ) can b e expressed a s

From Eq . (1.10 ) we can write

If th e value s o f ( 1 — 2v) an d m fro m Eqs . (g ) an d (h) , respectively , ar esubstituted i n Eq . (1.17) , the n

or


For smal l changes , a s encountered wit h elasti c strains, we can write

Equation (1.18 ) is of particular interest , not jus t becaus e i t represents amore familia r form o f the expressio n for the strai n sensitivit y factor , but fo rtwo othe r reason s a s well .

1. Th e relationshi p give n i n Eq . (1.18 ) ca n b e derive d independentl y o fthe relatio n give n by Eq . (1.10) .

2. Fo r an y particula r metal , Eq. (1.18) indicates the portion s o f the strainsensitivity facto r whic h ar e th e resul t o f geometrica l chang e an dresistivity change , respectively . The valu e (1 + 2v ) corresponds t o th egeometrical change , whil e (dp/p)/(dL/L) correspond s t o th e resistivitychange.

We see that whe n plastic deformation takes place, since v = 0. 5 and d p = 0,Eq. (1.18 ) als o indicate s a valu e of 2 fo r S t.

Small strain vs. large strain

Let u s now loo k int o the detai l o f the differenc e betwee n the expression s fo rthe instantaneou s an d approximat e value s o f th e strai n sensitivit y factors .The instantaneou s valu e o f S , is

while th e approximat e valu e of S , is

For smal l strains (less than 1 percent), a s developed i n the elastic rang eof metals , bot h expression s wil l yield , fo r al l practica l purposes , th e sam eresult. However , sinc e i t wil l b e mor e convenien t t o evaluat e th e strai nsensitivity, an d subsequentl y t o comput e strains , o n th e basi s o f change sfrom th e initia l condition , w e wis h t o kno w th e magnitud e o f th e larges tstrain tha t ca n b e handled i n thi s manner withou t running int o intolerabl ylarge errors .

Returning t o Fig . 1.5 , w e se e a comparison , base d o n Weibull' sexperimental observations , betwee n th e plo t o f AK/R 0 vs . L/L0 an d th elogarithmic plo t o f \n(R/R 0) vs . ln(L/L 0). Th e logarithmi c plo t show s a


straight lin e wit h a slope , S t, of 2.0, wherea s the plo t of R/R0 vs. L/L 0gives a long radiu s curv e whose initia l slope (fo r R = L = 0) is 2.0, butfor whic h the slope increases slightl y as the changes in length an d resistanc ebuild up .

Examination o f Fig. 1. 5 reveals that, for a graph o f this size and withi nthe limit s of error i n plotting th e points, the curve of R/R0 vs . L/L0 ca nbe represented b y a straigh t lin e u p t o value s o f about 1 0 to 1 5 percent o fL/L0. Fo r larger strains the departure fro m linearity , although not serious,can be noticed. However, we observe that the slope of the line (the indicatedvalue of Sr) is slightly greater tha n that o f the logarithmic plot. This explainswhy one can use post-yield gages up to strain levels in the range of 10 percentor more , on the basis of R/R0 an d L/L0, withou t introducing noticeabl eerrors a s a resul t of making a linea r approximation .

As thes e comment s hav e bee n develope d fro m experimenta l observa -tions, w e ca n no w examin e th e situatio n fro m a theoretica l poin t o f view .We star t by developing the relatio n betwee n resistance an d lengt h fro m Eq .(1.20) o n th e assumptio n tha t S t is a constant . W e can rewrit e Eq. (1.20) inthe followin g form :

Equation (1.22 ) can als o b e expressed a s

Integrating Eq . (1.23 ) results in

where C = constant o f integration.Since th e initia l value s o f resistanc e an d length , R 0 an d L 0, wil l b e

known, the constan t o f integration can b e written as

Substituting the valu e of C from Eq . (1.25 ) into Eq . (1.24 ) gives us

This expressio n ca n b e modified to rea d

22 TH E BONDE D ELECTRICA L RESISTANC E STRAIN GAG E

Equation (1.26 ) tell s u s tha t th e plo t o f ln(R/R0) vs . ln(L/L0) wil l givea straigh t lin e whos e slop e i s equa l t o S t. Thi s ha s bee n verifie d experi -mentally b y bot h Weibul l (13 ) and Shou b (14) .

From Eq . (1.26 ) w e ca n expres s th e relatio n betwee n resistanc e an dlength o f a meta l conducto r tha t ha s bee n straine d i n the plasti c range a s

Since th e valu e of S t fo r plasti c strai n ha s bee n predicte d theoreticall yas 2.0, as shown b y Eq. (1.12), and becaus e thi s value has bee n corroboratedby th e experiment s o f Weibull (13) and Shou b (14) , thi s is the numbe r tha twill b e use d fo r th e exponen t i n Eq . (1.27) . Thus , Eq . (1.27 ) ca n no w b ewritten a s

Because R = R0 + R an d L = L0 + L, Eq . (1.28) can be converted int oterms of R , L , R 0, and L0. Thus ,

Expanding th e right-han d sid e o f Eq . (1.29 ) result s in

Equation (1.30 ) presents the theoretical relationship between R/R 0 an dL/L0 fo r a meta l conducto r subjecte d t o plasti c strain . I t provide s th efollowing information :

1. R/R 0 i s a nonlinear functio n a t L/L 0.2. Fo r positive value s of L (tension) , R/R0 wil l alway s be larger tha n

2( L/L0).3. Th e slop e o f the curv e a t th e origi n i s 2.4. Th e deviatio n fro m th e tangen t (slop e = 2 ) through th e origi n i s given

by ( L/L0)2.

or


Item 4 indicate s bot h th e deviatio n fro m linearit y an d th e deviatio n fro mthe relatio n involvin g the instantaneous value s of R an d L .

It i s noteworth y tha t whe n L/L 0 i s 1 0 percent, th e deviatio n fro mlinearity i s only 5 percent. Thi s i s illustrated i n Fig . 1.7 , which shows a plo tof theoretica l value s of R/R0 vs . L/L0, a s computed fro m Eq . (1.30).

If an approximat e linea r relatio n i s set up b y using the secan t fro m th eorigin t o som e poin t o n th e curve , then th e erro r wil l b e zero a t th e poin tof intersection wit h the curve , and a t al l othe r point s th e erro r wil l b e lessthan that represented b y the deviation o f the secant fro m th e initial tangent .This i s due to the fac t tha t the curve lies between the secant an d th e tangent

0

FIG. 1.7 . Theoretica l relation between R/R0 an d L/L 0 fo r large strains.

24 TH E BONDE D ELECTRICA L RESISTANCE STRAIN GAGE

through the origin. For example , when L/L0 equal s 10 percent, the expectederror, a t an y point , wil l neve r b e mor e tha n 5 percent , a s a maximum . Ingeneral i t will probably no t excee d 2.5 percent, except for relatively low strainvalues where the numerica l magnitude of the error wil l be of less importance .Examination o f Fig . 1. 7 will hel p t o clarif y thes e points .

From Eq . (1.30) an expressio n can b e written for th e valu e of the strai nsensitivity factor :

The value of S, varies in accordance wit h the value of L/L0 an d correspond sto th e slop e o f th e secan t fro m th e origi n t o th e poin t whos e coordinate sare ( R/R0, L/L 0) o n the curve .

1.5. Wire strain gages

The unbonded wire strain gage

One o f th e earl y wir e gage s wa s th e unbonde d type . I n thi s typ e o finstrument, the strain-sensitiv e wire i s mounted, unde r tension , on mechani -cal support s (pins ) i n suc h a manne r tha t a sligh t relativ e motio n o f th esupports wil l caus e a chang e i n strain . This, i n turn , produce s a chang e i nelectrical resistance . This resistanc e change i s then a measur e o f the relativ edisplacement o f th e support s and , i n turn , ma y represen t a strai n o r som eother quantity.

With th e unbonde d typ e of gage, th e fac t tha t th e strain-sensitiv e wiresmust b e carrie d o n som e sor t o f mechanica l moun t give s ris e t o certai ndifficulties i n connection wit h attachment . Discrepancies , due t o inertia , maybe introduce d whe n dynami c observation s ar e made . Th e procedur e o fmaking observation s a t a n appreciabl e distanc e fro m th e surfac e o n whichstrain i s to b e determined may sometime s b e ope n t o question .

The bonded wire strain gage

The firs t majo r improvemen t i n th e wir e resistanc e strai n gag e cam e wit hthe realizatio n tha t man y o f th e difficultie s wit h th e unbonde d wir e gag ecould b e eliminate d b y bondin g a ver y fine strain-sensitive wire directly t othe surfac e o n whic h strai n i s t o b e measured . Th e filamen t ha s t o b eelectrically insulated an d th e bondin g perfec t fo r the strain-sensitive elementto follo w th e strai n o n th e surfac e to whic h i t i s attached. Onl y conductor sof smal l diameter ar e suitable , since the force necessary t o strai n th e sensin gelement mus t b e transmitte d throug h it s surfac e by shea r i n th e cement , o rbonding agent . Unles s th e surfac e are a pe r uni t lengt h i s larg e relativ e t othe cross-sectiona l area , th e shearin g stres s i n th e cemen t wil l b e to o hig h


to permi t faithfu l followin g o f th e strain s i n th e surfac e t o whic h th econductor i s attached .

Since th e surfac e are a (pe r uni t length ) o f small-diamete r wire s i senormously greater tha n th e cross-sectional area (for 0.001-in diameter wire,the rati o is 4000 to 1) , the bonding agen t i s able to forc e th e filament t o tak eup th e necessar y strai n withou t excessiv e stres s i n itself . Suitabl e cement scan actually force the small conductor into the plastic range (and back again )when necessary.

Chronologically, th e secon d majo r development , an d tha t whic h ha sactually bee n responsibl e for makin g th e bonde d strai n gag e commerciallyattractive, i s represente d b y th e concep t o f premounting th e strain-sensin gelement o n some suitabl e carrier tha t ca n be attached t o a surfac e relativelyeasily. Originally , the strai n gag e wir e was cemented directl y to th e surfaceon whic h strai n wa s t o b e measured , an d th e glu e o r cemen t acte d a sinsulation. A s fa r a s operatio n wa s concerned , thi s procedur e wa s satis -factory, bu t fro m th e poin t o f view o f gage installation , i t was inconvenient.The attachment o f the gage require d an inordinat e amoun t o f skill and timeon th e par t o f th e installe r i f consisten t result s wer e t o b e obtained . Th eintroduction o f a paper , plastic , metal , o r othe r typ e of carrier upo n whic hthe strain-sensin g wir e coul d b e premounted , unde r controlle d factor yconditions, represente d a tremendou s improvement . Wit h thi s for m o fpremounted filamen t strai n gage , muc h les s skil l and tim e ar e require d t oachieve satisfactor y installations givin g good an d consisten t results .

Most bonde d wir e strai n gage s ar e mad e fro m wir e o f approximately0.001 in diameter, o r less , and i n resistances varying from abou t 5 0 ohms t oseveral thousan d ohms . Th e filament s ar e mounte d o n carrier s mad e o fmaterials selected fo r th e particula r application s fo r which the gage s ar e t obe employed.

Since a length of several inches o f wire is usually needed to produce thenecessary tota l resistance , an d becaus e th e desire d gag e lengt h i s almos talways les s tha n th e require d lengt h o f wire , i t i s necessar y t o arrang e th ewire i n som e for m o f grid i n orde r t o economiz e o n space , an d thereb y t opermit reductio n o f th e gag e lengt h t o a suitabl e size . Figur e 1. 8 showsdiagrams o f typical grid configurations for wir e gages. There are , o f course,variations of these typical designs, as manufacturers' literature shows (17,18).

The fla t gri d i s probably th e mos t usefu l form . When th e gag e i s on aflat surface, the centre line of the entire sensing element lies in one plane thatis parallel t o th e surfac e of attachment. Du e t o th e end loops , ther e is someresponse t o strai n a t righ t angle s t o th e directio n o f the gri d axis . Usuallythe filamen t consist s o f on e continuou s lengt h o f wire ; however, for som eself-temperature-compensated gages , two elements , which possess opposing ,or compensating , temperatur e characteristic s ar e joined together .

An alternat e typ e o f constructio n originate d a s a n expedien t fo rmanufacturing gage s o f shor t gag e lengt h (0.25 0 in o r less ) prio r t o th edevelopment o f the technique s now use d t o mak e shor t fla t gri d gages . I n


FIG. 1.8 . Typica l wir e strai n gages , (a , b) Singl e elemen t gages , (c , d) Two-elemen t stacke drectangular rosettes , (e , f) Three-elemen t stacke d rectangula r rosettes , (g ) Two-element rectan -gular rosette , (h ) Three-elemen t rectangula r rosette . (Fro m ref . 18.).

the wrap-around construction , the sensin g element is wound tightl y aroun da smal l flat carrier whic h i s then encased betwee n two cover sheet s providinginsulation an d protection . A n alternativ e procedure i s t o win d th e sensin gelement on a small tubular mandrel (like a soda straw ) that is then flattenedand encase d betwee n th e cove r sheets .

For th e variou s type s o f bonde d wir e strai n gages , th e strai n i sdetermined fro m th e relatio n

where e = strain i n th e directio n o f the gag e axis

R/R = unit chang e in resistance

GF = manufacturer's gage factor

Due t o th e geometrical difference s betwee n a straigh t wire and a straingage grid, the value of the manufacturer's gage factor, GF, is generally slightlylower tha n th e strai n sensitivit y factor , S, , o f th e wir e fro m whic h the gri d


is constructed . Furthermore , th e magnitud e o f G F wil l var y slightl y withvariations i n grid design .

Gages containin g a singl e continuou s filamen t whic h i s woun d bac kand fort h wil l respond slightl y to th e effec t o f lateral strai n whic h is sensedby the end loops. This means tha t Eq . (1.32), although generally applicable ,is subjec t to som e erro r whe n th e strai n field in whic h the gag e i s actuallyused differ s fro m tha t of calibration. Usually the error caused by the responseto latera l strai n ca n b e neglected , bu t ther e ar e a few situations i n whic h itbecomes appreciable . Th e magnitud e o f the erro r cause d b y latera l effect sand, wher e necessary, the mean s o f correcting for thi s error , ar e discussedin detai l i n a later chapter .

Some specifi c example s o f the relatio n betwee n strain an d uni t changein resistance for complete wire gages are show n in Fig . 1.9 . In eac h case th eslope o f line relating the percen t chang e i n resistanc e to th e percen t strai nrepresents the gag e factor. One wil l note tha t the advanc e wir e (constanta ntype) gag e ha s th e sam e gag e facto r fo r bot h elasti c an d plasti c strains ,whereas the isoelastic and nichrome gages both show a change in gage factoras one proceeds from elasti c to plastic conditions. One should not be alarmedabout thi s chang e i n gag e facto r because w e ar e usuall y intereste d i nmeasuring elastic strain s i n metals , an d thes e occu r wel l below the chang epoints show n i n th e diagrams . Thi s i s especially so i n th e cas e o f isoelasticwire (whos e chang e poin t occur s a t approximatel y 0.7 5 percen t strain) ,because thi s material i s usuall y chosen t o tak e advantag e o f it s hig h gag efactor fo r measurin g very smal l strains.

Wire gage s wer e use d unti l th e earl y 1950s , whe n foi l gage s wer eintroduced. Some wire gages are stil l used today and ca n be purchased fro mseveral manufacturers.

Weldable wire gages

The first weldable wire gage was developed in the mid-1950s (19). Subsequentdevelopment fo r a quarter-bridg e circui t used a singl e filamen t o f nickel -chromium wir e tha t wa s chemicall y etche d s o tha t it s cente r lengt h wa sapproximately 1 mil i n diameter . Th e wir e wa s the n folde d i n hal f an dinserted into a stainless stee l tube . The tube wa s filled with a metallic oxidepowder whic h wa s compacte d s o tha t i t no t onl y electricall y isolated th efilament but mechanicall y coupled i t to th e tube in order t o transmi t strain.The constructio n i s shown in Fig . 1.10 .

In orde r t o minimiz e the apparen t strai n du e t o temperatur e changes ,the nickel-chromiu m filamen t i s hea t treated . Sinc e differen t level s o f heattreatment resul t in differen t value s of the therma l coefficien t o f resistivity, itis possible t o make thi s change equal in magnitude but o f opposite polarityto th e therma l coefficien t o f expansion .

To achiev e temperatur e compensation , a separat e compensating , o rdummy, gage can be mounted o n a stress-fre e piec e o f material identica l to

FIG. 1.9. Typica l gag e characteristic s i n tension . (Fro m ref . 11 , with permission o f HMSO. )

FIG. 1.10. Singl e active gag e construction . (From ref . !9. )


FIG. 1.11. 'True ' dummy gage construction. (From ref . 19.)

FIG. 1.12. Ni-C r half-bridg e gag e construction. (From ref . 19.)

the materia l o n whic h th e activ e gag e i s mounted. Th e tw o gage s ar e the narranged int o a half-bridg e circuit. This i s a satisfactor y metho d providin gthe materia l o n whic h the dumm y gage i s mounted i s completely stress fre eand tha t th e dumm y gage' s temperatur e i s identica l t o th e activ e gage .Because thes e condition s d o no t alway s prevail , a 'true ' dumm y gag e wa sdeveloped. The dumm y gag e filament , identica l t o th e activ e gag e filament ,is woun d i n a tigh t heli x of the prope r pitc h angle . Sinc e i t i s embedded i na strai n tub e wit h compacte d magnesiu m oxid e powder , th e sam e a s th eactive gage , i t ha s th e sam e heat-transfe r characteristics . Therefore , th edummy gag e ca n b e use d wit h a compensate d activ e gage t o minimiz e theapparent strain . Th e dumm y gage i s shown i n Fig . 1.11 .

The nex t ste p wa s to incorporat e th e singl e activ e gage an d th e 'true 'dummy gage into one strain tube and mounting flange assembly. This resultsin a half-bridge gage rather than a quarter-bridge gage. The half-bridge gageis shown i n Fig . 1.12.

The earl y weldabl e wir e strai n gag e ha s resulte d i n a lin e o f bot hquarter- and half-bridg e gages (20). Two wire types are used for the filament.The firs t i s a nickel-chromiu m tha t i s temperature compensate d an d use dfor stati c measurements up to 600°F (315°C). Because of excessive drift abov e600°F, th e gages are use d only for dynamic test s between 600°F and 1500° F(815°C). Th e secon d wir e typ e i s platinum-tungste n tha t ca n b e use d fo rstatic measurement s u p t o 1200° F (650°C) . Sinc e thi s wir e cannot b e hea ttreated for temperature compensation, th e half-bridge gage is recommended .

1.6. Foil strain gages

General characteristics

The foi l gag e operate s i n essentiall y th e sam e manne r a s a wir e gage .However, the sensing element consists of very thin metal foi l (about 0.0002 i n

30 TH E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

thick) instea d o f wire . I n contras t t o th e wir e gage , i n whic h th e sensin gelement possesse s a unifor m cross sectio n throughou t it s entir e length , th ecross section of the sensing element of the foi l gage may be somewhat variabl efrom on e en d t o th e other . On e o f the mos t importan t advantage s o f the foi lgage i s that th e rati o o f contact surfac e area t o th e volum e o f the resistanc eelement i s relatively high, whereas in the wir e gage, du e t o th e circula r cros ssection, thi s rati o i s a minimum.

The earl y foi l gages , introduce d i n Englan d i n 1952 , were mad e fro mfoil cemente d t o a lacque r sheet . The desire d gri d desig n fo r th e strai n gag ewas printe d o n th e foi l wit h a n acid-resistin g in k an d th e shee t wa s the nsubjected t o a n aci d bat h whic h removed al l metal excep t wher e th e printe ddesign protecte d it . Durin g th e intervenin g years, a tremendou s amoun t o fvery fruitfu l researc h ha s bee n carrie d o n wit h respec t t o foi l gages . Th ewell-established alloy s hav e bee n improve d an d ne w one s developed . I naddition, ther e has been a vast improvement i n the photographic technique scurrently use d i n th e photoetchin g proces s employe d t o manufactur e foi lgages. Th e degre e o f precision wit h whic h gages ca n no w b e produced , an dthe sharpnes s o f definitio n o f the boundarie s o f line elements , hav e made i tpossible t o mak e gage s possessin g a unifor m gage facto r fo r a larg e rang eof gage length s (previously, gage facto r varied slightly with gage length) . Theresult o f these improvement s ha s bee n t o exten d th e advantage s o f th e foi lgage t o a muc h wide r variet y of applications , includin g those a t ver y lo wand ver y high temperatures , an d especiall y for ver y precis e transducers .

Foil gage s ar e availabl e i n variou s gag e length s fro m 1/6 4 in t o 6 in,and i n a wid e variet y o f gri d configurations , including singl e gages , two- ,three-, an d four-elemen t rosettes , hal f bridges , an d ful l bridges . Figur e 1.13shows a fe w o f th e availabl e designs . Standar d alloy s suc h a s constantan ,isoelastic, nichrome , karma , an d platinum - tungsten, as wel l as a numbe r ofspecial proprietar y alloys , ar e use d i n th e sensin g elements .

In general , foi l gage s exhibi t a slightl y highe r gag e facto r an d lowe rtransverse response than their equivalent in wire. Since they are thinner, theyconform mor e easily to surface s with smal l radius of curvature, which meansthey ar e easie r t o instal l i n fillets . A s a resul t of thei r greate r contac t area ,they ca n dissipat e hea t mor e readil y and , i n consequence , i t i s possibl e t ouse higher operating current s (applied voltage) with foi l gages . The relativelylarge contac t area , especially a t th e end s o f the grid , reduce s shearin g stres sin the bondin g agent , an d consequently , foil gage s show comparatively littl ecreep an d hysteresis . Dependin g upo n th e carrier , th e alloy , and it s metal -lurgical condition , foi l gage s (generall y the large r sizes ) wil l measur e strain sprecisely into the rang e o f 1 0 to 1 5 percent. In term s of fatigue, suitabl e gage shave exhibited life i n exces s of ten millio n cycle s at strain s of + 150 0 uin/in.Foil gage s ca n b e obtaine d o n carrier s o f paper , epoxy , phenolic , glas sreinforced resins , an d othe r plastics .

By judicious choic e o f alloy and b y carefu l contro l o f the metallurgica lcondition (col d workin g and hea t treatment) , i t i s possibl e t o produc e foi l


FIG. 1.13. Foi l strai n gages, (a, b) Single-elemen t gages, (c) Stacked two-elemen t rectangularrosette, (d ) Stacke d three-elemen t rectangula r rosette , (e ) Three-elemen t delt a rosette , (f )Two-element rectangula r rosette torque gage. (Courtesy of Measurements Group, Inc .

with it s coefficien t o f linear expansio n an d resistance-temperatur e charac -teristic ver y closel y matche d t o th e coefficien t o f linear expansio n o f som earbitrarily selecte d material . B y this means, i t ha s bee n possibl e t o produc etemperature-compensated foi l gages whose response (within certain limits) is,for practica l purposes, independent of temperature, within a given tempera-ture range .

Weldable foil strain gages

For situation s i n which the conventional installatio n technique s may not beapplicable, weldable foil gages are available (18 , 20, 21). Single-element gagesand T-rosette s (two-element ) are mad e b y premountin g gage s o n a carrie rof stainless steel shim stock approximately 0.005 in thick. Surface preparationof the specime n requires solven t cleanin g and abrasio n wit h silicon-carbidepaper o r a smal l han d grinder . Th e uni t i s then attache d t o th e specime nwith a smal l spo t welde r designed specificall y for thi s purpose .

Sensing elements of constantan, nichrome , and high-temperatur e alloysare available . Th e norma l operatin g temperatur e range s fro m — 320°F t o570°F (-19 5 t o 300°C ) fo r stati c observations , althoug h unde r som econditions a single-loo p wir e (typically nichrom e V ) encased i n a stainles ssteel tub e may b e used t o 925° F (495°C ) o r higher .


1.7. Semiconductor gages (4 , 8, 9 , 22-25)

Within certai n limitations , semiconductor gage s ca n b e use d i n th e sam emanner as metallic gages. However , the semiconductor gage i s really a muchmore elaborat e devic e whose optimu m us e require s a knowledg e o f al l th evariables involved , and th e degre e t o whic h they influenc e th e performanc eof th e instrument . Th e compariso n betwee n th e use s o f meta l an d semi -conductor gage s i s somewha t paralle l t o th e differenc e betwee n playin gcheckers an d playin g chess. Bot h ar e goo d games , bu t ches s ha s a muc hbroader rang e o f opportunitie s fo r makin g move s and , correspondingly ,many mor e possibilitie s of getting into troubl e unless one consider s al l th evariables carefully .

The mai n attractio n o f th e semiconducto r is , of course, th e hig h strainsensitivity o f silicon , which i s th e favore d materia l fo r th e sensin g element .This mean s a relativel y larg e resistanc e chang e pe r uni t o f strain , whichcharacteristic i s helpfu l fo r bot h hig h and lo w value s o f strain.

1. Fo r hig h strains , th e larg e respons e enable s on e t o driv e indicatin gdevices directl y withou t intermediat e amplification . This provide s asimplification whic h is accompanied b y reduce d weigh t and expense .

2. Fo r lo w strains , which produce exceedingl y small changes i n resistanceof metal gages, the semiconductor gages wil l develop unit changes abou t50 time s greater , wit h th e resul t tha t th e indication s o f R/R ca n b emeasured convenientl y an d precisely .

As contrasted wit h th e abov e advantages , one mus t also recognize , andbe able t o cop e with , certain disadvantages .

1. Th e uni t chang e i n resistanc e (whic h i s based o n th e initia l resistance,R0, o f the unstresse d senso r a t temperatur e T 0) is a nonlinea r functio nof th e strain , althoug h fo r som e specia l condition s i t ca n b e take n a slinear fo r smal l strai n excursions.

2. Th e larg e resistanc e chang e pe r uni t o f strain , which i s the ver y thin gthat makes the semiconductor gage attractive, may also present a minorproblem du e to the fact that , in the process o f installation, the resistanc eof the gage may b e altered considerabl y from th e value which prevailedin th e unstresse d conditio n o f the sensin g element. O n thi s account, i tis necessar y t o determin e th e gag e resistanc e followin g installation s othat, i f necessary, an appropriat e correctio n ca n b e mad e fo r th e gag efactor.

3. Th e resistanc e o f the gag e wil l chang e wit h chang e i n temperature .4. Th e strai n sensitivity , o r gag e factor , wil l chang e wit h chang e i n

temperature.

Investigation o f silico n reveal s tha t bot h th e strai n sensitivit y an d th etemperature sensitivit y (change o f resistance with temperature) vary consider -ably wit h th e quantit y of impurity whic h i s present. I t i s also observe d tha t


high sensitivit y to strai n i s accompanie d b y hig h sensitivit y to change s i ntemperature. This suggests that som e compromise betwee n strain sensitivityand temperature response may be desirable, and perhaps essential, dependin gupon th e particular application .

Fortunately, b y suitable doping (introductio n of controlled amount s ofimpurities) durin g th e manufacturin g process , th e strai n an d temperatur esensitivities can be varied and adjusted (although not independently) to meetspecified requirements . Therefore, by suitable procedures i n the manufactur-ing process , i t i s possible t o achiev e a desired compromise whic h wil l resultin muc h improve d temperatur e characteristic s a t th e expens e o f a modes treduction i n strain sensitivity . Practical consideration s indicate tha t a goo dbalance i s achieved when the gag e facto r is about 120 .

Since semiconductor gages are available with both positive and negativegage factors, another approach , althoug h perhaps a more difficul t one , i s totake advantage of the characteristics of the electrical circuit of which the gageforms a part , and t o emplo y two simila r gages with gage factor s o f oppositesign.

Due t o th e relativel y larg e numbe r o f variable s involved , an d con -sequently th e somewha t mor e comple x procedur e require d fo r convertin gresistance chang e int o term s o f strain , i t seem s unlikely , a t leas t fo r th epresent, tha t semiconducto r gage s wil l replac e metalli c gage s fo r purpose sof stress analysis, excep t perhaps, unde r specia l circumstances involving th edetermination o f very smal l strains.

However, for transducers, in which gages can be installed under carefull ycontrolled factor y conditions , an d subsequentl y calibrate d i n complet ebridges, th e hig h outpu t o f th e semiconductor s make s the m exceedingl yattractive. I t seem s tha t semiconducto r strai n gage s wil l achiev e greates tsuccess and optimu m utilit y i n thi s type of application.

1.8. Some other types of gages

Temperature gages

Examination o f the characteristic s o f metal an d semiconducto r strai n gage sreveals tha t change s i n resistanc e occu r no t onl y a s a resul t o f changes i nstrain, bu t als o fro m change s i n temperature . Althoug h th e respons e t otemperature ma y complicat e th e determinatio n o f strain , i t nevertheles sprovides th e possibilit y o f making , an d using , temperatur e sensor s wit hessentially th e same technique s as those which are employe d i n the makin gand usin g of strain gages .

The choice of material for the sensing element, of course, will be differen tfor thes e tw o applications . Whe n i t i s desire d t o measur e strain , wit h aminimum influenc e fro m temperatur e changes , a copper-nicke l allo y o fthe constanta n typ e i s frequentl y employe d fo r temperature s i n th e rang efrom abou t -250° F t o abou t SOO T (155-260°C) . Fo r lowe r o r highe r


temperatures, i t i s necessary t o selec t anothe r typ e o f allo y (26) . However ,for a temperature sensor , i t is preferable to choose a material, such as nickel,platinum, o r a n iridium-platinu m alloy , whic h possesse s a muc h greate rresponse t o change s i n temperature . Fo r semiconducto r materials , th eprocessing i s varied t o produc e th e preferre d characteristics for either strai nor temperatur e sensing.

For a numbe r o f years , bonde d wir e temperatur e sensor s hav e bee ncommercially available , followe d mor e recentl y b y foi l temperatur e gage s(27, 28) . Foi l temperatur e gage s hav e severa l advantage s ove r wire-woundsensors i n tha t the y ar e les s expensive , no t a s fragile , an d thei r time -temperature response i s similar to tha t of a strain gage. Standar d strai n gageinstrumentation ma y als o b e use d wit h them .

For convenienc e i n makin g observations , sensor s an d thei r signal -conditioning networks have been designed t o produce signal s correspondin gto indication s o f 1 0 or 10 0 microstrain pe r degre e Fahrenheit . Therefore ,when th e strai n indicato r i s referenced t o som e temperature , on e i s able t oobtain a direc t readin g o f al l other temperature s withi n th e workin g rangeof the system. For example , if a temperature sensor an d networ k is used tha tprovides a n indicatio n o f 1 0 microstrain pe r degre e Fahrenheit , th e initia lbalance o f th e indicato r ma y b e adjuste d s o tha t th e readin g wil l b e 75 0microstrain whe n th e senso r i s actuall y 75° F (24°C) . Then , fo r an y sub -sequent observation , th e temperatur e i n Fahrenhei t wil l b e represente d b ythe indicato r readin g divided by 10 . If a subsequen t readin g turns out t o b e830, then th e temperatur e at th e senso r is 83 F (28 0C).

The obviou s advantage o f this method o f determining temperature liesin th e fac t tha t a standar d strai n indicatin g (an d recording ) syste m ca n b eemployed, without an y modificatio n at all , for the measuremen t o f tempera-ture a t strai n gag e locations , o r elsewhere , b y th e simpl e procedur e o fswitching the temperature sensor (wit h it s conditioning network) in and ou tof th e indicatin g circuit just a s i f it wer e another strai n gage .

Crack measuring gages

Another instrument incorporating certain features of the strain gage is knowncommercially a s th e Kra k Gage . It s mai n purpos e i s t o monito r th eprogression o f cracks whic h usually develop as a resul t o f fatigue cause d b yrepeated stressing . If the progres s o f a crack i s watched, a part can b e take nout o f service before a disaster occurs , which is a very valuable consideratio nin th e aircraf t an d man y othe r industrie s (29) .

A schemati c diagra m o f th e gage , show n i n Fig . 1.14 , i s produce d b yHartrun Corporatio n i n a variet y of different size s (30). I t possesse s certai ncharacteristics whic h ar e lik e thos e o f th e strai n gage , bu t it s us e i s verydifferent. Basically , the Kra k Gag e consist s o f a constantan foi l senso r 5 urnthick mounte d o n a n epoxy-phenoli c o r cas t epox y carrier , dependin g o nthe operating temperature . The carrier an d th e gage ar e cemented t o the tes t


FIG. 1.14 . Schemati c diagram o f a crack measuring gage. (From ref . 30.)

piece, o r machin e part , b y th e usua l strai n gag e bondin g procedur e a t alocation wher e a crack i s expected t o start , o r may already have started. Thepositioning o f the gag e i s such tha t i t wil l be cracked unde r it s centerline instep wit h the materia l underneat h it . The gag e is energized wit h a constan tcurrent, usually in the range between 0 and 10 0 milliamperes, and the changein potentia l dro p betwee n it s tw o inne r leads i s a measur e o f the distanc eby whic h th e crac k ha s advanced . Sinc e thes e gage s hav e a resistanc e o fabout 1 ohm before the crack commences, they cannot be used with ordinarystrain gag e equipment.

Another crac k detectio n gag e i s th e CD-Serie s produce d b y Micro -Measurements (31). This gage is used to indicate the presence of a crack, an dcrack growth rate may be monitored b y using several CD gages at a location.The C D gag e overcome s tw o o f the limitation s suffered b y th e us e o f thincopper wires . These ar e th e possibilit y o f a crac k progressin g beyon d th ewire withou t breakin g it , an d als o th e failur e o f the wir e b y fatigu e whe nlocated i n a regio n subjecte d t o cycli c strain o f large magnitude.

The gages consist of a single strand of high-endurance beryllium-coppe rwire on a tough polyimid e backing. A rigid high-modulu s adhesiv e i s usedto bon d th e senso r t o th e polyimid e backing . A crac k tha t i s growin gunderneath th e gage indices local fractur e o f the sensing wire and open s th eelectrical circuit . Bondin g o f the gages t o a structur e o r a machin e ca n b eaccomplished wit h conventiona l strai n gag e adhesive s tha t ar e compatibl ewith polyimid e backing.

Friction gages

For stres s probing , especiall y fo r vibratin g stresses , whe n a numbe r o fobservations ar e to b e made quickl y without taking tim e to instal l a large rnumber o f strain gages , a ver y usefu l typ e of gage ha s evolve d (18).

36 THE BONDED ELECTRICAL RESISTANCE STRAIN GAGE

This i s a conventiona l 120-oh m foi l gag e t o whic h strain i s transmittedby friction . Th e gag e i s bonde d t o on e fac e o f a rubbe r sheet , the n emer ypowder i s cemented ove r the gage face t o provid e a frictiona l surface . T o th eother sid e o f th e rubbe r i s cemente d a meta l backin g plate . Th e gag e i spressed agains t th e tes t membe r s o that the emergy powder contact s th e testsurface, wher e th e frictio n i s great enoug h t o transmi t th e surfac e strains t othe sensin g elemen t o f th e strai n gage . Thi s devic e ca n b e move d quicklyand easil y fro m plac e t o place , thu s enabling one t o mak e a rapi d surve ywith a minimu m amount o f equipment.

Embedment gages

Embedment gage s an d transducer s ar e designe d an d use d primaril y t omeasure curin g an d loadin g strain s i n concrete . The y ma y als o b e used ,however, wit h resins , ice , asphalt , an d othe r materials . Ther e ar e severa lvariations o f these gage s an d transducers .

One i s a polyeste r mol d gag e tha t ca n b e supplie d a s a singl e gage , atwo-element rectangula r rosette , o r a three-elemen t rectangula r rosette .Standard wir e gage s an d lea d wire s ar e hermeticall y sealed betwee n thi nresin plates , thu s waterproofin g th e unit . Th e uni t i s the n coate d wit h acoarse gri t t o enhanc e bondin g betwee n i t an d concrete . Excellen t electricalinsulation i s exhibited even afte r severa l months o f embedment (18).

A transducer i s available in either half- o r full-bridg e arrangements , thusgiving temperatur e compensation . Th e gage s i n thi s transduce r ar e mad e ofa specia l allo y foi l encase d i n a low-elastic-modulu s materia l i n orde r t oprevent swellin g an d t o minimiz e loadin g effects . A quarte r bridg e i s alsoavailable fo r temperatur e measurement (18).

Another embedmen t gag e use s nickel-chrom e wir e in a quarte r bridg eand come s i n gag e length s o f 2 , 4, an d 6 in (20) . The gag e wir e i s enclose din a 0.040-i n diamete r stainles s stee l tub e an d i s insulate d b y compacte dmagnesium oxid e powder . En d disk s wit h thre e equall y space d hole s ar eattached a t eac h en d o f th e stainles s stee l tub e fo r anchorin g th e gage .Anchoring i s accomplishe d b y tyin g wire s throug h th e hole s i n th e disks ,then pullin g the m radiall y outward an d tyin g the m t o th e structur e o r t oreinforcing bars . Th e wire s ar e pulle d tau t bu t shoul d no t loa d th e gag ealong it s axi s o r appl y a torque . Th e gag e lengt h should b e fou r time s th esize o f the larges t aggregat e i n orde r t o provid e strai n averaging , an d i t i simportant tha t concret e shoul d contac t th e gag e alon g it s entire lengt h fo roptimum bondin g an d strai n transfer . Figure 1.1 5 shows thi s gage.

1.9. Brittle lacquer coatings (3)

Brittle lacquer coatings hav e been mentioned earlie r as a tool i n experimentalstress analysis , and s o a few comments are in order, since these coatings hav ebeen use d quit e extensively. Thei r developmen t ha s evolve d ove r th e years ,having it s beginning in the observatio n tha t brittl e oxide coats o n hot-rolled


FIG. 1.15. Typica l embedment gage . (Courtesy o f Eaton Corporation. )

steel cracke d whe n a membe r wa s loaded. Thi s led , in th e earl y par t o f the1900s, t o th e us e o f varnish , lacquer , o r molte n resin s o n machin e o rstructural members . Whe n loade d i n th e elastic region , th e coatin g cracke din a direction normal to the maximum principal strain direction. In the 1930s ,Greer Elli s developed a brittl e lacquer whil e at th e Massachusett s Institut eof Technology . I t wa s subsequentl y markete d i n 193 8 unde r th e nam e o fStresscoat b y Magnaflu x Corporation o f Chicago, Illinois .

Brittle lacquers are sensitive to both temperatur e and relativ e humidity.For thi s reason , the y ar e mad e i n a numbe r o f formulation s fo r specifi ctemperature an d relativ e humidit y conditions. Whe n plannin g a test , on emust anticipat e th e temperatur e an d relativ e humidity at th e tim e th e tes tis t o b e conducted , an d the n choos e th e coa t accordingly . Whe n properl ychosen, the threshol d strai n o f the coatin g wil l be approximately 500 uin/in.If, however, the temperature o r the relative humidity increases, the thresholdstrain wil l increase and perhaps produc e no cracks within the loading range .Conversely, a decrease i n temperature o r relativ e humidity will decrease th ethreshold strain , resulting, in the worst case, in the coating becomin g craze d(cracking int o a rando m pattern) .

The applicatio n o f a brittl e lacque r consist s o f severa l steps . Th e tes tmember i s first thoroughly cleaned t o insur e that i t is free o f scale, dirt, an doils. The member is next sprayed with a coat of aluminum powder in a carriersolvent an d allowe d t o dr y fo r a t leas t 3 0 minutes. This undercoa t form s areflective coatin g tha t enable s on e t o se e cracks i n the lacque r mor e easily.Next, the brittle lacquer is applied in a number of thin coats until its thicknessis approximatel y 0.00 3 t o 0.00 6 in thick . A t th e sam e tim e tha t th e tes tmember i s coated, a t leas t fou r calibratio n bar s shoul d b e prepare d i n th esame manner an d kep t wit h the test member. The entire group, tes t memberand calibratio n bars , i s the n allowe d t o cur e fo r a t leas t 1 8 hours befor etesting.

The tes t membe r i s loaded i n increments, and a t eac h incrementa l load


the brittl e coa t i s examined fo r cracks . Th e tip s o f the crack s i n eac h are awhere the y appea r ma y b e outline d wit h a felt-ti p pen . A s th e loa d i sincreased, th e crac k growt h a t eac h are a i s marked, a s wel l as notin g othe rareas where new cracks appear . This proces s is continued unti l the maximumload i s reached. Whe n th e yiel d poin t o f the materia l i s attained i n any are aof th e tes t member , th e brittl e coa t wil l flake off .

Although th e brittl e coat crack s onl y unde r tensil e strains , i t ca n als obe use d t o determin e compressiv e strains . To accomplis h this , th e ful l loa dis applied t o th e tes t membe r an d hel d fo r a t leas t 3 hour s afte r th e tensil estrains ar e determined . Durin g th e hol d tim e th e brittl e coa t creep s an drelaxes. Th e loa d i s then remove d a s quickl y a s possible , wit h th e coa t the nreacting t o th e compressiv e strain s a s thoug h the y were tensil e strains .

At the start o f the test , a calibrator! ba r i s loaded int o a cantilever fixtureand on e end deflecte d a known amount. The ba r i s subjected t o strains alongits length, and th e minimu m strain at whic h a crack i s observed i s recorded ;this i s the threshol d strain . As the tes t progresses , particularl y over a perio dof time in whic h the temperatur e o r relativ e humidity may chang e an d thu schange th e threshol d strain , other calibration bar s ca n b e tested a t interval sin orde r t o determin e whethe r o r no t th e threshol d strai n ha s changed .

The brittl e coa t ca n b e treate d t o enhanc e th e cracks . Th e crack s ca nbe recorded b y photographing , markin g a drawing , o r som e othe r means . Iffurther testin g i s t o b e don e wit h strai n gage s (a s i s usuall y th e case) , th ecoat ca n b e strippe d of f i f the sam e membe r i s t o b e use d an d strai n gage sapplied. Sinc e th e principa l strain direction s ar e known , two strai n gage s ( atwo-element rectangula r rosette) ma y b e applie d i n these direction s and th eprincipal stresse s computed . Th e advantage s o f the brittl e coat are :

1. Th e brittl e coat an d it s crac k patter n allo w on e t o se e th e strai n (an dstress) distributio n over mos t o f the entir e tes t member .

2. Whe n strai n gage s ar e applie d i n the direction s o f the principa l strain sin th e variou s area s o n th e tes t member , onl y tw o gage s ar e require drather tha n th e thre e tha t woul d b e necessar y i f th e principa l strai ndirections wer e unknown . Thi s result s i n a savin g o f bot h tim e an dmoney.

3. Th e metho d i s relativel y inexpensiv e and i s extremel y usefu l fo r apreliminary investigatio n prio r t o a detaile d strai n gag e study .

Although brittle lacquers hav e been use d extensively in order t o observ ethe strai n distributio n o n th e surfac e o f a member , thei r mai n us e has bee nas a n ai d i n th e placemen t o f strain gages . On e shoul d b e aware , however ,of th e us e of photoelastic coating s tha t ca n b e applie d t o a structure . The ygive full-field dat a that accuratel y identif y area s o f high strain , and constitut ea nondestructiv e test . The member , unlik e brittle lacquer tests , can b e teste da numbe r o f times , wit h th e result s bein g recorde d o n fil m o r vide otape. Fo r mor e information , on e shoul d consul t eithe r manufacturer s o fphotoelastic equipmen t o r an y o f several book s o n th e subject .

FUNDAMENTAL CONCEPT S FOR STRAI N GAGE S 3 9

Appendix 1

Approximate strai n sensitivitie s of some metal s (11 )

Hard drawn Annealed

Metal

Sensitivity Sensitivity Change Sensitivity Sensitivity Changein in point in in point

low high (strain low high (strainrange range percent) range range percent)

SilverPlatinumCopperIronNickelFerry (60/4 0 Cu-Ni)Minalpha (Manganin )10 percent iridium-platinu m10 percent rhodium-platinum40 percent silver-palladiu m

2.96.12.63.9

Negative2.20.84.85.50.9

2.42.42.22.42.72.12.02.12.41.9

0.80.40.50.8-

0.50.60.40.50.8

3.05.92.23.7

Negative2.20.63.95.10.7

2.32.32.22.12.32.21.91.92.02.0

0.20.3-

0.5---

0.30.40.5

REFERENCES

1. "Brittl e Coatin g fo r Stres s Analysi s Testing, " Bulleti n S-109 , Measurement sGroup, Inc. , P.O . Bo x 27777 , Raleigh, NC 27611 , 1978 . (Now ou t o f print.)

2. "Genera l Instructions for the Selection and Us e of Tens-Lac Brittl e Lacquer an dUndercoating," Instructio n Bulletin 215-C, Measurements Group , Inc. , P.O. Box27777, Raleigh, NC 27611 , 1982 . (Now ou t o f print. )

3. "Usin g Stresscoat," Electri x Industries, Inc., P.O . Bo x J, Roundlake , IL 60073 .4. Sanchez , J. C. and W. V. Wright, "Recent Development s in Flexible Silicon Strain

Gages," in Semiconductor and Conventional Strain Gages, edited by Mill s Dea nIII an d Richar d D . Douglas, Ne w York , Academic Press , 1962 , pp. 307-345 .

5. Mack , Donal d R. , "Linearizing th e Outpu t o f Resistance Temperatur e Gages, "SESA Proceedings, Vol. XVIII, No. 1 , April 1961 , pp. 122-127 .

6. Thomson , W . (Lor d Kelvin) , "O n th e Electrodynami c Qualitie s o f Metals, "Philosophical Transactions o f th e Royal Society o f London, Vol . 146 , 1856 , pp .649-751.

7. Sette , W. J., L. D. Anderson, and J . G. McGinley , "Resistance-Strain Character -istics o f Stretched Fin e Wires, " Davi d Taylo r Mode l Basin , Repor t No . R-212 ,Sept. 1945 .

8. Dorsey , James , "Semiconducto r Strai n Gages, " Th e Journal o f EnvironmentalSciences, Vol. 7, No. 1 , Feb. 1964 , pp. 18-19 .

9. Dorsey , James , Semiconductor Strain Gage Handbook, Par t 1 . BLH Electronics ,75 Shawmut Road , Canton , MA 02021. (No w ou t o f print. )

10. Baker , M . A. , "Semiconducto r Strai n Gauges, " i n Strain Gauge Technology,edited b y A . L. Window an d G . S . Holister, Londo n an d Ne w Jersey , AppliedScience Publisher s Inc., 1982 , p. 274. Copyright Elsevie r Science Publishers Ltd .Reprinted wit h permission.


11. Jones , E . an d K . R . Maslen , "Th e Physica l Characteristics o f Wir e Resistanc eStrain Gauges, " R . an d M . No . 266 1 (12,357) , A.R.C . Technica l Report , He rMajesty's Stationer y Office , London , 1952 , Reproduced wit h th e permissio n o fthe Controlle r o f He r Britanni c Majesty's Stationer y Office .

12. d e Forest , A . V. , "Characteristics an d Aircraf t Application s of Wir e Resistanc eStrain Gages, " Instruments, Vol . 15 , No. 4 , Apri l 1942 , pp. 112-114 , 136-137.

13. Weibull , W., "Electrica l Resistanc e o f Wire s wit h Larg e Strains. " Nature. Vol .162, pp . 966-967 . Copyrigh t (r ; 194 8 Macmilla n Magazine s Limited.

14. Shoub , H. , "Wire-Resistanc e Gage s fo r th e Measuremen t o f Larg e Strains, "David Taylo r Mode l Basin , Report No . 570 , Marc h 1950 .

15. Biermasz . A . J. , R . G . Boiten , J . J . Koch , an d G . P . Roszbach , "Strai nGauges—Theory an d Application, " Philips Technical Library, Philips Industries,Eindhoven, Netherlands , 1952.

16. Meier . J . H. , "On th e Transverse-strai n Sensitivity o f Foi l Gages, " ExperimentalMechanics, Vol . 1 , No. 7 , July 1961 , pp. 39-40 .

17. "Strai n Gages , SR-4, " BL H Electronics , Inc. , 75 Shawmu t Road , Canton , M A02021, 198 5 Edition .

18. "TM L Strai n Gauges," E-10 I V and E-10 1 Y, Tokyo Sokk i Kenkyujo Co., Ltd. ,Tokyo, Japan , 1988 . Distribute d by Texa s Measurements , Inc. , P.O . Bo x 2618,College Station , T X 77841.

19. Gibbs , Josep h P. . "Tw o Type s o f High-temperatur e Weldabl e Strai n Gages :Ni-Cr Half-bridg e Filament s an d Pt- W Half-bridg e Filaments," Proceedings o fthe Second SESA International Congress on Experimenal Mechanics, Washington ,DC, Sept . 2 8 t o Oct . 1 , 1965, pp. 1-8 .

20. "Weldabl e an d Embeddabl e Integra l Lea d Strai n Gages, " Application s an dInstallation Manual , Eato n Corp. , Ailtec h Strai n Gag e Products , 172 8 Maple -lawn Rd. , Troy, M I 48084 , 1985 .

21. "Catalo g 500 : Par t A—Strai n Gag e Listings ; Par t B—Strai n Gag e Technica lData," Measurement s Group , Inc. , P.O . Bo x 27777 , Raleigh , N C 27611 ,1988.

22. Sanchez , J . C., "The Semiconducto r Strain Gage—A Ne w Tool fo r ExperimentalStress Analysis, " i n Experimental Mechanics, edite d b y B . E . Rossi , Ne w York ,The Macmilla n Company , 1963 , pp. 255-274.

23. Vaughn , John, Application o f B& K Equipment t o Strain Measurements, Brue l &Kjaer, Naerum , Denmark , 1975 , Ch. 10 .

24. "Semiconducto r Strai n Gages, " SR-4 Application Instructions, BLH Electronics ,Inc., 7 5 Shawmut Road , Canton . M A 02021 , 1986.

25. Dorsey , James , "Data-reductio n Method s fo r Semiconducto r Strai n Gages, "Experimental Mechanics, Vol . 4, No . 6 , June 1964 , pp . 19 A -26A.

26. Weymouth , L . J. , "Strai n Measuremen t i n Hostil e Environment, " AppliedMechanics Reviews, Vol . 18 , No. 1 . Jan. 1965 , pp. 1-4 .

27. "Cryogeni c Linea r Temperature Sensor, " Produc t Bulleti n PB-104-3 , Mesasure -ments Group , Inc. , P.O. Bo x 27777 , Raleigh, NC 27611 , 1983.

28. "Temperatur e Sensor s an d LS T Matching Networks," Produc t Bulleti n PB-105-7, Measurement s Group , Inc. , P.O . Bo x 27777 , Raleigh, NC 27611 , 1984 .

29. Liaw , Peter K. , W. A. Logsdon, L . D. Roth , and H . R . Hartmann, "Krak-Gage sfor Automate d Fatigu e Crac k Growt h Rat e Testing: A Review, " ASTM SpecialTechnical Publication No . 877. 1989 . pp. 177-196 . Copyrigh t ASTM . Reprinte dwith permission .


30. Hartmann , H. R. and R . W. Churchill, "Krak-Gage, a New Transducer for CrackGrowth Measurement, " presente d a t SES A Fal l Meeting , Keystone , CO , Oct .1981.

31. "CD-Serie s Crac k Detectio n Gages, " Produc t Bulleti n PB-118 , Measurement sGroup, Inc. , P.O . Bo x 27777, Raleigh, NC 27611 , 1984.

STRESS-STRAIN ANALYSIS ANDSTRESS-STRAIN RELATIONS

2.1. Introduction

The materia l i n Chapte r 2 should b e familia r fro m course s i n mechanic s ofmaterials an d design , an d s o serve s a s a review . Th e notatio n an d sig nconvention fo r both stress and strain follow that generally given in the theoryof elasticity.

Strain gage s ar e applie d t o a surfac e tha t i s usuall y stres s fre e i n adirection normal t o th e strain gag e surface . Fo r thi s reason, th e transforma-tion equations for plane stress are developed instead o f the more complicatedtriaxial stres s state . Th e necessar y equation s ar e derive d tha t enabl e u s t otransform fro m on e coordinat e syste m t o another . Furthermore , w e ca ncompute th e principa l stresses an d determin e thei r orientation relativ e to achosen coordinat e system .

Since we cannot determin e stres s experimentally by direct measurement ,we resor t t o measurin g strai n o n a surfac e throug h th e us e o f a strain -measuring device . I n orde r t o mak e us e o f th e experimentall y determinedstrains, transformatio n equation s fo r plan e strai n ar e generate d tha t ar esimilar i n form to th e transformation equation s for plane stress. Here w e seethat th e orientations o f the principal strains are identica l to the orientation sof th e principa l stresse s fo r th e chose n coordinat e system .

Although al l o f th e necessar y value s wante d ma y b e handle d throug hcalculation, i t i s ofte n desirabl e t o determin e th e value s graphically . T oaccomplish this , Mohr's circl e for stres s and fo r strai n are generated . Thes ediagrams allo w u s t o visualiz e th e transformatio n fro m on e coordinat esystem t o another , and , i f they ar e accuratel y drawn, wil l giv e satisfactoryanswers. Wit h th e availabilit y of hand-held calculators , though , i t i s muc heasier to draw the diagrams freehand, observe the required orientations, andthen calculat e th e answers . I n drawin g th e circles , not e th e definitio n forpositive shearin g stres s an d shearin g strain.

You wil l observ e tha t materia l propertie s d o no t ente r int o th edevelopment of the transformation equations. The transformatio n equationsfor stres s ar e base d o n th e stati c equilibriu m o f a n element , whil e th etransformation equation s fo r strai n ar e base d o n th e geometr y o f smal ldeformations o f the element . In orde r t o relat e th e two , material propertie s

2

STRESS-STRAIN ANALYSI S AN D STRESS-STRAI N RELATIONS 4 3

now enter . Th e stress-strai n relationship s ar e give n fo r th e differen t state sof stres s an d strain .

2.2. Basic concepts of stress

When a solid bod y i s acted upo n b y a system of forces, which may b e eitherexternal o r internal , o r bot h externa l an d internal , i t i s said t o b e subjectedto stress . I n general , thi s mean s tha t force s ar e transmitte d fro m on eelemental particle t o another within all or part o f the body. How these forcesare distribute d o n th e externa l surfaces , or throughou t the interio r body , isof vita l importance , sinc e the abilit y of the bod y materia l t o withstan d th eaction o f the forces depends upo n th e force intensity prevailing at each poin twithin th e material .

Usually w e think o f stress a s th e effec t o f force s o n part , o r all , o f th esurface o f a body , o r internall y a s th e influenc e which th e force s acting o none side o f a sectio n (usually a plane section ) through th e bod y exer t upo nthe materia l o n th e othe r sid e o f the section .

Since, from practica l considerations, the forces which act on solid bodiesmust, of necessity, be distributed over areas (o r throughou t the volume) , wemust b e rathe r specifi c regardin g ou r meanin g o f th e ter m stress . I t i ssometimes use d t o indicat e tota l force , and unde r othe r condition s implie sforce pe r uni t area . Bot h usage s ar e correct , bu t ever y now an d the n th eexact meanin g i s somewhat loosel y implied.

To b e technically correct, on e shoul d sa y "total stress " whe n referrin gto force , and "intensit y o f stress" or "uni t stress " whe n forc e per uni t are ais meant . However , whe n onl y on e o f th e tw o meaning s i s require d i n aparticular discussion , i t i s quite commo n t o us e th e wor d "stress " by itselfwith th e word s total , intensit y of , o r unit , bein g understood . Fo r ou rpurposes, th e ter m stres s wil l b e used t o indicat e forc e pe r uni t area .

Figure 2.1 a show s such a body acte d upo n b y forces P1, P2, P3, and F 4.An imaginar y cuttin g plane A B i s passed throug h th e bod y an d th e uppe rportion o f the bod y i s removed. In orde r fo r the lower portion to remai n i nequilibrium, a syste m o f forces , representing th e effec t o f the uppe r par t o fthe body , act s upo n th e cu t surfac e a s show n i n Fig . 2.1b . On e o f th eelemental force s is represented b y the force P actin g o n the incrementalarea A I f all such force s ar e summe d ove r th e entire area , the resultan twill b e a forc e (no t normal , i n general , t o plan e AB ) havin g th e prope rmagnitude an d directio n t o maintai n equilibrium.

We now turn our attention to the force P an d define stress at a point as

Since the loading o n th e body i n Fig . 2.1 is complex, w e expect the stres s t ovary in intensity from poin t t o point o n the cut surface . Thus, when we speak


FIG. 2.1. Bod y i n equilibriu m acte d upo n b y externa l forces .

of stress , w e mus t defin e th e poin t o n th e cu t surfac e on whic h i t i s acting .Furthermore, P wil l generall y no t b e norma l t o th e surfac e bu t wil l b einclined a t som e angl e 9 as show n i n Fig . 2.1c . The lin e N i s normal t o th esurface an d th e line S lie s on the surface , and s o P ca n be resolved int otwo component s alon g line s N an d S , namel y P n an d P S. Usin g th edefinition o f stress given b y Eq . (2.1) , w e now hav e th e tota l stres s resolve dinto tw o components ; w e call th e stres s directe d alon g N th e norma l stress ,a, and th e stres s directe d alon g S th e shearin g stress , T. The norma l stres s awill b e tensil e (+ ) i f it tend s t o separat e th e materia l o n opposit e side s ofthe section , o r i t wil l b e compressiv e ( —) i f i t tend s t o pus h togethe r th ematerial on opposit e side s of the section . The shearin g stress T has a tendencyfor th e materia l o n on e sid e o f th e sectio n t o slid e b y th e materia l o n th eother sid e o f the section .

When the force acting on the area i s distributed uniformly ove r the area ,each elemen t o f the are a wil l be subjecte d t o th e sam e intensit y of loading ,and th e magnitude of the stress at every point wil l be the same as the averag evalue, which is computed b y dividing the total forc e by the whole area. Thus,for uniforml y distribute d stress ,

We wil l generall y b e workin g with plan e stress . Suppose , i n Fig . 2.1c ,that all the elemental forces P wer e contained i n planes paralle l to the planedefined b y lines N an d S . The normal stresse s and th e shearing stresses wouldalso li e in thes e planes , an d s o n o stresse s woul d appea r i n plane s norma lto th e plan e containin g line s N an d S . Thi s conditio n give s u s th e plan estress state .

STRESS-STRAIN ANALYSI S AN D STRESS-STRAI N RELATIONS 45

FIG. 2.2. Elemen t subjected to plan e stress.

2.3. Biaxial stresses

Since mos t o f our problem s ar e plan e stres s problems , w e want t o develo pthe transformation equations for this stress state. Figure 2.2 shows the planestress state , where az — -cxz = iyz = 0. The sig n convention for th e stresse s isthe following: (1) the normal stress is positive (tensile) it it is directed outwardfrom th e plane , negativ e (compressive) i f directed inwar d toward th e plane ;and (2 ) the shear stresses are positive when directed in a positive coordinat edirection o n a plan e whos e outwar d norma l i s directe d i n a positiv ecoordinate direction, or when directed in a negative coordinate direction o na plane whose outward normal is directed in a negative coordinate direction .In Fig . 2.2 , al l stresse s ar e positiv e accordin g t o th e sig n conventio n jus tstated.

Suppose the element in Fig. 2.2 has a cutting plane, AB, passed throug hit a s show n in Fig . 2.3 . We now wan t t o determin e th e stresse s i n th e newx'y' system , where x' i s normal t o plan e A B an d / lie s in plane AB . Thenormal stres s actin g o n plan e A B i s designated a s a x., whil e th e shearin gstress o n tha t plan e i s rxy. I n orde r t o determin e th e ne w stress state , theequilibrium o f the elemen t in Fig . 2. 3 must be considered. I f forces i n th e x 'direction ar e summe d (takin g the distanc e in th e z direction , o r norma l t othe paper , a s unity) , th e following results:

From Fig. 2.3, we see that OA/AB = cos 9 and OB/AB = sin 9. Dividing eachterm b y A B an d usin g these relationships , <r x. ca n b e expresse d i n term s ofax, ff y, x xy, and 9 , Thus,

46 THE BONDE D ELECTRICA L RESISTANCE STRAIN GAG E

FIG. 2.3. Elemen t cut b y plan e AB .

By summin g force s in th e y ' direction , w e obtai n

The expression s fo r a x, an d T X.... ar e rewritte n as

Equations (2.5 ) and (2.6 ) allow the determination o f ax, an d r x,y, a t a poin tfor an y valu e o f the cuttin g plane angl e 0 .

Following th e sam e procedur e a s before , the expressio n fo r i x•,,. becomes

Equations (2.3 ) and (2.4 ) can b e expresse d i n term s o f 28 b y usin g thefollowing identities :

STRESS-STRAIN ANALYSI S AN D STRESS-STRAI N RELATION S 4 7

As th e angl e 2 9 varie s fro m 0 ° t o 360° , a x, wil l chang e i n value . Th eobjective i s to find its maximum and minimu m values. This i s accomplishedby differentiatin g ff x- wit h respec t t o 9 an d settin g th e resultin g equatio nequal t o zero . Fro m Eq . (2.5),

The directions of the principal stresses , and therefor e the principal axes,are determine d fro m Eq . (2.7) . Thus , tw o values , 90 ° apart , fo r 9 ar edetermined. On e valu e corresponds t o th e angl e measure d fro m th e x axi sto th e firs t principa l axi s (counterclockwis e i s positive ) alon g whic h th emaximum principa l stres s acts . Th e othe r valu e corresponds t o th e angl emeasured fro m th e x axi s t o th e secon d principa l axi s alon g whic h th eminimum principa l stress acts . Equation (2.7 ) by itsel f doe s no t allo w us t odistinguish betwee n th e tw o axes , an d s o w e must cal l i n anothe r trigono -metric relationship in order t o distinguis h between the two.

There ar e tw o quadrant s i n which tan 20 can hav e th e valu e given byEq. (2.7); these are the first and thir d quadrants. Considerin g first quadrantvalues fo r Eq . (2.7) , we have

Substituting th e value s o f si n 20 an d co s 20 give n b y Eqs . (2.8 ) an d (2.9) ,respectively, int o Eq . (2.5 ) result s i n G X, = a 1. Carryin g ou t th e require dalgebra,

If third-quadran t value s ar e used , si n 20 an d co s 20 ar e negative .

Dividing each ter m b y co s 29 results in


Substitution of these values into Eq. (2.5) gives the second principa l stres s as

We ca n no w writ e th e equatio n fo r th e principa l stresse s a s

Since Eqs . (2.8 ) an d (2.9 ) giv e th e value s o f si n 20 an d co s 20 fo r th eprincipal stresses , substitutio n of these value s int o Eq . (2.6) shows t o b ezero. Thi s tell s u s tha t ther e i s no shearin g stres s o n th e plane s containin gthe principal stresses . This is also apparent i f Eqs. (2.6) and (a ) are compared.

In orde r t o determin e th e orientatio n o f wit h respec t t o th e x axis ,two o f the thre e trigonometri c relation s give n b y Eqs . (2.7) , (2.8), and (2.9 )must b e used .

The sam e procedure ca n b e use d i n findin g th e maxmu m valu e o f T ,in the x y plane . Thi s is achieved b y differentiating . with respect t o 0 an dsetting th e resultin g equatio n equa l t o zero . Fro m Eq . (2.6),

Dividing each ter m by cos 28,

Note tha t Eq . (2.11 ) i s the negativ e reciproca l o f Eq . (2.7) . I n thi s case , th evalue o f ta n 2 0 give n b y Eq . (2.11 ) wil l b e negativ e i n eithe r th e secon d o rfourth quadrant . Takin g second-quadran t values ,

Substituting thes e value s o f si n 20 an d co s 29 int o Eq . (2.6 ) result s i nrx,y, = T max. Carrying ou t th e operation ,

STRESS-STRAIN ANALYSI S AND STRESS-STRAI N RELATION S 4 9

If fourth-quadran t value s ar e used , si n 29 is negative and co s 29 is positive.Substitution o f these value s into Eq . (2.6 ) yield s

Thus, th e maximum shearin g stres s ca n b e written as

If value s o f si n 26 an d co s 29, give n b y Eqs . (2.12 ) an d (2.13 ) an d wit happropriate sign s fo r eac h o f th e tw o quadrants , ar e substitute d int o Eq .(2.5), we will find that eac h plan e o f the maximu m shea r stres s element willbe subjecte d t o a norma l stres s tha t ma y b e tensile , compressive , o r zero .The valu e of the norma l stres s acting o n thes e planes is

It i s bes t i f th e maximu m shea r stres s i s considere d i n term s o f th eprincipal stresses . I n th e plan e stres s state , Eq . (2.14 ) ca n b e expresse d i nterms of an d b y using Eq. (2.10). If i s subtracted from , the result is

The transformation equation s hav e been developed for the biaxial stressstate b y taking = = iyz = 0. The biaxia l stres s equation s ca n be usedeven though az i s some value other than zero; that is , az i s the third principa lstress, makin g a z = a3. The shearing stresse s i xz an d i yz, however , must bezero, otherwise we would be obliged to use the more complx stress equation sfor th e triaxia l stress state . Figure 2. 4 shows such an element . If crz = <r 3 = 0 ,then th e elemen t o f Fig. 2. 4 reduces t o th e elemen t i n Fig . 2.2 .

We see there will always be three mutually perpendicular principa l stres saxes, an d correspondin g t o thes e direction s ther e wil l b e thre e principa lstresses whos e numerica l value s ma y b e positive , negative , o r zero . Th e

Thus,


FIG. 2.4. Elemen t wit h a , = a3 no t equa l to zero .

stresses ar e

algebraically

Figure 2. 5 shows triaxial , biaxial, and uniaxia l stress states . Note , however ,that while tensile stresses are shown, some or all could also be compressive.

FIG. 2.5. Triaxia l (a) , biaxia l (b) , and uniaxia l (c) stress states .


We tur n ou r attentio n onc e agai n t o th e shearin g stress . I f the stres sstate is triaxial, then no matter what the values of the three principal stressesare, th e maximu m shear stres s at th e point wil l alway s be

where amax and <r min ar e principa l stresses .Since w e ar e dealin g wit h th e biaxia l stres s state , th e thir d principa l

stress, <r 3, wil l b e zero . Th e maximu m shearing stres s a t th e poin t ma y o rmay no t b e in the x y plane , depending on the signs of a^ and a 2. Ther e are ,then, thre e cases t o consider .

(a) ff j an d a 2 hav e opposite signs ; a{ > 0, a3 = 0 , a2 < 0.

Figure 2.6 shows the thre e cases , wit h one o f the shea r plane s marke dfor eac h case . Th e secon d shea r plan e fo r eac h cas e i s a t 90 ° t o th e on eshown. Generally , i n th e cas e o f th e maximu m shea r stress , w e ar e no tconcerned wit h the orientatio n o f the element , but instea d wan t to kno w itsmagnitude.

Thus,

Thus

Thus

and greater than zero;

and less than zero;


FIG. 2.6 . Plane s of maximum shear fo r th e thre e cases of plane stress.

Summary of major equations


Determine th e principal stressss and thei r orientation relativ e to the x axis , thensketch th e principa l stress element . Comput e the maximum shearin g stres s a tthe point .

Solution. Th e principa l stresse s ar e determine d fro m Eq . (2.10) .

Use Eqs . (2.7 ) and (2.8 ) t o determin e th e orientatio n o f CTJ with respec t t o th ex axis.

Since the onl y match o f tan 2 9 and si n 26 is in the thir d quadrant , th e angl e 29lies in the third quadrant . Thus , 2 9 = 217.3°, or 9 = 108.7° , measured counter -clockwise from th e x axis. Figure 2.8 shows the orientation o f CTJ and a 2 relativeto th e x axis .

The maximu m principal stres s i s a l, an d th e minimu m principa l stres s isa2. Th e intermediat e principal stress i s <r3 = 0 . Using Eq . (2.17) , the maximumshear stres s i s

Example 2.1. A plane stress element, shown in Fig. 2.7, has the following stressesacting o n it :


FIG. 2.7. Biaxia l stres s elemen t fo r Exampl e 2.1.

FIG. 2.8. Principa l stres s elemen t fo r Exampl e 2.1.

2.4. Mohr's circle for stress

Equations (2.5 ) an d (2.6 ) ar e parametri c equation s fo r a circle , wit h th ecoordinate o f any poin t o n th e circl e being (o^. , T A . V ) . If these equations ar eplotted, th e curv e wil l advanc e i n a clockwis e directio n rathe r tha n i n th ecounterclockwise directio n tha t i s taken a s positive . Thi s conditio n ca n b ealleviated by redefining the sign of the shearing stress. The graphica l metho dwe use is known as a Mohr's circle , and i s named afte r Ott o Mohr , a Germa nengineer an d professor , who propose d i t i n 1880 .

The sig n conventio n fo r normal stresse s i s the sam e a s give n in Sectio n2.3. The shearin g stress , however, wil l b e defined as follows : a shearing stres swill be positive if the pair , acting on opposit e and paralle l face s o f an element,form a clockwis e couple .


FIG. 2.9. Mohr' s circle for stress.

Figure 2. 9 shows an elemen t and th e correspondin g Mohr' s circle . T oplot th e circle, construc t th e orthogonal a and i axes , the n star t wit h th eelement fac e norma l t o th e positive x axis . Here a x i s a positive stress , so itsvalue i s laid of f on th e a axi s a s OA . Next , th e shea r stres s o n thi s fac e i sseen t o for m a clockwise coupl e wit h the shea r stres s on th e fac e norma l t othe negativ e x axis . Thi s i s positiv e shea r stres s for Mohr's circle. Th emagnitude o f rxy i s plotted a s A B paralle l t o th e T axis. The coordinate s o fpoint B ar e (a x, r xy). Th e stresse s o n th e fac e norma l t o th e positiv e y axi sare plotte d next . Her e <r }, is positive and i s laid of f on th e a axis as OC. Th eshear stresse s o n th e face s norma l t o th e y axi s for m a counterclockwis ecouple an d thu s ar e negative . The magnitud e o f r xy o n thi s fac e i s plotte d


parallel t o th e T axis i n th e negativ e directio n a s CD . Th e coordinate s o fpoint D are (ay, —r xy). Point s B and D are connected b y a straight line whoselength i s the diamete r o f the circl e with it s center a t poin t E . The diamete rextended outwar d throug h poin t B represent s th e x axis , while the diamete rextended outwar d throug h poin t D represents the y axis . On th e circle , thesepoints ar e 180 ° apart, whic h corresponds t o 90 ° on th e element . The angl e29 is measured fro m B E ( x axis) as positive in the counterclockwise direction.Points F an d G ar e th e maximu m shea r stres s value s (i n th e x y plane) ,designated her e a s r]2, whil e points H and / ar e a 1 an d a 2, respectively.

Mohr's circl e is particularly usefu l wit h th e han d calculator . A freehandsketch o f Mohr's circle can b e mad e an d desire d orientation s taken fro m it .Stress magnitude s an d angle s ca n b e computed wit h th e calculator .

If we examine Mohr's circl e shown in Fig. 2.9, the following can b e seen:

1. Th e cente r o f th e circl e correspond s t o th e isotropi c (o r hydrostatic )component o f stress, a H. Thi s i s the stres s denne d b y Eq . (2.15).

2. Th e radiu s o f the circl e is T 12, given b y

Note her e tha t T 12 i s the maximu m shea r stres s i n th e x y plane , bu t i fai an d a 2 ar e o f opposite sign , then i t i s the maximu m shea r stres s a tthe point .

3. Fro m Mohr' s circl e we see that the principa l stresse s may b e expressedas

Note tha t Eqs . (2.23) an d (2.24 ) are anothe r for m o f Eq . (2.10).4. Mohr' s circle i s very helpfu l i n determining th e location o f a t. W e saw

in Sectio n 2. 3 that tw o trigonometri c relationship s wer e require d t olocate (r 1( while on th e circl e w e can locat e i t visually and comput e th eangle b y usin g

where 2 0 is the acut e angl e betwee n B E an d th e a axis.


5. Th e signs of the stress components ar e easily determined fro m th e circle.Normal stresse s presen t n o problem , bu t fo r the shea r stres s it is moreconvenient t o calculate the magnitude analyticall y and the n determinethe sig n (directio n o f the stress ) b y reference to th e circle .

While w e have develope d transformatio n equation s an d Mohr' s circl efor plan e stress , th e fac t tha t th e thir d principa l axi s exists should b e kep tin mind, even though the stres s in tha t directio n i s zero. As long a s we haveprincipal stresse s a 1, a 2, an d <r 3, thre e Mohr' s circle s ma y b e drawn . Thei rradii wil l be given by

For ou r cas e o f plane strain , & 3 = 0, and s o the radi i becom e

Figure 2.10 shows the case where al > a2 > 0 and a3 = 0. It is obvious fro mthe diagra m tha t th e maximu m shea r stres s a t th e poin t i s r max = T 13. If<73 = 0 had bee n ignored , w e might have been foole d int o thinkin g tha t T 12was the maximu m shear stress .

Example 2.2 . Th e elemen t i n Fig . 2.1 1 ha s th e followin g stresse s actin g o n it :

ax = 10000 psi, a y = 3000 psi, t xy = -8000 psi

(the sig n o f ixy conform s t o th e conventio n establishe d i n Section 2.3) .

Perform th e liste d tasks .(a) Dra w th e correspondin g Mohr' s circle .(b) Sketc h a n elemen t showin g th e principa l stresse s an d thei r orientatio n

relative to th e x axis .(c) Sketc h a n elemen t showin g the maximu m shear stres s i n the x y plan e an d

its orientation relativ e t o th e x axis .

Solution, (a ) Althoug h a free-han d sketc h o f Mohr' s circl e coul d b e made , i twill be drawn to scale, but pertinen t value s will be calculated. I n plotting Mohr'scircle, the followin g steps ar e taken :

1. Plo t a x = 1000 0 psi on the a axis as point A .2. Sinc e i xy form s a clockwis e coupl e o n th e paralle l face s norma l t o th e x

axis, i t i s plotte d a s positiv e fo r Mohr' s circle . Throug h poin t A , plo tTxy = 8000 psi i n th e positiv e i direction . Thi s give s poin t B , whos ecoordinates ar e (10000, 8000) .

3. Plo t CTJ, = 300 0 psi on th e a axi s a s poin t C .


FIG. 2.10. Three-dimensiona l Mohr' s circl e for plan e stress .

FIG. 2.11 . Stres s elemen t fo r Exampl e 2.2 .

4. Fro m poin t C , plot i xy = — 8000 psi in the negativ e T direction. Thi s give spoint D, whose coordinate s are (3000, - 8000) .

5. Joi n point s B and D t o ge t th e diamete r o f the circle . The intersectio n o fthe lin e B D wit h th e o axi s i s th e cente r o f the circl e E .


FIG. 2.12. Mohr' s circle fo r Exampl e 2.2 .

6. Dra w th e circl e an d measur e al l angle s fro m th e x axi s a s shown ,with the counterclockwise directio n bein g positive. The completed Mohr' scircle i s shown i n Fig . 2.12 .

From Eq . (2.21),

From Eq . (2.22),


(b) I n orde r t o determin e th e orientatio n o f ff 1 relativ e t o th e x axis , wego i n a clockwise directio n fro m th e x axi s o n th e circl e throug h th e angl e20, = 66.4° to reach CT, . The acute angle 20, can be computed usin g Eq. (2.25):

Since w e traveled in a clockwise direction o n th e circl e t o arriv e at <r, , we mustgo in the sam e directio n whe n locatin g CT, relative to th e x axi s for the element ;that is , Oj = 33.2 ° clockwise . The elemen t i s shown i n Fig . 2.13 . Note tha t w ecould also have gone in a counterclockwise direction o n the circle to arriv e a t<T,, thu s traversin g th e circl e throug h 20 , = 293.6° . O n th e element , th e angl ebetween th e x axi s an d CT, is 146.8° , whic h i s th e vecto r fo r <r , in th e secon dquadrant o f Fig . 2.13 .

(c) Th e maximum shear stres s in the xy plan e i s r,2. Fo r thi s problem, th emaximum shea r stres s a t th e poin t i s also T max = r 12 = 873 2 psi. To determin ethe orientatio n o f the maximu m shear stres s element , go in a counterclockwisedirection o n th e circl e unti l i 12 i s reached. Thi s i s the poin t whos e coordinate sare (6500 , 8732) . Sinc e i, 2 i s positive here , th e shea r stresse s o n opposit e face sof th e elemen t for m a clockwis e couple . Continu e movin g o n th e circl e i n acounterclockwise directio n unti l — T , , is reached. A t thi s point , th e coordinate sof (6500, —8732 ) ar e th e norma l and shea r stresse s o n th e elemen t face , whichis 90 ° fro m th e firs t face . Th e elemen t i s show n i n Fig . 2.14 . Th e angl e202 = 90° - 20, , and so 02 = 11.8" .

FIG. 2.13. Principa l stres s elemen t fo r Exampl e 2.2.

Equations (2.23 ) and (2.24 ) give a , an d a 2, respectively:


FIG. 2.14. Elemen t showin g Tma). in Exampl e 2.2 .

2.5. Basic concepts of strain

Accompanying stres s ther e i s usuall y some typ e o f deformatio n whic h w eregard a s strain. As in the cas e of stress, we find there ar e tw o genera l kind sof strain ; namely, linear strai n an d shea r strain .

Linear strai n i s represente d b y th e lengthenin g ( + fo r tension ) o rshortening ( — for compression) o f a straight line in the material. We assumethat al l longitudinal fibers of the bar elongat e identically , and tha t th e crosssections of the ba r tha t ar e originally plane and perpendicula r t o th e axi s ofthe ba r remai n s o during elongation . Suc h a bar i s shown in Fig . 2.15 , andits uni t strain s is given b y the expression

where 6 = total elongatio n o f the ba rL = original lengt h o f the ba r

If th e cros s sectio n o f th e ba r i s no t constant , o r i f th e loa d i snot uniforml y applied , al l longitudina l fiber s o f th e ba r wil l no t elongat euniformly, an d s o Eq . (2.29 ) represent s averag e strai n only . Thus, th e uni tstrain varie s from poin t t o poin t alon g th e bar . I n thi s case, th e uni t strainis determine d b y considerin g th e elongatio n d d o f a cros s sectio n o f length

62 THE BONDE D ELECTRICA L RESISTANCE STRAIN GAGE

FIG. 2.15. Elongatio n o f a bar.

dL. Th e uni t strai n a t a poin t ca n b e written as

If axia l compressio n i s applied , Eqs . (2.29 ) and (2.30 ) apply , wit h th e uni tstrain bein g negative.

When a bar , suc h a s th e on e i n Fig . 2.15 , is loaded uniforml y ove r th eend faces , onl y thos e cross sections norma l t o th e ba r axi s ar e subjecte d t ostress. Observatio n o f such test s show s that th e extensio n o f the ba r i n th eaxial directio n i s accompanied b y a latera l contractio n o f the bar . Poisson ,a French mathematician , demonstrated analyticall y that the axial and latera lstrains are proportiona l t o each othe r withi n th e rang e o f Hooke's law, andthe rati o i s constant fo r a give n material . Thi s rati o i s known a s Poisson' sratio an d i s expressed a s

Figure 2.1 6 shows a n element , give n b y abed prio r t o loading , whos ecorners ar e square . Th e elemen t i s then loade d b y the shearin g stres s shownand distort s int o ab'c'd. Sinc e th e angl e 7 throug h whic h line s a b an d c drotate i s very small , i t i s assumed tha t ab ' i s equal t o a b an d dc ' i s equal t odc. In thi s case, ta n y = bb'/ab, an d s o for small angles ta n 7 may b e replace dby 7 . Thus , th e shearin g strai n i s give n b y th e angl e 7 , whose valu e i s i nradians.

2.6. Plane strain

The transformatio n equations fo r plane stres s were developed i n Section 2.3 .There the stresses in the z direction wer e zero; that is, a, = tx, = TV, = 0 . Wenoted, however , tha t i f a z = ff 3(rx, = iy. = 0), then <r 3 could hav e a valu eother tha n zer o an d th e biaxia l transformation equation s wer e stil l valid .

Since ou r wor k wit h strai n gage s wil l generally involv e applying the mto th e surfac e o f a machin e element , w e wil l nee d th e transformatio nequations for plane strain. This implies that e . = y x. ~ y y. = 0, and we might

STRESS-STRAIN ANALYSI S AND STRESS-STRAI N RELATION S 63

FIG. 2.16. Elemen t subjected t o pur e shear .

assume tha t a plan e stres s stat e produces a plan e strai n state . This i s no tthe case , however , fo r biaxia l stresse s produc e a strai n i n th e z directio nbecause o f the Poisso n effect . Th e strai n produced, e z, wil l b e th e principa lstrain e 3, with y xz = yyz = 0. This wil l become apparen t whe n stress-strai nrelationships ar e examined.

The stres s transformatio n equation s an d th e strai n transformatio nequations hav e th e sam e form , and s o the strai n transformatio n equation scan be written directly by making the following substitution s into th e stresstransformation equations :

Making thes e substitution s int o Eqs . (2.5 ) an d (2.6 ) wil l yiel d th e strai ntransformation equations . Thus ,

The sign of the shear strain must be compatible with shear stress. Figure2.17 show s a n elemen t subjecte d t o positiv e shea r stress . Prior t o loading ,sides A B an d A C ar e a t righ t angle s t o eac h other . Afte r th e stresse s ar eapplied th e right angl e BAG will deform to angle B'AC', whic h is n/2 - y xy.Since thi s distortion i s produced b y positive shear stresses , th e shea r strain ,yxy, wil l be denned as positive whe n the angle betwee n tw o orthogonal line sdecreases.


FIG. 2.17. Elemen t subjecte d t o positiv e shear stresses.

The orientatio n o f a t an d E J with respec t t o th e x axi s i s the same . I norder t o determin e th e orientatio n o f <T , with respec t t o th e x axis , tw o o fthe thre e trigonometri c relation s give n b y Eqs . (2.7) , (2.8) , an d (2.9 ) wer eused. These thre e expressions may b e written i n terms of strain, wit h th e us eof tw o o f th e thre e equation s establishin g the orientatio n o f e ^ Th e thre erequired expression s i n term s o f strain ar e

The principa l strains , S L an d s 2, follo w directl y fro m Eq . (2.10 ) whe nthe appropriat e value s of strain ar e substitute d for stress . Thi s gives

The secon d ter m o n th e righ t sid e o f Eq . (2.37 ) is one-hal f o f the maximu mshear strai n i n th e plane . Therefore ,

Unlike stresses , whic h can onl y b e determine d indirectly , linear strain sare subjec t to direc t measurement . I f an x y referenc e syste m i s chosen, thenstrain measurement s ar e mad e i n thre e known direction s relativ e to th e x y


coordinate syste m throug h th e us e of three-element strai n rosettes . A delt arosette has three gages arranged a t 60° (or 120° ) intervals, while a rectangularrosette ha s thre e gage s arrange d a t 45 ° intervals . Eac h measure d strai n i sentered int o Eq . (2.32) as e x., and 0 is the angl e betwee n th e x axi s and th emeasured strain . Th e thre e strain reading s use d i n Eq . (2.32 ) produce thre eindependent equation s tha t ar e solve d simultaneousl y for E X, ey, an d y xy.Knowing th e componen t strain s e x, a,, , and y xy, w e can no w comput e th eprincipal strain s b y usin g Eq . (2.37) . The principa l strai n axe s ar e locate drelative t o th e x axi s by using any tw o o f Eqs. (2.34) , (2.35), and (2.36) .

Summary of major equations

Example 2.3 . Th e followin g strain s an d thei r orientatio n relativ e t o a n x ycoordinate system on a machine elemen t are given. (Note: The symbol /i stand sfor 1 x 1(T 6).

The arrangemen t o f the gage s giving these reading s i s shown i n Fig . 2.18 .

(a) Determin e e x, sy, and y xy.(b) Determin e e l an d e 2 an d th e orientatio n o f e^ realtive t o th e x axis .(c) Determin e (y/2) max in th e plane .


FIG. 2.18. Gag e arrangemen t fo r Exampl e 2.3.

Solution. I n orde r t o determin e «, an d e 2, we must firs t determine e x, e y, an dyxy i n th e x y coordinat e system . Thi s require s thre e independen t equations ,which w e obtain b y usin g Eq. (2.32) . Not e tha t e. x, becomes i n tur n e a, s b, an dEC, while 9 takes o n th e correspondin g value s of O a, 6h, an d Q c.

The thre e equation s reduc e to

Solving th e equation s simultaneously , w e obtai n E X = 80 0 uin/in; e y =— 300 /^in/in; }'xy = 1 200 uradians. The principa l stresses are computed usin g Eq.(2.37).

(a)

STRESS-STRAIN ANALYSI S AND STRESS-STRAI N RELATION S 67

FIG. 2.19. Orientatio n o f the principa l stresse s i n Exampl e 2.3.

(b) I n orde r t o determin e th e orientatio n o f e ^ wit h respec t t o th ex axis , Eqs. (2.34 ) and (2.35 ) will be used. Fro m Eq . (2.34),

The commo n quadran t i s the first, and s o 29 = 41.5°. Thus, e 1 lie s at a n angl eof 9 = 23.75° in a counterclockwis e direction fro m th e x axis . The orientatio nis show n i n Fig . 2.19 .

(c) Th e maximu m shearin g strai n i n th e plane i s obtaine d b y usin gEq. (2.38).

Note: Th e problem coul d also b e solve d b y aligning th e x ' an d / axe salong EO and e c, respectively, then throug h th e us e of Eq. (2.32), determining th ecomponent strain s i n th e x'y' system . Finally , th e transformatio n equation scould b e used t o ge t the desire d value s in the x y coordinat e system .

Rrom Eq. (2.35),

68 THE BONDE D ELECTRICA L RESISTANCE STRAI N GAG E

FIG. 2.20. Mohr' s circl e fo r strain .

2.7. Mohr's circle for strain

Mohr's circl e for strain is constructed in a manner simila r to tha t o f Mohr' scircle fo r stress . Figur e 2.2 0 show s th e Mohr' s circl e fo r strai n an d it sattendant element . Figure 2.20 a i s the elemen t use d fo r th e constructio n o fMohr's circl e fo r stress , whil e Fig . 2.20 b show s th e elemen t i n it s distorte dposition (greatl y exaggerated).

To plo t th e diagram, construct the orthogonal e and y/ 2 axes, then startwith th e x axi s and plo t e x as OA o n th e e axis. The shearin g stresses actingon the element faces ope n angl e DAB t o D'A'B', an d s o it is taken as positivefor Mohr's circle. This is the shearing strai n y xv, an d sinc e i t is positive, y xy/2,

Note tha t Eqs . (2.41 ) an d (2.42 ) are anothe r for m o f Eq. (2.37) .4. Mohr' s circl e i s very helpfu l i n determinin g the locatio n o f e^ W e saw

in Sectio n 2. 6 that tw o trigonometri c relationship s wer e require d t olocate e 1; while on th e circle we can locate i t visually and comput e th eangle b y usin g

where 2 9 is the acut e angl e between B E an d th e e axis.

3. Fro m Mohr' s circle w e see that th e principal strain s ar e

2. Th e radiu s o f the circl e is


is plotte d a s A B paralle l t o th e y/ 2 axis . Th e coordinate s o f poin t B ar e(ex, yxy/2). Next , plo t E y as O C on th e e axis and y xy/2 a s C D parallel to th ey/2 axis . The coordinate s o f point D are (E y, — yxy/2). Connec t point s B an dD wit h a straigh t line ; length B D i s the diamete r o f the circl e with it s cente rat point E . The diameter extende d outward through point B represents the xaxis, while the diamete r extende d outwar d throug h point D represents th e yaxis. Thes e tw o points , 180 ° apar t o n th e circle , correspon d t o 90 ° on th eelement. Th e angl e 2 6 is measured positiv e i n a counterclockwis e directio nfrom th e x axis . Point F i s the maximu m valu e of y/2 i n th e x y plane , an dits coordinates ar e [_(E X + s y)/2, y max/2]. Point G is the othe r maximum valueof y/2, and it s coordinates ar e \_(E X + sy)/2, — ymax/2]. Point s H an d / ar e th evaluesof the principa l strains , e^ and e 2, respectively. The value s of 29 on th ecircles fo r stres s an d strai n correspond . Fo r instance , th e angl e between a 1and th e x axi s o n th e circl e fo r stres s i s identica l t o th e angl e betwee n E ^and th e x axi s on th e circl e for strain .

If we examine Mohr's circle shown in Fig. 2.20, the following can be seen:

1. Th e cente r o f th e circl e correspond s t o a n isotropi c (hydrostatic )component o f strain, % .

70 TH E BONDE D ELECTRICAL RESISTANC E STRAI N GAG E

5. Th e signs of the strain components are easily determined from th e circle.Normal strain s present no problem , but fo r th e shear strain, i t i s moreconvenient t o calculat e its magnitud e an d the n determine its sig n b yreference t o th e circle .

Example 2.4. Th e strains on the surface of a machine element are the following:

EX = 100 0 uin/in , a v = —40 0 uin/in, y xy = 80 0 uradians

(a) Plo t Mohr' s circl e and determin e th e principa l strain s « , an d f:2.(b) Determin e th e orientatio n o f t', } relativ e to th e x axis .(c) Determin e f; v., e. y-, and y v ) . - fo r 0 = 3 5 .

Note: y xy i s positive by th e sig n conventio n established i n Sectio n 2.6 , an dso mus t b e plotte d a s negative fo r Mohr' s circle .

Solution, (a ) Mohr' s circl e fo r strai n wil l b e plotte d t o scale , bu t th e pertinen tvalues wil l b e calculated. Th e followin g step s ar e take n i n plotting the diagra mshown i n Fig . 2.21.

1. Plo t E X = 100 0 |iin/in o n th e e . axis as poin t A .2. Sinc e y xy = 80 0 uradians, i t mus t b e take n a s negative for Mohr's circle.

Thus, w e plo t "f xy!2 = —40 0 uradians fro m e, t parallel t o th e negativ e y/ 2axis. Thi s i s point B , whose coordinate s ar e (1000 , —400) .

3. Plo t e, y = —40 0 uin/in o n th e B axis as poin t C .4. Fro m e, y, plot y xy/2 = 400 uradians paralle l t o th e positiv e y/ 2 axis . Thi s

is poin t D , whos e coordinates ar e ( — 400,400).5. Joi n points B and D to get the diameter o f the circle. The intersection of line

BD wit h the c axis i s the cente r o f the circl e £ .6. Dra w th e circl e and measur e al l angles fro m th e x axi s a s shown , with th e

counterclockwise directio n bein g positive .

From Eq . (2.39),

From Eq . (2.40),

From Eq . (2.41),

STRESS-STRAIN ANALYSI S AN D STRESS-STRAI N RELATION S 71

FIG. 2.21. Mohr' s circl e for Exampl e 2.4.

From Eq . (2.42) ,

(b) Th e orientatio n o f E J relative t o th e x axi s ca n b e measure d directl yfrom th e circle . I t lie s i n a counterclockwise directio n fro m th e x axis . We ca nalso calculat e it s value usin g Eq . (2.43) :

Therefore, 9 = 14.87° . The orientation s of el an d s 2 ar e show n i n Fig . 2.22 .(c) Th e thre e value s can b e scaled directl y from th e circl e i f so desired. W e

can als o us e Eq . (2.32 ) t o determin e th e norma l strains , an d Eq . (2.33 ) t odetermine th e shearin g strain . Usin g Eq . (2.32 ) an d 9 = 35° , s x. ca n b edetermined.


FIG. 2.22. Principa l strain s fo r Exampl e 2.4.

For e y., the angl e is 0 + 90°.

Using Eq . (2.33) and 0 = 35 ,

Observe her e tha t y^ y ' s negativ e whe n compute d usin g th e transformationequation, whic h i s in keepin g with th e valu e o f - t>x,y, from th e Mohr' s diagram.All of the computed values can b e checked by usin g Mohr' s diagram, shown inFig. 2.21.

2.8. Stress-strain relationships

We hav e considere d stres s an d strai n separatel y a t a point . I n developin gthe transformation equation s fo r stress , the stati c equilibrium o f the elemen twas examine d an d th e resultin g equations wer e no t dependen t o n materia lproperties. Althoug h th e strai n transformatio n equation s wer e writte ndirectly from thos e o f stress, on e shoul d b e awar e o f the fac t tha t the y ma ybe develope d fro m th e geometr y o f smal l deformations , an d therefor e


material properties do not appear . Th e objective, now, is to relate stress andstrain, an d s o material propertie s wil l be the link .

Robert Hook e was the first to state th e relationship between stress andstrain. Fo r a tensile tes t i t can b e stated a s

where o x = the longitudinal stress

EX = th e longitudina l strain

E = a constant o f proportionality called the modulus of elasticity

In th e mos t genera l for m o f Hooke' s law , i t i s assume d eac h stres scomponent ha s a linear relationship with the six strain components, resultingin 3 6 elastic constants. Fo r a n isotropi c material , however , it can b e shownthat th e 36 constants ar e not al l independent and tha t onl y two independentconstants exis t (1).

By testing, three elastic constants ca n be determined fo r a given material.The elasti c constant s ar e th e modulu s o f elasticity , E , th e shea r modulu s(modulus o f rigidity) , G , and Poisson' s ratio , v . If a tensil e test i s conductedon a specimen with a rectangular cros s section, the stress <rx will be uniforml ydistributed ove r tw o paralle l section s norma l t o th e x axis , with th e x axi staken in the direction o f loading; the faces norma l to th e y and z axes wouldbe stress free. Observatio n o f such tests show the extension in the x directionis accompanied b y a lateral contractio n i n the y and z directions. The strai nin thes e tw o direction s i s negative and proportiona l t o th e strai n i n th e xdirection. Thus, a s show n b y Eq . (2.31),

where v is Poisson' s ratio .Since there are only two independent constant s fo r an isotropic material,

a relationshi p mus t exis t between E, v , and G . The shea r modulu s G can b eexpressed i n term s o f E an d v as

The si x equations relatin g strai n i n term s o f stress ar e


If Eqs . (2.47) are solve d fo r stres s i n term s o f strain, w e have

Equations (2.47 ) an d (2.48 ) represen t th e triaxia l stres s an d triaxia l strai ncase. Special stress and strain states may be determined fro m thes e equations.

Plane stress state: a , = yx. = yy, = 0. Th e plane, or biaxial , stres s cas e wasdeveloped i n Sectio n 2.3 . Since az = 0 , Eq. (2.48c) can b e used t o determin es, i n term s o f sx an d e Therefor e

By substitutin g the expressio n fo r K , given b y Eq . (2.49 ) int o Eqs . (2.48a )


and (2.48b) , we arrive a t

The correspondin g strai n equation s ca n b e obtained fro m Eqs . (2.47) :

The expression s fo r e z give n b y Eqs . (2.49 ) an d (2.51c ) giv e identica lresults, o f course . A s pointe d ou t earlier , eve n thoug h a plan e stres s stat eexists, th e strai n stat e i s triaxial.

Plane strain state: EZ = yxz = j yz = 0 . Sinc e ez = 0 , az can be written in termsof a x an d a y b y usin g Eq. (2.47c) . This gives

The valu e of az give n by Eq . (2.52 ) can b e substitute d into th e expressionsfor e x and e y, given by Eqs. (2.47a) and (2.47b) , respectively. The expressionsfor e _ and e v in term s o f stress then becom ey


The corresponding equation s for stress in terms of strain are, from Eqs . (2.48),

Once again, note that whe n we have plane strai n we do no t hav e planestress. Furthermore, th e values of a. given by Eqs. (2.52) and (2.54c ) produc eidentical results .

Uniaxial stress state: a y = a . = T X). = ix. = T V ,_ = 0 . I n the case o f a uniaxialstress state , Eqs . (2.47 ) reduce t o

We se e here tha t eve n thoug h a uniaxia l stress stat e exists , the strai n stat eis triaxial .

Our equation s hav e been writte n in terms of the xyz coordinat e system,but i f w e ar e dealin g wit h th e principa l stresse s an d strains , the n th esubscripts x , y , an d z ca n b e replace d b y subscript s 1 , 2, an d 3 t o pu t th eequations i n terms of principal stresse s and strains . In thi s case, ther e wouldbe n o shearin g strain s an d henc e n o shearin g stresses . Equation s (2.47 )become


Equations (2.48) become

Corresponding change s ca n b e made fo r the othe r stres s and strai n states .

2.9. Application of equations

The material developed can now be used in an application. Suppose we havean existin g steel machine elemen t o f such a shap e tha t th e stresse s canno tbe determined analytically. At a point in question a three-element rectangularstrain rosett e i s applie d i n orde r t o determin e th e strains . Th e gage s ar eapplied so that one gage is aligned along the chosen x axis , as shown in Fig .2.23. The gages are designate d a s a , b, and c . As the member i s loaded, eac hgage wil l b e strained . Our goa l i s to obtai n th e principa l stresse s and thei rorientation relativ e to th e x axis .

After testin g has been completed, the following information is presentedfor analysis :

efl = - 80 0 uin/in at 9a = 0°

£„ = -30 0 uin/i n at 8 b = 45°

ec = 120 0 uin/in a t 9 C = 90°

E = 30 x 10 6 psi an d v = 0.3

Before th e principa l strain s ca n b e calculated , e x, e y, an d y xy mus t b edetermined. Th e desire d value s can b e computed usin g the strai n reading sin conjunctio n wit h Eq . (2.32) . It i s obvious tha t e x — Ea and s y = e c. Thus ,we can us e th e readin g given by s b and Eq . (2.32 ) in orde r t o obtai n y xy/2:

where 9 = 45°. This gives


FIG. 2.23 . Three-elemen t rectangula r rosette .

Thus,

The principa l strain s ca n b e computed usin g Eq . (2.37) :

K! = 131 8 uin/in; i; 2 = —91 8 uin/i n

The orientatio n o f K, can b e determined b y using any tw o o f Eqs . (2.34) ,(2.35), an d (2.36) . Fro m Eq . (2.34) ,

From Eq . (2.35) ,


FIG. 2.24. Orientatio n o f principal strains .

Since th e onl y matc h i s third quadrant , 2 9 must b e in tha t quadrant ; thus ,29 = 206.6°, or 9 = 103.3° . Figure 2.24 shows the orientation o f the principa lstrains relativ e to th e x axis .

A Mohr' s circl e could als o hav e been used , and s o i t wil l b e draw n i norder t o check the values of the principal strain s and thei r orientation. Note ,however, that while yxy/2 fo r the transformation equation i s negative, its signmust be changed t o positive when plotting Mohr' s circle . Figure 2.2 5 showsthe diagra m fo r strain . Mohr' s diagra m show s quite clearly th e orientatio nof E! relative to th e x axis . From the circle , we see that w e could hav e gon ein a negative (clockwise) direction fro m th e x axi s to E I through a n angl e of29 = 153.4° . In Fig . 2.24, this would be the clockwise angle of 0 = 76.7 ° fromthe x axi s to E ± shown i n th e fourt h quadrant .

Since thi s i s a plan e stres s problem , a v an d a 2 ma y b e determined b yusing Eqs . (2.50a ) an d (2.50b) . Her e th e subscript s x an d y ar e change d t o1 an d 2 , respectively. Thus,

The orientatio n o f a1 an d cr 2 W'H; °f course , be the sam e a s e l an d e 2.


FIG. 2.25. Mohr' s circle for strain .

The tw o principa l strain s e^ and £ 2 have been calculated . Because of thePoisson effect , ther e will also b e a strain alon g the z, or 3 , axis and so e3 = s, .This strai n ca n b e compute d b y usin g eithe r Eq . (2.49 ) o r Eq . (2.51c) .Choosing Eq . (2.51c),

Since a 3 = 0 an d a ^ an d <7 2 are o f opposit e sign , th e maximu m valueof th e shea r stres s a t th e poin t i s

The proble m coul d hav e been approache d i n a differen t manner . Onc esx, E y, and }' xv were determined , a x, a y, an d t xy coul d hav e bee n compute dby using the stress-strain relationships, and th e problem complete d b y usingthe stresses .


2.10. Stress and strain invariants

The development o f the transformation equation s for stress has been limite dto th e biaxia l stress state . Th e derivation for the triaxia l stress state , a s wel las th e determinatio n o f the principa l stresse s and thei r orientatio n relativeto th e origina l coordinat e system , is more complex . Durin g thi s proces s acubic equatio n i s developed whos e root s are real and ar e th e value s o f theprincipal stresses , a^, a2, an d <J 3(1). The cubi c equation i s

where

The term s / 1? I 2, an d / 3 ar e calle d stres s invariants , sinc e the y ar econstants fo r any axi s transformation . Considering 7 t a s an example ,

Thus, th e su m o f the norma l stresse s fo r an y transforme d axe s wil l alwayshave the sam e value . On e ca n chec k Ex . 2.1, where <7 3 = 0 , and fin d tha t

A cubi c equatio n simila r t o Eq . (2.58 ) can als o b e develope d fo r th edetermination o f the principa l strain s (1) . I t i s

where

Again, th e root s o f Eq. (2.62 ) ar e th e value s o f the principa l strains , e t, e 2,and e 3. The terms / t, I 2, and /3 ar e called strai n invariants.


The stress states tha t we deal wit h are generally biaxial . The strain stat ewill be triaxial, but sinc e the strains that wil l b e measured ar e o n th e surfaceof a machin e element , th e strai n norma l t o th e surfac e at tha t poin t wil l b edue t o th e Poisso n effect . I n thi s case th e strain normal t o th e surfac e a t th epoint wil l b e constan t regardles s o f th e orientatio n o f th e axes . Therefore,for plan e strain , It ca n b e written as

/, = e x + Ey = £v, + « v, = E ] + i: 2 = a constan t

since e , = zz, = c3.It wil l b e pointed ou t late r in th e tex t how / , ca n b e used whe n strain

rosettes ar e considered .

Problems

In problems 2. 1 through 2.8 , determine th e principal stresse s and show , by sketching,their orientatio n relativ e t o th e x y coordinat e system . Dra w Mohr' s circl e fo r eac hand determin e th e maximu m shear stres s a t th e point .

For al l

Prob. No.2.12.22.32.42.52.62.72.8

problems i n

psi1000012500

-80009500

-150000

-1500012000

this chapter ,

psi3500

1250047009500

-85000

-15000-4000

use v = 0. 3 and E = 30 x 10 6 psi.

psi-5000

80005500

0-3000

7500-8000

0

2.9. Fo r th e cantileve r bea m an d th e loadin g show n i n Fig . 2.26, determin e th efollowing:

(a) Th e principa l stresse s a t poin t A an d B an d thei r orientatio n relativ e t othe x axis .

(b) Th e maximu m shea r stres s a t point s A an d B .

2.10. A closed-end tub e has a n insid e diamete r o f 2.000 in an d a n outsid e diamete rof 2.12 5 in. Th e interna l pressur e i s 75 0 psi an d th e tub e i s subjecte d t o atorsional momen t o f 3000 in-lb. Determine th e principa l stresses .

2.11. Tw o gear s ar e keye d t o a rotatin g shaf t a s show n i n Fig . 2.27. Forc e F D isapplied t o gea r D in th e y z plane . Determin e th e reactiv e forc e F c (als o i n th eyz plane) , an d th e bearin g reactions , assumin g frictionles s bearings . Dra w afree-body diagra m o f the assembly and determin e the maximum shearing stres sin th e shaf t betwee n th e gears .

2.12. A mechanic use s a torqu e wrenc h an d a n extensio n bar t o tighte n a nu t (Fig.2.8). I f the torqu e wrenc h read s lOOft-lb , determin e th e principa l tensil e stres sand th e maximu m shea r stres s a t th e sectio n show n o n th e extensio n bar.

FIG. 2.28.

FIG. 26

84 THE BONDE D ELECTRICAL RESISTANC E STRAI N GAG E

FIG. 2.29.

2.13. Figur e 2.2 9 show s a principa l stres s elemen t an d it s orientatio n relativ eto th e x axis . Determin e a x, a y, an d T X>,, the n sketc h th e elemen t showin g th estresses i n thei r prope r directions .

2.14. Figur e 2.30 shows a stress element. Using Mohr's circle, determine the following:

(a) Th e principa l stresses . Sketc h th e principa l stres s elemen t an d sho w it sorientation relativ e t o th e x axis .

(b) Th e maximu m shear stress . Sketc h th e maximu m shear stres s elemen t an dshow it s orientation relativ e to th e x axis .

(c) Verif y you r answers by analytica l methods .

In Problem s 2.1 5 through 2.24 , determine th e principa l strains , s t an d E 2 and> b y a

sketching, thei r orientatio n relativ e t o th e x y coordinat e system . Dra w a Mohr' scircle fo r eac h problem . Al l values are i n |iin/in .

Prob. No.

2.152.162.172.182.192.202.212.222.232.24

180016601035-140

00

1400-8001 150640

-800355

-26071000

400400

1 150430

1520-960770

-3902000-50080000

-1430


FIG. 2.30.

FIG. 2.31.

2.25. Th e followin g strain s ar e given :

= 2500 uin/in, = - 100 0 uin/ = 150 0 uradian s

(a) Determin e gj , s2, an d y max.(b) Determin e the gage readings fo r (i) a three-element rectangular rosett e and

(ii) a three-elemen t delt a rosette , assumin g gag e a i s aligne d alon g th e xaxis, as shown in Fig . 2.31 .

2.26. Th e three-elemen t rectangula r rosett e show n i n Fig . 2.3 2 give s th e followin gstrains:

= -80 0 uin/in , = 500 uin/in, 120 0 uin/in

Determine th e principal strain s an d thei r orientation relativ e t o th e x axis .


FIG. 2.32.

2.27. A three-element rectangula r rosett e give s the followin g readings:

Determine th e principa l strains an d thei r orientatio n relativ e to gag e a , whichis aligned alon g th e x axis.

2.28. A three-element delt a rosett e give s the followin g readings :

Determine th e principa l strain s an d thei r orientatio n relativ e to th e x axi s ifgage b is aligned alon g th e x axis .

2.29. Th e rosett e i n Fig . 2.3 3 i s attache d t o a machin e member . Determin e th eprincipal strain s and thei r orientatio n i f the following strain readings have bee nrecorded:

FIG. 2.33 .

STRESS-STRAIN ANALYSI S AN D STRESS-STRAI N RELATION S

2.30. Determin e , an d a 2 f° r Proble m 2.26 .2.31. Determin e a an d <7 2 fo r Proble m 2.27 .2.32. Determin e a and a2 fo r Problem 2.28.2.33. Give n th e followin g strains :

ex = 750 uin/in e y = — 800 u.in/in

ez = 450 uin/in y xy = 200 uradians

yyz = — 5000 (iradians y xz = 3000 ^radian s

87

Determine2.34. I f E! = 80 0 |iin/in and £ 2 = —20 0 (iin/in, determine the stress necessary to make

£3 = 0.2.35. Give n the followin g stresses :

Determine th e principal strain s s l, e 2, and E3.2.36. Th e strai n gag e rosette , Fig . 2.34 , i s mounte d o n th e surfac e o f a machin e

member. Previous calculations have yielded the following strains in the xy plan eat poin t 0 :

sx = 157 0 uin/in, e y = —47 2 uin/in, y xy = 141 6 uradians

(a) Determin e th e expected strai n rosette readings .(b) Determin e al l o f the principa l strains .(c) Determin e a x, a y, and t xy a t poin t 0.(d) Sketc h the principal stress element and it s orientation relativ e to the x axis .

Show th e value s of o ^ an d <7 2-

FIG. 2.34.


2.37. Fo r th e stres s elemen t show n i n Fig . 2.35 , determine th e following :(a) Th e stres s actin g alon g th e z axis that wil l mak e R . = 0 .(b) Wit h the stress from par t (a ) acting on the element, compute cx, ey, and

2.38. Th e 1.0-i n diameter stee l shaf t i s loaded a s shown in Fig . 2.3 6 and ha s a strai ngage rosette attached a t point A . Determine the expected values of «„, s,,, and c.

FIG. 2.35.

FIG. 2.36.


REFERENCES

1. Durelli , A . J. , E . A . Phillips, an d C . H . Tsao , Introduction t o th e Theoretical andExperimental Analysis o f Stress and Strain, New York , McGraw-Hill , 1958, Chaps.1, 2 , 4 .

ELEMENTARY CIRCUITS

3.1. Introduction

Since the change in resistance of a strain gag e is measured b y its effec t upo neither the current passing throug h th e gage o r th e voltage drop across it , thegage mus t form par t o f some kin d o f electrical circuit . Figure 3. 1 shows sucha strai n gage .

For initia l condition s w e can writ e

where E = voltage dro p across th e gag e

/ = curren t passin g throug h the gag e

Rg = gage resistanc e

When th e gage resistanc e change s fro m R g t o R s + AR9, either th e current ,/, o r th e voltage , E , or both , wil l b e changed .

It i s our purpos e no w t o explor e two simpl e circuit s and t o investigat ethe correspondin g effect s o f uni t changes , AR g/Rg, i n gag e resistanc e upo nvoltage and current . Schematic diagrams o f the two elementary circuits, eachcontaining a singl e strai n gage , ar e show n i n Figs . 3. 2 and 3.3 . The firs t o fthese indicate s a constant-voltag e sourc e connecte d t o th e gage , whil e th esecond represent s a constant-curren t circuit .

FIG. 3.1. Strai n gage.

3

ELEMENTARY CIRCUIT S 91

FIG. 3.2. Constan t voltage applie d t o gage .

FIG. 3.3. Constant-curren t circuit .

3.2. Constant-voltage circuit

For th e cas e show n i n Fig . 3.2 , th e applie d voltage , V , will b e th e sam e a s£, th e potentia l dro p acros s th e gage . Becaus e thi s circui t contain s aconstant-voltage sourc e (fo r example, a batter y o f sufficien t size) , there wil lbe n o chang e i n potentia l dro p acros s th e gag e eve n thoug h ther e ma y b evariations i n resistance .

Due t o th e applicatio n o f constan t potential , V , th e onl y thin g tha thappens whe n the gage resistanc e changes i s a chang e i n the current . Thus ,when th e gag e resistanc e change s fro m R g t o R g + ARg, th e correspondin gchange i n curren t i s from /t o 7 — A/.

We can now examine how the change in current, A/, is related t o strain ,or t o th e uni t change i n gage resistance . Initially,


After a change in gage resistance of ARg, th e current changes by — A/, and s o

Expanding th e righ t side,

Since E = IRg, Eq . (3.2 ) becomes

or

Equation (3.3 ) may now b e expressed i n terms o f unit change s in resistanceby dividin g each ter m by R g. Thi s give s

Solving fo r A I produce s

We know, however, that

where £ = strain , in/i n

GF = gag e factor

Substituting th e valu e of ARg/Rg give n b y Eq . (3.5 ) into Eq . (3.4 ) gives

We can now write the expression for the change in current per unit of strain as

ELEMENTARY CIRCUIT S

The curren t can b e written in term s o f the applie d voltag e as

93

Using thi s value of /, Eq . (3.7 ) become s

Equations (3.4 ) an d (3.6 ) indicat e tha t th e chang e i n current , A/ , i s anonlinear functio n o f the uni t chang e i n gag e resistance , o r th e strain . O nthis accoun t i t i s sometime s mor e convenien t t o expres s Eq . (3.7 ) i n th efollowing modifie d form :

where n , the nonlinearit y facto r fo r thi s case, i s

Figure 3. 4 shows a curv e representin g th e value s o f th e nonlinearit yfactor, n , a s give n i n Eq . (3.10) , i n term s o f th e chang e i n gag e resistance ,

FIG. 3.4. Nonlinearit y factor vs . unit change in gag e resistance .


From th e relationshi p expresse d i n Eq . (3.9), we can dra w tw o interest -ing conclusions :

1. Th e circui t indication , o r output , pe r uni t strain , A//e , i s a nonlinea rfunction o f the current / an d th e gage facto r G t.

2. Th e maximu m indication , o r output , i n term s o f curren t chang e pe runit o f strai n wil l occu r fo r ver y smal l (theoreticall y zero) strains . I nquantitative term s thi s wil l b e represented b y th e produc t o f the initia lcurrent an d th e gage factor , 7GF.

Since th e maximu m curren t (roughl y 3 0 milliamperes ) tha t ca n b ecarried b y th e gag e depend s upo n th e abilit y t o dissipat e heat , ther e wil ldefinitely b e an uppe r limi t fo r th e outpu t pe r uni t strain , A//e , tha t ca n b eachieved for any particular installation. For example , if the maximum currentIm i s 30 milliamperes an d th e gag e facto r G F is 3.5, then th e maximu m valueof A// e wil l b e

= (0.030)(3.5) = 0.10 5 amps/unit strai n

or

= 0.10 5 microamps/microstrain

It shoul d b e note d that , fo r th e indicatio n o f strain , thi s i s a current -sensitive circuit . A s such , i t possesse s certai n characteristic s tha t i t share swith othe r type s o f current-sensitive circuits used wit h strai n gages .

Since A//e , th e outpu t pe r uni t strain , varie s directl y wit h th e gag ecurrent, the highes t possible curren t consisten t wit h the limitations impose dby heatin g effect s shoul d b e employed . Equatio n (3.8 ) indicate s tha t thi sobjective ca n b e achieve d eithe r b y th e us e o f low-resistanc e gage s o r b yemploying high values for the applie d voltage . However , because considera -tions of safety an d convenienc e impose a n uppe r limi t on the applie d voltag eV, i t wil l b e desirabl e t o selec t strai n gage s o f low resistanc e an d hig h gag efactor, fo r thi s typ e o f circuit , i n orde r t o achiev e th e maximu m possibl eindication fo r a give n strain .

3.3. Constant-current circuit

An alternativ e t o th e circui t whic h applies a constan t voltag e t o th e gag e i sthe circui t deliverin g a constan t curren t t o th e gage . Figur e 3. 3 shows th ecircuit. I n thi s cas e A / = 0 a t al l times . However , du e t o th e fac t tha t th ecurrent i s constant, ther e wil l be changes i n the voltag e drop across the gag eas it s resistance changes . W e are able , therefore , to determin e th e resistanc echange b y measuring th e chang e i n voltag e drop across th e gage .

ELEMENTARY CIRCUIT S 9 5

We no w investigat e th e relationshi p betwee n th e uni t chang e i n gag eresistance and th e corresponding chang e in voltage drop acros s th e gage forthe constant current circuit . In making th e analysis, w e refer to Fig . 3.3 . Theinitial condition s are agai n give n by Eq . (3.1).

When th e resistance of the gage changes from Rg to R g + ARg, w e can writ ethe corresponding expressio n for the voltag e drop acros s th e gag e as

Substituting th e valu e o f E from Eq . (3.1 ) into Eq . (3.11) , we have

Equation (3.12 ) can be written in terms of s and GF by using Eq. (3.5). Thus,

The potentia l dro p acros s th e gag e pe r uni t of strain may b e written as

From thes e equation s w e ca n dra w th e followin g conclusions fo r th econstant curren t circuit:

1. Th e chang e i n potentia l dro p acros s th e gage , E , wil l b e a linea rfunction o f the strai n (o r th e uni t chang e i n gage resistance , AR g/Rg).

2. Th e indication , or output , per uni t strain is a linear function o f each ofthe thre e quantities : (a ) gag e curren t 7 , (b) gag e resistanc e R g, an d(c) gage facto r G F, as wel l as thei r product .

3. Th e maximu m output pe r uni t strain , ( E/e)max, wil l occu r whe n th eproduct IR gGF reache s a maximum.

4. Fo r thi s typ e o f circuit , whic h i s voltage-sensitive , th e precedin gindicates tha t the maximum output will be achieved wit h high-resistancegages with high gage factors. This i s in direct contras t t o th e constant -voltage (current-sensitive ) circuit fo r whic h th e maximu m outpu t i sachieved wit h low-resistance gages possessin g hig h gag e factors.

5. Sinc e the gage resistance R g an d th e gag e factor G F are both propertie sof the gage, and becaus e th e maximum curren t I m is determined b y thegage's abilit y to dissipate heat, the maximum attainable output per unitstrain, (A£/e)max, depends entirely upon the characteristics of the gage.

96 TH E BONDE D ELECTRICA L RESISTANCE STRAI N GAG E

Item (5 ) is rather importan t becaus e i t tells us that the maximum possibleoutput, pe r uni t o f strain, i s dependent onl y upo n th e propertie s o f the gag eand no t upo n th e characteristic s o f the circui t t o whic h i t is attached. I f theefficiency o f the electri c circui t i s 10 0 percent, the n th e maximu m possibl eoutput ca n b e achieved .

The efficienc y o f th e electri c circui t i s expresse d a s th e rati o o f th emaximum output fro m th e circuit per uni t of strain divided by the maximumpossible output fro m th e gage pe r uni t o f strain. According to thi s definition,the constant-curren t circui t i s 10 0 percen t efficien t sinc e i t deliver s th emaximum possibl e outpu t fro m th e gage .

3.4. Advantages of the constant-current circuit

The tw o circuit s just discusse d represen t differen t approache s t o th e deter -mination o f th e sam e thing , namely , strain . The y ar e bot h specia l case s ofthe potentiometri c circuit , which i s o f a mor e genera l nature , an d includ ecertain o f the advantage s o f each o f these two elementar y forms .

The linear relatio n betwee n strai n an d the output , E, o f the constan tcurrent circui t i s a tremendou s advantage , i f no t a necessity . Fo r metalli csensors thi s characteristi c i s desirabl e bu t no t s o importan t sinc e th eresistance change s ar e small . However , fo r smeiconducto r gage s subjecte dto any appreciable amoun t of strain this is practically a necessity, particularlysince th e degre e o f nonlinearit y varies wit h change s o f referenc e o r initia lreading.

Since th e constant-curren t circui t gives its indication o f strain i n term sof a change i n voltage , i t i s ideally suited for us e with numerous well-knowntechniques an d standard instruments alread y develope d (fo r other purposes)to measure smal l voltage changes precisely. This represents both convenienc eand economy .

About th e only rea l advantage o f the constant-voltage (acros s th e gage )circuit lies in its ability to use a simple, inexpensive battery as a power supply .However, thi s advantag e i s als o possesse d b y th e potentiometri c circuit ,which ca n alway s approximate , an d sometime s achieve , constant-curren tconditions, wit h th e correspondin g advantage s o f th e constant-curren tcircuit.

There was no constant-current powe r supply commercially availabl e forstrain gage use until the early 1960 s (1). Although the constant-current powe rsupply cost s considerabl y mor e tha n a battery , nevertheless , i t i s no t a nexpensive instrument . Referenc e 1 contain s a lis t o f it s specification s an dcharacteristics, whil e Fig . 3. 5 is a schemati c diagra m o f a constant-curren tcircuit.

Some additiona l advantage s o f a constant curren t circui t are a s follows :

1. Withi n the powe r capabilit y o f th e constant-curren t source , ther e wil lbe n o effec t fro m lon g lead s o f appreciabl e resistance , sinc e th e lea d

ELEMENTARY CIRCUIT S 97

FIG. 3.5 . Schemati c diagram o f constant-current circuit .

resistance wil l no t alte r th e curren t flow . Sinc e a high-impedanc ereadout instrument , whic h draw s essentiall y n o current , mus t b eemployed, appreciabl e resistanc e i n th e lead s t o i t wil l caus e n otrouble, fo r practically n o curren t flows in these leads .

2. Fo r th e sam e reason s give n in Ite m (1) , i f the arrangemen t show n i nFig. 3.5 is employed, variations i n contact resistanc e at switche s or sliprings wil l have no effec t o n th e indicate d outpu t fro m th e gag e a s lon gas th e curren t sourc e ca n respon d rapidl y enoug h t o th e resistanc echanges t o maintai n th e constan t current .

3.5. Fundamental laws of measurement

In th e foregoin g circui t analysi s i t ha s bee n assume d tha t th e readou tinstrument woul d dra w n o current . Thi s bring s u s t o th e consideratio n o ftwo fundamenta l concept s whic h apply no t onl y t o strai n gage s an d thei rassociated electrica l instrument s bu t i n genera l t o measurement s o f al lkinds (2) .

These concept s ar e frequentl y referre d t o a s th e fundamental laws o fmeasurement an d ca n b e briefl y state d a s follows :

1. Th e instrument , o r device , use d t o mak e a measuremen t shoul d hav eno (o r negligible ) effec t upo n th e quantit y bein g measured .

2. Th e quantity being measured shoul d hav e no (or negligible) effec t upo nthe instrument , o r device , used t o mak e th e measurement .

Numerous example s of violations o f these laws may b e cited . However , th efollowing examples taken from strain gage studies will serve as illustrations:

1. A larg e strai n gag e o n a stif f carrie r i s use d t o measur e strai n o n aslender specime n o f low-modulus material . Thi s violate s th e firs t la wbecause th e stiffenin g effec t o f th e gag e mask s th e tru e valu e o f th estrain tha t i s to b e measured .

FIG. 3.6 .

REFERENCES

1. Stein , Pete r K. , "Th e Constan t Curren t Concep t fo r Dynami c Strai n Measure -ment," Strain Gage Readings, Vol . VI , No . 3 , Aug-Sept . 1963 , pp . 53-72 . Als oBLH Measurement Topics, Vol . 6 , No. 2 , Spring 1968 , pp . 1-2 , an d Instruments &Control Systems, Vol . 38 , No. 5 , May 1965 , pp . 145-155 .


2. A hig h gag e facto r strai n gage , whic h ha s bee n develope d t o measur every smal l strains , an d whic h wil l suffe r a los s i n gag e facto r i foverstrained, i s inadvertentl y use d t o measur e strain s wel l int o acondition o f yielding . Thi s violate s th e secon d la w becaus e a charac -teristic o f th e gage , namel y th e gag e factor , ha s bee n change d b y th estrain tha t th e instrumen t is endeavoring t o measure .

In estimatin g conformit y wit h th e law s o f measurement , sinc e mos tmeasuring devices have some influenc e (althoug h perhaps ver y small) on th equantity bein g measured , i t i s necessar y t o determin e t o wha t degre e thi sinfluence i s taking place an d whethe r or not thi s can b e considered negligibl efor th e particula r se t o f conditions a t hand . Unde r on e se t o f conditions agiven effec t migh t b e quit e negligible , wherea s i n othe r circumstance s th esame thin g migh t b e ver y important . Fo r example , fo r stres s analysi s on ecan often safely neglect errors which are exceedingly importan t when relatingto loa d cell s an d othe r weighin g devices .

Problems

3.1. Plo t A// / vs . ARa/Rg fo r value s o f ARg/R,, between 0 an d 1.0 .3.2. A strai n gag e wit h R a = 12 0 ohms an d G F = 2.5 i s bonde d t o th e simpl y

supported beam shown in Fig. 3.6 . A constant voltage of V = 2.4 volts is appliedacross th e gage . The bea m i s restrained in such a manne r that i t i s free t o ben dbut no t t o buckle . Determine (a) th e gag e curren t afte r loadin g an d (b ) th enonlinearity factor .

3.3. Red o Problem 3.2 for Rg = 350 ohms but al l other factors remaining the same.3.4. I n Fig . 3.2 , a resisto r is shunted across th e strai n gage, R a, i n orde r t o simulat e

a hig h strain . If Rg = 12 0 ohms, G F = 2.15 , V - 3 volts, an d th e shun t resistoris R p = 100 0 ohms, determine the final curren t I f, an d th e nonlinearit y factor .

3.5. I n the constant-current circui t shown in Fig. 3.3, a resistor. Rp, i s shunted acrossRg. If / = 0.02 5 amperes, Rg = 120 ohms, Gf = 2.0 , and E = -0.0 6 volts , deter-mine th e valu e o f R p.

ELEMENTARY CIRCUIT S 9 9

2. Stein , Pete r K. , Measurement Engineering, Stei n Engineerin g Services , Inc. , 560 3East Mont e Rosa , Phoenix , A Z 85018-4646 , Vol . II , 1962 , Chap . 24 . Vol . I I is :The Strain Gage Encylopaedia. Chap . 2 4 i s o n Circuits for Non-Self-GeneratingTransducers.

4THE POTENTIOMETRIC CIRCUIT

4.1. Introduction

The potentiometri c circui t i s als o know n a s th e ballas t circuit , o r serie scircuit. Because , i n effect , i t correspond s t o hal f a Wheatston e bridge , i t i ssometimes referred t o a s the hal f bridge. The circui t is represented schemati -cally i n Fig . 4.1.

In it s elementar y form , a s applie d t o strai n gages , th e potentiometri ccircuit contain s th e followin g three majo r components :

1. A power supply , usually a battery , which wil l furnis h constan t voltag eV t o th e circuit.

2. A strai n gage o f initia l resistanc e R a.3. A ballas t resistance , o f initia l valu e R h, t o contro l th e curren t i n th e

circuit. Sometimes the ballas t resistance consists of a second strai n gagewhich, dependin g upo n th e particula r condition s prevailing , may o rmay no t b e identica l to R fl.

In additio n t o th e abov e components , ther e mus t als o b e som e mean s o fobtaining a measure , o r readout , o f the chang e i n voltag e dro p acros s th egage (o r ballast resistance). This provides an indicatio n o f the change i n gageresistance, A7? g, which, in turn, represents a measur e o f the strain . The exac tnature o f the readou t device , or system , wil l depen d upo n th e magnitud e o fthe signal , A£ , an d th e precisio n wit h whic h i t i s desire d t o mak e th eobservation.

A stud y o f Fig . 4. 1 reveal s tha t th e potentiometri c circui t i s reall y acompromise betwee n th e tw o simpl e arrangement s describe d i n Chapte r 3on elementar y circuits . Both o f th e elementar y circuits are actuall y specia lcases o f this somewha t mor e generalize d form . Th e followin g concepts wil lassist i n clarification:

1. R b = 0: If the ballas t resistanc e i s reduced t o zero , w e have th e cas e ofthe strain gage directly connected t o a constant-voltage power supply.

2. R b - > oo: In this case R h is very large relative to R g. Le t us consider wha thappens a s th e ballas t resistanc e i s increased an d th e applie d voltage ,V, is correspondingly steppe d u p t o maintai n some desired initia l valueof gage current before strain takes place at the gage (i.e., when AR9 = 0) .

As Rb becomes progressivel y larger, the gage resistance Rs assume s

THE POTENTIOMETRI C CIRCUI T 101

FIG. 4.1. Th e potentiometri c circuit .

a smalle r proportio n o f th e tota l resistanc e i n th e circuit . I n con -sequence, any changes in gage resistance, AK 3, will have a progressivelysmaller influenc e o n th e flo w o f current , unti l ultimately , whe n R b i slarge enough , th e effec t o f change s i n gag e resistanc e wil l hav e a ninsignificant effec t o n th e current . Whe n thi s conditio n ha s bee nreached, for practical purposes , we have essentially achieved a constant-current circuit.

Subject t o the limitation of keeping the applied voltag e within safeworking limits, the potentiometric circui t may be made to approximat ea constant-curren t circui t t o an y degre e o f precision . Unde r thes econditions we might think of the power supply and th e ballas t resisto ras being combined int o a singl e unit providing, within specified limits ,a constant-curren t sourc e connecte d t o th e strai n gage .

3. AR b = — AK9: Sometimes i t i s possibl e t o var y th e ballas t resistanc e(for example , when it consists of a second strai n gage) i n such a mannerthat it s change, A/?6, is equal i n magnitude, but o f opposite sign , to th echange i n gag e resistanc e AR 9. Unde r thes e condition s th e tota lresistance in the circui t remain s constant . Thus , for a constant applie dvoltage, w e hav e a tru e constant-curren t circui t possessin g al l th eadvantages indicate d previously.

4.2. Circuit equations

In orde r t o obtai n a n immediat e insigh t int o the properties o f the potentio -metric circuit , th e circui t equation s an d som e discussio n o f the m ar epresented here . Th e complet e derivation s wil l b e develope d late r i n th echapter.

For convenience , the relationship between the ballast resistor , Rb, an dthe gag e resistance , R g, i s expressed a s a dimensionles s ratio as follows:


The incrementa l output fro m th e circuit , when writte n in term s o f unitchanges i n ballas t an d gag e resistances , and th e rati o a , is expressed a s

where n i s a nonlinearit y facto r give n as

For a singl e activ e gage , R b i s a constan t an d AR b = 0, s o Eqs . (4.1 )and (4.2 ) reduc e t o th e following:

Since AR g/Rg = strai n time s gag e facto r = eG f , Eqs . (4.3 ) and (4.4 ) ca n b erewritten a s

Characteristics of the circuit

Some discussio n o f th e equation s fo r th e potentiometri c circui t i s no w i norder.

1. Difference o f tw o strains — Eq. (4.1) . Whe n th e ballas t resistanc e i svariable, th e chang e i n outpu t voltage , A£ , i s directly proportiona l t othe algebraic differenc e betwee n the unit changes in the gage and ballas tresistances, providin g the nonlinearit y factor , n , can b e neglected . This

THE POTENTIOMETRI C CIRCUI T 10 3

means the circuit is capable of providing a reading directly proportiona lto th e algebraic difference between the strains a t tw o gage locations. Ifthe gages hav e positive and negativ e gage factors , then th e reading wil lbe the algebrai c sum .

2. Magnification o f th e strain gage signal — Eq. (4.1) . Whe n strain s o fknown rati o bu t o f opposite sig n prevai l a t tw o locations , th e signa lARg/Rg ca n b e increased b y usin g a secon d activ e gage fo r the ballas tresistor, Rb. Fo r example , if the strain at the gage comprising th e ballas tresistor, R b, i s equa l bu t o f opposit e sig n t o R g, th e outpu t wil l b edoubled. I n thi s particula r cas e w e hav e a constant-curren t circui twhose output will be linear with strain, and o f the maximum obtainabl evalue pe r uni t o f strain .

3. Linearity — Eqs. (4.2 ) and (4.4) . Basically, the incrementa l outpu t o f thecircuit, A£, i s a nonlinea r functio n of the strain .

Single gage. Fo r a singl e gag e wit h a fixe d ballas t resistance , R b,nonlinearity i s alway s th e case . Nevertheless , th e nonlinearit yfactor, n , ca n b e mad e negligibl y smal l b y havin g R b larg erelative t o R g.

Two gages. When th e ballas t resistance , R b, consist s o f a secon d strai ngage (which is not necessaril y required t o have the same resistanc eor gag e facto r a s R g), th e optimu m conditio n i s achieved whe n

When thi s situatio n prevail s w e hav e a constant-curren t circui tthat give s a linea r outpu t o f maximum attainabl e valu e pe r uni tstrain; tha t is ,

where I m i s the maximu m permissible gag e current .

When th e ballas t resistanc e i s fixed and severa l like gages are connecte din serie s i n th e adjacen t arm , show n i n Fig . 4.2 , th e change i n voltage dropacross all the gages will correspond to the average of the strains experiencedby th e gages. I n othe r words , A £ represent s th e averag e strain , e , for thi sarrangement.

For lik e gase s i n series , th e previou s equation s ca n b e use d wit h th efollowing modifications :


FIG. 4.2. Gage s in serie s i n th e sam e arm.

What was previously expressed as the unit change in gage resistance is now

Substituting th e value s o f a give n b y Eq . (4.8 ) and th e uni t chang e i n gag eresistance given by Eq . (4.9 ) into Eq . (4.3) , we arrive at

Similarly, th e valu e of n i s

Applications

The potentiometri c circui t shown in Fig . 4.1 will work equally well for stati cor dynamic strains , o r combination s thereof . However, the means employe dto measur e A £ impos e certai n limitation s which determine its applicabilit yfor stati c strai n a s wel l as dynamic strai n observations .

If th e resistanc e chang e i n th e strai n gag e i s larg e enoug h s o tha t a ninstrument employed t o measure E , the initial voltage drop across the straingage (fo r zero strain) , i s also capabl e o f measuring th e change , E , t o th edesired degre e o f precision, the n bot h stati c an d dynami c observation s ca nbe made (a s long a s the dynami c response o f the instrumen t i s suited t o th efrequencies o f th e strai n signals).

unit change

THE POTENTIOMETRI C CIRCUIT 105

FIG. 4.3. Potentiometri c circuit with filter to eliminate steady component, E, from the output.

In general , however , E wil l be too small to be measured conveniently,and wit h th e desire d precision , o n th e sam e scal e a s tha t use d fo r th emeasurement o f E . I n orde r t o overcom e thi s difficulty , i t i s customar y t ouse a filter tha t wil l eliminat e th e stead y voltage , E , so tha t E ca n beamplified an d measure d b y itself . Th e metho d work s ver y wel l fo r th edetermination o f dynamic strains , but th e filter which eliminates the stead y(zero-frequency) component , E , als o eliminate s an y othe r zero-frequenc ysignals and, i n consequence, preclude s the possibility o f making stati c strai nobservations. Th e potentiometric circui t with a filter to eliminate the steadycomponent, £ , i s shown schematicall y i n Fig . 4.3 .

Due t o th e relativel y smal l signal s produce d b y metalli c strai n gages ,and th e consequen t us e of the filter to eliminat e £ , th e us e of the potentio -metric circuit has, in the past , bee n limited to dynamic strain measurements .As a result of the developmen t o f semiconductor strai n gages with high gagefactors, and th e availability of four- o r five-place digital voltmeters , it is likelythat thi s circui t wil l also be use d fo r numerou s static applications .

Advantages and limitations of the potentiometric circuit

Among th e advantage s o f th e potentiometri c circui t th e followin g may b eincluded:

1. Extrem e simplicity.2. Abilit y t o approach , an d i n som e case s t o reach , th e linearit y an d

optimum outpu t o f the constant-curren t circuit .3. Th e circui t is able t o us e a simpl e constant-voltage , ripple-free , power

supply fro m a batter y (dr y cell ) and , a t th e sam e time , t o provid e a noutput i n the form of a voltage chang e tha t ca n be measured relativel yeasily.

4. Th e circuit , the readou t instrument , an d associate d amplifie r (i f one isused), can al l be connected t o a commo n ground .


Among th e limitation s of the potentiometri c circuit we find the following :

1. Inabilit y to measur e stati c strain s wit h metalli c or othe r strai n gage sproducing ver y low-leve l signals . This i s no t reall y a limitatio n of th ecircuit, bu t o f the readou t equipmen t associated wit h it .

2. Th e strai n signal , A£ , i s directly proportiona l t o th e batter y voltage ,V. I f the batter y run s down , th e strai n signa l wil l b e influenced . Fo rdynamic measurement s whic h ca n b e complete d i n a shor t spac e o ftime, thi s wil l probabl y caus e n o difficulty , bu t i f observations ar e t obe made over a considerable time interval, the n periodi c checks of thebattery conditio n shoul d b e made.

4.3. Analysis of the circuit

For th e purpos e o f analyzing the potentiometri c circuit, refer agai n t o Fig .4.1 an d conside r th e cas e wher e the resistance s of both th e strai n gage , R g,and th e ballas t resistor , R h, underg o changes .

Under initia l conditions , befor e any change s tak e place, th e expressio nfor th e voltag e dro p acros s th e gag e is

If R h an d R g chang e t o R b + ARb an d R g + A.Rg, respectively , the voltag edrop acros s the gage become s £ + A £ and i s expressed a s

By subtractin g th e valu e of £ give n b y Eq . (4.12 ) fro m bot h side s o f Eq .(4.13), th e chang e i n voltage dro p acros s th e gage , A£ , is

Now divid e th e numerato r an d denominato r o f th e right-han d side o f Eq .(4.14) by Rg. This puts all of the quantities into terms of dimensionless ratio sand uni t changes . Thus ,


Again w e can writ e the rati o o f the ballas t t o gag e resistanc e as

and s o

Insertion o f th e value s o f a an d l/R g, give n b y Eqs . (4.16 ) an d (4.17) ,respectively, into Eq . (4.15 ) gives

We observe that , i n Eq . (4.18) , all of the quantitie s involving resistanceare expressed either in terms of the dimensionless ratio, a , or as unit changesin gag e an d ballas t resistances . I f we assum e th e ballas t resistanc e t o b e astrain gage , the n th e uni t change s i n th e resistance s ar e directl y relate d t ostrain, since

Equation (4.18 ) can be reduced t o a simpler and mor e convenient form.We star t b y slightl y rearrangin g th e term s an d puttin g everythin g over acommon denominator . Thi s gives

Clearing th e parenthese s i n the numerator , Eq . (4.19) become s


This reduce s t o

Multiplying th e numerato r an d denominato r o f Eq . (4.20 ) b y ( 1 + a )results i n

Equation (4.21 ) indicate s tha t E wil l no t b e a linea r functio n o f thedifference between the uni t change s in gage and ballas t resistance s unles s

This wil l take plac e whe n

which give s u s

This mean s

Since R g i s usually not equa l t o — R h, i n general E wil l b e a nonlinea rfunction o f th e differenc e betwee n th e uni t change s i n gag e an d ballas tresistances.

We now examine the deviation from linearity , represented b y the symboln, in a modifie d versio n o f Eq . (4.21) . It ma y b e rewritte n a s

where n i s th e nonlinearit y factor.


Since Eqs. (4.21) and (4.26) represent the same quantity, E, w e see that

We now solve for the nonlinearity factor, n, in terms of the uni t change sin th e ballas t an d gag e resistances , R g/Rg, R b/Rb, an d th e rati o o f th eballast t o the gage resistance , a = Rb/Rg. Rearrangemen t o f Eq. (4.27) showsthat

Putting th e right-han d sid e ove r a commo n denominato r give s

This reduce s t o

Dividing th e numerato r an d denominato r o f th e right-han d sid e b yARg/Rg + a R b/Rb, w e arrive a t

Thus, fo r a give n set o f conditions, th e deviatio n fro m linearit y can b edetermined fro m Eq . (4.29) . Not e tha t th e nonlinearit y i s a functio n o f a ,

Rg/Rg, an d R b/Rb.


Limitation on applied voltage V

Since th e signal , E , i s directl y proportiona l t o th e applie d voltage , th emaximum outpu t wil l b e achieve d fo r th e larges t valu e o f V . This wil l b elimited b y th e followin g tw o practica l considerations :

1. Th e maximu m current , I m, tha t th e gag e ca n carry . Frequentl y th emaximum curren t wil l b e limite d to abou t 3 0 milliamperes, bu t i t ma ybe less , depending o n th e particula r conditions prevailing .

2. Th e maximu m voltag e tha t ca n b e safel y handle d fo r a give n applica -tion. U p t o 30 0 volts have bee n use d i n som e cases , bu t i t i s preferre dto kee p V dow n t o 9 0 volts. I f i t wer e no t fo r thi s restriction , an ypotentiometric circuit , fo r practica l purposes, coul d b e mad e linea r b ymaking th e rati o a = R h/Rg indefinitel y large .

From th e firs t limitation , whe n l m i s th e maximu m permissibl e gag ecurrent, th e maximu m allowabl e voltag e wil l b e

However, the secon d restrictio n o f safety ma y requir e the us e of a somewha tlower value .

Use with a single gage

When onl y on e strai n gag e i s use d i n th e circuit , th e ballas t resistanc e wil lbe fixed. Under thi s condition, R h i s constant an d R h = 0 . Equations (4.26 )and (4.29 ) then reduc e t o Eqs . (4.3 ) and (4.4) . The latte r two equation s wil lbe renumbere d i n thi s section fo r convenience ; the y ar e

If i t i s more convenien t t o dea l i n term s o f strain rathe r tha n i n term sof uni t change s i n gag e resistance , the n Eqs . (4.5 ) an d (4.6 ) ma y b e used .Again, renumberin g give s

and

THE POTENTIOMETRI C CIRCUIT 11 1

and

Circuit efficiency

The circuit efficiency, , of a particula r circui t may b e expressed a s the rati oof it s maximu m output , pe r uni t o f strain , t o th e correspondin g valu e fo rthe constant-current circui t tha t produce s th e maximum obtainable output .Thus,

By expressing Eq . (4.33 ) in term s o f current, w e can readil y determin ethe efficienc y o f a given circui t wit h a singl e gage. Sinc e

we can substitut e this value of V into Eq . (4.33 ) to obtai n

From Eq . (4.35) ,

Equation (3.14 ) give s the potentia l dro p across th e gag e pe r uni t o f strain,for a constant-curren t circuit , a s

and s o


It is interesting to note that, for the potentiometric circuit , the maximumvalue of E/E occur s whe n n = 0 , which corresponds to zero strain . For th econstant-current circuit , whos e outpu t i s linea r wit h strain , n = 0 fo r al lvalues o f strain.

Since « i s a functio n o f strain , al l comparison s shoul d b e mad e o n th esame basi s o f strain, o r fo r th e sam e valu e of n . However, th e onl y valu e ofn commo n t o th e constant-curren t circui t an d al l potentiometri c circuit s isn = 0 . Hence thi s valu e must b e employed fo r th e previou s analysis.

Gages in series

Sometimes i t is desirable t o obtai n th e average value of the strains a t severa ldifferent locations . Thi s ca n alway s b e don e b y measurin g th e individua lstrains a t eac h locatio n an d subsequentl y calculating the average value . Fo rstatic observation s ther e i s n o proble m becaus e w e merel y emplo y aswitching devic e t o connec t eac h gage , i n turn , t o th e strain-indicatin ginstrument. Bu t fo r dynamic observations , i n orde r t o determin e th e strainsat al l location s simultaneously , it i s necessar y t o hav e a complet e channe lof instrumentatio n fo r eac h strai n gage , o r t o hav e a high-spee d scannin gdevice.

Unless we need t o know the individua l values of the strain a t each gage ,time, equipment , an d effor t ca n b e save d i f a readin g o f th e averag e valu ecan b e obtained directly . Fortunately , w e are abl e to d o thi s b y connectin ga numbe r o f like gages i n series , as show n i n Fig . 4.4 .

Equations (4.31 ) throug h (4.34 ) stil l appl y fo r gage s i n series , bu t th evalues o f the symbol s wil l b e somewha t different . I n th e ar m o f th e circui tcontaining th e strai n gages , th e resistanc e wil l no w b e mad e u p o f the su mof th e resistance s o f the individua l gages. Thus ,

where x = the numbe r o f like gages . The tota l resistanc e change i n this ar m

FIG. 4.4. Potentiometri c circui t wit h strai n gage s i n series .


consists o f the su m o f the change s i n th e individua l gages. Thus ,

Equation (4.31 ) now become s

and Eq . (4.32 ) i s

For x like gages of resistance Rg, Eq . (4.37) may be modified as follows :

The rati o a i s expressed a s

Using AR g/Rg = sGF, E ca n be written in terms of strain:

We see that th e averag e strai n i s


and s o

Maximum applied voltage

When gage s ar e adde d i n series , th e rati o a mus t b e kep t fixed in orde r t omaintain a give n conditio n o f linearity . This mean s th e ballas t resistanc emust be stepped u p proportionately. However, to obtain th e greatest outpu tper unit of strain, the gage current must be maintained a t it s maximum value,lm. Th e applie d voltag e mus t the n b e increase d i n proportio n t o th e tota lresistance i n th e circuit . Therefore , subjec t t o th e limitation s o f safety , th emaximum applie d voltage , V m^, fo r gages i n serie s wil l b e give n by

Static vs. dynamic measurements

An examinatio n o f th e schemati c diagra m Fig . 4. 1 fo r th e potentiometri ccircuit, and Eqs. (4.1), (4.3), and (4.5) for the incremental output E, indicate sthat al l we need for a strain measurement is to observe the change in voltagedrop across the gage . Thi s applie s t o eithe r static o r dynami c conditions .

This i s perfectl y correct . However , whe n w e begi n t o loo k int o th epractical aspect s o f selecting a suitabl e measurin g instrument , we run int othe difficult y tha t E ma y be ver y smal l relativ e to E . In thi s case , i f theinstrument ha s a readou t scal e suitabl e fo r measurin g E , it ma y b e entirelyunsuited for the measurement of E, o r vice versa. We should therefore makesome preliminar y estimate o f the approximate value s of E and E i n orde rto decid e upon a n instrument, or readou t system , which wil l determin e E(or th e strain ) t o th e desire d degre e o f precision.

To illustrat e this point , we take u p tw o examples . The first considers asemiconductor strai n gage, the second a metallic strain gage. Fo r simplicity .

Equation (4.39 ) tell s u s tha t i f « i s smal l enoug h t o b e neglected , the nE x eav.

Equation (4.38 ) become s

THE POTENTIOMETRI C CIRCUIT 115

consider tha t a = Rb/Rg wil l be large enough s o the outpu t an d linearit y ofthe constant-current circuit are closely enough approximated for all practicalpurposes.

Example 4.1. Th e followin g value s ar e give n fo r a potentiometri c circui t usinga semiconducto r strai n gage :

Gage resistance, R s 12 0 ohmsGage factor, GF 10 4Gage current, / 2 0 milliamps

Solution. Th e voltage , £ , acros s th e gage is

E = IRg = (0.020)(120) = 2. 4 volts

For optimu m condition s (constan t current ) and assumin g the gag e ha s linea rresponse,

or

When s = 400 0 microstrain = 4000 uin/in, the n

A£ = 250f i = (250)(400 0 x 1 0 ~6) = 1 volt

We ca n loo k a t ho w thi s migh t b e represente d o n a D C voltmete r (o rrecorder) wit h a linear scal e 5 inches lon g marke d of f in inches and subdivide din tenth s o f inches . Thi s i s represented graphicall y i n Fig . 4.5 . Not e tha t th evoltmeter shoul d hav e hig h impedanc e t o avoi d loadin g th e circuit.

FIG. 4.5. Voltmete r reading s o n linea r scale , semiconducto r gage .


FIG. 4.6. Reading s o n digita l voltmeter , semiconductor gage .

For zer o strain the pointer wil l come t o the position fo r 2.4 volts (2.4 inchesalong th e scale) . I f the gag e facto r i s positive, th e pointe r wil l mov e t o th e righ twhen strai n i s applied (fo r positive strain) or t o th e lef t (fo r negative strain) b yan amoun t o f 1 inch fo r ever y 400 0 microstrain , o r 0. 1 in fo r ever y 40 0microstrain. I f we ca n mak e observation s o f the positio n o f the pointe r t o th enearest half-divisio n o n th e scale , th e reading s wil l b e goo d t o th e neares t200 microstrain.

For strain s o f th e orde r o f 400 0 t o 500 0 microstrain, a readin g t othe neares t 20 0 microstrain represent s th e neares t 4 o r 5 percent , whic hin man y case s i s good enough . However , i f we are dealin g wit h magnitude s o f400 o r 50 0 microstrain, the n a readin g t o th e neares t 20 0 microstrain ( + 50percent) i s no t nearl y goo d enough , an d s o a differen t typ e o f voltmete r i srequired.

Let us see how the same situatio n appears o n a high-impedance, four-placedigital voltmete r capable o f measuring from 0 to 9.99 9 volts . Initially, the mete rwill rea d 2.40 0 volt s fo r zer o strain , a s show n i n Fig . 4.6 . Fo r a strai n o f+ 400 microstrain, A £ wil l b e 0.100 , s o th e mete r wil l rea d 2.50 0 volts , o r achange o f 0.100 t o th e neares t 1 in 10 0 or th e neares t 1 percent.

This exampl e indicate s tha t fo r thi s particula r semiconducto r gage ,operating under the stated conditions, the potentiometric circuit can be usedfor stati c o r dynami c (u p t o th e frequenc y limit s o f the instruments ) strai nmeasurements a s follows :

1. Wit h th e simpl e mete r fo r strain s o f 4000 microstrain an d above .2. Wit h the four-place digital voltmeter for strains above 200 microstrain.

For dynami c observation s a t frequencie s highe r tha n thos e t o whic h thes emeters wil l .faithfully respond , a differen t syste m wil l hav e t o b e used .


Example 4.2. The followin g value s are give n for a metalli c strai n gag e use d i na potentiometri c circuit :

Gage resistance , Rg 12 0 ohmsGage factor, Gt 2.0 8Gage current, / 2 0 milliamps

Solution. Th e voltage , E , across th e gag e is

E = IRg = (0.020)(120 ) = 2. 4 volts

For optimu m condition s (constan t current) ,

A£ = IR gGFs = (0.020)(120)(2.08) e = 5e

Thus, whe n e = 400 0 microstrain = 400 0 uin/in,

E = 5s = 5(400 0 x 1 0 ~6) = 0.02 volts

or

A£ = 0.0 1 volts for 2000 microstrai n

On th e mete r scal e illustrate d i n Fig . 4.5, this woul d b e equivalen t t o20000 microstrain fo r one minor division, as shown in Fig. 4.7. For 400 0 micro -strain th e pointe r woul d mov e 1/ 5 of a minor division , and fo r 400 microstrainthere woul d hardl y b e an y perceptibl e motio n a t all . Obviously, thi s kin d ofmeter canno t b e used wit h metalli c gages becaus e th e ratio A£/£ i s too small .

We no w conside r wha t wil l happe n wit h a four-plac e digita l voltmeter .This is indicated i n Fig . 4.8 , where we observe tha t a reading t o th e neares t200 microstrain i s possible . I f the instrumen t ha d fiv e places , however , wecould obtai n a readin g t o th e neares t + 20 microstrain. Thi s woul d b eadequate fo r essentiall y al l requirements . Therefore , fo r th e metalli c gage ,

FIG. 4.7. Voltmete r reading s o n linea r scale , metalli c gage .


FIG. 4.8. Reading s o n digita l voltmeter , metalli c gage.

the simpl e voltmete r i s unsuitable , bu t a four-plac e digital voltmete r mightbe use d fo r relativel y roug h reading s ( + 5 percent ) o f hig h strains . Afive-place digita l voltmete r woul d b e capabl e o f indication s dow n t o+ 20 microstrain, an d fo r al l reading s abov e 20 0 microstrain capabl e o fachieving a precisio n o f 1 percent o r better .

Static strains

The preceding example s indicat e that static strain measurements can be madewith th e potentiometri c circui t provide d tha t w e hav e a suitabl e readou tinstrument an d tha t th e strain s ar e sufficientl y large .

When large strain s ar e to be measured wit h a semiconductor strai n gag epossessing a high gage factor , the change i n the voltage drop across the gag ewill b e large enough , wit h respec t t o th e initia l value , to permi t satisfactor yobservations wit h a n inexpensiv e meter . However , whe n semiconducto rgages ar e subjecte d t o smal l strains , o r fo r metalli c gages , th e chang e i npotential drop across the gages wil l be so small, relative to the ambient value ,that a comparativel y expensiv e digita l voltmete r wil l b e require d t o obtai na reasonabl y precis e strain indication .

Since static strain measurements requiring the use of an expensive digitalvoltmeter can b e obtained equall y well b y other method s wit h les s expensiveinstruments, th e us e o f th e potentiometri c circui t fo r stati c reading s i s no tvery attractive .

Dynamic strains

For dynami c strai n measurement, the simplicity of the potentiometric circui tand th e convenienc e o f using a commo n groun d fo r th e circui t an d relate dcomponents, make i t ver y attractive , in spit e o f the fac t tha t ther e ar e othe rcicuits tha t ca n als o b e use d t o determin e time-varyin g strain. Wher e stati c


FIG. 4.9. Potentiometri c circui t a s applied t o dynami c strain measurements .

observations are not required , the potentiometric circuit is very popular an dwidely used .

The usua l arrangemen t fo r usin g a potentiometri c circui t t o measur edynamic strains i s shown i n Fig . 4.9 . This include s a filter (condenser) tha twill eliminate the steady component , E , but wil l pass th e dynamic part, E(representing th e strain ) o f th e potentia l dro p acros s th e gage . Whe nthe signal , E , ha s bee n isolated , i t ca n b e amplifie d an d show n a s afunction o f time on some readily available instrument suc h as a cathode-rayoscilloscope.

The resul t o f eliminating E i n orde r t o observ e E i s to impos e th efrequency limitation s (both uppe r an d lower ) of the filter and th e amplifie rupon the final output signal . Since the filter was put int o the syste m for thespecific purpose of eliminating the steady component, E , it will also eliminateany stead y strain signals.

We shoul d not e tha t i t i s th e filter , whic h i s a par t o f th e readou tapparatus, tha t make s th e syste m unsuite d fo r stati c strai n measurement .The potentiometric circuit itself responds t o both stati c and dynamic strains,although w e can onl y observ e dynami c strains wit h this particular metho dof obtaining th e indication .

4.4. Linearity considerations

We ca n no w examin e th e deviatio n fro m linearit y o f the signal , E , wit hrespect t o strain , or AR g/Rg, i n a potentiometri c circui t with a singl e strai ngage an d a fixed ballas t resistanc e R b. Fo r thi s purpose , i t wil l b e bes t t oexpress the signal E a s a fraction of £, the initial potential dro p across thegage. We know that


where

The initia l potentia l drop acros s th e gag e i s

The applie d voltag e acros s th e circui t i s

Substituting th e valu e o f V given by Eq . (4.43 ) into Eq . (4.31 ) results in

Figure 4.1 0 show s E/ E fro m Eq . (4.44 ) plotte d versu s AR 9/R9 fo rvarious value s o f a = R hjRa. Fo r comparativ e purposes , th e linea r relatio nfor th e constant-curren t circuit , representing th e optimum , is also shown .

From Fig . 4.10 , we observe th e followin g characteristics :

1. Th e deviatio n fro m linearit y becomes large r a s AR g/Re increases .2. Fo r a give n valu e o f A/? 9/Rg, th e deviatio n fro m linearit y i s les s fo r

larger value s o f a and approache s zer o a s a become s ver y large .

Since i t i s necessary t o kno w the applie d voltage , V , that i s required fo ra given potentiometric circuit , we can choose the gage resistance and curren tfrom whic h E , the voltag e drop acros s th e gage , ca n b e computed b y usin gEq. (4.42). Using Eqs . (4.42 ) an d (4.43) , the rati o V/ E i s

The applie d voltag e V can no w b e calculated usin g Eq . (4.46) .

Since the circui t efficiency give n by Eq . (4.36 ) is r\ = aj(\ + a) , Eq. (4.44 ) ca nbe rewritte n as


FIG. 4.10. Rati o A£/E a s a function o f R/K .

Linearization with variable ballast resistance

The general case. Le t us now look into what may be achieved when the rati oEg/Eb, a t tw o locations , i s known from th e physica l conditions which prevail,especially whe n eithe r s g o r s b i s tensio n an d th e othe r i s compression . B ythe use of one, or more , strai n gage s for the ballas t resistance , R b, i t may bepossible t o produc e a constant-curren t circui t wit h a constan t applie dvoltage. T o accomplis h this , the tota l resistanc e i n th e circui t mus t remai nconstant a t al l times , so that , in symbols , one ca n write

This wil l tak e plac e whe n AR b = — ARg, s o tha t th e tota l resistanc e o f thecircuit, neglectin g lead resistance , is given by th e expressio n

Let u s now examin e th e relationshi p betwee n AR b an d — ARg tha t wil lpermit u s to hav e a constant-curren t circui t with constan t applie d voltage ;that is , when AR b = -AR g. Fro m th e basi c strai n gag e relatio n give n inChapter 1 , we can writ e


SO tha t

Since th e rati o R h/Rg wil l alway s be positive, Eq. (4.49) indicates tha t eithe rsb/£g o r (G F)b/(Gp\ mus t b e negativ e i n orde r t o achiev e constant-curren tconditions.

It shoul d b e note d tha t i f R a an d th e ballas t gage , o r gages , ar e no toperating unde r identica l lateral strain conditions, th e term (G F)b/(GF)g ma yhave to b e modified slightly to tak e into account th e difference s i n the ratio sof latera l strai n t o axia l strain on eac h gag e (latera l effect s ar e discusse d i na late r chapter) . However , thi s proble m ca n b e completel y eliminate d b yselecting gages , for bot h R a an d R h, whic h hav e transverse sensitivity factorsequal t o zero .

For semiconducto r gages , whic h ca n b e manufacture d wit h eithe rpositive o r negativ e gage factors , this means tha t thes e gage s ca n b e used i nlocations o f strain o f eithe r th e sam e o r opposit e sign . However , fo r gage swith metal sensors , fo r which the gage factors are only positive, one i s limitedto th e requiremen t tha t R g mus t b e locate d i n th e regio n o f tensil e strai nwhile R b mus t b e subjecte d t o compressiv e strain , or vic e versa.

Let u s assum e fo r th e momen t tha t (G F)b = (G F)g = G F. Fo r th econstant-current circuit s the nonlinearit y factor, n , becomes zero . Thi s ca nbe show n b y referrin g t o eithe r Eq . (4.2 ) or Eq . (4.29) . Thus ,

From this ,

and

Therefore, th e ratio —AR h/ R !l ma y be written as


If th e ballas t resistanc e consist s o f a strai n gage , whic h i s subjecte d t o th eappropriate amoun t of strain with respect t o that occurrin g at Rg, then , eventhough R b an d R g ma y no t b e equal, i t i s stil l possible , b y suitable choice ofrelative strain , gage factor , an d resistance , to produce a situation suc h tha tARb = — AR9, and thereb y t o achiev e linearity between incremental output ,E, an d the strain, a s well as the maximum signal per uni t o f strain.

The analysis of this is done by using Eq. (4.26), then taking ARb = — ARgand n = 0. Thus,

This reduce s t o

Since

then

Equation (4.50 ) is valid fo r al l value s o f Ri,/Rg = a.The relation expressed in Eq. (4.50) indicates that whe n one has selected

the gage resistance , R g, th e gage current , / , an d determine d th e gage factor ,GF, on e ha s establishe d th e valu e o f output pe r uni t strain , sinc e

For a give n gag e curren t an d resistance , / an d R g, th e choic e o f theballast resistance, R b, wil l determine the necessary applied voltage , V , or viceversa.

Comparison with fixed ballast resistance

It is of interest to compare the output pe r uni t strain for this variable-ballastconstant-current circuit , given by Eq . (4.51) , wit h th e correspondin g circui t

Using becomes


containing a fixed ballast, Rb. Fo r the fixed ballast, A£ is given by Eq. (4.35):

From this , th e outpu t pe r uni t strai n i s

The relative output of the fixed-ballast circuit to th e constant-curren t circuitis obtaine d b y takin g the rati o o f Eq . (4.52 ) to Eq . (4.51) . Calling thi s ratioRrel, w e obtain

or

where r \ = circui t efficiency .Figure 4.1 0 als o show s th e sam e informatio n i n term s o f th e rati o o f

the ordinate s o f th e curve s t o th e correspondin g ordinate s o f th e straigh tline for the constant-current circuit. In this figure we see two points commo nto al l th e curves , including the straigh t lin e fo r constan t current . Th e firs tpoint correspond s t o the origin , or zero valu e for &Rg/Rg. Th e second poin tcorresponds t o AK 9/Rg = — 1. Thi s latte r poin t represent s a somewha ttheoretical concept , sinc e i t correspond s t o a reductio n i n gag e resistanc eequal t o the original value , Rg. Thi s means that th e gage resistance has beenreduced t o zero , whic h cannot b e achieve d i n actua l practice wit h conven -tional strai n gages , althoug h i t migh t b e possibl e wit h a slide-wir e deviceunder short-circuit conditions. I f ARg/Rs = — 1 could be achieved, this wouldmean th e voltag e dro p acros s th e gag e ha d bee n reduce d t o zer o an dconsequently E/ E = — 1.

Let u s now examin e two differen t situation s involving the linearizationof the potentiometri c circuit wit h variabl e ballast resistance. I n th e firs t casethe gag e an d ballas t resistance s wil l b e equal , R h = Ra, an d i n th e secon dcase the y wil l b e unequal , Rh = £ R,.


Example with equal ballast and gage resistances

A usual case of this nature is represented by the use of two like gages mountedback to back o n a uniform beam of rectangular cros s section an d subjectedto simpl e bending , a s show n in Fig . 4.11 . In thi s particula r case , R b = R g,(GF)b = (GF\, and , due to th e characteristic s of the beam , e fc = — eg, so thatthe genera l equatio n fo r the rati o o f changes i n th e resistanc e o f the ballas tto tha t o f the gag e fo r thi s special cas e reduce s t o

Therefore, thi s rati o fulfill s th e requirement s fo r a constant-curren t circui t(when supplie d wit h constant voltage) . Thus, the equation fo r E, give n bythe genera l expressio n o f Eq. (4.26) , is

Since, for this case, V = 21R, th e value o f A£ is

From this ,

FIG. 4.11. Bea m in bending with potentiometric circuit, (a) Wiring diagram, (b) Space diagram.


and

This circui t i s automaticall y temperature-compensated , a s a n active -dummy system , a s lon g a s th e tw o gage s ca n b e maintaine d a t equa ltemperatures.

Gages with positive and negative gage factors

Another method o f achieving the sam e electrical characteristics is to us e twogages of equal resistance having gage factors of equal magnitude but oppositesign. The gage s ar e the n installe d sid e b y side, either independentl y o r o n acommon carrier . Thus , fo r an y strain , positive o r negative , th e increas e i nresistance of one gage is just equal to the decrease in resistance of the other ,and constant-curren t (optimum ) condition s wil l therefor e prevail . Thi ssystem ha s th e advantag e tha t bot h gage s ar e subjecte d t o th e sam e strain .The circui t is shown i n Fig . 4.12.

For gage s wit h metalli c sensors, th e concept o f employing elements withpositive an d negativ e gag e factor s i s somewha t academic , a s fe w metal spossess negativ e strai n sensitivity . Mos t o f thos e tha t d o posses s thi scharacteristic hav e othe r propertie s tha t mak e the m undesirabl e fo r straingages. However , sinc e th e adven t o f semiconducto r gages , whic h ca n b eproduced wit h a n infinit e variet y o f gage factors running from abou t — 100to abou t +200 , thi s concep t o f linearizin g th e circui t ha s becom e ver yimportant. Sanche z an d Wrigh t (1) give excellent quantitative information .

Example with unequal ballast and gage resistances

The approach to linearization under the special conditions of Rb = Rg, whileconvenient, is not a n essentia l condition. W e now loo k int o the genera l cas eto determine the relationship actually required between Rb and R . We know

FIG. 4.12. Gage s wit h ( + ) and ( —) gage factor s place d sid e b y side .


that fo r conditions o f constant curren t (V assumed fixed) the total resistanc ein the circui t mus t remai n constant . To achiev e this , i t i s necessary tha t

If strain gages ar e use d for both R b an d R g, then , since e = ( R/R)/GF fo r astrain gage , we can write

and

The subscripts b and g refer t o quantities related to the ballas t an d th e gage ,across whic h E i s being measured , respectively .

In orde r t o satisf y Eq . (4.25) , we must have

This means we can have any physicall y possible value s for the si x quantitiesin Eq . (4.56 ) a s lon g a s w e satisf y th e equation . Th e rati o Rt/R g ma y no wbe expressed as

Since R b an d R g mus t always be positive for strain gages , Eq . (4.57) tells usthat i f the tw o gag e factor s have th e sam e sign , then th e strain s mus t hav eopposite sign , or vic e versa.

Theoretically, there i s a wid e choice of values for th e quantitie s i n Eq .(4.57). Fro m a practica l poin t o f view, however, there ar e som e limitations .For example , whe n tw o gage s hav e bee n chosen , R b/Rg i s fixed as wel l a sthe ratio o f the gage factors , which do no t hav e to b e the same. This mean sthat th e gage s mus t b e installe d a t location s suc h a s e a/e6 wil l satisf y Eq .(4.57). I f this can b e don e conveniently , we have a mean s o f adjustment fordifference betwee n gag e factors .

When (G F)e = (GF)b, Eq . (4.57) reduces t o

Equations (4.57 ) and (4.58 ) indicat e the possibilit y o f linearizing the circui twith a pai r o f unlik e strain gage s whe n eg/eb =^ — 1.


Example 4.3. Tw o gage s o f unequa l resistance , bu t o f equal gag e factor , ar e t obe use d o n a cantileve r beam , a s show n i n Fig . 4.13 . Th e gage s ar e arrange dalong th e longitudina l axis , to p an d bottom , an d th e purpos e i s to desig n th ebeam cros s sectio n s o th e potentiometri c circui t i s linearized.

Solution. Sinc e a = R^/R g, w e choose « b = -K g/a. Fro m Eq . (4.54 ) we hav e

From Eq . (4.55) ,

With(G f)h = (G F)9 ,then AK b = — AK9 , and s o the nonlinearity factor, n, is zero.The signa l fro m th e circuit , A£ , i s given b y Eq . (4.1) :

and s o

We also kno w tha t

Fif;. 4.13 . Cantileve r bea m wit h strai n gages .


Substituting thi s value of V into th e expression for E, w e obtain

Thus, under these particular conditions , E i s a linear function o f strain.To complet e th e problem, R b and R g mus t be chosen, and the n a n

appropriate bea m cros s sectio n determine d s o tha t th e necessar yrequirements wil l b e met . Tw o gage s readil y availabl e wit h th e sam egage facto r hav e resistances o f 12 0 ohms an d 35 0 ohms. Mino r differ -ences betwee n th e gage s ca n b e expected , o f course , bu t the y wil lprobably be less than 1 percent. For thi s problem, choose R b = 35 0 ohmsand plac e i t o n th e to p o f the beam , the n R g = 12 0 ohms i s placed o nthe bottom of the beam directly underneath R h. This produces the valueof a and th e relationship between eb and e g. Determining the beam crosssection i s lef t a s a homewor k problem .

4.5. Temperature effects

Whenever th e mechanica l strai n varie s rapidl y i n relatio n t o chang e o ftemperature, i t i s perfectl y permissibl e t o emplo y a singl e strai n gage , a sshown in Fig . 4.14, and t o neglec t th e effec t o f the temperatur e change upo nthe signa l fo r th e time-varyin g part o f th e strain , eve n thoug h th e sensin gelement o f th e strai n gag e ma y hav e a hig h respons e t o change s i ntemperature. This procedure is quite appropriate when the mechanical effec ttakes place in such a relatively short interva l of time that the accompanyingchange i n temperatur e i s to o smal l t o caus e a n appreciabl e erro r i n th eindication o f th e dynami c componen t o f th e strain . However , i t i s alwaysdesirable to make an estimate of the approximate error anticipated fro m thi sprocedure a s applied t o a particula r se t of conditions.

Let us now look into what may be expected from a strain gage of knowntemperature response , a s mounte d o n a particula r material , whe n a give namount o f strai n i s t o b e measure d a t som e particula r frequenc y i n th epresence o f a varyin g temperature.

To illustrat e the point , conside r th e following :

Strain magnitude 50 0 microstrainFrequency 6 0 cycles/secGage response t o temperatur e chang e 150microstrain/° FRate of temperature change 12°F/mi n

or


FIG. 4.14 .

What wil l b e th e error , durin g on e cycle , o n th e trac e o f a recordin goscillograph?

Time fo r on e cycl e = 1/6 0 secTemperature chang e i n 1/6 0 sec = (1/60)( 12/60) = 1/300 CFGage respons e fo r 1/30 0 F = (1/300)(150 ) = 0. 5 microstrainPercent erro r i n signa l = 0.5/50 0 = 0. 1 percent

For stres s analysis , i n general , a n erro r o f 1 percen t ca n safel y b eneglected. However , w e mus t realiz e that th e precedin g calculatio n applie sonly t o th e dynami c strain signal for a singl e cycle, which is completed i n avery shor t interva l o f time. It give s no informatio n in regar d t o th e gradua lchange i n referenc e or zer o shift .

Let u s no w loo k int o th e questio n o f th e lengt h o f time fo r recordin ga transien t strai n withou t exceedin g a specifie d amoun t o f erro r du e t ochange i n temperature . W e assum e th e sam e numerica l value s use d i n th epreceding example , and determin e the tim e to develo p a 1 percent erro r du eto zer o shif t o r referenc e change. Figur e 4.15 wil l hel p t o illustrat e wha t i staking place .

Limiting erro r i n microstrai n = 1 percent o f 500 = 5 microstrainChange i n temperatur e t o develo p thi s error = 5/15 0 = 1/30° FTime for temperature change of 1/30°F to take place = (1/30)/12

= 1/36 0 min= 1/ 6 sec

This neglect s an y error s produce d b y temperatur e change s i n th e lea d wire sand soldere d joints . I t als o assume s tha t th e amplifier s transmit th e strai nsignals faithfull y a t thes e frequencies .

Slowly varying strains vs. temperature change

We hav e just discusse d th e measuremen t o f dynamic strai n withou t regardto temperature . W e no w conside r th e measuremen t o f dynamic strai n whe nthe influence s of temperatur e chang e canno t b e neglected . Therma l effect s


FIG. 4.15.

can produc e intolerabl y larg e error s no t onl y whe n th e mechanica l effec tvaries slowly in th e presenc e o f moderate variatio n in temperature , bu t als ofor certai n combination s o f high-frequency strain s and violentl y fluctuatingtemperature, suc h as migh t occur i n certain part s o f gas turbines . However,since th e importan t facto r t o conside r i s represented b y th e relativ e rates ofchange o f strain and temperatur e wit h respect t o time , the sam e method s ofcombating th e temperatur e effect s ma y b e applie d t o eithe r o f thes e tw oconditions. Some approaches t o this problem, with comments on the relativemerits, wil l no w b e noted .

Self-temperature-compensated strain gage

When applicable , on e o f the mos t effectiv e way s of minimizing the influenc eof temperatur e chang e i s t o emplo y a self-temperature-compensate d strai ngage. However , i t is necessary tha t th e environmenta l conditions be suitableand, i n general , w e have to conside r th e followin g points.

1. On e ma y hav e t o b e satisfie d wit h a gag e o f lowe r factor , sinc e th eusual self-temperature-compensate d gage s hav e gag e factor s o f abou t2.0 as contraste d wit h isoelastic gages wit h a gag e facto r o f about 3. 5(or platinum-tungsten alloy s and pure platinum, whose strain sensitivi -ties ar e abou t 4 and 6 , respectively).

2. Th e maximu m temperatur e (o r minimu m temperatur e fo r cryogeni capplications) a t whic h th e strai n gag e i s expecte d t o operat e wil ldetermine whethe r a conventiona l self-temperature-compensate d gag ecan b e employe d a t all , o r i f i t wil l b e desirabl e t o us e a gag e tha tpermits adjustmen t to sui t a particula r se t o f conditions .

3. Th e averag e operatin g temperatur e wil l hav e t o b e considered s o tha tone ma y selec t a gag e wit h th e bes t compensatio n fo r th e operatin gconditions. Thi s is due to th e fac t tha t th e gage's temperatur e respons eper uni t chang e i n temperatur e varie s wit h temperature . A gag epossessing th e bes t compensatio n o n a particula r materia l a t roo mtemperature may not b e nearly as well-compensated a t higher or lower


temperature a s som e othe r gag e wit h a muc h poore r performanc e a troom temperature .

4. Th e rang e o f temperature variatio n i s naturally of utmost importance ,since i t i s th e magnitud e o f th e temperatur e chang e whic h determinesthe erro r fro m thi s source . I f temperature variatio n coul d b e reduce dto zero , ther e woul d b e n o erro r fro m thi s cause . Zer o shif t migh t stil loccur i f the operatin g temperatur e i s differen t fro m ambien t tempera -ture, but thi s wil l no t appea r i n th e dynami c signal from th e potentio -metric circuit unless the temperature change fro m ambien t to operatingconditions take s plac e ver y rapidly .

5. Lea d wir e errors mus t also b e considered. Eve n though a self-tempera-ture-compensated strai n gag e ma y b e employe d wit h grea t succes s t ominimize the effect s o f temperature changes withi n the gag e itself , therestill ma y b e appreciabl e error s arisin g from th e temperatur e change soccurring i n th e lea d wires , especiall y i f th e rang e o f temperatur evariation i s large . Th e lengt h o f lea d wir e subjecte d t o temperatur echange will , o f course , b e important . Th e commo n metho d fo r eli -minating th e erro r cause d b y changes o f temperature in th e lea d wiresis t o us e th e three-wir e system shown i n Fig . 4.16.

The junction , C i n Fig . 4.16 , betwee n th e gag e an d th e tw o lead sindicated a s ballas t an d gag e mus t b e mad e righ t a t th e gage . Th e ballas tand gag e lead s mus t be brought ou t t o th e res t o f the circui t in close contac twith eac h othe r s o the y wil l b e subjecte d t o th e sam e temperatur e effects .For convenience , th e commo n lea d i s usuall y brought ou t i n contac t wit hthe othe r two , bu t thi s i s no t essential , sinc e i t carrie s n o curren t (high -impedance readou t instrumen t assumed ) an d doe s no t for m a par t o f th eactual strai n gag e circuit . W e no w loo k int o th e require d relationship sbetween th e resistance s o f the thre e leads: (a ) th e commo n lead , an d (b ) th eballast an d gag e leads .

FIG. 4.16. Three-wir e system for potentiometri c circuit.


The common lead

The sole purpose o f the common lea d i s to sens e the change in voltage dropacross th e gage . Because it carries n o curren t and i s not reall y a par t o f thestrain gage circuit , its resistance, and an y change s thereof , will not influenc ethe indicated outpu t from th e circuit. For convenience , this lead is frequentlyidentical t o on e o r bot h o f th e othe r two , bu t thi s i s no t essential , no r i sit necessar y tha t th e commo n lea d shoul d b e subjecte d t o th e sam etemperature condition s a s the othe r two .

The ballast and gage leads

We no w conside r mean s o f eliminatin g (o r minimizing ) errors cause d b ytemperature change s i n th e othe r tw o leads . Sinc e th e ballas t lea d an d th egage lea d ar e i n serie s with the ballas t an d gag e resistances , any chang e ofresistance produced i n eithe r o f the lead s by a change i n temperatur e wil lappear t o th e readou t instrumen t as a chang e i n gag e resistanc e (strain) .This result s in a n erro r i n th e strai n indication . However , b y appropriat eproportioning o f th e resistances , i t i s possibl e t o mak e th e temperature -induced error s cance l eac h othe r s o tha t th e outpu t signa l fro m th e circui tis independent o f thi s temperature effect .

Because the tota l indicated outpu t fro m th e circuit may be obtained b ysuperposition o f th e effect s i n th e lead s o n th e indicatio n fro m th e strai ngage, i t i s i n orde r t o conside r th e resistanc e change s i n th e lead s b ythemselves an d t o determin e unde r wha t condition s th e indicate d outpu tfrom the m can b e reduce d t o zero .

The output of the potentiometric circuit, A£, is given by Eq. (4.1). It is

Here w e observe that th e outpu t i s proportional t o th e differenc e betwee nthe uni t changes i n gage an d ballas t resistances . Thus,

Equation (4.59 ) indicates tha t ther e wil l b e n o outpu t fro m th e circui twhen th e uni t change s i n gag e resistanc e an d ballas t resistanc e ar e equal .Therefore, i f this condition ca n b e fulfille d whe n the lead resistanc e changes ,the effec t wil l no t b e seen by th e readou t instrument , and th e circui t outputwill b e independen t o f temperatur e effect s i n th e leads . Thus , w e se e b yinspection that , i f th e uni t change s i n lea d resistanc e ar e equal , th e


temperature effect s wil l balanc e out . Th e leads , then , shoul d b e selecte d s othat

where R hL = ballas t lea d resistanc e

RgL — gag e lea d resistanc e

To prov e th e statemen t mathematically , we make tw o assumptions :

1. Fo r th e tim e bein g th e ballas t an d gag e resistance s wil l remai n fixed ,while th e resistance s o f th e lead s underg o changes . Th e resistanc echanges i n th e circui t wil l the n b e R hl o n th e ballas t sid e an d R, lLon th e gag e side .

2. Bot h lead s wil l hav e the sam e temperatur e coefficient , s o tha t eac h wil lexhibit th e sam e percentag e chang e i n resistanc e pe r uni t chang e i ntemperature. Thus ,

From Eq . (4.59) we see that fo r zero output , E = 0 , and so

For th e particula r situatio n a t hand , whe n lea d resistanc e i s take n int oaccount, w e can writ e

If we substitute the value s of ARhI an d &R aI_, give n b y Eqs . (4.60 ) and (4.61) ,respectively, into Eq . (4.63) , we have

This reduce s t o


By inversion , we have

which give s u s

Equation (4.64 ) is the necessary relation betwee n the resistances o f the lead sin orde r t o permi t cancelin g ou t o f th e effect s o f th e temperatur e change supon them . This equatio n als o tells us that, for the specia l cas e in which theballast an d gag e resistance s ar e equal , the tw o lead s shoul d b e alike.

The mai n poin t o f the analysi s i s to dra w attentio n t o th e fac t tha t i tis necessary t o conside r th e circui t parameters i n order t o achiev e complet eelimination o f the error s arisin g fro m change s i n temperatur e o f the leads .For example , if we were to use identical leads with a ratio of Rh/Rg = 5 , only20 percent o f the erro r woul d b e eliminated.

Temperature compensation with two active strain gages

Whenever ther e i s a know n fixed ratio betwee n th e strain s a t tw o nearb ylocations o n th e sam e membe r (o r betwee n strain s i n tw o direction s a t asingle location), i t may be possible t o achiev e temperature compensation b yusing a secon d activ e strai n gag e fo r th e ballas t resistance . Successfu lapplication o f thi s metho d o f temperatur e compensatio n requires th efollowing:

1. Th e tw o gage s mus t alway s b e maintaine d a t equa l temperature s i nspite of fluctuation s i n the temperature of the member upo n whic h theyare mounted .

2. Th e temperature characteristics of the gages must be matched as closelyas possibl e ove r th e operatin g rang e o f temperature .

3. Th e sign s o f the gag e factor s an d th e sign s and relativ e magnitudes ofthe strain s must be compatible .

Equal strains of opposite sign

The method o f equal strains of opposite sig n is best suited, although certainlynot limited , t o condition s involvin g tw o strain s o f equal magnitud e bu t o fopposite sign , such as encountered o n opposite side s of a beam o f rectangularcross sectio n unde r th e influenc e o f bending . Unde r thes e conditions , tw oidentical strai n gage s ar e used . Th e gage s ma y b e connecte d t o th e circui tby tw o identica l pair s o f leads, o r b y th e three-lea d syste m wher e identicalballast an d gag e lead s ar e used . The circuit s are show n in Fig . 4.17 .

136 THE BONDED ELECTRICAL RESISTANC E STRAI N GAG E

FIG. 4.17. Two identical gages.

With thi s arrangement, changes i n temperatur e wil l caus e equa l resist -ance change s i n bot h th e ballas t an d gag e side s o f th e circuit . Du e t o th etemperature change , thi s result s in ARh = AR a. Sinc e the gages are identical,/?,, = R g, an d so , due t o temperatur e change.

Thus, the changes in temperature wil l no t affec t th e outpu t fro m th e circuit ,as ca n b e see n fro m Eq . (4.1) . an d i t wil l respon d onl y t o th e influenc e o fthe mechanica l strain s on th e gages .

The propertie s o f th e circui t fo r thi s particula r situatio n ma y b esummarized a s follows:

1. Temperatur e compensation , a s just shown.2. Linearit y betwee n output , A£ , an d th e mechanica l strain , K . For th e

mechanical effect , R fl = — R h, henc e n = 0 .3. Maximu m obtainabl e outpu t pe r uni t o f strain . This i s equa l t o th e

output pe r uni t strai n from a constan t curren t circuit .

This come s abou t becaus e AK, ; = — R h, n = 0 , R s = R h, an d a = 1.Therefore, fro m Eq . (4.1) ,


For th e conditions a t hand ,

V=1R9(\ + a) = 2IRg

Substituting thi s valu e o f V into th e precedin g equatio n fo r A£ , an d the ndividing both side s by e , gives u s

This neglect s lea d resistance .It i s interestin g t o observ e that , fo r an y fixe d temperature , thi s i s a

constant-current circui t because th e resistanc e change s i n the two gages justbalance eac h othe r an d th e tota l resistanc e i n th e circui t remains constant .However, whe n th e temperatur e changes , th e tota l resistanc e i n th e circui tchanges an d consequently , fo r constan t applie d voltage , V , th e curren tchanges. Du e t o th e constan t voltage , though , th e curren t chang e jus tcompensates fo r th e overal l resistanc e change , s o tha t eve n whe n th etemperature i s changin g (an d th e curren t i s varying ) th e behavio r o fthe circui t i s th e sam e a s tha t o f a constant-curren t circui t a t constan ttemperature.

Unequal strains of opposite sign

When th e strain s ar e o f opposit e sign , i t i s alway s possibl e t o obtai ntemperature compensation b y using a second active strain gage for the ballastresistance provide d tha t

1. Th e gag e factor s of both strai n gage s hav e th e sam e sign .2. Th e tw o gage s hav e identica l temperatur e characteristics.

When tw o identical gages ar e used , equa l change s i n temperatur e wil lproduce equa l change s i n resistanc e an d therefor e equa l uni t change s i nresistance (becaus e th e gage s ar e identical) . Thi s mean s ther e wil l b e n oinfluence o n th e outpu t fro m th e circuit . I n othe r words , temperatur ecompensation ha s bee n achieved . However , th e outpu t an d linearit y o f thecircuit wil l depend upo n th e rati o o f the strains .

In th e analysi s dealing wit h strain s o f equa l magnitude , bu t opposit esign, excep t fo r th e sig n o f th e circui t output , i t wa s unimportan t t odistinguish between the ballast gage correspondin g t o Rh and the active gagecorresponding t o R g, sinc e bot h wer e equall y active . However , whe n th estrains to which the gages are subjected are unequal , it is necessary to specif ywhether R g correspond s t o th e numericall y larger o r smalle r strain .

In accordanc e wit h thi s requirement , w e wil l conside r tha t th e activ egage, R g, i s subjected t o th e numericall y larger strain , and th e ballas t gage ,

13S TH E BONDE D ELECTRICA L RESISTANC E STRAI N GAG E

Rh, experience s th e numericall y smalle r strain . Accordin g t o thi s specifica -tion, the ratio o f the strain o n th e ballas t gage, Rh, t o th e strain o n th e activegage, K 9, wil l li e between 0 , when ther e i s no strai n o n th e ballas t gage , an d- 1 , when th e strain o n th e ballas t gag e i s just equa l (bu t opposit e i n sign )to tha t o n th e activ e gage . Unde r thes e conditions , th e circui t wit h tw oidentical gage s wil l exhibi t th e followin g characteristics :

1. Temperatur e compensatio n wil l b e achieved .2. Th e sign o f the output wil l correspond t o the sign of the strain on R g.3. Th e magnitud e o f th e outpu t wil l alway s b e large r tha n tha t availabl e

from a singl e gage wit h a fixed ballas t o f equal initia l resistance . A s th estrain rati o approache s — 1. the outpu t wil l approac h a valu e twic e a slarge a s thi s latte r figure .

4. Th e outpu t wil l alway s b e a t leas t 5 0 percen t o f th e maximu mobtainable (wit h a constant-curren t circuit) , and , a s th e strai n rati oapproaches — 1, will actuall y approac h thi s optimum value .

5. Th e nonlinearit y o f the output , with regar d t o strain , wil l disappea r a sthe strai n rati o approache s - 1.

It i s interestin g t o not e that , sinc e non e o f th e precedin g propertie sdepends upo n an y specifi c valu e o f the rati o o f the strain s o n th e tw o gages ,the circui t wil l wor k equall y well for al l strai n ratio s betwee n 0 and — 1. Theactual valu e of the strai n rati o prevailin g under a particula r se t of condition swill, o f course, b e reflected , either directl y o r indirectly , i n th e calibratio n interms o f the strai n o n R, r

When th e strai n rati o approache s — 1, the outpu t an d linearit y may b esufficiently clos e t o th e optimu m (constant-curren t conditions ) fo r th eparticular requirement s a t hand . If , however, th e strai n rati o i s nearer t o 0 ,it ma y b e preferabl e t o conside r a n alternativ e metho d o f temperatur ecompensation tha t wil l yield highe r outpu t an d bette r linearity .

Use of more than two identical yayes or two similar yayes of unequal resistance

When E b/eg, th e rati o o f the strain s o n R h an d R a. i s small, i t wil l b e possibl eto improv e th e linearity , an d t o increas e th e outpu t fro m th e circuit , b ymaking th e ballas t resistance . R h, large r tha n th e gag e resistance , R a. Fo rbest result s with thi s approach , the rati o >- h/t:a mus t b e known , an d it s valuemust remai n fixed.

As shown previously , the optimu m condition s wil l prevail , for gage s ofequal gag e factor , whe n th e rati o

This i s a necessar y requiremen t whe n dealin g wit h gages whos e temperatur eresponse an d othe r characteristic s mus t b e matche d a s nearl y a s possible .


Thus, unde r th e condition s expressed i n Eq. (4.58), we will have a constant -current circuit , with al l the advantages , for a negative strain rati o whe n th eratio R b/Rg i s numerically equa l t o th e invers e ratio o f the strains .

How wil l th e differenc e i n resistanc e betwee n R b an d R s influenc e th etemperature compensation ? Fortunately , thi s differenc e betwee n R b an d R awill no t alte r th e temperature-compensatin g characteristic s a s lon g a s th egages use d for Rh and R g hav e matched temperatur e characteristics, becauseunit change s i n resistance s are involved ; whereas, to establis h th e constant -current circuit , we had t o conside r tota l resistanc e change s i n the tw o arm sof the circuit. We can best illustrate this by considering th e situation in whichwe have a single gage for R g an d a number of gages, x, all identical with R g,connected i n serie s t o for m R h whic h wil l thu s be x time s as larg e a s R g. I fthere i s a chang e i n temperature , the n w e have the following :

Change in resistance

Unit change in resistance

Hence, fo r a change i n temperature , the outpu t fro m th e circuit , A£, wil l b ezero becaus e

This means w e still have temperature compensatio n eve n through R b > R g.Furthermore, since x is not required to be an integral number, the compensa -tion ma y b e affecte d eithe r b y usin g integra l number s o f identical gage s o rby employing any two gages having the appropriate resistanc e ratio , as longas the gag e facto r an d temperatur e characteristic s ar e th e same .

We ma y therefor e summariz e th e characteristic s o f thi s particula rarrangement o f the potentiometri c circuit by sayin g tha t

1. Temperatur e compensation ca n always be achieved as long as the gagescorresponding t o R b and R g have the same temperature characteristics ,even thoug h the y have differen t resistances .

Furthermore, when Rb/Rg — — Ks/eb, th e followin g additional propertie swill b e exhibited:

2. Th e output, E , wil l be linear with strain .3. Th e magnitud e o f the outpu t wil l correspond t o tha t obtainabl e fro m

a constant-curren t circuit , i.e.,


It shoul d b e noted tha t i f the resistance o f the ballas t gage , R b, i s mad egreater tha n tha t require d t o produce constant-curren t conditions , th eoutput ca n be increased somewha t more, but a t the expense of linearity.The writer s fee l i t woul d b e bette r t o hav e R a large r i n th e firs t plac eand t o kee p R h i n the prope r proportio n t o produc e constant-curren tconditions, an d henc e linearity , between strai n an d circui t output .

Strains of one sign only

When th e strain s a t al l location s wher e gage s ca n b e mounte d ar e o f on esign only (eithe r all tension o r al l compression) , i t i s still possibl e t o achiev etemperature compensatio n b y usin g a n activ e ballast . However , unles s th estrain on the ballas t gages i s relatively small, this is not a n attractiv e metho dfor eliminatin g th e temperatur e effect , fo r th e followin g reasons :

1. Whe n th e gage s correspondin g t o R h an d R g bot h hav e th e sam e sig nfor th e gag e factor , the resul t o f strains of like sign actin g o n the m wil lbe t o reduc e th e circui t outpu t belo w tha t availabl e fro m R g actin galone i n conjunctio n wit h a fixe d o r inactiv e ballast . I n certai n type sof transducers , however , the advantag e o f temperatur e compensatio nmore tha n offset s a sligh t loss i n sensitivity .

2. Wit h change s o f resistance o f the sam e sig n i n bot h side s of the circuit,it i s impossibl e t o achiev e constant-curren t conditions , an d th e cor -responding linearit y betwee n strai n an d th e output . Actually , th edeviation fro m linearit y wil l b e greate r tha n tha t fo r fixe d R b an dvariable R g.

3. Althoug h the us e o f gages wit h positive and negativ e gage factor s ma ybe very attractive for increasing th e circuit output a t constant tempera -ture, ther e ma y b e considerabl e difficult y i f the temperatur e changes .The magnitud e o f the difficult y wil l depend upo n th e precisio n desire dand th e magnitud e o f the temperatur e fluctuation .

Since the temperatur e respons e o f a gag e depend s upo n th e effec tof temperatur e upo n th e gag e factor , th e temperatur e coefficien t o fresistance o f the materia l o f the sensin g element , an d th e differenc e i ncoefficients o f expansion o f the sensin g elemen t an d th e materia l upo nwhich i t i s mounted, i t i s very difficult t o mak e al l thes e effect s balanc eout, excep t a t on e o r tw o temperatur e levels , because the y are actuallynonlinear function s o f temperature .

As an example , le t u s imagine tha t th e coefficient s o f expansion o fthe sensin g elements o f both gages ar e th e sam e bu t differen t fro m tha tof th e materia l upo n whic h the y ar e mounted . I f there i s a chang e i ntemperature, bot h gages wil l fee l an expansion or contraction. However ,since thi s effec t wil l b e indistinguishable from simila r strain s producedby direc t mechanica l action , i t wil l appea r i n th e for m o f a n outpu tfrom the circuit unless the temperature change also produces compensa -ting changes i n th e resistance s o f the sensin g elements.


We migh t thin k o f a specia l cas e i n whic h temperature compensatio nmight b e achieve d as follows :

1. Imagin e that th e coefficien t o f expansion of both gage s is matched wit hthe materia l upo n whic h the y ar e mounted . Unde r thes e conditions ,when there i s a change in temperature , th e sensin g elements wil l movefreely wit h the base material and n o resistance change takes plac e a s aresult o f differential expansio n o r contraction .

2. I f the temperature coefficients o f the two sensing elements are the same,then equa l uni t change s i n resistanc e wil l appea r i n bot h side s o f th ecircuit an d ther e wil l be n o effec t o n th e output .

3. Th e effec t o f temperatur e o n th e valu e o f th e gag e factor s shoul d b enegligible (or compensating) , since otherwise a change in temperature,although producin g n o direc t outpu t fro m th e circuit , ma y hav e th einconvenience o f changing the calibration .

Further details in regard to temperature effects an d methods of allowingfor the m ar e give n by Hines an d Weymout h (2), and Wnu k (3) .

4.6. Calibration

In orde r t o determin e wha t th e signa l fro m th e circui t represent s i n term sof strain , som e typ e o f calibratio n i s require d (4) . There ar e a numbe r o fdifferent way s in whic h thi s can b e done , an d eac h metho d wil l hav e som especial advantage s wit h respec t t o som e particula r application . Fo r th epurpose here , however , one usua l method wil l b e discussed i n som e detail .

We wil l conside r th e shun t calibratio n metho d a s applie d t o a singl egage wit h a fixed ballast . Thi s involve s the simulatio n o f a chang e i n gag eresistance b y th e introductio n o f a larg e know n resistanc e i n paralle l wit hthe gage , and calculatio n of the equivalen t strain which corresponds t o th ecircuit output. Theoretically, we should be able to employ a series resistance,but i n genera l thi s wil l b e s o smal l tha t variation s i n contac t resistanc e a tswitches ar e likel y t o impai r th e accuracy . From a practica l poin t o f view,it is better to use a large parallel resistanc e because the variations in contac tresistance a t switche s will the n b e reltively insignificant.

Figure 4.1 8 represent s a potentiometri c circui t wit h a fixe d ballas tresistance, R b, a strain gage , R g, a calibrating resistor , R c, an d a switch , S,to bring Rc into the circuit. Although not shown in the diagram, there shouldbe some means (chopper ) o f opening an d closin g the switch , S, at approxi -mately th e sam e frequenc y a s the strai n gag e signal .

Let u s conside r that , fo r th e moment , th e strai n gag e i s a t res t unde rzero strain . Whe n th e switch , S , i s closed , th e readou t devic e wil l sens e achange in resistance, AR C, which produces a chang e i n voltage , AEC, at th eoutput terminals . Thi s chang e i n resistance , AR C, correspond s t o th edifference betwee n th e gag e resistance , Rg, an d th e combine d effect , R cg, o f


FIG. 4.18. Potentiometri c circui t wi t h calibration resistor .

R an d R c i n parallel . Fo r th e paralle l resistances ,

This give s R cfl a s

The chang e i n resistance , AR C , i s

Since strai n i s represented b y uni t change i n resistance , w e no w divid eboth side s o f Eq . (4.66) b y R g, s o tha t

Solving fo r R c, th e calibratin g resistance , give s

Since th e readou t devic e canno t determin e whethe r th e chang e i nresistance tha t i t sense s come s fro m strai n i n th e gag e o r th e introductio nof th e paralle l calibratin g resistance , a s fa r a s i t i s concerne d AR C/R9


represents

or

Substituting this equivalent value of strain into Eq. (4.68), we may now writethe expressio n for th e siz e of the calibratin g resistanc e t o represen t a givenstrain a s

In som e cases , however , i t wil l b e necessar y t o determin e th e strai nsimulated b y a calibratin g resistanc e o f som e arbitrar y o r predetermine dvalue. Unde r thes e conditions Eq . (4.69 ) is used t o solv e for s , which gives

From Eq . (4.70 ) w e ca n comput e th e strai n simulate d b y a calibratin gresistance o f some particula r magnitude .

Special case for uniaxial stress

For uniaxia l stres s conditions , when the gage axis is lined u p with the stres saxis, Eqs . (4.69 ) an d (4.70 ) can convenientl y be expresse d directl y i n term sof stress . This i s due t o th e fac t tha t fo r uniaxia l stres s

where a = stress

E = modulus o f elasticity

Substituting thi s value of e into Eq . (4.69 ) produce s


Thus,

Problems

4.1. Verif y Eq . (4.21) .4.2. Verif y Eq . (4.29) .4.3. I n Eq . (4.29) , le t AR h = 0 an d ^R g/'Ra = G f.e, wher e G F = 2.0. Plo t th e

nonlinearity factor , n , vs . the strain , R , on log-lo g pape r fo r value s of a = 1 , 2,5, and 9 i n orde r t o sho w th e dependenc y o f n o n th e strai n level , E.

4.4. Usin g the dat a i n Proble m 4.3 , on semilo g paper plo t n vs. the rati o a in orde rto sho w th e dependenc y o f n on th e rati o a .

4.5. Th e followin g dat a ar e availabl e fo r th e potentiometri c circuit : V = 3 5 volts,Rg = 12 0 ohms, Rh = 108 0 ohms, and G F = 2.0 . Determine the circuit efficienc yand th e strai n tha t wil l resul t i n a 2 percen t erro r i n A£ . Wil l th e curren t i nthe circui t exceed 0.0 3 amperes ?

4.6. Red o Proble m 4. 5 i f the strai n gag e i s change d t o R g = 35 0 ohms, al l othe rfactors remainin g the same .

4.7. I f th e voltag e i n Proble m 4. 6 i s increase d s o tha t th e circui t curren t i s0.03 amperes, wil l th e erro r b e affecte d I f the erro r i s no t t o excee d 2 percent ,compute th e change in E .

4.8. A stee l tensio n lin k o f rectangula r cros s sectio n i s subjecte d t o a n axia l loa dthat varie s betwee n 0 an d 3 3 750 Ib. Th e loa d i s offse t fro m th e longitudina laxis o f th e bar , a s show n i n Fig . 4.19 . Fou r gages , arrange d alon g th elongitudinal axis , ar e bonde d a t th e cente r o f each fac e o f th e ba r an d wire din serie s t o for m th e potentiometri c circui t o f Fig . 4.4 . I f R s = 35 0 ohms,GF = 2.5 , R h = 700 0 ohms , V = 60 volts, an d E = 30 x 10 6 psi,

(a) Determin e th e strai n on eac h gage .(b) Determin e th e maximu m value of A£ considerin g n = 0.(c) Comput e th e nonlinearit y term, n . Is i t wort h considering ?

4.9. A steel beam i s subjected to a bending moment of M = 1 2 500 in-lb and a tensileforce of F = 1800 0 Ib , as shown in Fig. 4.20. Using £ = 3 0 x 10 6 psi, GF = 2.08 ,Rh = R g = 12 0 ohms, an d V = 25 volts,

(a) Determin e th e strai n o n eac h gage .(b) Determin e th e chang e i n resistance o f Rg an d R h.(c) Determin e th e value of E .(d) I s thi s a constant-curren t circuit?(e) I f the loa d F i s eliminated, will thi s b e a constant-curren t circuit?

4.10. Usin g a T cross section , determine suitable dimensions for the beam i n Example4.3 i f £ = 15 0 Ib, a = 1 5 in, an d L = 1 8 in.


FIG. 4.19.

FIG. 4.20.

REFERENCES

1. Sanchez , J. C. and W . V. Wright, "Recent Development s i n Flexible Silicon Strai nGages," in Semiconductor and Conventional Strain Gages, edited by Mills Dean II Iand Richar d D . Douglas , Ne w York , Academic Press , 1962 , pp. 307-345 .

2. Hines , Frank F. and Leo n J. Weymouth, "Practical Aspects of Temperature Effect son Resistanc e Strai n Gages, " i n Semiconductor an d Conventional Strain Gages,edited b y Mill s Dean II I an d Richar d D . Douglas , Ne w York , Academi c Press ,1962, pp. 143-168 .

3. Wnuk , S. P. Jr., "Strain Gage s for Cryogenics," IS A Journal, Vol . 11 , No. 5 , May1964, pp . 67-71 . Reprinted b y permission. Copyright © Instrument Society o fAmerica 1964. From Strain Gages o f Cryogenics, S . P. Wnuk , Jr .

4. Geldmacher , R . C. , "Ballast Circui t Design," SESA Proceedings, Vol. XII , No . 1 ,1954, pp. 27-38 .

WHEATSTONE BRIDGE

5.1. Introduction

Although th e potentiometri c circuit , show n i n Fig . 5. 1 an d discusse d i nChapter 4 , possesses man y desirable characteristic s fo r use with strain gages ,nevertheless, i t doe s presen t th e difficult y tha t th e strai n signal , E , mus teither b e measure d i n combinatio n wit h a ver y muc h large r voltage , E , o rfirst isolated an d the n measure d b y itself.

When E i s determined b y measuring th e combined quantit y E + E ,and notin g th e chang e fro m a comparable indicatio n o f E, one runs int o th eproblem tha t i f E i s relatively small with respec t t o E , a smal l erro r i n theobservation o f either E o r E + E ma y produc e a n excessivel y large an dintolerable percentag e erro r i n th e comparativel y smal l change , E . Th eimportance o f th e readin g error , o f course , wil l depen d upo n th e relativ emagnitudes o f E an d E , an d th e instrument s availabl e fo r makin g th eobservations.

For larg e signals from semiconducto r gages , there ma y be no proble min obtainin g sufficientl y precis e value s of E fro m reading s o f E + E. I ngeneral, however , for strain gage wor k i t will be preferable to isolate E an dmeasure thi s quantity entirely by itself . Thi s approac h i s much mor e direct ,since i t involves making an observatio n immediatel y upon th e quantity thatis the rea l measur e o f the strain .

For dynami c strains , E ca n be isolated b y using a filter (condenser)

FIG. 5.1. Potentiometri c circuit.

5

WHEATSTONE BRIDG E 14 7

that wil l block th e stead y componen t £ bu t stil l transmi t th e time-varyingsignal, E. Fo r stati c strains thi s syste m will not work because the filter wil lnot transmi t an y constant valu e of E. Therefore , anothe r approac h mus tbe sought .

The rea l difficult y encountere d i n makin g stati c strai n measurement swith the potentiometric circuit is caused by the wide divergence in the relativemagnitudes between E and E , an d s o we now look int o th e possibility ofovercoming this problem. W e can achieve our objectiv e either by increasingE wit h respec t t o E (usin g semiconductor gage s wit h large strains ) or byreducing E relative to E. Th e latter approach must be followed to develo pa metho d fo r metalli c gages , o r fo r semiconductor s whe n the strai n leve l i slow. If a scheme for reducing E relative to E ca n be worked out, and i f itcan b e carried t o th e ultimat e s o tha t E i s finally reduced t o zero , the n wehave achieve d a mean s o f isolating E s o tha t th e strai n signa l ca n bemeasured directly by itself. The ideas just expressed are presented graphicallyin Fig . 5.2 , which indicates qualitativ e relations between £ an d E .

Figures 5.3 and 5. 4 show various stages in the development of a methodfor reducin g E t o zero , an d thereb y facilitating the direc t measuremen t ofthe strain signal, E, b y itself. The fundamental idea is to change the referencelevel fro m whic h E + E i s measured s o that th e numerical value of E wil lbe reduced and , ultimately , brought t o zero .

Instead o f measuring E + E a s the potentia l dro p acros s th e gage ,shown as points A and C in Fig . 5.1 , we will establish a reference other thanC wit h a stead y potentia l leve l muc h nearer , o r perhap s equal , t o tha tprevailing at A . I f an auxiliar y battery with voltage V l (whic h is slightly lessthan the voltage drop across the gage) is introduced and connecte d a s shownin Fig . 5.3a , the n b y measuring the voltag e dro p across terminals A an d B

FIG. 5.2. Qualitativ e relations between E an d A£ .


FIG. 5.3 . Method s o f reducing £ wit h respec t t o A£ .

FIG. 5.4. Wheatston e bridge.

instead o f across A an d C , the steady-stat e component , E , wil l b e reduce dby an amoun t equa l t o V v. Even though thi s is a move in the righ t direction ,since E ma y b e exceedingl y small , especiall y fo r metalli c gages , th eintroduction o f th e auxiliar y batter y ma y no t reduc e E sufficientl y wit hrespect t o E. Consequently , we strive for something better .

An improved techniqu e i s shown in Fig . 5.3b . Her e a n auxiliar y batter ywith voltage , V 2, that i s greater tha n th e potentia l dro p acros s th e gage , i sconnected t o a potentiometer wit h which we can var y the voltage drop, V BC,between point s B and C . Thus, w e can no w contro l th e voltag e a t termina lB. Furthermore , since , for zer o strai n conditions , th e differenc e i n potentia lbetween A an d B represent s th e steady-stat e component , E , o f th e output ,


control o f th e voltag e a t B als o provide s contro l o f E . Therefore , b yadjusting th e potentiomete r unti l there i s no potentia l differenc e betwee n Aand B , w e ca n mak e £ equa l t o zer o an d thereb y eliminat e i t fro m th eoutput. Whe n thi s ha s bee n done , an y chang e i n th e gage' s resistanc e wil lproduce a chang e i n potentia l a t termina l A . Thi s chang e i s equal t o A£ ,which ca n b e measured directly , and b y itself , agains t th e referenc e voltageat termina l B .

What has actually been accomplished b y making the initial adjustment,which bring s th e potentia l differenc e acros s A- B t o zero , i s t o mak e th evoltage drop, V BC, from B to C equal t o th e potentia l drop across th e gage ,VAC. Then , when terminal B i s used for reference we have, in effect , change dthe leve l o f reference voltage fro m th e leve l at C t o th e origina l leve l a t A .We can now read A£ independently (because we are using E as the referenc elevel o f voltage).

Theoretically, this method provides us with a direct means o f observingthe strai n signal , A£ , fo r bot h stati c an d dynami c strains . However , fo rpractical reason s thi s procedur e i s no t convenien t t o us e (especiall y fo rlong-time stati c readings ) becaus e i t i s subjec t t o error s arisin g fro mdifferences i n rat e o f deca y (voltag e drop ) betwee n th e tw o batteries .Fortunately, thi s difficult y ca n b e eliminated very easily.

Let u s now se e how th e difficult y involve d with th e secon d batter y canbe overcome . Th e onl y requiremen t i n regar d t o th e voltage , V 2, o f th eauxiliary batter y i s that i t shoul d b e large r tha n th e potentia l dro p acros sthe gage . Sinc e the presence of Rb require s that th e voltage , V , must also b elarger than the potential drop across the gage, it appears tha t a single batterycan b e use d t o powe r bot h circuits , whic h ca n b e connecte d togethe r a sshown i n Fig . 5.4a . Thi s i s th e well-know n Wheatstone bridge , whic h i sshown i n mor e conventiona l for m i n Fig . 5.4b . Whe n th e terminal s A an dB o f the Wheatstone bridg e are brough t to th e sam e potential , th e bridge issaid to be balanced (E = 0). However, since it is quite possible that the bridgemight be initially unbalanced, the output indicated in Fig. 5.4 has been shownas £ + A£ , wher e £ represent s th e potentia l differenc e betwee n A an d Bresulting from initia l unbalance, and A£ corresponds t o the change in outputdue t o th e chang e in gage resistance.

5.2. Elementary bridge equations

As with the potentiometric circuit, the Wheatstone bridge circui t equations ,and som e discussio n o f them , wil l b e presente d first . Figur e 5. 5 shows a nidealized Wheatston e bridge in which all four arms may contain strain gages.The bridg e i s supplie d wit h a constan t voltage , V , (fro m a sourc e o f zer ointernal resistance ) at terminal s D and C . The output voltag e across A-B i smeasured with an instrumen t of infinite impedanc e which draws no current .Although this represents a theoretical situation , nevertheless , there are timeswhen i t ca n b e very closely approximated .


FIG. 5.5 . Idealize d Wheatston e bridge .

For a n initiall y balanced bridge , E = 0 . Thus,

From thi s

When th e gage s ar e strained , th e incrementa l bridg e outpu t i s given as

where n = the nonlinearit y facto r which , fo r thi s case , i s ver y closel yapproximated b y

When th e gage s ar e al l alike and o f initial resistance R a, the n

For thi s case a = I an d Eqs . (5.3 ) and (5.4 ) simplif y t o

WHEATSTONE BRIDG E 151

FIG. 5.6. Generalize d bridge .

and

Equations (5.1 ) through (5.7 ) are the elementary bridge equations . Fo ra more general concept of the Wheatstone bridge , we examine Fig. 5.6 . Hereallowance i s made fo r th e interna l resistance , R s, o f the powe r suppl y an dthe fac t tha t th e meter resistance , Rm, ma y not b e infinite. Sinc e R s i s treatedmerely as a resistance in series with th e bridge , this might include resistanc eof leads , a voltag e control , o r an y othe r resistance , includin g th e actua linternal resistanc e o f the powe r suppl y itself . Whe n th e resistanc e i n serie swith the power supply and th e resistance o f the meter (or galvanometer) ar etaken into account, th e expression for the incremental output, A£ 0, fro m a ninitial conditio n o f balance i s given by

where R BI = bridge inpu t resistanc e a s see n betwee n terminal s D an dC (no t includin g R s)

RBO = bridge outpu t resistanc e a s see n b y th e mete r acros sterminals A an d B (thi s includes the serie s resistance R s)


Analysis o f the circui t shows tha t fo r th e unbalance d bridge ,

and

where

If each arm o f the bridge now contains one o f four identica l strain gage swhose initia l resistance i s Rg, a s given by Eq . (5.5), and th e bridge is initiallybalanced, then , unde r thi s special condition ,

This mean s tha t whe n th e change s i n gag e resistanc e ar e small , a s usuall yoccurs wit h metalli c gages , w e ca n writ e th e expressio n fo r AE 0 t o a verygood approximatio n a s

From Eq . (5.16) , we see that th e maximu m output, (A£)max, wil l occu rwhen R s = 0 and R m = oc.

Galvanometer current

For th e unbalance d bridge , i t ca n b e show n tha t th e curren t throug h th emeter (galvanometer ) can b e expressed a s


When R s = 0, Eq. (5.17) reduces t o

Ways of using the Wheatstone bridge

There ar e thre e differen t way s i n whic h th e Wheatston e bridg e i s usuallyemployed t o obtai n indication s fro m strai n gages : the nul l balance system ;the unbalanc e system; the referenc e system.

The null balance system. I n thi s syste m there i s provision fo r adjustin g th eresistance i n one o r mor e arms o f the bridg e to compensat e fo r the effec t o fchange i n gag e resistance . Th e bridg e i s brough t t o initia l balanc e b ymanipulating th e adjustabl e resistances . Then , afte r th e gage s hav e bee nsubjected t o strain , a further adjustmen t o f the variable resistances is madeto restor e th e condition o f balance. The amoun t o f the adjustmen t requiredto reestablis h the balanc e i s a measur e of the chang e i n gag e resistance , o rthe strain . This metho d ha s th e advantag e o f giving an indicatio n indepen -dent o f variations in bridg e supply voltage and, unde r certain conditions , itwill eliminate some nonlinearities.

On th e othe r hand , it s us e i s limite d t o static , o r exceedingl y low-frequency dynamic , observations . Thi s i s du e t o th e fac t tha t i t take sappreciable time to rebalance the bridge and, in consequence, it is impossibleto follo w rapidl y fluctuatin g changes . Furthermore , dependin g upo n th emanner in which the rebalancin g of the bridg e is accomplished, th e readou tmay b e a nonlinea r quantit y requirin g a conversio n char t fo r determiningstrain. In th e event that al l four bridg e arm s contain strai n gages, i t may beimpossible to avoid an appreciable amount of desensitization (loss in effectiv egage factor) caused b y the balancing network . If a direc t calibration ca n b emade, thi s should no t presen t a seriou s difficulty .

The unbalance system. Th e bridge is directly connected to the readout device ,which ma y b e a galvanometer , a cathode-ray oscilloscope , o r som e typ e ofrecording oscillograp h producin g a recor d o f th e strai n signa l (usually ,although not necessarily) as a function o f time. This system has the advantagethat it is suited for both stati c and dynami c observations. However, since itsindication i s directl y proportiona l t o th e applie d voltage , a stabl e powe rsupply i s required. For measurement s conducted ove r long periods o f time,this i s particularly important.

The reference system. Ther e ar e certai n instrument s combinin g th e advan -tages o f both th e nul l balance an d th e unbalanc e systems , an d a t th e sam etime eliminating som e o f the undesirabl e feature s o f each procedure . Thes einstruments incorporat e a n interna l bridge tha t i s separate fro m th e strai ngage bridg e bu t powere d fro m th e sam e source . Provisio n i s mad e fo r


FIG. 5.7 . Schemati c diagram o f referenc e bridg e wit h gag e facto r control.

adjustment o f the resistance s i n the interna l bridg e so that it s output ca n b eset a t som e fixe d valu e o r controlle d t o matc h th e outpu t o f th e externa lbridge. I t i s thus possibl e t o emplo y th e interna l bridg e a s a referenc e fro mwhich t o establis h th e strai n indication . A schemati c diagra m o f a circui twith a referenc e bridge i s shown i n Fig . 5.7 .

When th e outpu t o f the interna l bridge i s calibrated, then , by compari-son, on e i s abl e t o evaluat e th e indicatio n fro m th e externa l bridge . Th ecomparison an d evaluatio n ca n b e carrie d ou t b y on e o r th e othe r o f th efollowing tw o arrangements .

1. Th e null balance reference bridge. Wit h thi s system , th e outpu t o f th ereference bridg e is initially adjusted to cancel th e output fro m th e straingage bridge . An y subsequen t chang e i n outpu t fro m th e strai n gag ebridge wil l requir e a readjustmen t o f th e referenc e bridg e i n orde r t orestore equalit y o f output s fro m th e tw o bridges . Th e amoun t o f th ereadjustment o f th e referenc e bridg e (i n orde r t o restor e equalit y o foutputs fro m th e tw o bridges ) is a measur e o f the chang e i n strain , o rother indications , fro m th e strai n gage bridg e (1).

2. Th e unbalance reference bridge. Th e referenc e bridg e i n thi s syste mis initiall y adjuste d s o tha t it s outpu t jus t cancels , o r balances , th eoutput fro m th e strai n gag e bridge . An y subsequen t chang e i n th estrain gage bridg e wil l the n b e indicated b y an unbalanc e o r differenc ein outpu t betwee n th e tw o bridges . This unbalanc e i s a measure o f thechange whic h has take n plac e i n th e strai n gage bridge . Calibratio n o fthis signa l can b e achieved by making a known change i n the referencebridge, and the n comparing th e signa l from th e strain gage bridg e wit hthe signa l produce d b y th e chang e i n th e referenc e bridge (2).

With bot h th e nul l balanc e an d unbalanc e referenc e bridges , w e ar emerely comparin g th e outpu t o f th e strai n gag e bridg e wit h a calibrate dreference. Fro m this , then, th e indicatio n from th e strai n gag e i s evaluated .


The nul l balanc e referenc e syste m i s suite d t o stati c an d low-frequenc ydynamic conditions . Th e unbalanc e referenc e syste m can b e use d fo r bot hstatic and dynami c observations .

Some of the advantages of the reference bridge methods are as follows:

1. Th e strai n indicatio n i s independent o f the powe r suppl y voltage tha tis connected t o th e tw o bridges . I n th e cas e o f the unbalanc e referenc emethod, i t i s necessary that th e calibratio n indicatio n shoul d b e mad ewith the same applied voltage as that employed for the strain indication.

2. Th e syste m lend s itsel f convenientl y to th e inclusio n o f a gag e facto radjustment.

3. Th e referenc e bridg e ca n b e se t up an d calibrated , then lef t alone .4. Th e strain gage bridge ca n be closed, and , since its output i s compare d

with that from th e reference bridge, it is not necessary to provide furtheradjustment b y addin g serie s o r paralle l resistanc e i n an y on e o f th earms. Thi s i s a grea t convenienc e whe n al l fou r arm s o f th e bridg econtain strai n gages , becaus e i t overcome s th e necessit y fo r includingtrimming resistance s t o achiev e initia l balance .

Summary of properties of the Wheatstone bridge

1. Fo r strai n gage applications, probabl y the most attractive characteristicof th e Wheatston e bridg e i s it s abilit y t o provid e th e mean s fo rmeasuring bot h stati c an d dynami c strains , o r combination s thereof ,conveniently.

2. I n compariso n wit h th e potentiometri c circuit , the Wheatston e bridg eis more elaborate . Thi s i s to b e expected sinc e i t actually contain s tw opotentiometric circuit s connecte d together . Furthermore , du e t o th enature o f th e Wheatston e bridge , a measurin g syste m employin g i tcannot hav e al l component s connecte d t o a commo n ground . I f on eside of the inpu t is grounded, the n the outpu t must be floating, or vic eversa. This requires complete isolatio n o f one part o f the system relativeto th e remainder .

3. Temperatur e compensation . Unde r suitabl e conditions the Wheatston ebridge wil l provide an electrica l method fo r temperature compensatio nof strai n gage s a s wel l a s man y othe r convenien t propertie s o f th epotentiometric circuit . One wil l observ e tha t fo r a singl e active strai ngage, the equations representing the output from th e Wheatstone bridgereduce t o exactl y the sam e for m a s th e correspondin g expression s fo rthe potentiometri c circuit.

4. Optimu m bridge ratio (for a single gage). When the Wheatstone bridg eis to b e used wit h a single gage, we have the opportunit y o f making a narbitrary decision regarding the choice o f the bridg e ratio , whic h is theratio o f the resistance s i n th e hal f bridg e connecte d acros s th e powe rsupply an d containin g th e strai n gage . Thi s rati o i s represented b y thesymbol a in Eqs. (5.3) and (5.8) . Examination of the relations expresse d


in the equations fo r the bridg e outpu t indicate s tha t th e valu e of the bridg eratio, a , necessary fo r optimum outpu t pe r uni t change i n resistance (or pe runit chang e i n strain ) wil l depen d upo n th e characte r o f th e powe r suppl yas follows :

Character of power supply Value of ratio a formaximum output perunit strain

a. Fixe d voltage (V = constant) a= \b. Variabl e voltag e (max. gage current = constant ) a = 1

For th e fixed-voltage power supply , i t ca n b e prove n analyticall y thatfor optimu m output , a = \ . However , for the variable-voltage power supply ,since theor y predict s th e optimu m bridg e outpu t fo r th e larges t possibl evalue o f a , w e wil l hav e t o procee d fro m practica l consideration s i n orde rto establis h a definite an d convenien t valu e for the bridge ratio.

We commenc e b y selectin g a strai n gag e o f resistance R g an d decidin gupon th e maximum permissible gage current and th e maximum voltage tha tcan be safely employed. From the maximum permissible current and voltage ,the tota l resistanc e i n th e hal f bridge , R ± + R 2, ca n b e computed . I f R gcorresponds t o #, , th e bridge ratio , a = R2/Ri, ca n be calculated .

Since approximatel y 9 0 percent o f the ultimat e output can b e achievedwith a bridge rati o o f 10 , there is little incentiv e to mak e th e bridg e ratio , a,larger than 1 0 because the required increase in applied voltag e goes up muchfaster tha n th e gai n i n output . Man y investigator s prefer t o us e a valu e ofabout 5 for the bridge ratio , since this will yield an outpu t of about 8 5 percentof th e ultimate . Correspondingly, th e voltag e require d i s onl y abou t thre etimes a s grea t a s tha t neede d whe n a = 1.

If a carrie r syste m i s employed , th e powe r requirement s wil l usuall ynecessitate keepin g the valu e o f the bridg e rati o nea r on e (3).

5. Computin g characteristics. Equations (5.8) and (5.16 ) also indicate that,by appropriat e contro l o f the parameters , th e Wheatston e bridg e ca nbe employed t o perform certain additions, subtractions, multiplications,and divisions . The relationship s ca n b e summarize d b y th e followin gstatements:

Subject t o th e possibility of some nonlinearities, the bridge output,E, wil l be:

a. Directl y proportiona l t o th e applie d voltage .b. Directl y proportional to the sums and difference s o f the unit change s

in th e resistance s i n th e fou r arms .c. Directl y proportional t o th e produc t o f the applie d voltag e and th e

net uni t change i n resistanc e o f al l fou r arms .1 Practica l consideration s wil l usuall y plac e a n uppe r l imi t o f abou t 1 0 o n th e maximu m usabl e

value o f the bridg e ratio .


d. Inversel y proportiona l t o function s involvin g resistanc e i n serie swith th e bridg e an d powe r supply , an d th e resistanc e o f th einstrument which is used to determine the output voltage or current.Stein (4 ) gives a detailed discussion.

5.3. Derivation of elementary bridge equations

Figure 5.8 shows a n elementar y an d idealize d Wheatston e bridge i n whic hall fou r arm s ma y contai n strai n gages . I n th e succeedin g analysis , th efollowing assumption s hav e been made :

1. Th e bridge is supplied wit h a constant voltage, V , from a source whoseimpedance i s negligible .

2. Th e resistance s o f the lead s fro m th e powe r suppl y t o th e bridge , an dof al l the lead s connectin g th e interna l components o f the bridge , ca nbe neglected .

3. Th e outpu t fro m th e bridg e i s represented b y th e differenc e i n voltag ebetween terminals A and B. The instrument used to measure the outputhas infinite impedance an d draw s n o current .

The bridg e outpu t i s the differenc e i n voltage betwee n A an d B , whichis als o th e voltag e dro p fro m A t o C minus the voltag e drop fro m B t o C .According t o assumption (3) , no current flows from A to B; thus, current / tflows through R 1 an d R2, whil e current / 2 flows through R 3 and R4. Sincethe voltag e aroun d eac h loo p mus t su m to zero , we can write

FIG. 5.8. Elementar y Wheatston e bridge .


From thes e

Let u s no w conside r th e situatio n in whic h all fou r arm s o f the bridg econtain strai n gage s whos e initia l resistance s ar e R ^ R 2, R 3, an d R 4, a sshown i n Fig . 5.8 . The correspondin g initia l output, E , is then

Substituting th e value s o f 7 j an d I 2, give n b y Eqs . (5.21 ) an d (5.22) ,respectively, into Eq . (5.23) , w e have

or

If each gage undergoes a change in resistance such that R^ - > Kj + AR15

#2 -* R2 + &R2, R 3 - > R3 + AK3> an d R 4 ^ # 4 + AK 4, the n th e bridg eoutput wil l change fro m E to E + E . Equatio n (5.25 ) can be written, usingthe ne w resistances an d ne w output , as

With the ful l bridge , just as in the case of the half bridge (potentiometri ccircuit), we can show that the change in output, E, i s a function of the unitchanges i n gage resistance , o r th e strain s in the materia l t o whic h the gage sare attached .

The valu e o f E ca n no w be determined , i n term s o f resistances , b ysubtracting E , or it s equivalen t a s expresse d b y Eq . (5.25) , from bot h side sof Eq. (5.26) . This result s in


Equation (5.27 ) is a perfectly genera l expression for the change in bridgeoutput from an y initial condition. I t specifie s no particular relatio n betwee nthe initia l resistances o f the bridg e arms , bu t unfortunatel y i t i s somewha tcumbersome t o handle .

For th e specia l situatio n i n whic h th e bridg e i s initially balanced , th einitial output , E , wil l b e zero an d th e expression fo r the chang e i n output ,A£0, wil l be much simpler than th e genera l relation give n by Eq . (5.27).

When th e bridge i s initially balanced, th e initia l output is

This means tha t

Equation (5.29 ) indicate s that , fo r a balance d bridg e (outpu t = 0) , adefinite relatio n mus t exis t amon g th e resistance s o f th e fou r arms . Thi srelationship can b e expressed i n th e thre e followin g ways :

1. Fro m Eq. (5.29), we see that the cross products of the resistances in thearms mus t be equal. Thu s

2. W e also se e tha t

Equation (5.31 ) indicates tha t th e ratio s o f the resistance s i n th e tw ohalves o f the bridge , whic h are i n serie s with the powe r suppl y (DACand DE C i n Fig . 5.8), must b e equal . Thi s ratio , frequentl y calle d th ebridge ratio , i s equivalen t t o th e rati o o f ballas t resistanc e t o gag eresistance in the potentiometric circuit . It i s represented b y the symbola. Hence ,

3. I f we divide the bridg e into tw o halve s with respec t t o th e tw o outpu tterminals (ADB and ACB in Fig . 5.8), the rati o of the resistance s in

Bridge ratio


these tw o halve s mus t als o b e equal . Lettin g th e symbo l b represen tthis ratio , w e have

When th e value s of the ratio s a and b hav e been chosen , an d als o th eresistance i n on e o f th e bridg e arm s (fo r example , Rj) , th e othe r thre eresistances ca n b e computed .

For any values of a and b

For any values when a = b

For any value of a when b = 1

For any value of b when a — \

When a = b = 1

(equa l arm bridge )

Choice of ratios a and b

a = b = 1 Sinc e i t i s frequently desire d t o us e strain gage s i n tw o an d fou rarms o f th e bridge , th e equal-ar m arrangemen t i s probabl y th emost usual , in spite of the fac t that , for a singl e gage, it s efficienc yis onl y 5 0 percent .

a > 1 Fo r operatio n wit h a single gage, and unde r some conditions wit htwo gages, the efficiency ca n be improved b y increasing th e bridg eratio. Ther e i s relativel y littl e t o b e gained , however , b y goin gbeyond a rati o o f abou t 10 , which wil l yiel d approximatel y 9 0percent o f th e ultimate . Man y investigator s prefe r t o us e amaximum value of 5, which allows considerably lowe r voltage forthe powe r suppl y wit h a n efficienc y tha t i s above 8 0 percent .

b / 1 Th e choic e o f the valu e o f b is not critical . I t i s ofte n take n a sunity for convenience. We should avoid making thi s ratio s o large

WHEATSTONE BRIDGE 161

that th e relativel y low resistanc e o f one sid e o f the bridg e cause san exceedingl y heavy curren t deman d o n th e powe r supply .

Output of the initially balanced bridgeWhen the bridge is initially balanced, th e expression fo r the change in outputfrom th e initia l condition i s simplified. By referring t o Eq . (5.27), we see thatthe second ter m corresponding t o the initial output drops out , because E = 0for th e conditio n o f balance .

In orde r t o b e specifi c w e wil l us e th e symbo l A£ 0 fo r th e chang e i noutput fro m th e initia l condition o f balance. Thi s make s a distinctio n withrespect to the symbol E whic h has been use d for the change in output fro many initia l condition. Therefore , A£0 correspond s onl y to th e specia l case ofinitial bridg e balance . Sinc e th e outpu t o f th e bridg e i s usuall y nonlinear ,this distinction betwee n th e genera l cas e an d a particula r cas e i s necessary .Furthermore, it becomes more important wit h larger resistance changes, suc has those tha t ma y b e encountered wit h semiconductor gages .

We now rewrite Eq. (5.27) in the simplified form corresponding t o initialbridge balance . I t become s

Since th e strai n gag e indicate s strai n i n term s o f uni t change s i nresistance, w e now procee d t o conver t Eq . (5.34 ) int o term s o f ratio s an dunit changes. If both numerator and denominator ar e divided by the produc tRiR3, Eq. (5.34) can b e rewritten as

As we are now dealing with conditions o f initial bridge balance, we introducethe relation s give n by Eqs . (5.29 ) and (5.32) . They ar e

These relation s ar e now substituted into Eq . (5.35) to arriv e a t


Multiply th e numerato r an d denominato r b y a to obtai n

This reduces t o

In orde r t o pu t Eq . (5.37) into a more desirabl e form, let

Equation (5.37 ) become s

Next, Eq . (5.36 ) can furthe r b e rearranged .


The bracketed ter m i n Eq . (5.38) is the nonlinearit y factor , and s o A£0 ca nnow b e written as

where th e nonlinearit y factor , ( 1 — n), is

Equation (5.40 ) i s exac t an d wil l yiel d correc t value s o f n , o r ( 1 — n),for al l value s o f th e uni t change s i n resistanc e o f th e bridg e arms . I t is ,however, somewhat inconvenien t t o handle .

When th e uni t changes i n resistanc e ar e smal l relativ e to unit y (let u ssay les s tha n 1 0 percent), thei r product s wil l b e eve n smalle r (les s tha n 1percent) an d ca n b e neglected . From Eq . (5.40) w e can therefor e develop amuch simple r an d ver y good approximat e relationshi p i f we disregard th esecond-order quantitie s i n th e numerato r an d i n th e expansio n o f th edenominator. This procedur e wil l give us

Equation (5.41 ) can b e solved for n by letting

Thus,


The relatio n fo r n become s

If w e compar e Eq . (5.39 ) for th e bridg e outpu t an d Eq . (5.42 ) for th enonlinearity factor with the corresponding expressions for the potentiometriccircuit, a marked similarit y will be observed. Furthermore , if the bridge armscorresponding t o R 3 an d R 4 contai n fixed resistors, AjR 3 an d AR 4 wil l bot hbe zero . Equation s (5.39 ) an d (5.42) , then, becom e identica l wit h thos e o fthe potentiometric circuit. In additon, Eq. (5.42) loses its approximate natureand become s exact.

Equations (5.39 ) an d (5.42 ) ca n b e writte n i n term s o f strain , sinc eR/R = G FE. With like gages i n al l fou r bridg e arms , a = 1 , and Eq . (5.39)can b e written as

Equation (5.42 ) becomes

When measurin g elastic strains in metals , the erro r du e t o nonlinearityis generall y smal l an d i s usuall y ignored . A s a rul e o f thumb , th e error , i npercent, i s approximately equa l t o th e strain , i n percent .

When nonlinearit y mus t b e take n int o account , it s influenc e fo r an ybridge arrangement ca n b e readily computed throug h the us e of Eqs. (5.39)and (5.40) . To illustrate this, a quarter-bridge circuit can be examined, whereARl/Ri = GF£. Usin g thi s valu e of ARl/Rl an d a = 1 , Eq. (5.39 ) produce

The nonlinearit y factor, ( 1 — n), i s obtained fro m Eq . (5.40) . Thus ,


Substituting th e valu e of (1 — n) given by Eq . (5.46 ) into Eq . (5.45 ) yields

This expression ca n b e rewritten as

The strain , e , i n thes e equation s mus t b e entere d a s e x 1 0 6 in/in. Th esecond ter m i n th e denominato r o f Eq. (5.47 ) produces th e nonlinearit y inA£0/K Thus , a compressive strai n will produce an indicated valu e of A£0/Kthat is too large in magnitude, while a tensile strain wil l produce an indicatedvalue that i s too lo w in magnitude.

Reference 5 gives a tabulatio n o f th e effec t o f nonlinearity for variousbridge arrangements. Furthermore, it also gives the ratio of the actual strain,e, to th e indicated strain, e;. In orde r t o sho w this, we know that ( A /K)/e ;is equal t o th e constan t G f/4, an d s o the followin g ca n b e written:

From this , the indicate d strai n is

Solving Eq . (5.48 ) fo r E produces

The rati o o f - can b e writte n a s

or

166 THE BONDE D ELECTRICAL RESISTANC E STRAI N GAG E

In th e followin g tw o exampl e problems , tw o bridg e arrangement s ar edeveloped, whil e other s are lef t a s problems a t th e en d o f the chapter .

Example 5.1. A cantileve r bea m wit h fou r gage s arrange d i n a ful l bridg e i sshown in Fig. 5.9 . Each page wil l read th e sam e magnitude o f strain, with gage s1 an d 3 i n tension , an d gage s 2 an d 4 i n compression . Usin g Eqs . (5.39 ) an d(5.40), determine E 0. Als o determine , .

Solution

bridge rati o = 1 ,

From Eq . (5.39),

Equation (5.40 ) is

Substituting th e gage facto r and appropriat e strain s fo r R/R, w e have

FIG. 5.9. Cantileve r bea m wit h strai n gage s aligne d paralle l t o th e longitudina l axis .

WHEATSTONE BRIDG E

The circui t i s linear, an d s o the outpu t is

167

Since th e circui t i s linear an d ( 1 — n) = 1 , then

Example 5.2. A round ro d i n tension ha s four gage s mounted o n i t in order t oform a ful l bridge . Gages 1 and 3 are mounted in the axia l direction 180 ° apart .Gages 2 and 4 are mounted transvers e to gage s 1 and 2 , respectively, as shownin Fig . 5.10 . Determin e E 0, usin g Eqs . (5.39 ) and (5.40) , a s wel l a s E/E t.

FIG. 5.10. Tensio n member wit h strai n gages .

Solution

From Eq . (5.39) ,

bridge ratio = 1


From Eq . (5.40),

Substituting i n th e gag e facto r an d appropriat e strains , we have

Multiplying th e expressio n for A£ 0 b y ( 1 — n), w e obtai n

The valu e of the indicate d strain can b e written as

Equating thi s to th e valu e of A£ 0 whe n nonlinearit y i s considered gives

Thus,

Solving fo r E ,

In term s o f the rati o o f th e actua l strain , s . to th e indicate d strain, E,. , w e have

Thus,

Other bridg e arrangement s ca n b e handle d i n the sam e manner .


Alternate method for the derivation of elementary bridge equations

An alternat e metho d fo r developin g th e expressio n fo r th e output , A£ 0, o fan initiall y balance d Wheatston e bridg e wil l b e shown . Conside r th epossibility o f connecting tw o potentiometri c (half-bridge ) circuits togetherin parallel , a s shown in Fig . 5.11 .

The initial resistances are Rlt R 2, R3, and R^. Since the two half-bridgesare t o b e joined together , the y wil l bot h b e subjecte d to th e sam e voltage,V. The potential drops acros s R t an d #4 ar e represented a s £2-1 an d £3-4 ,respectively, and ca n b e expressed as

where

When th e tw o half-bridge s ar e pu t togethe r t o for m a Wheatston ebridge, a s shown in Fig. 5.12 , and the n initially balanced, th e voltage drops

FIG. 5.11 . Tw o potentiometric circuits (or tw o hal f bridges) .


FIG. 5.12. Wheatston e bridge formed fro m tw o hal f bridges.

across resistance Rt an d R 4 mus t be equal. Thus, from Eqs . (5.52) and (5.53),

From this , it is evident that a 2 - i = a a-4- This means tha t

where

For initia l bridge balance , th e ballas t rati o mus t b e th e sam e fo r bot hsides, a s expresse d b y Eq . (5.54) . When change s tak e plac e i n eac h ar m b ythe appropriat e A# , the potentia l dro p across R t an d R 4 wil l be

The bridg e output , A£ 0, wil l b e equa l t o th e differenc e i n voltag e betwee nA and B . Therefore,

For the condition of initial balance, however, £


From th e relations fo r the potentiometri c (half-bridge ) circuit , a s givenby Eq . (4.21) ,

Rearranging,

In a lik e manner , A£ 3 _ 4, i s written

Note that , in Eqs. (5.57 ) and (5.58) , R2 an d R 3 ar e the ballast resistances . Ifthe values of A£2-i an d A£ 3_4 given by Eqs. (5.57 ) and (5.58) , respectively,are substitute d into Eq . (5.56) , the outpu t voltag e wil l be

If th e brackete d ter m onl y i s considered, i t can b e expressed a s


If th e numerato r i s expanded, i t become s

Combining al l terms , th e outpu t voltag e is

Equation (5.60 ) i s exactly the sam e a s Eq . (5.37).

5.4. General bridge equations

We wil l no w conside r a somewha t mor e elaborat e arrangemen t o f th eWheatstone bridge . Thi s wil l includ e the followin g items tha t wer e omitte din th e previou s section :

1. Th e effec t o f resistance i n serie s wit h th e bridge . Thi s wil l includ e th einternal resistanc e o f the powe r suppl y as wel l a s th e resistanc e o f th eleads connectin g th e bridg e to th e energ y source . I n th e analysis , bothof thes e resistance s wil l b e lumpe d togethe r an d considere d a s thoug hthey presente d a singl e combine d resistanc e in serie s wit h th e bridge .

2. Th e influenc e o f the mete r (o r galvanometer ) resistanc e on th e bridg eoutput voltage. In the previous section, the analysis of the bridge outputwas mad e o n th e assumptio n tha t th e mete r presente d a n infinit eimpedance and , in consequence, would draw no current from th e bridge.We wil l no w examin e th e situatio n i n whic h th e mete r ha s a finit eimpedance an d draw s som e curren t fro m th e bridge .

Fortunately, th e result s o f th e analysi s o f th e idealized , o r simplified ,bridge circui t ca n b e use d i n buildin g up th e genera l case , whic h include sthe precedin g considerations .


Effect of resistance in series with the bridge

The bridge input resistance , R BI, an d the bridge outpu t resistance , R BO, ar egiven b y Eqs. (5.9 ) and (5.10) , respectively. Here w e will outlin e the metho dof computin g them . Sinc e Thevenin' s theore m wil l b e used , i t i s state d a sfollows (6) :

Any two-termina l netork o f fixed resistances and source s o f e.m.f. may b ereplaced b y a singl e sourc e o f e.m.f . havin g an equivalen t e.m.f . equa l t othe open-circui t e.m.f . a t the terminal s of the origina l network and havingan internal resistance equal to the resistance looking back into the networkfrom th e tw o terminals , an d wit h al l source s o f e.m.f . replace d b y thei rinternal resistance .

The resistanc e i n serie s wit h th e bridg e wil l includ e th e interna lresistance of the power supply as well as the resistance of the leads connectingthe bridge to th e source of energy. In the analysis , both of these resistancesare lumped together and considered a s a single combined resistance in serieswith th e bridge . Th e effec t o f th e serie s resistanc e i s t o reduc e th e voltag eactually receive d a t th e bridg e compare d wit h tha t availabl e a t th e powe rsupply, sinc e th e tota l voltag e mus t b e apportione d acros s th e serie s an dbridge resistances rather than being applied entirely to the bridge. The circuitis again show n in Fig . 5.13 .

In orde r t o comput e R BI, th e circui t is opened a t point s D and C . Theresistance, R s, i s no longe r i n th e circui t bein g considered , an d neithe r arethere energy sources . Lookin g int o the bridg e fro m point s D and C , we seea circui t with resistance s R it R 2, R 3, R 4 an d R m. Sinc e th e circui t is not acombination o f series and paralle l resistances , it must b e changed int o suc ha combination . Figur e 5.14 a show s th e origina l circui t bein g considered ,while Fig . 5.14 b show s the converte d circuit.

FIG. 5.13. Wheatston e bridge with supply resistance an d mete r resistance.


FIG. 5.14. Origina l circui t (a ) an d equivalen t circui t (b).

The mean s o f obtainin g th e circui t of Fig . 5.14 b wil l b e outlined . Th eresistances R 2, #3 , and R m form a Delta networ k tha t mus t b e converted toa Wye network consistin g of resistances R A, R B, an d R c. Th e resistances inthe Wy e network (6 ) are give n as

Referring t o Fig . 5.14b , th e resistance s R B + R^ an d R c + R4 ar e i nparallel, and thei r equivalent resistanc e is then in series with R A. Th e bridgeinput resistanc e i s then

Equation (5.64 ) ca n b e expressed i n term s of the origina l resistance s show nin Fig . 5.14 . Although considerable algebr a i s involved, the fina l resul t i s


If th e resistance s J? 1; R 2, R 3, an d R 4 ar e increased b y thei r individua lR values , the n Eq . (5.65 ) become s Eq . (5.9) , th e expressio n fo r R BI.Furthermore, if R! = R2 = R3 = R4 = Rg, the n Eq. (5.65 ) reduce s toRCD = Rg> regardless o f the valu e of R m.

Since R s i s in serie s with the bridge , th e bridg e voltage , V DC, is

Equation (5.66) shows that when a resistance i s in series with the bridge, thevoltage mus t be multiplied by the desensitizatio n factor , 1/( 1 + RJR BI), i norder t o determine th e actua l bridg e voltage .

Since the bridge outpu t i s directly proportional t o th e applie d voltage ,the voltage , V DC, can b e substitute d fo r th e voltage , V , i n Eq . (5.39) . Th evalue o f A£0 the n become s

It shoul d b e note d i n Eq . (5.67 ) tha t R BI i s not a constant , sinc e i t varieswith th e R quantities . I f the uni t change s i n resistanc e ar e large , then ,depending o n the relative magnitude of Rs, som e allowance fo r the variatio nin R BI ma y be required .

Influence of meter resistance

So fa r w e hav e examine d th e bridg e outpu t voltag e whe n th e meter , o rindicating device , wa s considere d a s havin g infinit e inpu t impedance . W enow loo k a t wha t happen s whe n the mete r (o r galvanometer ) ha s a finiteresistance and draws current from th e bridge. To do this, the circuit is openedbetween the mete r and on e o f the outpu t terminal s o f the bridge , a s show nin Fig . 5.15 . Thevenin' s theore m wil l the n b e applie d i n orde r t o ge t a nequivalent circuit.

According to Thevenin's theorem, we first find the open-circuit potentia lbetween point s A an d B . In orde r t o d o this , the loop, o r mesh , equationscan be written by referring to Fig. 5.15. As we see, there will be two equations .They ar e


FIG. 5.15. Wheatston e bridg e wit h suppl y resistanc e and outpu t mete r disconnected .

Rearranging, we have

Solving Eqs. (5.68) an d (5.69 ) simultaneousl y fo r 1 ^ and / 2 result s i n

Substituting th e value s o f / , an d I 2 give n b y Eqs . (5.70) an d (5.71) ,respectively, int o Eq . (5.72) , w e hav e

Thus, Eq . (5.73 ) i s the voltag e sourc e applie d t o th e equivalen t circuit .The interna l resistanc e o f th e equivalen t circui t mus t b e determined .

This i s accomplishe d b y lookin g bac k int o th e networ k fro m terminal s Aand B wit h th e potential , V , shorted. Th e interna l resistanc e o f V i s added

The potential , E , across AB i s


FIG. 5.16. Origina l circui t (a ) an d equivalen t circuit (b).

to th e resistanc e R s. Th e origina l networ k an d th e equivalen t networ k ar eshown i n Fig . 5.16 . Figure 5.16 a show s R s acros s terminal s D and C so tha tresistances R±, R2, an d R s for m a Delta network tha t i s to be converted t othe Wy e network, show n b y resistance s R D, R E, an d R F i n Fig . 5.16b . W esee that R D i s now i n serie s with the paralle l resistanc e forme d by R F + R 3and R E + R 4. Th e Wye resistances are

The equivalen t resistance, R AB, i s

The resistanc e R AB ca n b e expressed in terms of the origina l resistance sshown i n Fig . 5.16a . Carryin g ou t th e necessar y algebra, th e final resul t is

Again, if resistances R t, R 2, R3, and K4 are increased b y their individual Rvalues, the n Eq . (5.78 ) become s Eq . (5.10) , th e expressio n fo r R B0. Also , ifRt = R2 = R3 = R 4 = R g, the n Eq . (5.78) reduces t o R AB = Rg, regardles sof th e valu e o f R, .


The circuit can now be drawn as shown in Fig. 5.17. The voltage source,E, is given by Eq. (5.73). The curren t flowing through the circui t is the mete rcurrent, / galvo. Thus , w e can write

Equation (5.79 ) can b e rewritte n as

where E m i s the voltag e drop across th e meter .If w e conside r th e specia l cas e i n whic h th e bridg e ha s bee n initiall y

balanced, the n Eq . (5.80 ) ca n b e expressed a s

Rearranging Eq . (5.81 ) to obtai n A£ m0, th e chang e i n voltag e drop acros sthe mete r fro m a conditio n o f initial balance, w e have

Equation (5.82 ) show s th t th e outpu t i s furthe r desensitize d b y th e facto r1/(1 + Rthe R quantities . If the unit changes in resistance are large, depending upo nthe relativ e magnitudes o f Rm, som e allowanc e for variation in R BO ma y b erequired.

FIG. 5.17. Equivalen t circui t fo r th e Whealston e bridge.

Also note that RBio is not a constatnt, since it varies with


There ar e tw o desensitizatio n factor s involved , on e concernin g th eresistance i n th e powe r supply , R s, an d th e othe r concernin g th e mete rresistance, Rm. Multiplyin g the right sid e of Eq. (5.67) by the desensitizatio nfactor containin g R m, th e chang e i n voltag e dro p acros s th e meter , A£ m0,from a condition o f initial balance, become s

If R! = R 2 = R3 = K4 = Rg, then RBI = RBO = Rg, and the bridge ratiois a = 1 . Using &R/R = GFe, Eq. (5.83 ) become s

Meter current

The curren t draw n b y th e meter , o r galvanometer , ca n b e compute d b yreferring t o Fig . 5.17 . The voltage , £, i s given by Eq. (5.73). If the resistancesin Eq. (5.73) are increased b y the R quantities , a s per Eqs. (5.11 ) through(5.14), t o mak e i t compatibl e wit h R BO, the n th e galvanomete r curren t fo rthe unbalance d bridg e is

When expanded , Eq . (5.85) becomes Eq . (5.17). For th e balance d bridge ,

Example 5.3. A full bridg e is made u p o f four 120-oh m gages, each wit h a gagefactor o f GF = 2.05. The gages ar e mounte d o n a cantilever beam, wit h gage s 1and 3 o n th e to p surfac e an d gage s 2 an d 4 o n th e botto m surfac e directlyunderneath. Thus , e l = e3 = s an d s 2 = £4 = — e. Assum e tha t n ma y b eneglected.

(a) Usin g a n instrumen t such that R m - > oo and R s = 0, determine A£ m0.(b) Usin g an instrument such that Rm = 350 ohms and Rs = 0, determine A£m0.


Solution, (a ) Equatio n (5.84 ) reduce s t o Eq . (5.43) . Thus ,

(b) Sinc e R s = 0 and R B, = R BO = Rq, Eq . (5.84) is

The outpu t signa l i s reduced b y approximatel y 2 5 percen t whe n a mete r wit hR = 350 ohms i s used .

5.5. Effect of lead-line resistance

When strain gages are located a t a test area remot e from th e instrumentation,lead-line resistanc e densensitize s th e syste m an d produce s strai n read -ings lowe r tha n thos e actuall y occurring . Thes e resistance s wil l no t onl ydesensitize the circuit, but they wil l affec t calibratio n and ma y als o introduc ea temperature-compensatio n problem . Th e objectiv e no w i s t o examin eseveral common circui t arrangement s an d determin e to wha t exten t each i sdesensitized b y lead-lin e resistance .

Full bridgeAs pointe d ou t i n Sectio n 5.4 , th e interna l resistanc e o f th e powe r supply ,Rs, coul d als o hav e bee n include d i n the lead-line resistance tha t i s in serie swith th e powe r supply . Reserving now th e symbo l R s fo r the powe r suppl yinternal resistance , ther e i s in series wit h it the lead-lin e resistance , 2R sL, asshown i n Fig . 5.18 . Whil e no t state d explicitl y i n Sectio n 5.4 , th e mete rresistance, Rm, could als o have included the lead-line resistance on the outputside o f the circuit . Again, this i s evident i n Fig . 5.18 . The resistances , R sL o nthe powe r sid e an d R mL o n th e outpu t side , could als o contai n switch , andother, resistances .

Lead-line resistanc e ca n b e accounte d fo r withou t a ne w analysi s b yreplacing R s wit h R s + 2RsL an d R m wit h Rm + 2RmL i n Eq. (5.83). Thus,


FIG. 5.18. Wheatston e bridge wit h lead-lin e resistance.

If /? ! = R 2 = R 3 = K4 = R g, the n R B} = RBO = R g, th e bridge rati o isa = 1 , and usin g R/R = GFE, Eq. (5.87) can b e rewritten as

For th e case in which .Rm is very large, there is no correction for lead-lineresistance o n th e outpu t side . Thus , fo r a syste m wher e R m — > oo (ope ncircuit) and R s i s negligible, Eq. (5.88 ) reduces t o

Therefore, fo r th e remot e ful l bridg e th e outpu t signa l i s desensitize d(attenuated) by the factor R g/(Rg + 2R sL).

In th e circuit s that follow , th e interna l resistance i n th e powe r supply ,Rs, wil l b e considered negligibl e and th e mete r resistance , R m, wil l b e largeenough s o tha t th e outpu t sid e is taken a s open.

Half bridge —four wire

In thi s arrangement , R j an d R 2 ar e th e activ e gage s an d ar e locate d a t adistance fro m th e instrument . Each lea d ha s a resistanc e o f RL. Th e circui tis shown i n Fig . 5.19 .


FIG. 5.19. Hal f bridge wit h fou r lea d wires .

If th e loo p equation s ar e writte n and the n solve d fo r th e currents , / jand / 2, th e resul t is

The potentia l differenc e betwee n points A an d B is

Substituting th e value s o f 1 ^ an d / 2 give n b y Eqs . (5.90 ) an d (5.91) ,respectively, into Eq . (5.92 ) produces

Equation (5.93 ) gives the initia l output , E, for the unbalance d bridge .If gages R l an d R2 underg o a change in resistance such that R t change s

to R t + ARi an d R 2 change s t o R 2 + AR 2, the n th e bridg e outpu t wil lchange fro m E to E + E, an d so Eq. (5.93) becomes


If we start wit h an initiall y balanced bridge , th e initia l output , E , is

From this ,

Thus, Eq . (5.94 ) can b e rewritten, for an initiall y balanced bridge , as

Equation (5.97 ) can b e written in terms of unit changes in resistance bymultiplying and dividin g A/?j by R 1 an d AK 2 by R2. Doin g this , and usin gEq. (5.96) , the en d resul t is

v

The resistance R 4 ca n b e eliminated fro m Eq . (5.98) by again usin g Eq.(5.96). Making this substitution and carrying out the intervening algebra, Eq.(5.98) can finally be rewritten as

Equation (5.99 ) can b e put int o a more desirabl e for m i f we let


Using Eqs . (a) , (b) , and (c) , Eq . (5.99 ) become s

The brackete d term i n Eq. (5.100 ) i s the nonlinearit y factor, (1 — n), an dso b y substitutin g the value s of A , B , and C give n by Eqs . (a) , (b) , an d (c) ,respectively, back int o Eq . (5.100) , the output , A£ 0, become s

where

Letting R t = R2 = Rg and knowing R/R = G FE, the output, AE0, from Eq .(5.101) an d n , from Eq . (5.102), can b e written in terms o f strains. These tw oequations the n becom e

For thi s half-bridge arrangement, the output i s desensitized (attenuated )by th e facto r R g/(Ra + 2RL). Althoug h R l an d R 2 wer e considere d activ egages, on e coul d b e activ e and th e othe r use d a s a compensatin g (dummy )gage fo r temperatur e compensation . Th e dumm y gag e i s mounte d o n a nunstrained piec e o f materia l simila r t o tha t o n whic h th e activ e gag e i smounted, wit h both gage s subjecte d t o th e sam e temperature .

Half bridge —three wire

In thi s circuit , 7? j and R 2 ar e locate d som e distanc e fro m th e instrument ,but R j an d R 2 ar e joined a t A ' s o tha t onl y on e lea d i s brought fro m thi sjuncture t o th e instrument . Each lea d ha s a resistanc e o f R L. Th e circui t isshown i n Fig . 5.20 .


FIG. 5.20. Hal f bridge wit h thre e lead wires.

If th e loo p equation s ar e writte n and the n solve d fo r th e currents , / jand / 2, w e obtai n

The potentia l differenc e betwee n point s A an d B is

Substituting th e value s o f / t an d I 2 give n b y Eqs . (5.105 ) an d (5.106) ,respectively, into Eq . (5.107) gives the output , £ , fo r the unbalanced bridge .Thus,

If gages R 1 and R2 underg o a change in resistance such that Rl change sfrom R l t o R1 + ^Rl an d R2 change s from R 2 to R2 + AR2> then the bridgeoutput wil l change fro m E t o E + A£. Equatio n (5.108 ) then become s

If we start wit h an initiall y balanced bridg e an d writ e the output , A£ 0,in term s o f the uni t changes i n resistance , th e fina l resul t is


Equation (5.110 ) ca n b e pu t int o a mor e desirabl e form , and s o i t ca n b erewritten a s

where

Letting R i = R2 = Rg and knowing R/R = GF£, AE0 and n from Eqs .(5.1 11) and (5.112) , respectively , becom e

The output, A£ 0, o f this circuit is desensitized b y the factor Rg/(Rg + R L)\thus, we see that th e desensitizatio n o f the three-wir e half bridge differ s fro mthe four-wire half bridge. Thi s circuit ca n be used i n the same manne r a s thecircuit wit h fou r wires . Table 5. 1 compares the desensitizatio n factor s o f thetwo circuits .

Table 5.1. Comparison o f desensitizatio n fac-tors fo r three-wir e an d four-wir e hal f bridge s

Wire resistance, Three-wire, Four-wire,RL Rgl(Re + R,) Re/(Rg + 2RL)

05

1015202530

00.9600.9230.8890.8570.8280.800

00.9230.8570.8000.7500.7060.667


Quarter bridge —three wire

In thi s circuit R 1 i s the onl y activ e gage an d i t is located a t a distance fro mthe instrument. Three lead s o f resistance R L ar e used in this circuit, with thethird lea d bein g brought fro m th e gage t o the center poin t connection , A , atthe instrument , as show n i n Fig . 5.21 . The tw o lea d wire s in adjacen t arm sshould b e of the same lengt h an d maintaine d a t the same temperature . Thi sthree-wire circui t i s th e standar d metho d fo r a singl e activ e temperature -compensated strai n gag e in thi s arrangemen t (7) .

If th e loo p equation s ar e writte n an d the n solve d fo r th e currents , / jand I 2, w e have

The potentia l differenc e betwee n point s A an d B is

Substituting th e value s o f / j an d I 2 give n b y Eqs . (5.115 ) an d (5.116) ,respectively, into Eq . (5.117) gives the output , E , for the unbalance d bridge .Thus,

If gage R! undergoe s a change in resistance fro m R 1 t o R1 + AR1, the n

FIG. 5.21. Quarte r bridge with thre e lead wires .


the bridg e outpu t wil l chang e fro m E t o £ + E . Equatio n (5.118 ) the nbecomes

If we start fro m a n initiall y balance d bridg e and writ e the output , A£ 0,in term s o f unit change s i n resistances , w e hav e th e fina l resul t a s

where

Letting R ± = R2 = Rg an d usin g R/ R = G FK, A£0 an d n fro m Eqs .(5.120) an d (5.121) , respectively, becom e

In thi s circui t th e resistor , R 2, i s equa l t o R g an d i s locate d a t th einstrument. The equations ar e identical to those for the three-wire half bridg eif R 2 i n tha t circui t i s a dumm y gage . I n tha t cas e AR 2/R2 an d £ 2 are zero ;thus, Eqs . (5.111) , (5.112), (5.113), and (5.114 ) reduce t o Eqs . (5.120) , (5.121),(5.122), an d (5.123) , respectively .

Quarter bridge —two wire

As in the three-wire quarter bridge , R^ i s the only active gage and i t is locate dsome distanc e fro m th e instrumen t b y tw o lea d wires , eac h havin g aresistance o f R L. I n thi s circuit , temperatur e compensatio n i s lost , an d fo rRL o n th e orde r o f 0.5 ohms th e bridg e wil l no t balance , an d s o th e initia lreading wil l b e tha t fo r a n unbalance d bridge . A valu e o f R L o n th e orde r


FIG. 5.22. Quarte r bridge with tw o lea d wires .

of several ohms wil l generally be out o f the instrument' s rang e an d reading scannot b e obtained . Th e circui t i s shown in Fig . 5.22.

As before, the potential difference between points A and B is found. It is

If R I undergoe s a chang e i n resistanc e fro m R 1 t o R ± + A,R l5 th e bridg eoutput wil l chang e fro m £ t o E + E. Thus ,

If w e star t wit h a balance d bridge , th e output , A£ 0, ca n b e writte n i nterms o f the uni t change in resistance , an d s o th e fina l resul t is

where

If R 1=R2 = Rg, the n AE 0 an d n , fro m Eqs . (5.126 ) an d (5.127) ,

190 TH E BONDE D ELECTRICA L RESISTANC E STRAIN GAGE

respectively, ca n b e written as

If th e lead-lin e resistanc e i n a particula r circui t i s known , th e outpu tvoltage, A£ 0, ca n b e correcte d b y multiplyin g i t b y th e reciproca l o f th edesensitization facto r fo r tha t circuit . Corrections fo r th e circuit s discusse dare listed , where AE 0c i s the correcte d outpu t voltage .

Full bridge

Half bridge—four wire

Half bridge —three wire

Quarter bridge —three wire

Quarter bridge —two wire

Figure 5.2 3 shows th e influenc e of lead-line resistanc e o n a half-bridg efour-wire circuit. The information plotted i s from a cantilever beam test , withone gag e o n to p o f the bea m an d th e secon d gag e o n th e botto m directl yunderneath.


FIG. 5.23. Influenc e o f lead-line resistance, RL, o n a half-bridge , four-wir e circuit .

Figure 5.2 4 show s the influenc e o f lead-line resistance on a half-bridg ethree-wire circuit . Th e sam e cantileve r bea m wa s used , bu t i t i s apparen tthat th e attenuatio n o f thi s circui t i s les s tha n tha t o f th e fou r wires. Acomparison o f Eqs . (5.131 ) and (5.132 ) show s the reaso n fo r this .

Example 5.4. Th e linea r driv e tub e o f a machin e ha s fou r 120-oh m gages ,forming a ful l bridge , mounte d o n it in order t o determine the longitudinal forc eacting on th e tube . Gages 1 and 3 are aligne d paralle l t o th e longitudina l axi sand ar e 180 ° apart , whil e gage s 2 an d 4 ar e mounte d transvers e t o th elongitudinal axis . The bridge i s connected t o th e instrumentation , located i n acontrol booth , wit h 100f t o f No . 2 6 coppe r wir e havin g a resistanc e o f4.081 ohms/100 ft. Figur e 5.2 5 show s th e driv e tub e an d bridg e arrangement .Determine th e outpu t voltage .

Solution. Wit h gages 1 and 3 in opposite arms, as well as gages 2 and 4 , bendingstrains wil l b e cancele d an d onl y longitudina l compressiv e strain s wil l b erecorded. Furthermore , th e nonlinearit y facto r wil l b e smal l an d ca n b e

FIG. 5.24. Influenc e o f lead-line resistance , R L, o n a half-bridge , three-wir e circuit .

FIG. 5.25. Driv e tub e wit h bridg e arrangemen t fo r measurin g axia l force.


disregarded. Thus,

Ignoring the lead-line resistance for the moment, Eq. (5.43 ) ca n b e used t ocompute A£ 0:

This resul t show s that th e signa l wa s reduced by approximately 6. 8 percent. Inpassing, note tha t i f 350-ohm gage s wer e used , the correctio n facto r woul d b e

Thus, the signa l would b e reduced b y approximately 2.3 percent, and so , if longlead line s ar e used , i t woul d b e bette r t o us e higher-resistance gages .

5.6. Circuit calibration

The tw o basi c method s o f calibrating a strai n gag e circui t are mechanica land electrica l (8 , 9) . Th e mechanica l calibratio n method , whil e goo d fo restablishing the validit y of the measurin g system, is inconvenient and costlyfor regula r use. In thi s section, electrical calibratio n onl y will be considered ,where a calibratio n resistor , R c, i s shunte d acros s on e o f th e gages .Furthermore, i t wil l be assumed tha t th e permissibl e error wil l be such tha tthe nonlinearit y o f th e Wheatston e bridg e ca n b e neglected . A s a furthe rrestriction, only arm R t wil l be shunted, as shown in Fig. 5.26. For a detailedanalysis o f shun t calibration , fo r bot h smal l an d larg e strains , Referenc e 9is recommended .

When th e resisto r R c i s shunte d acros s R ls wher e R ^ = Rg, th e tota lresistance in tha t ar m i s reduced. Th e equivalen t resistance is

The voltage , A£0, can b e corrected by usin g Eq . (5.130).

and


FIG. 5.26. Wheatston e bridge with calibratio n resistor.

The chang e i n resistanc e i n th e bridg e ar m i s

Dividing bot h side s o f Eq. (5.136 ) by R q gives

Since R/ R = G Fe, th e equivalen t strai n produce d b y shuntin g R c acros sR1 i s

The negativ e sig n tell s u s tha t thi s calibratio n metho d produce s a nequivalent strai n tha t i s compressive i n sense. Precisio n calibratio n resistor scan be purchased, usin g G F = 2.0 , that wil l give microstrains o f even values ,such a s 500 , 1000 , etc . Thi s metho d ca n b e employe d whethe r o r no t aquarter-, half- , o r full-bridg e circui t i s bein g used . Knowin g th e bridg earrangement, th e surfac e strai n a t th e primar y gag e ca n b e foun d b ycalculation. It shoul d b e noted that the shunt is applied at the gage and no tat th e instrument.

Example 5.5 . Determin e the valu e o f Rc tha t wil l produc e an equivalen t strai nof -50 0 uin/i n whe n G F = 2.0 and R g = 12 0 ohms.


FIG. 5.27. Circui t wit h calibratio n resistor , R c, shunte d across resistor

Solution. Solvin g Eq. (5.138 ) for R c produce s

The calibration of a circuit with gages mounted remote from th e instrumentand tha t hav e equa l resistance , R L, i n eac h lea d lin e wil l b e considered .Equations (5.130 ) throug h (5.134 ) sho w th e factor s b y whic h th e indicate doutput voltage (or indicated strain ) wil l have to be multiplied in order t o obtainthe true output value.

Figure 5.2 7 shows a half-bridg e arrangemen t wit h R r a n activ e gag eand R 2 bein g eithe r a n activ e o r a dumm y gage . Th e calibratio n resistor ,Rc, ca n b e locate d a t eithe r R ^ o r bac k nea r th e instrument , bu t i n eithe rcase it s leads als o hav e the sam e resistance , R L. I n general , fo r high values ofRc, it s lead resistance s wil l have little effect o n th e calibratio n strain . When R Lis now shunte d across R lt th e gai n (gag e facto r setting ) o f the instrumen t canbe adjuste d s o tha t th e indicate d strai n read s th e calibratio n strain . Fo rsubsequent loading, the instrument will now read th e strains directly. Althougha hal f bridge has been shown, the method als o applie s to a quarter, half , o r ful lbridge.

5.7. Comments

In th e developmen t o f the bridg e equations , th e outpu t o f the bridg e ha sbeen i n term s o f voltage , specifie d eithe r a s AE 0 o r A£ m0. I n th e strai ninstrumentation generall y used , the instrument is calibrated to read directl yin strain . Furthermore , i f a ful l bridge i s considered, a s show n in Fig . 5.28 ,we have learned, startin g wit h ar m 1 , that th e arm s alternate i n sign . Thus,


FIG. 5.28. Wheatston e bridge showing th e sign s o f the respectiv e arms.

if a gag e connecte d i n ar m 2 i s subjecte d t o a compressiv e strain , th e sig nwill b e change d an d th e indicato r wil l giv e a positiv e value . Because o f thisproperty o f the Wheatston e bridge , bridg e circuit s ca n b e arrange d i n suc ha manne r tha t w e can isolate , fo r instance , th e effec t o f a n axia l forc e an dnull ou t th e effect s o f bending . Th e bridg e arrangemen t i n Fig . 5.10 , a s a nexample, wil l d o jus t tha t b y cancelin g bendin g strain s an d producin g th estrains o f the direc t axia l force.

Many times , a numbe r o f strain gages , use d i n quarter-bridg e circuits ,may b e bonde d a t variou s location s o n a structure . Becaus e i t woul d b etime-consuming and awkward to connect each strain gage, in turn, to a strainindicator and the n load th e structure, a switching and balancin g uni t i s usedin conjunction wit h the strai n indicator . A typical multichannel applicatio nis show n i n Fig . 5.2 9 wit h a strai n indicato r an d it s companio n switchin gand balancin g unit .

In thi s application , a numbe r o f gages ar e connecte d t o th e switchin gand balancin g uni t which, in turn, is connected t o th e strain indicator . Here ,six o f a tota l o f ten channel s are used . Th e switc h is turned t o eac h channe land th e strain indicato r i s balanced b y using the balancin g potentiomete r o fthe individua l circuit. Then, a t ever y load level , the switc h is turned t o eac hchannel an d tha t strai n recorded .

Since th e strai n gage s ma y no t al l b e alik e (singl e gage s an d rosette smay b e mixed) , there wil l b e severa l differen t gag e factors . I n thi s case , se tone valu e of G F on th e strai n indicato r an d correc t th e indicate d strai n b ycalculation. Since R/ R wil l be the sam e regardles s o f the valu e of GF used,we can writ e


FIG. 5.29. Multichanne l arrangemen t usin g a switchin g an d balancin g unit . (Courtes y o fMeasurements Group , Inc.)

This i s rearranged t o

where G' F = gage facto r set on th e strai n indicator

GF = gage factor of the strai n gage

e' = indicate d strai n

e = correcte d (actual ) strai n

Therefore, onc e al l o f the indicate d strain s are recorded , Eq . (5.140 ) can b eused t o determin e the actua l strains.

Problems

5.1. A full bridge , made u p of 120-ohm gages , has a constant-voltage power supplyof 1 0 volts. The following resistors ar e shunted , in turn, across ar m R^ . 11 9 880,11 880, 1080 , 360 , 120 , 40, and 1 0 ohms. Using Eqs. (5.39) and (5.40) , plot A£ 0vs. A.R,/Kj .

In Probs. 5. 2 through 5.7 , use Eqs. (5.39) and (5.40) to determine an expression for

5.2. I n Fig . 5.9 , gage 1 is the onl y active gage, so tha t 5.3. I n Fig . 5.10 , gages 1 and 2 are th e activ e gages, thus and5.4. I n Fig . 5.9 , gages 1 and 2 are the activ e gages. I n thi s case and 5.5. I n Fig . 5.9 , gages 1 and 3 are activ e gages, so tha t

198 THE BONDE D ELECTRICAL RESISTANC E STRAI N GAGE

FIG. 5.30 .

5.6. A cantileve r bea m ha s gage s arrange d a s show n i n Fig . 5.30 . Gage s 1and 2 are longitudina l gages, mounte d to p an d bottom , respectively . Gage 3 ismounted o n th e botto m transvers e t o gag e 2 , while gag e 4 i s mounted o n th etop transvers e t o gag e 1.

5.7. Th e gage s o f th e cantileve r bea m i n Fig . 5.3 0 ar e rewire d int o th e bridg earrangement show n i n Fig . 5.31.

5.8. A smal l assembl y machin e ha s th e dimension s show n i n Fig . 5.32 . Gage s 1and 2 ar e bonde d a t th e inne r an d oute r radius , respectively , i n a longi -tudinal direction . Eac h gag e i s rea d individually , with e, ^ = 108 3 uin/in an de,2 = —65 2 uin/in. Determine th e stresse s a t eac h gag e locatio n a s wel l a s th eload actin g o n th e machine . Th e materia l i s steel.

5.9. Tw o stee l sleeve s are shrunk together , a s shown in Fig . 5.33 . The nomina l radi iare a = 2.00 in, b = 2.7 5 in, an d c = 3.2 5 in. Afte r assembly , a strai n gag e i sbonded t o th e oute r cylinde r i n th e hoo p (tangential ) direction , th e strai nindicator i s balanced , an d the n th e inne r cylinde r i s pushe d out . Afte rdisassembly, th e strai n gag e give s a readin g o f —84 0 uin/in. Determin e th eshrink-fit pressur e an d th e amoun t o f interference .

FIG. 5.31 .


FIG. 5.32.

FIG. 5.33.

5.10. A cantilever beam, shown i n Fig . 5.34 , ha s a width o f 2 in and a thicknes s of0.250 in. A weight o f 25 Ib can b e positione d a t an y poin t betwee n 1 0 in an d18 in from th e support . Strai n gage s are t o b e placed a t 1 in and 8 in from th esupport.

(a) Sho w tha t th e differenc e i n th e moment s a t th e strai n gag e location s wil lbe th e sam e fo r an y positio n o f th e loa d withi n it s range ; tha t is ,AM = M! -MS.

(b) Determin e a suitabl e full-bridg e arrangemen t tha t wil l giv e th e strai nassociated wit h A M and determin e its magnitude.

FIG. 5.34 .


5.11. A thick-walle d cylinder o f stee l wit h cappe d end s i s subjecte d t o a n interna lpressure. The inner radius is 2 in and th e oute r radius is 3.125 in. On th e outsid esurface a t mid-length , tw o strai n gage s ar e bonded . Gag e 1 is i n th e circum -ferential (hoop ) directio n an d gag e 2 i s i n th e longitudina l direction . Afte rpressurization th e followin g reading s ar e obtained:

KI = 59 0 u.in/in, e. 2 = 13 9 uin/in

Determine th e stres s stat e and th e interna l pressure.5.12. Figur e 5.3 5 show s a cantileve r bea m wit h offse t loading . Fou r longitudina l

strain gage s ar e bonde d t o th e bea m a t sectio n A- A an d the n arrange d int othe bridg e circuit s illustrated in A , B, C, and D . Beneat h eac h bridg e circui t isthe strai n indicato r reading . Determin e th e loads , F x, F y, and F, , as wel l a s th etotal strai n a t eac h gage .

5.13. Whe n a shaft i s in pure torsion, the principal stresses , and therefor e the principa lstrains, lie at ±45 ° to th e longitudinal axis. If a pai r o f strain gages ar e bondedto th e shaf t i n thes e direction s an d anothe r pai r ar e bonde d diametricall yopposite, then , if they are arrange d int o a prope r ful l bridge , onl y the torsiona leffect wil l b e measure d b y th e bridge . Furthermore , i f the shaf t i s subjected t obending moments o r axia l forces, their effect wil l be canceled. Figur e 5.3 6 showsa sectio n o f the shaft .

FIG. 5.35 .


gages c and d diametrically opposite gages a and b

FIG. 5.36 .

(a) Sketc h a Mohr' s circl e and verif y th e strai n directions.(b) Sho w how th e gage s shoul d b e arranged int o a ful l bridge .(c) Explai n why strain s due t o bendin g moments or axia l force s wil l cancel .

5.14. Th e dies on a two-post castin g machine are to b e set so that eac h pos t ha s a nequal axia l force. Tw o gage s ar e bonded , 180 ° apart, t o each pos t a s shown inFig. 5.37 . The gage s ar e arrange d i n tur n t o for m th e bridg e circuit s shown,along wit h thei r respectiv e reading s afte r loading . I f th e post s ar e 3. 0 in i ndiameter, determine the following :(a) Th e axia l force i n each post .(b) Th e bending momen t i n each pos t i n the plane containin g gages .

5.15. A round tensio n lin k made o f steel carries a maximum load o f 50 000 Ib.(a) Arrang e four strai n gages into a ful l bridg e so that temperature compensa -

tion i s achieved an d onl y tensile loading i s measured .

FIG. 5.37 .


(b) Determin e th e lin k diamete r i f n o individua l strain gag e i s subjecte d t omore tha n 1500uin/in .

(c) Usin g a gage facto r o f 2.0, determine A£0/K a t th e maximu m load.

5.16. A circuit has th e followin g resistances :

If R ] an d R 2 chang e b y 1 5 percent, wha t i s the percentag e chang e i n R BI15.17. Comput e th e bridge resistanc e fo r arm resistance s of Rt, R 2, R? , and R 4 whe n

Rs = 0 and R m = GO . Us e Thevenin's theorem.5.18. A Wheatstone bridg e ha s th e followin g resistances :

R1 = R} = 12 0 ohms, R 2 = R4 = 600 ohms, R m = 500 ohms,

Rs = 7 ohms

If th e bridg e i s initiall y balance d an d V = 1 0 volts, determin e A£ m0 fo r th efollowing conditions :

(a) Resistance s R ^ an d R 3 increas e b y 1 percent.(b) Resistance s R l an d R 3 increas e by 1 5 percent.

5.19. I f /?] , R2, .R 3, and R 4 i n Proble m 5.1 8 each increase by 1 5 percent, determin eA£m0-

5.20. Usin g the value s given in Problem 5.1 8 for R lt R 2, K 3, K 4, and V , let R l hav ethe followin g percentag e changes : 0.5 , 1.0 , 2.0, 5.0, 10.0 , an d 15.0 .

(a) Fo r R s = 0 and R m = oo , plot AE m0 vs . the percentage change in Rj .(b) Fo r Rs = 0 and Rm = 75 0 ohms, plot E m0 vs . the percentage change in R l.

5.21. A n aluminum cantilever beam, shown in Fig. 5.38, has four strain gages bonde dto it . Gages a and b are on the top of the beam, wit h gage a being a longitudinalgage an d gag e b bein g a transvers e gage . Gag e c (longitudinal ) and gag e d(transverse) are directl y underneath. The followin g dat a ar e given:

R9 = 12 0 ohms, G F = 2.08 , E = 1 0 x 10 6 psi, v = 0.33 ,

Rs = 0 , K = 1 0 volts

(a) Arrang e the gages into a full bridg e in order to get the maximum reading.(b) Whe n the en d o f the bea m i s deflected 0.225 in, determine A£m0 i f Rm = oo;

if R m = 300 0 ohms; i f Rm = 450 ohms.(c) I s i t worthwhile considering th e change i n RBO o r t o comput e n l

5.22. A round, hollo w shaf t o f steel ha s fou r 120-oh m gage s bonde d t o it . The gage sare arrange d i n a ful l bridg e i n orde r t o functio n a s a torqu e mete r (se e Fig .5.36). The shaf t ha s a n oute r diameter of 1.500 in, an inner diameter o f 1.125 in,and i s subjecte d t o 700 0 in-lb o f torque . I f V = 1 0 volts, G F = 2.07 , an d th elead-line resistanc e i s 2.0 ohms, determin e A£ 0.

5.23. A weight , W , i s a t res t a s show n i n Fig . 5.39 . When th e weigh t i s release d i tfalls ont o th e stop , wher e a latc h i s engaged tha t keep s i t fro m rebounding .

R = R2 + 120 ohms, R3 + R4 + 500 ohms, Rm = 750 ohms,

Rs = 0


FIG. 5.38.

FIG. 5.39.

Strain gages a and b are bonded longitudinall y to the vertical bar and wired intoopposite arm s o f a ful l bridge , wit h the gage s i n adjacen t arm s being dummygages. Eac h lea d o f the bridge has a resistance o f RL =1. 5 ohms . Assuming thestress i s uniformly distribute d throughou t th e length of the vertica l bar, deter-mine W i f the maximu m strai n recorde d b y th e bridg e i s 1520uin/in , d =0.505 in, v = 0.3 , E = 30 x 10 6 psi, R g = 120 ohms, h = 1 8 in, and L = 3 0 in.

5.24. Th e dummy gages in Problem 5.2 3 are replaced wit h gages bonde d transvers eto gage s a an d b . The lea d wire s ar e als o extende d s o tha t eac h lea d ha s aresistance of RL = 2.5 ohms. Using the value of f^from Proble m 5.23 , determinethe maximum indicated strain that the meter would record for a repeated test .

5.25. Fou r 120-oh m gage s ar e bonde d t o a machin e elemen t an d individuall yconnected t o a strai n indicato r throug h a switchin g an d balancin g unit ,using th e three-wir e quarter-bridg e circuit show n i n Fig . 5.21 . Th e followingdata ar e given:


Gage No. G y R L K, /.tin/in1 1.9 5 0 195 02 2.07 5 0 124 53 2.0 0 0 -50 04 2.1 5 4 97 5

A gag e facto r o f 2. 0 i s se t o n th e strai n indicator . Determine th e actua lstrain a t eac h gage .

REFERENCES

1. "Portabl e Digital Strain Indicator P-350A," Bulleti n 130-A , Measurements Group,Inc., P.O . Bo x 27777 , Raleigh , NC 27611 , 1980. (No w ou t o f print.)

2. "Portabl e Strain Indicato r P-3500, " Bulletin 245, Measurements Group, Inc. , P.O .Box 27777 , Raleigh, N C 27611 , 1983.

3. Handbook o f Experimental Stress Analysis, edite d by M. Hetenyi, New York , Wiley,1950, pp . 191-193 .

4. Stein , Pete r K. , "Strain-Gage-Based Computers, " Strain Gage Readings, Vol . IV ,No. 4, Oct.-Nov. 1961, pp. 17-50 . Also, Chap. 26 in Th e Strain Gage Encyclopaedia,Vol. I I o f Measurement Engineering, b y Pete r K . Stein , 1962 , 2 d edition , Stei nEngineering Services , Inc. , Phoenix , A Z 85018-4646 . (No w ou t o f print. ) (1960 ,1st edition. )

5. "Error s Du e to Wheatstone Bridg e Nonlinearity," TN-507, Measurement s Group ,Inc., P.O . Bo x 27777 , Raleigh, NC 27611 , 1982.

6. Herber t W . Jackson an d Presto n A . White, III , Introduction t o Electric Circuits,7e, (j j 1989 , pp . 213 , 236 . Adapte d b y permissio n o f Prentice-Hall , Englewoo dCliffs, Ne w Jersey .

7. "Studen t Manua l fo r Strai n Gag e Technology, " Bulleti n 309B , Measurement sGroup, Inc. , P.O . Bo x 27777 , Raleigh, NC 27611 , 1983 , p . 24.

8. Handbook o n Experimental Mechanics, edite d b y A . S . Kobayashi , Englewoo dCliffs, Prentice-Hall , 1987 , pp. 102-104 .

9. "Shun t Calibratio n o f Strai n Gag e Instrumentation, " TN-514 , Measurement sGroup, Inc. , P.O . Bo x 27777 , Raleigh , NC 27611 , 1988 .

6SENSITIVITY VARIATION

6.1. Introduction

Reasons for varying strain sensitivity

Why shoul d on e desire t o vary the sensitivit y of strain gages , or the circuitsof whic h they for m a part ? I n general , thi s requirement stem s fro m a nee dto pu t th e indication s fro m tw o o r mor e strai n gage s o n a commo n basis ,or in the correct relativ e proportions. There are numerous special situationswhich ma y sho w up . However , a fe w of the mor e commo n case s requirin gsensitivity variatio n are liste d as follow s (1) :

1. T o allo w fo r difference s i n gag e facto r amon g individua l gage s whenthe readou t fo r al l gages i s to b e made directl y in term s of strain on asingle scale . Fo r example , th e gag e facto r dia l adjustmen t o n strai nindicators.

2. T o combine the indications from severa l strain gages in different relativeproportions. Fo r example , th e direc t an d automati c computatio n ofsome quantit y whos e indication depend s upo n a combinatio n o f two,or more , strai n indication s i n specified relative proportions .

3. T o facilitat e th e us e of an instrumen t whic h has a limite d inpu t rang ewith a strai n gag e tha t develop s a n outpu t whic h i s large r tha n th emaximum tha t ca n b e accepte d b y th e instrument . Fo r example , th euse o f a standar d strai n indicato r designe d fo r metalli c gage s wit h asemiconductor gag e tha t i s subjected to a reasonably large strain . Th esame sor t o f situation ma y als o prevai l when a meta l gag e i s used t omeasure post-yield strains o f several percent.

4. T o adjus t th e calibratio n facto r o f a transduce r t o som e convenien tround number . Fo r example , t o adjus t t o a readou t o f 100 0 o n th eindicator scal e fo r 100 0 unit s o f th e quantit y bein g measured , a scontrasted wit h a n indicato r readin g o f 98 1 pe r 100 0 unit s bein gmeasured.

5. Fo r automaticall y correctin g a n indicatio n fo r som e uncontrolle dvariable whic h ma y chang e b y unknow n amounts . Fo r example , th ecompensation o f a load cel l or torque meter indication for the influenc eof temperature changes on the modulus of elasticity of the load-carryingmember.


6. Fo r producin g a direc t readou t o f som e quantit y which i s indicate dby th e produc t o f tw o independen t quantities . Fo r example , th emeasurement o f th e instantaneou s valu e o f powe r bein g trans -mitted b y a circula r shaft . Thi s can b e accomplishe d b y usin g a strai ngage bridge to sense the torque and energizing it with a variable appliedvoltage (variabl e sensitivity ) tha t i s proportiona l t o th e spee d o frotation.

Indicated strain vs. actual strain

One wil l recal l tha t strai n i s sense d throug h a chang e i n gag e resistanc eaccording t o th e following relationship:

Provided there are no inactive resistances in series (or parallel) with the gage,the readou t instrumen t will b e able t o indicat e th e correc t valu e of strain i naccordance wit h Eq . (6.1) . However, if there are inactiv e resistances (relativeto strain ) in series and (or ) parallel with th e gage , these will , to some extent,mask th e observatio n th e instrumen t i s makin g s o tha t th e indicate dstrain being read out i s only a fraction o f that actually prevailing at the gage .The correspondin g relatio n fo r th e indicate d strain i s given by

The reason for the desensitization, or reduction in indicated strain, whenseries an d paralle l resistance s ar e connecte d t o th e gage , i s tha t thes eadditional resistance s contribut e nothing to th e change in resistance in spiteof the fac t tha t the y have a n influenc e o n th e tota l overal l valu e as see n b ythe indicatin g device . Thi s desensitizatio n becam e apparen t whe n lead-lin eresistance wa s considere d i n Sectio n 5.5 .

where e , = th e indicate d strai n

Q, = th e desensitization factor, whose numerical value is less than 1

From Eq . (6.2) ,

SENSITIVITY VARIATIO N 20 7

Kinds of desensitization

Strain gag e desensitizatio n du e t o th e effect s o f resistance s i n serie s an din paralle l wit h th e gag e ca n b e considere d fro m tw o point s o f view ,depending upo n whether the effec t represent s an inconvenience that must beovercome or a n advantag e tha t can b e employed for some specifi c purpose .One may therefore look upo n desensitization as falling into one or the otherof th e tw o categorie s tha t follow .

1. Parasitic desensitization. Thi s i s cause d b y suc h thing s a s lead-wir eresistance an d paralle l resistance s whic h are brough t int o th e circuitfor trimmin g and balancin g purposes . Thi s i s something tha t mus t b eaccepted. Usuall y (althoug h no t always ) th e parasiti c desensitizatio nproduces a small deviation fro m th e theoretical calibratio n factor . Th eimportant thin g is to appreciat e tha t thi s condition prevail s and t o beable to make a reasonably good estimate of the magnitude of its effect .

2. Planned desensitization. This involve s the understandin g o f the factor swhich contribute to desensitization and th e deliberate manipulation ofthem in order to produce certain desired results, such as those indicate din th e introduction in th e reason s fo r varyin g strain sensitivity.

Other approaches to sensitivity variation

Since th e resistanc e chang e o f a strai n gag e i s actuall y determine d b y th ecorresponding effec t o n voltag e o r current , w e ma y als o approac h th eproblem o f sensitivity variation by control of the applied voltage, or the gagecurrent.

One may consider the use of a resistance network connected t o the gageas a primar y mean s o f achievin g sensitivit y contro l sinc e thi s produce s adirect effec t upo n th e indicate d relatio n betwee n strai n an d uni t change i ngage resistance , independentl y o f gage curren t o r applie d voltage .

On th e othe r hand , variatio n o f sensitivity throug h contro l o f applie dvoltage, o r gag e current , means tha t w e have t o expres s th e indicatio n ofsensitivity i n term s o f voltag e o r curren t change s pe r uni t o f strain .Furthermore, fo r thos e systems , suc h a s nul l balanc e an d som e o f th ereference bridge arrangements, which produce an indication tha t is indepen-dent o f variations in applied voltag e or gage current, this method o f varyingthe sensitivit y i s inapplicable .

6.2. Analysis of single gage desensitization (1, 2)

Resistance in series

Figure 6.1 shows a strain gage , Rg, desensitize d b y placing a resistor, R s, inseries wit h it . The initia l tota l resistance i s


FIG. 6.1. Resistanc e in serie s wit h gage.

After a chang e i n gag e resistance , AR S, w e have

Dividing al l terms b y R , an d rearranging , th e resul t is

or

Since R t = R + Rs, thi s reduce s t o

If th e numerator an d denominato r o f the right-han d sid e are divide d by R g,then

Letting R S/R<, = s, then

In this case, the nonlinearity factor, n, is zero, and th e desensitization factor is

SENSITIVITY VARIATION

FIG. 6.2. Desensitizatio n o f a singl e gage with series resistance .

Figure 6.2 shows the value of the desensitization factor, Q, as a functionof th e ratio , s , of series resistance t o gag e resistance .

Resistance in parallel

Figure 6. 3 shows a resistor , Rp, i n parallel wit h the strain gage , Rg, i n orde rto desensitiz e the strai n gage . Initially , the tota l resistanc e i s

After straining ,

Dividing both side s b y R, result s in

209

FIG. 6.3. Resistanc e in paralle l wit h gage.

From this ,

Expanding th e right-han d sid e o f Eq. (6.13) result s i n

or

For simplicit y i n writing , le t p = R p/Rg. Usin g this , Eq . (6.14 ) i srewritten a s

Dividing the numerato r an d denominato r o f the right-han d sid e by (1 + p )results i n

From Eq . (6.15),

THE BONDED ELECTRICAL RESISTANCE STRAIN GAGE210

SENSITIVITY VARIATIO N 211

FIG. 6.4. Desensitizatio n o f a single gag e wit h paralle l resistance .

and

Also,

The valu e of Q, the desensitizatio n facto r exclusiv e o f nonlinearities, isshown as a function o f p in Fig. 6.4. Here it is seen that for values of p greaterthan 100 , th e desensitization wil l be less than 1 percent.

Equation (6.18 ) indicate s that , a s long as the paralle l resistance , R p, i sgreater tha n th e gag e resistance, th e nonlinearit y facto r wil l b e les s tha n

Combination of series and parallel resistances

An examination o f the work covering resistances in series and resistance s inparallel reveals that R, is greater than R g when series resistance is employed,and R , i s less than R g whe n parallel resistanc e is used.


FIG. 6.5. Tw o arrangement s of connecting series an d paralle l resistances to a gage .

In th e even t that i t i s desired t o desensitiz e a strai n gag e i n on e ar m o fa bridg e whe n th e resistanc e i n eac h o f the othe r arm s correspond s t o R g,the tw o previou s method s o f desensitizatio n ar e unsuitabl e becaus e th ebridge cannot b e initially balanced. W e wil l no w investigat e how serie s an dparallel resistances may be combined so that R , = R g, whic h conditio n willpermit initia l balanc e o f th e bridge . Figur e 6. 5 illustrate s tw o alternativ emethods fo r connectin g serie s an d paralle l resistance s t o a strai n gage . I nthe followin g analysis , the arrangemen t i n Fig . 6.5 a wil l b e analyzed .

The initia l resistance, R, , a s seen b y th e readou t instrument , is

After straining , th e gag e resistanc e change s t o R s + ARg, an d s o Eq . (6.19)becomes

If both side s o f Eq. (6.20) are divide d b y R, , th e resul t i s

Divide th e numerato r an d denominato r o f eac h brackete d ter m o n th eright-hand sid e b y R g, the n


Since s = R sjRg and p = Rp/Rg, Eq. (6.21) can be rewritten as

For initia l bridge balance , however , R, = R g, s o Eq. (6.19) become s

In term s o f ratios ,

If bot h side s o f Eq. (6.24 ) ar e divide d b y R p, the n

Equation (6.25 ) show s tha t th e las t brackete d ter m o n th e right-han dside o f Eq. (6.22 ) is equal t o p , and s o Eq . (6.22 ) is rewritten a s

The expression fo r the uni t change i n resistance a s seen by the readou tinstrument ca n now b e written as

Expanding Eq . (6.27) produce s


From Eq . (6.25), (1 + s) p = 1 + s + p , and s o

Equation (6.25 ) also shows tha t

and

Substituting th e valu e of 1 + 5 + p given by Eq . (6.30 ) int o Eq . (6.28) ,

This expressio n ca n als o b e writte n as

Further rearrangemen t give s

Equation (6.31 ) tell s us tha t th e desensitizatio n facto r is


Neglecting nonlinearities ,

The nonlinearit y factor , n, is

Together with the knowledge that s and p must always be positive, Eqs.(6.29), (6.33) , and (6.34 ) provide u s with some interesting facts .

1. Fro m Eq . (6.29 ) one see s that p must alway s be large r than 1 , becauses becomes large r a s p becomes smalle r an d woul d have to b e infinite ifp becam e unity . Also, i f p were les s tha n unity , s woul d b e negative ,which i s impossible .

2. Equation s (6.32 ) an d (6.33 ) indicat e tha t th e desensitizatio n facto rapproaches zero as p approaches unity . This is to be expected, of course,because s is approaching infinit y a s p approaches 1 , and consequentl yany change s i n gage resistance have less overal l influence .

3. Th e nonlinearti y factor, n, approaches zer o as p approaches 1 , and als oas p becomes ver y large. B y differentiation w e fin d tha t th e maximumvalue occurs fo r p = 2, so tha t

So far th e desensitizatio n facto r has bee n determine d fo r a give n valueof p, or s . There are , however , other situation s in whic h it wil l b e necessaryto determin e th e value s of p and s that wil l b e required to produc e a givendesensitization. Fo r thi s purpos e w e wil l nee d t o fin d p an d s i n term s ofQ. From Eq . (6.33) ,

From thi s


From Eq . (6.29),

or

Substituting th e valu e o f p give n b y Eq . (6.35 ) int o Eq . (6.34) , th enonlinearity factor , n , can b e writte n as

From Eq. (6.37) we see that n = 0 when Q equals 0 or 1 , and by differentiatio nwe fin d tha t th e maximu m valu e occurs whe n Q = 0.25 ( p = 2) , which, aspreviously, produces th e maximum nonlinearity represented a s

For rapi d evaluation of the ratio s p and s that ar e require d to produc ea give n desensitization, Fig . 6.6 , in which the value s are plotte d i n term s of2, wil l be found helpful. Sinc e Fig. 6.6 neglects the effec t o f nonlinearity, thevalue o f >/2(l — v2) na s bee n plotte d a s a functio n o f Q, in Fig . 6.7 . Inmost cases , however, it wil l be sufficien t i f we know that the maximu m valueis 0.25.

A note on temperature effects

The derivation s o f thi s sectio n al l assum e tha t th e temperatur e remain sconstant. However , i f ther e i s a temperatur e change , a fals e indicatio n o fstrain wil l b e produce d unles s al l o f th e followin g ar e independen t o ftemperature changes: (1 ) the gag e resistance , (2) the resistanc e o f the leads ,(3) the auxiliary series and parallel resistances. Theoretically, these condition scan b e fulfille d b y usin g a self-temperature-compensate d strai n gag e wit hleads an d auxiliar y resistance s havin g a zer o temperatur e coefficien t o fresistance. Obtainin g a suitabl e strain gage shoul d presen t n o problem , bu tacquiring lead wire (including soldered joints), and auxiliary resistances, withzero respons e t o temperatur e ma y presen t a difficul t problem . O n thi saccount i t wil l b e preferable t o tak e another approac h usin g th e half-bridgearrangement a s discusse d i n the followin g section.

FIG. 6.6. Serie s an d paralle l resistance s fo r singl e gage desensitization . (Fro m ref . 2.)


FIG. 6.7. ,/£>( ! - JQ } a s a function o f Q.

6.3. Analysis of half-bridge desensitization

In thi s section, methods of eliminating the effect s o f temperature changes byemploying activ e an d dumm y gages i n adjacen t arm s o f a hal f bridge wil lbe discussed.

Duplicating the system for a single gage

The mos t direc t approac h i s to se t u p duplicat e arrangement s i n th e tw oadjacent arm s o f th e hal f bridg e an d t o mak e sur e tha t correspondin gcomponents are subjecte d to exactl y the sam e temperatur e conditions.

When th e temperature variation at th e gag e is greater tha n tha t a t th ereadout instrument , it will be best to locate the series and parallel resistancesnear th e instrument and t o run th e leads out t o the gages, making sure thatthe lead s from th e paralle l resistance s around th e gages ar e equa l i n lengthand tie d int o th e syste m a t equivalen t locations i n bot h arm s o f th e hal fbridge.

In th e even t tha t th e hal f bridg e i s to b e connecte d acros s th e powe rsupply, i t wil l no t b e necessar y t o us e bot h paralle l an d serie s resistanc ebecause th e rati o o f the tota l resistanc e in eac h ar m ca n b e maintaine d a tunity for either series or paralle l resistance connected t o th e gage . If the hal fbridge is connected acros s the bridge output, depending upon the resistancesin the othe r tw o arms , the rati o o f the resistance s o f the arm s in series wit hthe power supply may , or ma y not , be unity .

When th e hal f bridge i s connected acros s th e powe r supply , althoughthe adjustment may be a little more difficult, i t will be preferable to desensitizewith serie s resistanc e alon e becaus e th e outpu t wil l b e linea r an d th ecomplication o f the extr a lead s fro m th e paralle l resistance s can b e elimi -nated. This means that th e standard four-lea d active-dummy system can beemployed wit h a pai r o f equal serie s resistor s in eac h ar m adjacen t t o th ereadout instrument , as lon g a s th e tota l resistanc e i n eac h ar m doe s no texceed th e capabilit y o f the instrument .

The concep t o f desensitizatio n using serie s resistanc e alon e i n a hal fbridge become s eve n mor e attractiv e whe n on e wishe s t o us e a singl e


FIG. 6.8. Desensitizatio n with temperatur e compensation.

self-temperature-compensated strai n gage in the active arm, because on e canthen us e the three-wir e system with al l the resistanc e (exclusiv e of leads) i nthe inactive arm in, or at, the readout instrument , and stil l maintain freedomfrom th e influenc e o f temperature.

Figure 6. 8 shows a schematic layout for one arrangement o f half-bridgedesensitization wit h temperature compensation .

An alternate and superior method of desensitization

An alternate method o f desensitization, whic h uses the half bridge to provid etemperature compensation , i s shown schematically in Fig. 6.9 . This arrange -ment employ s a common paralle l resisto r i n both arms .

Some o f the advantage s o f this system ar e a s follows:

1. Th e tota l effectiv e resistanc e i n eac h o f the tw o desensitize d arm s ca nbe mad e equa l t o th e gag e resistance , R g, i f desired. Thi s i s merely aconvenience. Th e onl y requiremen t i s that , initially , th e effectiv eresistance should be the sam e i n both arms.

FIG. 6.9. Alternat e method o f half-bridge desensitization.


FIG. 6.10. Physica l connection s o f gages t o indicator . Note: O n som e indicator s th e relativepositions o f the terminal s fo r th e activ e and compensatin g gage s ar e reverse d wit h respec t t othis diagram .

2. On e les s resisto r i s required tha n fo r th e previou s method .3. Th e equations for computing th e serie s and paralle l resistances , R s an d

Rp, are simpler .4. Th e networ k o f resistances require d i s simple and eas y t o instal l at th e

strain indicator , a s show n i n Fig . 6.10 .5. Al l the advantages an d simplicit y of the standard active-dumm y system

are retained . No additiona l lea d wire s are required .

Limitations

In th e analysi s tha t follows , tw o limitation s wil l becom e evident ; however,these shoul d caus e n o difficult y i f one i s cognizan t o f them . Fo r emphasi sthe limitation s wil l b e summarize d here .

1. Th e inpu t impedanc e o f the instrument connected t o th e bridge outpu tmust b e extremely high.

2. Th e pai r o f gage s (hal f bridge ) containin g th e desensitizin g networ kmust b e connected acros s th e bridg e powe r supply .

Analysis

Let u s refe r t o Fig . 6.9 , whic h show s a half-bridg e diagra m wit h a pai r o flike gage s an d a desensitizin g networ k consistin g o f tw o serie s resistance sand a singl e paralle l resistanc e commo n t o bot h gages .

The firs t ste p i n analyzin g th e networ k wil l b e t o determin e th eequivalent o f the combine d gag e an d paralle l resistance s whic h shoul d b econsidered i n each o f bridge arm s 1 and 2 . This ca n b e don e b y mean s o f aDelta-Wye transformation , a s show n i n Chapte r 5 .

Figure 6.1 1 show s th e Delt a networ k forme d b y th e strai n gages , R gl


FIG. 6.11. Delta-wy e transformation .

and Rg2, an d the parallel resistor , R p. Als o in the same figure is the equivalentWye network.

From thi s transformation ,

where Rel = equivalent resistanc e i n arm 1

Re2 = equivalent resistanc e i n arm 2

Re0 = equivalen t resistanc e i n the output circui t

The equivalen t total resistances , R ^ an d R 2, i n arm s 1 and 2 o f thebridge ma y no w b e expressed a s

Examination o f Fig. 6.12 indicates that the transformation ha s facilitatedsetting up relatively simple expressions fo r the equivalent resistance s i n arm s1 and 2 of the bridge . However , i t also indicate s tha t ther e i s resistive effect ,


FIG. 6.12. Electrica l equivalen t o f Fig . 6.9.

represented b y R e0, i n the outpu t circuit . This latte r influence , Re0, mus t b eallowed fo r i n some manner .

One wa y o f allowin g fo r R e0 i s t o mak e i t ineffectiv e b y usin g a ver yhigh-impedance readou t devic e s o tha t essentiall y n o curren t flows acrossthe outpu t fro m th e bridge . Thi s i s th e reaso n fo r statemen t (1 ) unde rLimitations. Ite m (2) of the limitations can b e explained b y considering wha twould happe n i f the hal f bridg e containin g th e gage s wer e no t connecte dacross th e powe r supply . In thi s case , wit h th e hal f bridg e containin g th egages across th e output , th e effec t o f R e0 wil l b e th e sam e a s tha t o f an yother resistanc e i n serie s wit h the bridg e acros s th e powe r supply . Tha t is ,a desensitizatio n o f th e entir e bridg e wil l tak e place . Thus , t o avoi d thi soverall desensitizing effect, th e hal f bridg e wit h th e gage s mus t b e connecte dacross the powe r supply .

The derivatio n o f the expressio n fo r th e desensitizatio n facto r wil l no wbe considered . Not e tha t eve n thoug h tw o identica l strain gage s (bot h o fresistance R e) ar e use d i n th e hal f bridge , thei r resistance s hav e bee ndesignated separately by the symbols, R9, an d R s2, t o indicate their respectivelocations i n th e bridge . Thi s i s necessar y becaus e th e tw o gage s wil l hav esomewhat differen t function s i f one i s t o d o th e strai n measurin g an d th eother t o provid e temperatur e compensation .

Since the series and paralle l resistances , R s and R p, ca n be so chosen a sto provid e fo r a wide range o f resistances in the bridg e arms , le t us conside rthat thei r value s will b e s o chosen that , numerically,

Thus, fro m Eqs . (6.41) , (6.42) , an d (6.43) , Eqs . (6.44 ) an d (6.45 ) wil l resul t


when initia l condition s ar e used :

and

Let u s no w se e what happen s whe n th e gag e i n ar m 1 is strained an dchanges it s resistance t o R gl + &Rgl. Since R gl appear s in the expression sfor bot h R i an d R 2, thi s chang e wil l influenc e bot h arm s 1 and 2 o f th ebridge and, consequently , w e will have to conside r change s i n both o f themsimultaneously. Hence , afte r th e change , Eqs . (6.44 ) an d (6.45 ) wil l becom e

From th e relation s expresse d i n Eq . (6.43) , Eqs . (6.48 ) an d (6.49 ) ca n b esimplified t o

and

and

The uni t change s i n resistanc e i n th e arm s o f the bridg e ca n no w b efound by dividing Eqs. (6.46) and (6.47) by R1 an d R2, respectively, so that

and


Since th e bridg e outpu t i s proportiona l t o th e algebrai c differenc ebetween th e uni t change s i n resistanc e o f adjacen t arms , on e ca n obtai n ameasure o f thi s b y subtractin g Eq . (6.51 ) from Eq . (6.50) . Thi s mean s tha twhat th e instrumen t indicate s is

Equation (6.52 ) simplifie s t o

From Eq . (6.53 ) i t ca n b e see n tha t th e desensitizatio n factor , Q t, i sgiven b y

If th e numerato r an d denominato r o n th e right-han d sid e o f Eq. (6.54 ) ar edivided by Rg, and the ratio Rp/Rg i s expressed by the single symbol p, then

This expressio n ca n b e rewritte n as

If eithe r Eq . (6.41 ) or Eq . (6.42 ) i s use d wit h th e value s given i n Eq . (6.43) ,then w e obtain

From Eq . (6.55),


and th e nonlinearit y factor is given by

which wil l always be les s than (0.5)( R g/Rg).Sometimes i t wil l b e necessar y t o determin e th e size s o f th e serie s

and paralle l resistance s whic h wil l b e require d t o produc e a give n desen -sitization. Thi s can b e done by solvin g Eq. (6.56 ) fo r p an d s in term s o f Q.This result s in

and

For convenience , the value s o f p and s have bee n plotte d agains t Q inFig. 6.13 . From Eq . (6.57 ) th e valu e of the nonlinearit y factor , n , has bee ndetermined i n term s o f Q a s

Example 6.1. A cantileve r bea m ha s fou r longitudina l strai n gage s (R g =120 ohms) bonded to it that ar e arranged int o a ful l bridge . When a 5-lb weightis placed o n th e beam , th e strai n indicato r read s 214 0 uin/in.

(a) Us e a series resistance in arms Rt an d R2, as shown in Fig. 6.1 , to desensitizethe syste m so that th e readin g i s reduced t o 150 0 uin/in.

(b) Us e a paralle l resistanc e i n arm s R { an d R 2, a s show n i n Fig . 6.3 , t odesensitize the syste m so tha t th e readin g i s reduced t o 150 0 uin/in.

Solution, (a ) For eac h gage ,

The tw o arm s no t desensitize d wil l rea d a tota l o f 107 0 (iin/in. The othe r tw oarms must read 150 0 — 1070 = 43 0 uin/in, or 215 |iin/in per arm. Equation (6.8 )is no w used , bu t i f both side s ar e divide d b y G F, the n A.R,/(G FRt) = £> , thedesired indicate d strain , an d A.R g/(GFRs) = E, the actua l strain . I n term s o fstrain, then , Eq . (6.8 ) is


FIG. 6.13. Ratio s p an d s as function s o f Q.

or

From this , s — 1.488, and s o

Rs = 1.488K 9 = 1.488(120 ) = 178.5 6 ohms

Therefore, us e a serie s resistor o f 178.56 ohms i n arms R t an d R 2.(b) Refe r t o Fig . 6. 3 for th e paralle l arrangement . Again , tw o arm s wil l

read 53 5 uin/in, whil e th e tw o desensitize d arm s wil l eac h rea d 21 5 uin/in.Considering th e nonlinearit y portion t o b e unity , Eq . (6.15 ) gives

or


From this , p = 0.672 , an d s o

Rp = 0.612R S = 0.672(120) = 80.6 4 ohm s

Therefore, us e a paralle l resisto r o f 80.64 ohms i n arms R t an d R 2.

Example 6.2. A torque meter (four active arms with R s = 12 0 ohms each) read s1420 uin/in when subjected t o a torsional momen t o f 1200 in-lb. Desensitize on earm, usin g parallel-series resistances i n order to hav e the strai n indicato r rea d1200 uin/in .

Solution. The mete r readin g mus t be reduce d by 142 0 — 1200 = 220 uin/in.Since eac h ar m read s 1420/ 4 = 35 5 uin/in, the n th e ar m tha t i s desensitize dmust rea d 355-220 = 13 5 uin/in. Thus , R,/(G f R,) = 135 uin/in an d &R J(GFRg) = 35 5 uin/in. Again , considering th e nonlinearit y portio n t o b e unity ,Eq. (6.31 ) gives

or

From this , p = 2.609, an d s o

Rp = 2.609Rg = 2.609(120) = 313.0 8 ohm s

From Eq . (6.29) ,

Thus,

Rs = 0.622K, = 0.622(120 ) = 74.6 4 ohm s

Use a serie s resisto r o f 74.64 ohms an d a paralle l resisto r o f 313.08 ohms .

6.4. Analysis of full-bridge sensitivity variation

There ar e certain situations in which it is desirable t o var y the sensitivit y ofan entir e bridge . Fo r example :

1. T o compensat e th e outpu t fro m loa d cell s fo r change s i n modulu s ofelasticity of the load-carrying element due to variations in temperature.

2. T o permi t a standard strai n indicator, whic h has been designed fo r usewith metallic gages, to be employed with a semiconductor bridge whoseoutput i s in excess of the rang e o f the instrument.


3. T o perform some computation automaticall y in order to obtain a direc treadout o f som e desire d quantity , as i n th e cas e o f th e torqu e mete rthat i s mad e t o indicat e powe r transmitte d b y makin g th e excitatio nvoltage proportiona l t o th e spee d o f rotation .

Method of approach

Since th e bridg e outpu t i s directly proportional t o th e applie d voltage , th esensitivity ca n b e varie d b y mean s o f voltag e control . Frequentl y thi s i saccomplished b y usin g a powe r suppl y wit h a fixed voltage tha t i s greate rthan tha t neede d t o energiz e th e bridg e an d the n reducin g thi s t o th enecessary leve l by including a fixed, or variable , resistance i n serie s wit h th ebridge, accordin g t o th e particula r requirement s a t hand . Th e arrangemen tis show n i n Fig . 6.14.

Limitation

Since thi s metho d o f sensitivit y variatio n depend s upo n controllin g th evoltage actually applied t o the bridge, it is unsuited for use with a null balancesystem wher e indicatio n i s independen t o f th e magnitud e o f th e applie dvoltage. Likewise , i t wil l no t wor k fo r certai n type s o f referenc e bridg einstruments whic h hav e als o bee n designe d t o produc e reading s tha t ar eindependent o f supply voltage , or fluctuation s therein .

Derivation of equations

The voltag e acros s th e bridg e ca n b e expressed a s

FIG. 6.14. Bridg e wit h resistanc e i n series .


where V P = powe r suppl y voltag e

Rs = resistance i n series with the bridg e

RBI = input resistanc e o f the bridge , excluding R s

The bridge output (assuming initial balance and neglecting nonlinearity)is expressed a s

Note that, in Eq . (6.62), infinite impedance i s assumed a t th e bridge output .Therefore, wit h respec t t o th e voltag e o f the powe r supply ,

For a constant-voltage powe r supply , this means tha t

In othe r words , the desensitizatio n facto r fo r the entir e bridg e is given by

This valu e of Q assumes th e bridg e resistanc e remain s constant .When th e bridg e resistanc e remain s constant , a s i n th e cas e o f certain

transducers, such as torque meters, in which the resistances in adjacent bridg earms chang e b y equal amount s bu t o f opposite sign , o r whe n change s ar eproportionately ver y small , as i s usually th e cas e wit h metalli c strain gages ,Eq. (6.65 ) is directly applicable. Als o observe tha t Eq . (6.65 ) i s o f the sam eform a s Eq . (6.9 ) whe n the symbo l s is used t o represen t th e rati o R S/RBI.On thi s account, Fig . 6. 2 may be used to determine no t onl y the desensitiza -tion factor for a single gage with resistance in series, but also the correspond -ing effec t fo r a n entir e bridg e wit h a resistanc e i n series .


Effect of changes in bridge resistance

When there is an appreciabl e chang e i n the bridg e resistance, Eq . (6.65) wil lhave to be modified by considering the actual bridge resistance, RBI + AR BI,at any particular instant. In this case, Eq. (6.65) can be written in the modifiedform

Except fo r th e factor , 1/[1 + AR B,/RBI] i n th e denominato r o f Eq . (6.66) ,Eqs. (6.65 ) an d (6.66 ) are alike .

The erro r i n Q can no w b e examined i f the chang e i n bridg e resistanc eis neglected. Examination o f Eq. (6.66) indicates that th e erro r produce d b yneglecting th e chang e i n bridg e resistanc e wil l b e smal l fo r smal l ratio s o fboth R S/RBI an d AR B,/RBI.

To ge t some ide a o f the numerica l value of the error , we can investigatea particular situation for approximate values . The following values are given:

R — 40 percent fo r a single active arm

Neglecting th e chang e i n bridg e resistance ,

The change i n bridge resistance can now be included. If the single activearm change s b y 4 0 percent , then , fo r fou r equa l arms , AR BI wil l b e abou t10 percent. This can b e verified b y assuming 120-oh m gage s and computin gthe bridg e resistanc e wit h R m = oo . In thi s case,

Thus, for the conditions given, the error in Q caused by neglecting the chang ein bridg e resistanc e wil l b e less than 2 percent .

For smalle r ratio s o f RJRB, an d AR B,/RBI, th e variation s will be even


less and, consequently , fo r a grea t man y cases, we are justified i n neglectingthe effec t o f change s i n tota l bridg e resistance . Nevertheless , i t i s alwaysdesirable t o chec k t o b e sure tha t th e probabl e erro r fro m thi s sourc e wil lfall withi n tolerable limits .

Discussion

Use o f a bridge with unequal arms. Th e precedin g exampl e suggest s tha t i na bridg e containin g a singl e strain gag e (i f one ha s th e choice) , ther e ma ybe som e advantage s t o b e gaine d b y havin g tw o o f the arm s o f somewhathigher resistance than th e strain gage . In addition t o improving the linearityand increasin g th e outpu t pe r uni t strain , thi s procedur e wil l enabl e u s t oreduce the rati o AR B,/RBI, eve n for large values of resistance chang e i n th eone activ e arm, an d thereb y cu t dow n o n th e variatio n Q with chang e i nbridge resistance .

Temperature effects. A note o f caution, especiall y i n respec t t o transducer sinvolving fou r activ e arm s containin g semiconducto r gages , wil l b e men -tioned wit h regard t o the total change i n bridge resistance . Even though thegages al l change b y exactly the sam e amoun t an d n o bridg e outpu t result sfrom this , nevertheless, as far as the total bridg e resistance i s concerned, thiseffect wil l be additive an d wil l have some influenc e o n th e valu e of Q. If th eratio RJR BI i s small, th e effec t ma y no t b e noticeable , bu t fo r larger ratio sof serie s to bridg e resistance , th e influenc e on Q should b e checked .

Increasing an d decreasing the sensitivity. Equation s (6.63 ) an d (6.64 ) sho wthat th e maximum output wil l occur when Rs = 0. For thos e applications inwhich on e may wis h to b e able t o increase , or decrease , th e sensitivity fromsome usua l valu e (such a s modulu s compensatio n o f load cells) , i t wil l b enecessary to design the system to provide for normal operation a t somewhatless than th e maximu m outpu t s o that i t wil l b e possible t o decreas e R s b ythe necessar y amoun t i n order t o achiev e the desire d increas e i n sensitivity .When R s ha s bee n reduce d t o zero , th e maximu m possibl e sensitivit y willhave bee n achieved .

Problems

6.1. A singl e strai n gag e record s 125 6 nin/in . I f G F = 2. 0 an d R g = 12 0 ohms,determine th e valu e o f the serie s resistor , R s, tha t i s required i n order t o hav eR,/R, = 0.002. If this gage is used in a quarter-bridge circui t an d R 2 = R$ =R4 = 12 0 ohms, can th e bridg e b e initiall y balanced ? I f one i s fre e t o choos eresistors R2, R 3, an d R4, ca n the bridge be initially balanced ?

6.2. Repea t Proble m 6. 1 using a paralle l resistor , R p.6.3. Repea t Proble m 6. 1 using serie s and paralle l resistor s so that R t = R g.6.4. Develo p equation s fo r th e series-paralle l arrangemen t show n i n Fig . 6.5b .

Follow th e metho d use d fo r Fig . 6.5a .


6.9.

6.5. I n Fig . 6.8 , a half-bridg e circuit i s show n wit h fou r lea d wires . Th e bridg e i sdesensitized wit h serie s resistor s alone . I f th e activ e gag e o f 12 0 ohms i ssubjected to a strain of s, determine the value of Rs needed t o make the indicatedstrain, e ;, equal t o 0.75c.

6.6. I n Proble m 6.5 , th e dumm y gag e become s a n activ e gage . I f B I = e , an de2 = — VB, will th e valu e o f R s chang e i f th e indicate d strain , e, , i s t o b e 7 5percent o f the tota l strain?

6.7. I n Exampl e 6.1 , desensitiz e arm s R , an d R 2 b y usin g a combination o f seriesand paralle l resistances .

6.8. Figur e 6.1 5 shows th e smal l assembl y machin e use d i n Proble m 5.8 . I n orde rto measur e th e loa d o n th e machine , add tw o gage s o n th e centerlin e A-A s othat gag e 3 is transverse to gag e 1 and gag e 4 i s transverse to gag e 2 . To ge tthe strains in Problem 5.8 , gages wer e used wit h GF = 2.0 8 and R s = 12 0 ohms.Using F = 34 0 800 Ib fo r th e strain s obtaine d i n Proble m 5.8 , perfor m th efollowing tasks :

(a) Arrang e the gage s int o a ful l bridg e i n orde r t o ge t the maximu m output .Sketch th e bridg e arrangement .

(b) Sinc e the sensitivity of the circuit can be altered by adjusting the gage factor,set the gag e facto r so tha t a n indicate d strai n o f 1 uin/in represent s a forceof 10 0 Ib. The gag e facto r settin g range s fro m 1.1 5 to 3.50 .

Using th e dat a give n i n Proble m 6.8 , desensitiz e th e circui t b y addin g serie sresistors in the two arms with gages 1 and 3 . A force of 200 Ib is to be representedby a n indicate d strai n o f 1 (lin/in.

6.10. Rewor k Proble m 6. 9 but us e paralle l resistors .

FIG. 6.15.


6.11. Tw o 120-oh m gage s are arranged i n a half-bridge circuit for tempera-ture compensation . Th e gag e in arm 1 is active while the gag e in ar m2 i s a dummy . Upon loading , th e activ e gage read s 269 5 uin/in. I t i sdesired to desensitize the system using the arrangement show n in Fig .6.9 so that th e indicated strai n is 2000 jxin/in . Determine R s an d R p.

REFERENCES

1. Murray , Willia m M . an d Pete r K . Stein , Strain Gage Techniques, Lecture s an dlaboratory exercise s presente d a t MIT , Cambridge , MA : Jul y 8-19 , 1963 , pp .249-286.

2. Stein , Pete r K. , "Individua l Strai n Gag e Desensitization, " Lette r t o th e Editor ,SESA Proceedings, Vol . XIV, No. 2 , 1957 , pp . 33-36 .

7LATERAL EFFECTS IN STRAIN GAGES

7.1. Significance of strain sensitivity and gage factor

Strain sensitivit y i s a genera l ter m relatin g uni t chang e i n resistanc e an dstrain i n an electrica l conductor accordin g t o th e followin g expression :

In symbols , this is

where S = strai n sensitivity

R = initial resistance

R = change in resistance

e = strai n

The numerical value of the strain sensitivity will depend upo n th e condition sunder whic h it ha s bee n determined .

For a straigh t conducto r o f unifor m cros s sectio n tha t i s subjected t osimple tension , o r compression , i n th e directio n o f it s axis , and unstraine dlaterally, th e strai n sensitivit y is a physica l propert y o f th e material . Th enumerical value will be represented by S,, which is determined by the relation

where L i s the initia l length.

The transverse effect i n strain gages (1-11). Whe n a conductor i s formed intoa gri d fo r a strai n gage , th e relationshi p betwee n unit chang e i n resistanc eof th e conducto r an d th e strai n become s muc h mor e complicated , an d th e

LATERAL EFFECT S I N STRAI N GAGE S 235

numerical valu e o f th e strai n sensitivit y i s influence d b y a variet y o fconditions. Th e mos t importan t ar e th e following:

(a) Th e strai n sensitivity o f the materia l o f the sensin g element .(b) Th e geometr y o f the grid .(c) Th e strai n field in which th e gag e i s used .(d) Th e directio n o f the strai n use d i n makin g the computatio n o f the

numerical valu e o f the strai n sensitivity .

In addition , there ar e als o a numbe r o f other smalle r effects .

Special cases of strain sensitivity. Sinc e th e strai n sensitivity is influenced byso man y factors , i n statin g a numerica l value , th e condition s unde r whichthis ha s bee n determine d shoul d als o b e known. Figur e 7. 1 shows a strai ngage mounte d o n a surfac e which ha s referenc e axes, O A an d ON , scribe don it . The referenc e axes ar e paralle l an d normal , respectively , to th e gag eaxis. Th e corresponding strains in the axia l and transvers e (normal ) direc -tions wil l b e represente d b y £ a and £„ , respectively .

Lateral effects in strain gages

Although ther e i s a n infinit e variet y o f conditions unde r whic h th e strai nsensitivity o f a strain gage might b e determined, for practical purposes thereare onl y three specifi c situation s wit h whic h one mus t b e concerned , a s al lother condition s ca n b e represente d i n term s o f thes e thre e specia l cases ,which ar e a s follows :

Fa = axial strai n sensitivity

Fn = norma l strai n sensitivity

GF — the manufacturer' s gag e factor

FIG. 7.1. Singl e strain gage aligned along axis OA .


These thre e specifi c values o f strain sensitivit y are define d i n th e followingways:

Axial strain sensitivity

when th e norma l strai n i s zero. Thi s can b e written as

when £ „ = 0 .

Normal strain sensitivity

when th e axia l strain i s zero. Thi s can b e written as

when e = 0 .

The manufacturer's gage factor. Th e manufacturer' s gag e factor , a s deter -mined i n accordanc e wit h AST M Standar d E2 5 1-86(12), mean s th e strai nsensitivity, wit h reference t o th e axia l strai n o n th e gag e whe n th e gag e i smounted i n a uniaxia l stress field, with the gag e axi s in th e directio n o f thestress axis , and o n a piece o f material of known Poisson rati o (v 0 = 0.285) .This procedur e correspond s t o calibratin g the gag e i n a biaxia l strai n fieldin whic h th e latera l strain , e n, i s equa l t o — v0ea. I n symbols , th e manu -facturer's gag e facto r can b e expressed as

when

7.2. Basic equations for unit change in resistance

Since strai n gages , i n general , chang e thei r resistance s fo r bot h axia l an dnormal strains , let us proceed towards establishing a general relation for unit

LATERAL EFFECT S I N STRAI N GAGE S 23 7

change i n resistanc e b y considerin g eac h o f thes e effect s alone , an d the nadding th e individua l influenc e t o determin e th e resul t o f bot h axia l an dtransverse strain s acting simultaneously .

Derivation. T o develo p th e require d expressions , w e commence b y writingEq. (7.1 ) in term s o f change i n resistanc e a s follows:

Equation (7.6 ) i s general bu t need s furthe r specification when applie dto an y particula r conditio n t o whic h the gag e ma y b e subjected . Equatio n(7.6) wil l be applied to the determination of the change in resistance producedunder th e following two conditions :

(a) Whe n e a ^ 0 and en = 0(b) Whe n s a = 0 and e n ± 0

The two change s ar e the n adde d togethe r t o determin e th e overal l chang ein resistanc e resultin g from th e combine d effec t o f th e strain s paralle l an dnormal to th e gage axis .

For th e first condition, wher e there i s strain only i n the direction o f thegage axis, the symbols of Eq. (7.6) will take on the following particular values:

R = ARa Chang e of resistanceS = F a Strai n sensitivit y (by definition)

£ = £ „ Strai n

Substituting these values into Eq . (7.6 ) gives

For th e secon d condition , whe n ther e i s strai n only i n th e directio nnormal t o th e directio n o f the gag e axis , th e symbol s o f Eq. (7.6 ) wil l takeon th e following particular values :

R = ARn Chang e i n resistanceS = F n Strai n sensitivit y (by definition )E = e n Strai n

Again, substituting these value s into Eq . (7.6) yields


When th e gag e i s subjected, simultaneously, to strain s i n the axia l an dnormal directions , th e expressio n fo r th e tota l chang e i n resistanc e ca n b ewritten b y adding Eqs . (7.7 ) and (7.8 ) together t o giv e

Thus,

From this , th e overal l unit chang e i n resistanc e ca n b e foun d b y dividingboth side s o f Eq. (7.10 ) by R , th e resistanc e o f the gage , so that

The transverse sensitivity factor, K . Althoug h Eq. (7.11 ) present s th e funda -mental relation between uni t changes in resistance and th e axia l and latera lstrains, i t i s no t i n a convenien t for m fo r th e user , sinc e th e manufacturer sdo no t giv e the value s of F a and F n directly. Instead, the y provide th e user swith th e equivalen t information in term s o f gage facto r (determine d unde runiaxial stress ) an d th e transvers e sensitivit y factor . Transvers e sensitivit yfactor i s a poorly chosen name, since it can easily be mistaken for the norma lstrain sensitivity represented b y the symbol Fn. The meaning of the transversesensitivity factor , whic h wil l b e represente d b y th e symbo l K , ca n no w b eexamined. In orde r t o d o this , Eq. (7.11 ) can b e rewritten as

If the ratio F n/Fa i s represented b y the singl e symbol K, the n th e uni t changein resistanc e is expressed a s

This means tha t th e transvers e sensitivity factor fo r a strai n gag e i s definedas th e rati o o f th e norma l sensitivit y t o th e axia l sensitivity . I t ca n b eexpressed a s


The significanc e o f the numerica l valu e of the transvers e sensitivit y factor isthat it indicates the proportion (or percentage) by which the transverse straincontributes t o the total indicated strai n fro m th e gage. Tabl e 7. 1 lists valuestaken from the literature for the gage factor (approximate) an d the transversesensitivity facto r fo r SR- 4 wir e gage s (1) . Table s 7. 2 and 7. 3 are gag e an dtransverse sensitivity factors for foil gages from tw o manufacturers (13,14).

Table 7.1. Typica l value s o f gag e facto r an dtransverse sensitivit y facto r fo r SR- 4 gage s

Gage type Gftapprox.) K(%)

A-lA-5A-6A-8A-llA-12A-14A-18C-lC-5C-8C-10

2.02.02.01.82.12.02.01.93.53.33.13.2

2.03.51.75

-2.00.51.0

-0.75-2.0

1.754.0

-2.0-0.75

Source: referenc e 1 .

Table 7.2 . Typica l value s fo r gag e facto r an dtransverse sensitivit y coefficients "

Gage type G F K(%)

FAE-03-12FAE-03-35FAE-06-35FAE-12-12FAE-12-100FAE-25-12FAE-50-35FAB- 12- 12FAB-12-35FAP-03-12FAP-06-12FSM-03-12FSM-12-12FSE-06-35FSE-25-35

1.901.882.021.982.042.072.022.022.031.871.961.942.001.992.03

1.3-0.3

0.7-0.8-0.6

0.0-1.7-1.2

0.50.0

-0.70.4

-2.7-1.4-1.7

Source: referenc e 13 ." The liste d value s ar e typica l only . Actua l G F an d Kvalues t o b e use d depen d o n foi l lot , an d ar e provide don th e engineerin g data for m provide d with eac h packageof gages.


Table 7.3 . Typica l value s fo r gag e facto r an d trans -verse sensitivit y coefficient "

Gage type Gf K(%)

EA-06-0625AK-120EA-06-125BT-I20WA-06-250BG-120CEA-06-250UW-120CEA-06-250UW-350EA -06-03 1CF- 120ED-DY-031CF-350CEA-06-125UN-120CEA-06-125UN-350WA-06-500AE-350WK-06-500AE-10CEA-06-500BH-120SA-06-250BK-10CSK-06-250BK-30CSK-06-031EC-350

Source: referenc e 14.

2.025 + 0.5°2.085 + 0.5°2.040 ± 0.5"2.045 + 0.5"2.085 ± 0.5"2.000± 1.0"3.25 ± 3.0"

2.060 ± 0.5"2.090 + 0.5°2.065 + 0.5"2.04+ 1.0"

2.060 ± 0.5°2.065 ± 0.5"2.06 + 1.0"1.99 + 1.0"

0.80.7

-1.10.60.41.4

N/A1.00.5

-1.4- 5.9

0.1-0.5-1.9

0.5

"The liste d value s ar e typiea l only . Actua l Ci F and K value s t o h eused depen d o n foi l lot , an d ar e provide d o n th e engineerin g dat aform provide d wit h each packag e of gages.

For th e standar d type s o f gages , th e numerica l value s o f K will , i ngeneral, b e les s tha n abou t 4 percent , an d fo r man y gage s th e K facto r i sless tha n 2 percent . Fo r comparabl e gag e size , foi l gage s usuall y exhibitsmaller value s of K tha n wir e gages, an d som e eve n indicate K equal s zero .Flat-grid wir e gages wil l alway s have a positiv e valu e o f K . Wrap-aroun dconstruction fo r wire gages produces negative values of K, due to the Poissoneffect withi n th e gage . Foi l gages , dependin g upo n th e materia l o f th e foil ,can exhibi t either positive o r smal l negativ e values of K .

Relations between gage factor an d the axial and normal strain sensitivities. Th erelation betwee n th e axia l strai n sensitivity , Fa, o f a strai n gage , an d th emanufacturer's gage factor, GF, can now be investigated. Since the gage factoris determine d unde r uniaxia l stres s conditions , wit h th e gag e axi s i n th edirection o f th e stres s axis , w e wil l conside r th e genera l aspect s o f thi scondition first, and the n tak e u p th e specia l situatio n whic h prevails whenthe gage s ar e calibrated . Figur e 7. 2 shows a gag e i n a uniaxia l stress field.

For uniaxia l stress i n th e directio n o f the gag e axis,

where v = Poisson' s ratio fo r the materia l upon whic h th e gag e i s mounted .For thi s situation, the expression fo r uni t change i n resistance , given by Eq .

LATERAL EFFECT S I N STRAI N GAGES 241

FIG. 7.2. Strai n gag e in a uniaxia l stress field.

(7.13), ca n b e writte n as

When the gage factor i s being determined, the Poisson rati o correspondsto v 0 = 0.285 , whic h i s th e valu e fo r th e ba r o n whic h th e manufacture rmakes th e calibration . Therefore , for conditions o f calibration ,

For th e same conditions , however , the manufacture r tells us that

From thi s we can write

Since th e uni t change i n gag e resistance , R/R, i s independent o f themathematical relation s whic h ar e use d t o expres s it , Eqs. (7.17) an d (7.19 )represent th e sam e thing , so tha t

From this ,


or

Since F n = KF a, then

7.3. Determination of gage factor and transverse sensitivity factor (12)

Several methods o f determining the gage facto r fo r bonded resistanc e straingages wil l b e outlined . The tw o method s considere d wil l b e a bea m i n pur ebending and a constant-stress cantileve r beam.

Beam in pure bending

Figure 7. 3 shows a typica l system. The tes t bea m i s loaded b y dead weightsin such a manner tha t the beam i s subjected to pure bending. The test beam ,of a suitable material, has minimu m dimensions of 0.75 in by 1. 0 in by 3 0 in,and the minimum distance between the pivot points on the supports is 96 in.The assembl y is symmetrical about a vertica l line at it s midpoint.

The pivots and weights are adjusted to give a strain on the beam surfac eof 100 0 + 5 0 uin/in. The strain over the usable portion o f the tes t beam ma ynot var y by mor e tha n 1 percent o f the strai n a t th e referenc e point. Th eneed fo r measuring th e strai n directl y can b e eliminate d by maintainin g acalibration o f th e syste m wit h a Clas s A extensomete r (15) . However , th estrain a t th e referenc e poin t ma y als o b e measure d wit h a permanentl y

FIG. 7.3. Constant-bending-momen t method for gage-factor determination . (From ref. 12 withpermission. (C D ASTM. )


mounted strai n gage that has been calibrated by spanning i t with a Class Aextensometer.

The usabl e portio n o f the bea m i s to b e at leas t one-half of its exposedlength. Measurements ove r each test station ar e made with the extensometerin orde r t o verif y th e strai n distributio n ove r th e bea m width . Gages ar einstalled o n th e unstraine d tes t sectio n an d the n th e bea m i s loaded thre etimes to the required strai n leve l to 100 0 ± 5 0 uin/in. The gage factor of theindividual gage is determined b y dividing the uni t change i n gage resistanceby the strai n valu e determined fro m th e bea m calibration .

Constant-stress cantilever beam

A typica l syste m usin g a constant-stres s cantileve r bea m i s show n i n Fig .7.4, whil e the bea m detail s ar e give n i n Fig . 7.5 . The siz e and arrangemen tof th e equipmen t mus t b e suc h tha t th e bea m ca n b e deflecte d i n eithe rdirection t o produc e a strai n o f 120 0 uin/in. Two o r mor e referenc e straingages ma y be permanently bonde d t o the beam an d calibrate d b y spanningthem with a Class A extensometer. The constant-stress area i s also exploredwith th e Clas s A extensometer i n orde r t o determin e th e are a wher e th estrain i s th e sam e a s tha t o f th e referenc e gages . Onl y area s wher e th edifferences i n strain between the extensometer an d the reference gage do no texceed 1 0 uin/in ar e t o b e used .

Test gage s are installe d i n th e satisfactor y areas, wit h the activ e axis ofthe gag e paralle l t o th e cente r lin e o f the beam . Th e bea m i s deflected s othat th e surfac e strain i s 100 0 + 5 0 uin/in, an d th e uni t resistanc e chang erecorded. Thre e suc h reading s ar e taken, wit h the gage facto r computed fo reach loadin g cycle.

FIG. 7.4. Constant-stres s cantilever beam method for gage-faclor determination . (From ref. 12with permission. © ASTM.) .


NOTE I—AM dimensions are in inches (I in - 25.4 mm).Noit 2—Surfaces "A" and "D" lo be parallel 10 0.0005 T1R and flai to 0.0002 TIR.Note 3—Sides of beam must form iriangje at apex as shown. Maximum allowable deviation of beam sides from correct IIIMI

±0.001 in. in active area, 0.003 in. elsewhere.

FIG. 7.5. Constant-stres s cantileve r beam . Al l dimension s are i n inche s ( 1 in = 25. 4 mm).Surfaces A an d B to b e paralle l t o 0.0005 TIR an d fla t t o 0.0002 TIR. Sides of beam must for mtriangle a t ape x a s shown . Maximu m allowabl e deviatio n o f bea m sides from correc t lin e i s+ 0.001 in i n activ e area , 0.003 in elsewhere . (From ref. 12 with permission . © ASTM. )

Transverse sensitivity

Strain gag e transvers e sensitivit y results in a n undesire d signa l induce d b ystrains i n directions othe r tha n the one being measured. The errors inducedin the plane o f the gages depend on the stress distribution in the gaged areas .Figure 7. 6 shows a typical test rig for determining transverse sensitivity, whileFig. 7.7 gives the test beam detail s and gag e arrangements . The control gag emay b e eithe r a Clas s A extensomete r o r a permanentl y installe d an dwaterproofed resistanc e strai n gag e temperatur e compensate d fo r the bea mmaterial and calibrate d b y a Class A extensometer.

The sid e plate s fastene d t o th e bea m ar e loade d a t thei r lowe r edg ethrough th e us e o f the cran k mechanism , a s show n i n Fig . 7.6. Thi s placethe bea m i n compressio n a s wel l as i n bending . Th e transvers e direction i sin the lon g directio n o f the beam , an d so , on th e to p surface , th e transversestrain du e t o th e compressiv e loa d i s a tensil e strain , whil e the transvers estrain du e t o bendin g i s a compressiv e strain . Th e dimension s o f th eapparatus ar e chosen so that these two strains cancel each other , thus leavinga plan e strai n conditio n acros s the beam .

The tes t beam ha s 1 6 defined stations . The differenc e betwee n the strainmeasurements b y the control gage and th e actual strain at each station , bothparallel an d perpendicula r t o th e principa l strai n direction , mus t b e deter -mined. The strain perpendicular to the principal strain (the transverse strain)must b e les s tha n 4 uin/in o r 0. 5 percen t o f th e principa l strain , wit h amaximum principa l strai n o f 1000 + 5 0 uin/in.

LATERAL EFFECTS IN STRAIN GAGES 245

FIG. 7.6. Transverse-sensitivit y test rig . (From ref . 12 with permission . © ASTM. )

FIG. 7.7. Testin g stations and gag e arrangemen t fo r transverse-sensitivit y test . (From ref . 12with permission . © ASTM. )

A tes t require s a minimum of five identical gages o f one type . At leas tthree gages are mounted perpendicula r to the principal strain direction anda minimu m o f tw o gage s ar e mounte d paralle l t o th e principa l strai ndirection. After gag e installation , the beam i s loaded t o about 100 0 uin/in atleast thre e time s befor e reading s ar e taken . Afte r thes e thre e loa d cycles ,readings from th e control gag e and th e tes t gages are taken i n the unloade dcondition, the n th e bea m i s loaded s o tha t th e surfac e strai n i s 100 0 uin/inand reading s take n again . Thi s i s repeate d fo r thre e loadin g cycles . Th etransverse sensitivit y is computed a s


where R,/R, 0 = uni t resistance chang e i n transverse gag e

ARL/RLO = uni t resistanc e change i n gage paralle l t o th e principa lstrain directio n

The rang e o f all values obtained i s to b e reported , whil e the transvers esensitivity o f a gag e typ e i s taken a s th e averag e of all value s recorded .

7.4. Use of strain gages under conditions differing from thosecorresponding to calibration

If a strai n gag e i s use d unde r biaxia l condition s whic h diffe r fro m thos eprevailing durin g calibration , theoretically , ther e wil l b e a n erro r i n th eindicated valu e o f the axia l strain . Fortunately , thi s error i s usuall y rathersmall an d ca n b e neglected . I t ca n b e shown tha t fo r gages whos e K facto ris les s tha n 3 percent , th e maximu m error wil l no t excee d abou t 4 percen tas long a s the numerica l value of the norma l strai n does no t excee d tha t ofthe axia l strain.

The exact valu e of this error can no w be examined, along wit h a simplemeans o f correctin g fo r i t unde r an y conditio n o f biaxia l strain . Fo r thi spurpose, it wil l be convenient to represen t th e ratio o f normal t o axia l strainby a singl e symbol . Thus,

From Eqs . (7.13 ) an d (7.21) , a n expressio n fo r th e uni t chang e i nresistance ca n b e written as

From Eq. (7.24), ^„ = ae a. Substituting this value of £„ into Eq. (7.25) yields

Solving Eq . (7.26 ) for s a gives

The significanc e o f th e resul t give n i n Eq . (7.27 ) is represente d b y th efollowing observations :


1. Th e quantit y ( R/R)/GF correspond s t o th e indicatio n o f strai n a sdetermined by th e manufacturer . That is,

2. Th e ter m ( 1 — v0K)/(i + tzK) represent s a modifyin g facto r whos evalue depends upon a , the rati o betwee n the normal an d axia l strainson the gage. When the gage is employed in a stress field correspondingto calibration conditions , a = — v0 and the modifying expression revertsto unity , since th e indicated strain , fo r this case, represents th e correc tvalue.

3. Sinc e the value of K wil l be small with respect t o unity (less than abou t4 percen t fo r standar d gages ) fo r mos t gages , a precis e knowledg e ofthe exac t valu e of a i s not required . Th e rati o o f the indicate d norma land axia l strains shoul d b e good enoug h without corrections; however,if a bette r valu e o f th e modifyin g facto r i s desired , the n a furthe rcorrection ma y be obtained b y taking the ratio of the initially correctedvalues.

Some special cases

Correction factor,

Strain relations Ratio, a.

1. Tw o equa l an d lik e principal strains :

2. Tw o equal bu t unlik e principalstrains:

3. Uniaxia l stres s with the gag e axisin th e directio n o f the stres s axis :

limits: v = 0 t o + 5

4. Uniaxia l stres s with th e gag e axi sperpendicular t o th e stres s axis :

limits : v = 0 to + ^

We can write an expression for the error that result s when a single straingage i s used in a biaxial stress field. The actua l strai n along th e gag e axi s isEa, while the actua l strain norma l t o th e gag e axis is £„. From Eq . (7.27), the


Table 7.4 . Erro r i n strain s whe n usin g a uniaxia l gag e i n abiaxial field

True strain , £„ Tru e stain , £„ a = £„,£ „

t;a 5c 0 5sa 3t: a 3K 2i: 2e.a i: a Ica 0 0<-„ - K a - 1f.a - 3<: 0 - 3£„ -5e, - 5£„ -10E 0 -1 0

n W18.711.68.14.50.0

-2.5-9.6

-16.7-34.3

strain indicato r wil l rea d

The percent error, q, between the meter reading, E'a, and the actual strain, sa, is

Table 7.4 shows the resulting error between the strain indicato r reading ,s'a, an d th e actua l strain , s a, fo r a gag e wit h a transvers e sensitivit y ofK = 0.035.

7.5. Indication from a pair of like strain gages crossed at right angles

We assume tha t the strai n gradient i s so small that bot h gages are subjectedto th e sam e strai n condition . Le t u s now examin e th e tota l uni t chang e i nresistance o f both gage s whe n the y are connecte d i n series . Sinc e th e gage sare oriente d a t righ t angles , conside r the m t o b e aligne d parallel , an dperpendicular, t o th e referenc e axes, OA an d ON , a s show n in Fig . 7.8 , an dthat they make any angle 9 (or 0 + 90°) wit h respec t t o th e directions o f theprincipal axes.

The strai n gages whose axes are paralle l an d perpendicula r t o O A an dON ca n now be examined. Subscripts a and n will refer th e various quantitiesto thes e axes , respectively . When th e tw o gage s ar e connecte d i n series , R T

This reduce s t o


FIG. 7.8. Strai n gages crossed a t righ t angles.

is the tota l resistanc e o f both gages , whil e AR T i s the chang e i n resistanc eof both gages . Thus ,

or

where ft = Rn/Ra = ratio o f resistance o f gage N t o resistanc e o f gage A .

From Eq . (7.11),

This ca n b e rewritten as


Also, fro m Eq . (7.21) , for a singl e gage w e have

For eac h gage , then,

The uni t change i n resistance for the two gage s in series , given by Eq . (7.31) ,can b e rewritten by substituting the values of &Ra/Ra an d AR n/Rn, give n byEqs. (7.32 ) and (7.33) , respectively, into Eq . (7.31) . This produce s

Equation (7.34 ) is a general expressio n for the uni t change in resistanc eof bot h gage s i n term s o f variabl e strains , th e gag e factors , transvers esensitivity factors , an d resistances , al l o f which may b e differen t fo r eac h o fthe tw o gages .

In it s presen t form , Eq . (7.34 ) i s no t ver y convenient ; however , it ca nbe reduce d t o workabl e condition s fo r certai n specia l situations . Fo rexample, i f the tw o crosse d gage s ar e alik e in all respects , the y wil l have th esame gag e factors , th e sam e transvers e sensitivity , an d equa l resistances .Thus,

(GF)a = (G F)n = G F, K a = Kn = K

Ra = Rn = R, 0 = RJRa = 1

For thes e conditions , Eq . (7.34) reduces t o

This furthe r simplie s t o

LATERAL EFFECT S IN STRAI N GAGE S 25 1

An eas y wa y t o accoun t fo r th e transvers e effec t i s t o adjus t th e gag efactor dia l o n th e strai n indicato r (th e scale factor ) i n orde r t o correc t fo rit. Also , a specia l situatio n o f interes t involving the solutio n o f Eq . (7.34)will b e shown when stress gages are discussed.

Problems

7.1. Th e following data are given for a thin-walled pressure vessel: diameter = 9 6 in,wall thicknes s = 2 in, interna l pressur e = 100 0 psi, v = 0.3 , E = 3 0 x 10 6 psi.Two identica l strain gages , wit h G F = 2.1 and K = 3.5 percent, are bonde d t othe vessel , on e in the longitudinal directio n an d on e in the hoop direction.

(a) Determin e th e actua l strains .(b) Determin e th e strai n indicato r reading s fo r each gage .(c) Determin e th e percen t erro r i n each reading .

7.2. A t a point on a machine element the stresses are ax= — 8000 psi, af = 4700 psi,and TJ.J , = 550 0 psi. A technician bonds two identical strain gages, with G F = 2.0 4and K = — 1.1 percent, along what he believes are the principal stress directions.Gage a is located 55 ° CCW fro m th e x axis , while gage b is located 90 ° CC Wfrom gag e a .

(a) Hav e th e gages been properly located?(b) Determin e th e actua l strai n a t eac h gag e if v = 0. 3 and E = 3 0 x 10 6 psi.(c) Determin e th e indicate d strai n fo r each gage .

7.3. A single strain gage, with GF = 1. 9 and K = 2. 5 percent, is bonded to a membersubjected t o a uniaxial stress. The gage axis is aligned along the principal stres saxis. I f th e maximu m stres s i s 3 0 000 psi, v = 0.3, an d E = 3 0 x 10 6 psi,determine the valu e of the indicate d strain .

7.4. In Proble m 7.1 , the two gage s are wire d in serie s in orde r to hav e the strai nindicator read 100 0 uin/in when the vesse l is pressurized to 100 0 psi. What gagefactor settin g must be use d to accomplis h this ?

7.5. A t a poin t on a machine element , a x = 25 000 psi, a y = — 5000 psi, an d i xf =12 000 psi. Th e membe r i s loade d i n suc h a manne r tha t th e principa l stres saxes remai n fixe d i n direction . A strai n gag e i s bonde d alon g eac h principa lstress axis , the n the y ar e connecte d i n series . Th e gage s ar e identical , wit hGF = 2.04 an d K = —1. 1 percent. Determin e th e gag e facto r settin g o n th estrain indicato r s o tha t th e readin g wil l b e 50 0 uin/in whe n th e su m o f th eprincipal stresses , a l + a2, equals 20 000 psi.

REFERENCES

1. Baumberger , R. and F. Hines, "Practical Reduction Formulas for Use on Bonde dWire Strain Gages in Two-dimensional Stress Fields," SESA Proceedings, Vol. II,No. 1 , 1944, pp. 113-127 .

2. Bossart , K . J . an d G . A . Brewer , " A Graphica l Metho d o f Rosett e Analysis, "SESA Proceedings, Vol. IV, No . 1 , 1946, pp. 1-8 .

3. Campbell , William R., "Performance Test s of Wire Strain Gage s IV—Axial andTransverse Sensitivities, " NACA Technical Note No . 1042, 1946.


4. Handbook o f Experimental Stress Analysis, edite d b y M . Hetenyi , Ne w York ,Wiley, 1950 , pp . 407-^10.

5. Meier , J. H. , "On th e Transverse-strai n Sensitivit y o f Foil Gages," ExperimentalMechanics, Vol . 1 , No. 7 , July 1961 , pp. 39-40 .

6. Wu , Charle s T. , "Transverse Sensitivit y o f Bonded Strai n Gages, " ExperimentalMechanics, Vol . 2, No. 11 , Nov. 1962 , pp. 338-344 .

7. Meyer , M . L. , " A Unifie d Rationa l Analysi s fo r Gaug e Facto r an d Cross -Sensitivity of Electric-Resistance Strain Gauges," Reprinted b y permission o f theCouncil o f th e Institutio n o f Mechanica l Engineer s fro m Journal o f StrainAnalysis, Vol . 2 , No . 4 , 1967 , pp . 324-331 . O n behal f o f th e Institutio n o fMechanical Engineers .

8. Meyer , M. L., "A Simpl e Estimate for the Effec t o f Cross Sensitivity on EvaluatedStrain-gage Measurement, " Experimental Mechanics, Vol . 7 , No. 11 , Nov. 1967 ,pp. 476-480.

9. "Error s Due t o Transvers e Sensitivit y i n Strain Gages," TN-509, MeasurementsGroup, Inc. , P.O . Bo x 27777, Raleigh , N C 27611 , 1982 .

10. Measurement s Group , Inc. , "Error s Du e t o Transvers e Sensitivit y i n Strai nGages," Experimental Techniques, Vol. 7, No. 1 , Jan. 1983 , pp . 30-35 .

11. Handbook o n Experimental Mechanics, edited by Albert S. Kobayashi, EnglewoodCliffs, Prentice-Hall , 1987 , pp. 51-54 .

12. 1986 Annual Book o f ASTM Standards, 191 6 Race St. , Philadelphia , P A 19103 ,"Performance Characteristic s o f Bonde d Resistanc e Strai n Gages, " Vol . 03.01 .Designation: E251-86 , pp. 413^428. Copyright ASTM . Reprinte d wit h permission.

13. "SR- 4 Strai n Gage Handbook," BLH Electronics , Inc., 75 Shawmut Rd., Canton ,MA 02021 , 1980 .

14. Dat a furnishe d b y Measurement s Group , Inc. , P.O . Bo x 27777 , Raleigh , N C27611, 1989 .

15. 1986 Annual Book o f ASTM Standards, 191 6 Race St. , Philadelphia , P A 19103 ,"Verification an d Classicatio n o f Extensometers, " Vol . 03.01 . Designation :E83-85, pp . 267-274 . Copyright ASTM . Reprinte d with permission.

8STRAIN GAGE ROSETTES AND DATA ANALYSIS

8.1. Reason for rosette analysis

We sa w in Chapte r 2 tha t fo r an y poin t o n a fre e (unloaded ) surfac e o f asolid it is necessary to know three independent quantitie s in order t o specif ythe state of stress completely. These quantities are the magnitudes of the twoprincipal stresses, a 1 an d a 2, and thei r directions, 9 or 9 + 90°, with respectto som e reference .

For isotropic elastic materials these values can be calculated from strainsmeasured on the surface at the point in question, and since three independentquantities are to be determined, in general, it will be necessary to make threeindependent measurement s o f strain . Ther e are , however , som e specia lsituations in which one o r tw o observation s o f strain wil l suffic e t o provid ethe informatio n necessary fo r completely establishing th e stat e o f stress.

It wil l be well , at thi s time, to dra w attentio n t o th e fac t that , althoughwe refe r t o th e stres s conditio n a t a point , th e manne r o f measuring th estrain gives the average ove r a smal l distance. Therefore , from th e practica lpoint o f view , th e result s o f a se t o f rosette observation s wil l approximat ethe averag e conditions ove r a smal l area . Thi s i s not objectionabl e a s lon gas th e lengt h ove r whic h th e strai n is measured i s shor t enoug h tha t ther eis relativel y little chang e fro m on e en d t o th e other . Th e gag e lengt h wil ltherefore depen d upo n the strain gradien t an d may run from smal l (^ i n toYg in) value s to severa l inches o r more .

8.2. Stress fields

Stress fields were examined in Chapter 2 , where the severa l stress states werediscussed. In general, the concern has been with plane stress, and transforma-tion equation s wer e developed t o enabl e th e determinatio n o f plane stres sat a poin t i n an y directio n relativ e to a chose n coordinat e system . Thes econcepts wil l b e reexamine d her e in th e developmen t o f rosette analysis.

Special case of uniaxial stress (simple tension or compression)

In the case of simple tension o r compression , on e knows that the direction sof the principal stress axes will be parallel and perpendicular to the directionof the applie d force , o r load , an d tha t th e magnitud e of the principa l stres s


whose direction i s at righ t angles t o th e loa d wil l b e zero. Thi s mean s tha ttwo of the three quantities are known from th e prevailing physical conditions.On thi s account , i t wil l therefor e b e necessar y t o mak e onl y a singl eobservation of the strain along the direction of the load i n order t o determin ethe on e remainin g unknow n quantity . Fo r a n elasti c body , th e stres s i scalculated a s

where a = th e stres s intensity

E = the modulu s o f elasticity of the materia l

£ = th e measure d strai n (positiv e fo r tensio n an d negativ e fo rcompression)

It shoul d be noted here that i f the stress is tension, a represents <r l, thealgebraically large r principa l stress , and a 2 — 0- I f the stres s i s compression ,ffj = 0 and a corresponds t o <r 2, th e algebraicall y smalle r principa l stress .

Special case of biaxial stress (principal stress directions known)

In a few special case s i n which th e directions of the principa l stres s axes (theangle 0 ) ca n b e establishe d b y auxiliar y means , suc h a s condition s o fsymmetry o r throug h a previou s application o f a brittl e coat, there are onlytwo unknowns, cr, and a 2, the principa l stress magnitudes, to be determined .These can be found b y measuring the corresponding principa l strains , E{ and£2, i n th e direction s o f the principa l stres s axes , an d calculatin g th e value sfrom Eqs . (2.50a) and (2.50b) . I n these equations th e subscripts , x and y , arechanged t o 1 an d 2 , respectively . Equation s (2.50a ) an d (2.50b ) ca n b erewritten a s

where CT J = th e algebraicall y large r principa l stres s

<72 = th e algebraicall y smalle r principa l stres s

£t = th e algebraically large r principa l strai n

£2 = th e algebraicall y smalle r principa l strai n

E = the modulu s of elasticity of the materia l

v = Poisson' s rati o

STRAIN GAG E ROSETTE S AND DAT A ANALYSI S 255

For late r us e it wil l b e more convenien t t o expres s the principa l stres svalues in the following form:

where

The general case

In man y instances , neithe r th e magnitude s of the principa l stresse s no r th edirections o f thei r axe s wil l b e known . Thi s mean s tha t fo r a complet edescription o f the stat e o f stress, at an y particula r point , three independentquantities must be found. In consequence, i t wil l be necessary t o make threemeasurements o f linear strai n i n differen t direction s (se e Sectio n 2.6) , an dfrom thes e three observations, to compute the two principal stress magnitudesand th e direction s o f the axes .

Figure 8. 1 illustrates a pai r o f orthogonal referenc e axes , O X an d O Y,and thre e othe r axes , OA , OB , an d OC , makin g angle s 9 a, 9 b, an d 6 C,respectively, with respect t o th e reference s axis OX. Th e axe s OA, OB, andOC for m wha t is described a s a rosette , an d i f corresponding linea r strains ,sa, £ b, an d e c, ar e measure d i n thei r respectiv e directions , th e linea r an dshearing strains , e x, e y, and y xy, correspondin g t o th e O X an d O Y axe s ofreference, ca n b e calculated.

The value s o f e x, e y, an d y xy ar e calculate d i n term s o f th e measure dstrains, e a, e b, an d e c, b y th e us e o f Eq . (2.32) . I t i s repeate d her e an d

FIG. 8.1. Referenc e axe s OX-OY wit h rosette axes.

the hydrostatic component o f strain and corres -ponds to th e cente r o f Mohr's circlethe shear componen t o f strain and correspond sto th e radiu s of Mohr's circle


renumbered a s Eq . (8.6) . Thus,

In Eq . (8.6) , th e subscript , x' , take s o n th e value s o f th e thre e measure dstrains in turn, and 8 has the value associated wit h its particular strain . Thisgives thre e independen t equation s tha t ca n b e solved simultaneousl y for e x,F,y, an d y xr Th e thre e equations s o forme d ar e

When ex, ey, and y xy have been determined b y the simultaneous solutionof Eqs. (8.7) , (8.8), and (8.9) , the principa l strains may b e found by using Eq .(2.37). Thus , th e principa l strain s ar e

The orientatio n o f the principe l stresses relativ e to th e referenc e axes isthe sam e a s th e principa l strai n axe s relativ e to th e reference s axes . Thus ,the orientation o f the principal axe s may b e obtained fro m Mohr' s circl e forstrain, or , analytically , by usin g Eq . 2.34 and eithe r Eq . (2.35 ) or (2.36) .

8.3. Rosette geometry

Theoretically, th e relativ e directions o f strai n measuremen t (th e angle s 9 a,9b, an d 6 C) are o f n o particula r importance . However , fro m th e practica lconsideration o f solvin g th e equations , on e find s tha t certai n preferre d

(corresponds t o th e radiu s o f Mohr'scircle)

where

or

(corresponds t o th e cente r o f Mohr's circle )

STRAIN GAG E ROSETTE S AN D DAT A ANALYSIS 257

FIG. 8.2. Three-elemen t rectangular rosette arrangements.

FIG. 8.3. Delt a rosett e arrangements.

orientations permi t a muc h simple r reductio n o f the strain s int o term s ofstress. At the present tim e there are fou r generall y accepted arrangement s ofthe gage axes for strain rosettes . Basically , there are just two arrangements ,the rectangula r an d th e equiangular , bu t eac h o f these has a modificationinvolving a redundan t fourt h observatio n o f strain.

Basic arrangements involving three observations of strain

Figure 8. 2 shows tw o arrangement s o f a three-elemen t rectangula r rosette .The thre e gag e axe s in arrangement (a ) are lai d ou t a t 45 ° and 90 ° to eac hother. I n arrangemen t (b) , gag e B forms a 135 ° angle wit h gage s A an d C .

The equiangular or delta rosette has the three gage axes laid out parallelto th e side s o f a n equilatera l triangle . Thi s typ e o f rosett e ha s th e mos tdesirable orientation s o f th e direction s o f strai n observation , bu t th eequations fo r computing stress value s are not quit e so simple as those of therectangular rosette . Fo r thi s reason, the rectangula r rosett e i s preferred bymany. Figur e 8. 3 illustrates two arrangement s fo r th e delt a rosette .

Modified arrangements involving four observations of strain

The T-rectangula r rosett e ha s fou r gage s wit h axe s 45 ° apart, a s indicate din Fig . 8.4 . Although the fourt h observatio n i s theoretically unnecessary , i t


FIG. 8.4, T-rectangula r rosette.

FIG. 8.5. T-delt a rosette .

nevertheless provides a convenient means of checking the strain observations,since th e su m o f the strain s in any tw o direction s a t righ t angles shoul d b ea constan t fo r a give n set o f conditions. Thus,

The T-delt a rosett e i s essentially the sam e a s th e equiangula r arrange -ment wit h the addition of a fourth observatio n which is made a t righ t anglesto th e directio n o f on e o f th e othe r three . I t i s claime d tha t thi s for m o frosette ha s al l the desirabl e characteristic s of the equiangula r type plus th eadvantage of a little more precise determination o f the hydrostatic componentof strain a t th e referenc e point, i f this coincides wit h th e intersectio n o f twoperpendicular gag e axes . The arrangemen t i s shown in Fig . 8.5 .

8.4. Analytical solution for the rectangular rosette

This analysi s i s for th e three-elemen t rectangular rosette , an d i s starte d b ytaking the O A axis of the rosette in Fig . 8.1 as the reference axis and makin git coinciden t wit h th e O X axis . Sinc e th e three-elemen t rectangula r rosett eis being considered, th e strain gage axes wil l be those shown in Fig. 8.2 . Fo r

STRAIN GAG E ROSETTE S AN D DAT A ANALYSI S 25 9

this arrangement , then , on e has th e following angles :

Since the transformatio n equation give n by Eq. (8.6) is written in termsof twic e the angle , th e require d value s o f the trigonometri c function s ar e

These value s ca n b e substitute d int o Eqs . (8.7) , (8.8) , and (8.9 ) t o giv e th ethree simultaneou s equations necessar y t o determin e e x, ey, and y xy. Thus ,

From Eq . (8.13) , i t i s seen tha t


The shearin g strain , y xy, ca n b e determine d i n term s o f the strai n reading sby substitutin g th e value s o f s x an d s y give n b y Eqs . (8.16 ) an d (8.17) ,respectively, into Eq . (8.14) . I n doin g this ,

Solving fo r y xy,

Thus, Eqs . (8.16) , (8.17) , an d (8.18 ) give the value s of ex, e y, and y xy i n termsof the strai n gag e readings .


By substituting these values of EX, ey, an d y xy int o Eq . (8.10) , th e value sof the principal strains are determined directly in terms of the strain reading sfrom th e rosette . Consequently ,


where

The value s of EI and £ 2 given by Eq . (8.19) may no w b e substituted intoEqs. (8.2 ) an d (8.3 ) in orde r t o determin e the principa l stresses , a l an d <r2.In mos t cases , however , the numerica l value s o f the principa l strain s nee dnot b e known, since the values of A and B , given by Eqs. (8.20a) and (8.20b) ,respectively, can be substituted into Eqs. (8.4) and (8.5 ) in order to determinefTj an d a 2 directly in terms of the strain observations on the rosette. Carryingout thi s operation yields

Equations (8.21 ) and (8.22 ) are no t i n th e simples t for m bu t th e formgiven lends itself better to the determination o f the directions of the principa lstress axes.

Determination of the principal stress axes directions

In orde r t o determin e th e orientatio n o f th e principa l stres s axes , th eorientation o f the principa l strai n axes may b e found instead , sinc e th e axe sof each coincide . I n Sectio n 2.6 , the orientatio n o f the principa l strai n axe srelative to the reference axes were found analyticall y by the use of Eqs. (2.34),(2.35), an d (2.36) . Note , also , tha t th e angl e B i s alway s measure d i n acounterclockwise direction fro m th e positive OX axis to the positive 01 axis ,which correspond s to th e directio n o f et, an d therefor e a^.


Equations (2.34) , (2.35) , an d (2.36 ) ca n no w b e expresse d i n term s ofthe strai n readings of the rosette . Thus,

To establis h the angula r relationship , 0 t, betwee n the O l axi s and th eOA axi s (OX an d O A ar e coincident) , two o f the thre e equation s ar e used .In fact , i f Eq. (8.23 ) is chosen, the n w e need onl y t o determin e th e sig n ofeither si n 29 o r co s 26 t o obtai n th e matchin g quadrant . Fo r example , iftan 28 is negative, then 26 must be in either the secon d o r fourt h quadrant .If sin 26 is negative, then 2 6 could be in either the thir d o r fourth quadrant .The matching quadrants ar e th e fourth , an d s o the angl e 29 must be in th efourth quadrant . Fro m this , then , th e orientatio n o f axi s O l ca n b edetermined relativ e to axi s OA .

Fortunately, a check can always be made by sketching a Mohr' s circle .Three rule s for determining th e angle , 9 lt betwee n the O A an d th e O l axi swill b e stated .

Proof of rulesFigure 8. 6 show s a Mohr' s circl e fo r a three-elemen t rectangula r strai nrosette. The three strains , e a, efc, an d e c, are represented b y points A , B, andC, respectively , o n th e circumferenc e o f th e circl e an d a t th e end s o f th eradial lines that ar e 90° apart and take n i n the same sequence as the rosett eaxes, which are 45° apart.

lies between 0 and

lies between 0 and

262 THE BONDE D ELECTRICA L RESISTANC E STRAI N GAGE

FIG. 8.6. Mohr' s circle fo r th e rectangula r rosett e wit h thre e observation s o f strain.

If point A lies anywhere along the semicircumference below the abscissa ,then angle 20 ̂wil l be positive and hav e values between 0° and 180° , so tha t0! wil l be between 0° and 90° . If point A happens to li e on th e semicircumfer-ence abov e th e abscissa , the n angl e 20 1 wil l li e between 0 an d —180 ° an d0! wil l b e between 0 ° and -90° .

How ca n w e tell whether point A i s above o r belo w the absciss a o n th eMohr diagram ? A stud y o f Fig . 8. 6 shows tha t poin t A wil l li e belo w th eabscissa wheneve r poin t B i s t o th e righ t of the cente r o f the circle ; that is ,when e,b > (ea + eJ/2. Poin t A wil l be above the abscissa when eb < (s a + e c)/2.and wil l li e on th e absciss a whe n e. b = ^ a + ec)/2. Fro m this , th e followin grules ca n b e se t down:

1. Th e angle , 9 lt wil l lie between 0° and +90 ° when e.b > (ea + e c)/2. Thisis shown i n Fig . 8.7.

2. Th e angle , 0 t, wil l lie between 0° and -90 ° whe n eb < (ea + ec)/2. Thi sis illustrated i n Fig . 8.8 .

3. Figur e 8. 9 shows tha t th e angle , O l, wil l b e zer o whe n £ b = (ea + ec}/2and £ fl > e c. Fro m th e figure , i t i s evident tha t e a = e l7 th e maximumprincipal strai n i n th e plane . Figur e 8.1 0 shows tha t th e angle , 0 1; wil lbe 90° when e fc = (t. a + e c)/2 and e a < e.c. It i s apparent fro m th e figur ethat e. a = £2, the minimu m principal strai n i n th e plane .

FIG. 8.7. Moh r diagra m fo r



FIG. 8.10, Moh r diagra m fo r

and

and


The differenc e i n th e sig n o f the shearin g strain , y xy, shoul d agai n b ereviewed i n Sectio n 2. 6 i n orde r t o mak e th e analytica l solutio n fo r 6 1

compatible with the solution fro m Mohr' s circle.

Example 8.1. A three-elemen t rectangula r strai n rosett e give s th e followin greadings:

ea = 135 0 uin/in, e b = — 500 uin/in, e c = 560 |iin/in

(a) Determin e the principal strains .(b) Determin e 9 l analytically .(c) Sketc h th e orientatio n of the principa l axe s relative to OA .(d) Dra w a Mohr' s circl e and chec k th e position o f 9^(e) Determin e a l an d <r 2 using v = 0. 3 and E = 30 x 10 6 psi.

Solution, (a ) Th e principa l strain s are given by Eq . (8.19).

(b) Equation s (8.23 ) and (8.24 ) will b e use d t o determin e 0j .

From this , 2 9 may b e in eithe r the secon d o r fourt h quadrant .

The numerato r o f si n 29 is negative , a s ca n b e see n from ta n 29 , and s o onlythe sig n o f si n 20 i s needed . Sinc e si n 26 i s negative , 20 ma y b e i n eithe r th ethird o r fourt h quadrant . Sinc e th e fourt h quadran t i s the matchin g quadran tin each, 29 is a fourth-quadrant angle . Thus ,

20j = 360 - tan ~ '| - 3.68 3 54| = 285.2°

0, = 142.6°

The angle , 9 l, i s measured i n a counterclockwis e directio n fro m th e O A axi sto th e O l axis .

(c) Figur e 8.11 shows the orientation of the Ol axi s relative to the OA axis.


FIG. 8.11. Orientatio n o f the principa l strai n (an d stress) axe s relativ e t o th e referenc ecoordinates in Exampl e 8.1.

This ca n als o b e verified by the us e of Rule 2 , since e,b < (e a + ec)/2. In thi s case,fJi lie s between 0 ° and —90° , which is the acut e angl e betwee n the O A axi s an d£1( goin g i n a clockwis e (negative) direction .

(d) I n orde r to dra w Mohr' s circle, KX, e. , and y xy ar e compute d usin g Eqs .(8.16), (8.17) , an d (8.18) .

Ex = e, a = 135 0 uin/i n

£y = £ c = 56 0 uin/in

yxy = 2eh - (e a + sc) = 2(-500) - (135 0 + 560) = -291 0 uradian s

Figure 8.12 gives Mohr's circle . Note that yxy i s negative from the transformationequation, bu t fo r th e Mohr' s circl e i t mus t b e plotte d a s positive . Thi s i s i naccordance wit h th e sig n conventio n establishe d i n Chapte r 2 . I f th e circl e i straversed i n a counterclockwis e directio n fro m th e x axis , the n 20 l = 285.2°,which i s the angl e compute d i n par t (b) .

(e) Equation s (8.2 ) an d (8.3 ) ca n b e use d t o determin e a l an d a 2,respectively.

STRAIN GAG E ROSETTE S AN D DAT A ANALYSI S 267

FIG. 8.12. Mohr' s circl e fo r Exampl e 8.1.

8.5. Analytical solution for the equiangular or delta rosette

The procedure use d for the rectangular rosett e wil l also be used for the delt arosette. The O A axi s of the rosette , Fig . 8.1 , is taken coincident wit h the OXaxis o f reference. For thi s arrangement , then,

and

Equations (8.7) , (8.8), and (8.9 ) can be used with these trigonometric valuesto for m th e thre e simultaneou s equation s neede d i n orde r t o determin e ^ x,ey, and y xy. Thi s result s in



Substituting th e valu e o f e x give n b y Eq . (8.29 ) int o Eqs . (8.27 ) an d (8.28 )yields

Equations (a ) an d (b ) can b e solve d simultaneousl y for e y and j xy. Thus ,

The principa l strain s i n term s o f th e rosett e reading s ma y no w b edetermined b y substitutin g thes e value s o f £ x, e y, an d j xy int o Eq . (8.10) .Consequently,

The value s of el an d £ 2 given by Eq. (8.32) may no w b e substituted int oEqs. (8.2 ) an d (8.3 ) in orde r t o determin e th e principa l stresses , CT, and a 2,


where


in term s o f the rosett e strai n readings . Thi s gives

As with the three-element rectangular rosette , an y two of the three equation smust be use d in orde r to establis h 9^. If Eq. (8.36 ) is chosen, the n onl y thesign of sin 28 or cos 29 need be determined in order to establish the matchingquadrant.

As before , a chec k ca n alway s b e mad e b y sketchin g a Mohr' s circle .Three rule s for determining th e angle , 0 t, betwee n the O A axi s and th e O laxis wil l b e stated .

1. lies between 0° an d +90 °2. lies between 0° and -90 °

Determination of the principal stress directions

As with any rectangula r rosette , th e orientatio n o f the principa l strai n axe swill b e determine d analyticall y through th e us e o f Eqs . (2.34) , (2.35) , an d(2.36). Once again, note that the principal strai n axes and the principal stressaxes coincide , an d tha t th e angl e 9 i s measure d i n a counterclockwis edirection fro m th e positiv e O X axi s t o th e positiv e 0 1 axis , whic h cor -responds t o the direction o f el5 and therefor e t o a t.

Equations (2.34) , (2.35) , an d (2.36 ) ca n no w b e expresse d i n term s ofthe strai n readings o f the rosette . The y ar e

3.


Proof of rules

Figure 8.1 3 shows a Mohr' s circl e for the delt a rosette . Sinc e th e gag e axe sof the equiangular rosette are inclined at 120 ° (or 60° ) relative to each other ,the point s representin g th e correspondin g strain s o n th e circumferenc e ofMohr's circle are locate d a t th e vertice s of the equiangula r triangle ABC, a sindicated i n th e figure . A study of the diagra m reveal s that a s th e strain s £„,Eh, an d e c vary , th e triangl e AB C wil l rotat e abou t it s centroid , whic h i slocated a t th e cente r o f the circle.

Before continuing , however , attentio n i s draw n particularl y t o th eobservation that if one starts at point A and follow s around the circumferenceof Mohr' s circl e i n th e counterclockwis e direction, th e nex t statio n reache dwill b e point C . On firs t thought , this might appear t o b e an error , sinc e ingoing around th e rosette axes in the same direction , axis B follows axi s A, asshown i n Fig. 8.13a . The apparen t discrepanc y is caused b y the fac t tha t th e

FIG. 8.13. Gag e axe s and Moh r diagram for equiangular rosette.


FIG. 8.14. Cas e in which

angular displacement s ar e double d i n Mohr' s diagram . I f on e extend s th eaxis O C into the positio n OC ' shown in Fig . 8.13b , then th e reaso n fo r th erelative positions o f the point s A , B, and C on th e circumference of Mohr' scircle should b e clear .

If point A happen s t o fal l a t th e extreme left o f the circumference of thecircle, Fig . 8.14 , then, since the centroi d o f ABC lie s o n th e abscissa , C B i sat righ t angle s t o OA , whic h means tha t s c = e,, . Also, because A i s a t th eextreme lef t o f the circle, ea = £ 2, which is the algebraicall y smalle r principa lstrain. Fro m th e diagra m i t i s see n tha t 20\= +180° , an d therefor e6l = +90° , which substantiates Rul e 3(b).

If th e relativ e values of ea, e fc, and e c are no w change d s o tha t triangl eABC rotate s i n a counterclockwise direction from th e position i n Fig . 8.14 ,sb wil l becom e smalle r than s c and poin t A wil l mov e o n t o th e lowe r hal fof the circumferenc e of the circle . Under thes e conditions th e angl e 26 1 wil lbe between 0° and +180° , and 0 ^ will be between 0° and +90° , a s shown inFig. 8.1 5 and state d i n Rul e 1.

When th e triangl e AB C ha s finally rotated throug h 180° , point A wil lhave moved along the entire lower semicircumference of the circle and take nup the position show n in Fig . 8.16 , such that 20 1 = 0°, 9t — 0°, sa = EV, andsince A is again on the abscissa , E C = eb. This tim e ea > ec = e b and Rul e 3(a)is satisfied .

When th e strain s ar e furthe r altere d s o tha t th e continue d rotatio n ofthe triangl e causes point A t o mov e on t o th e semicircumference above th eabscissa, then , accordin g t o definition , 26 1 become s negativ e an d wil l li ebetween 0° and —180° . Strain e b will be larger than e c until A return s to th eposition corresponding t o e 2, where equality is again established between eband e c. This establishe s Rule 2 and is indicated in Fig . 8.17 .

and

FIG. 8.15. Cas e i n whic h


and

and


Example 8.2. Th e followin g readings wer e obtained fro m a three-element deltarosette:

(a) Determin e th e principa l strains .(b) Determin e O j analytically and chec k usin g the rule s listed.(c) Determin e CTJ and a 2, usin g v = 0.3 and E = 3 0 x 10 6 psi.

Solution, (a ) Th e principa l strain s ar e give n by Eq . (8.32).

(b) Equation s (8.36 ) and (8.37 ) wil l be used to determin e 9 l.

This value of tan 2 6 shows that th e angle may be in either the second o r fourt h

FIG. 8.17. Cas e i n which and


FIG. 8.18. Orientatio n o f the principa l strai n (an d stress ) axe s relativ e to th e referencecoordinates i n Exampl e 8.2 .

quadrant.

The numerato r o f sin 20 is

Since th e numerato r i s positive , si n 20 ha s a positiv e value , meanin g tha t29 ma y b e i n eithe r th e firs t o r secon d quadrant . Th e matchin g quadran tis the second , an d s o 2 9 is a second-quadran t value . Thus ,

The angle , 6 l, i s measure d i n a counterclockwis e directio n fro m th e O Aaxis t o th e 0 1 axis , a s show n i n Fig . 8.18 . T o chec k th e orientatio n o f01; Rul e 1 applies , sinc e e c>sb, an d s o 0 , lie s betwee n zer o an d +90° .This checks wit h Fig . 8.18 .

(c) Equation s (8.2 ) an d (8.3 ) ma y b e use d t o determin e a l an d <r 2,respectively.

Equations (8.34 ) an d (8.35 ) could als o hav e been use d to comput e a ^ and a 2.


8.6. Rosettes with four strain observations

These rosettes have been briefl y described earlier. The y are the T-rectangula rrosette, whic h ha s fou r gage s wit h axe s 45 ° apart , an d th e T-delt a rosette ,which has a fourth gag e at righ t angles to the axis of one of the equiangulargages. Th e equation s fo r th e principa l strains , g j an d £ 2, and th e principa lstresses, <T I an d a 2, i n term s o f the fou r gag e observations , wil l b e give n foreach configuration .

The rectangular rosette with four observations

Figure 8.1 9 show s thi s arrangement . Th e fourt h observatio n o f strai n i sredundant, bu t i t does provide a check since , within the limits of making thestrain readings ,

FIG. 8.19. Rectangula r strai n rosett e wit h fou r gages .

In thi s case , i t wil l b e simple r t o stat e th e expression s fo r th e principa lstrains, principa l stresses , an d th e angl e 0 l5 an d the n t o prov e the mgraphically with Mohr's diagram. Th e principa l strain s ca n be written as

Equation (8.40 ) ca n also be expressed a s

where


The directio n o f the principa l axe s ma y b e found fro m th e rati o o f thequantities unde r th e radica l suc h tha t

Insertion o f th e value s o f A an d B give n b y Eqs . (8.41a ) an d (8.41b) ,respectively, int o Eqs . (8.4 ) an d (8.5 ) produce s th e expression s fo r th eprincipal stresses . This give s

The rule s for determinin g 6 ^ are a s follows :

1. I f eb > £ d: 261 lie s between 0° an d + 180°0i lie s between 0 ° and +90 °

2. I f eb < ed: 20 i lie s between 0° and - 180 °$, lie s betwee n 0 ° and —90 °

3. I fe t = e, :

(b) I f

The abov e rule s an d Eqs . (8.39 ) throug h (8.44 ) ma y b e prove d b yrecourse t o Fig . 8.20 , whic h shows Mohr' s diagra m fo r thi s typ e o f rosette .Since th e direction s o f strai n measuremen t i n th e rosett e ar e incline dsuccessively a t 45° , th e radia l line s t o th e point s A , B , C , an d D , whic hrepresent th e strain s o f Mohr' s circle , wil l b e incline d successivel y a t twic e45°, or 90°. Therefore, A, B, C, and D will be located a t the corners o f a squareinscribed i n a circle .

Since th e intersectio n o f the diagonal s o f the squar e wil l coincid e wit hthe cente r o f the circle , and becaus e th e positio n o f the cente r o f the squar ecorresponds t o th e averag e o f the fou r corners , therefor e

Let u s no w determin e B , th e radiu s o f the circle , in term s o f ea, s b, ec,and ed, the horizontal distance s fro m th e ordinate throug h 0 t o the cornersof the square . Thi s wil l requir e th e followin g construction :

(a) If f

(b) Ifand

and

STRAIN GAG E ROSETTE S AN D DAT A ANALYSI S 277

FIG. 8.20. Mohr' s circle fo r rectangula r rosett e wit h fou r observations .

Let P b e th e cente r o f th e circl e an d dro p perpendicular s A m an d Bn ,respectively, fro m A an d B on t o th e absciss a a t m and n . Then fro m th eright-angled triangle s APm and BPn,

P = BPLPmA = LPnB

(radius o f the circle )90°

Since

LBPA = 90°LBPn = LPAm (90° - 20 t)

Therefore, triangle s AP m an d BP n are equal, s o tha t


Also,

The hypotenuse , PA , i s the radiu s o f the circle , and s o

The valu e o f ta n 2 9 j i s

T-delta rosette

If this rosette arrangement , shown in Fig . 8.21 is considered a s containing a

FIG. 8.21. T-delt a rosette .

delta rosett e wit h th e additio n o f a fourt h gag e whos e axi s D i s a t righ tangles t o th e axi s A , then , although th e fourt h observatio n i s redundant, avariety o f solutions ca n b e obtaine d utilizin g all four strai n readings .

Meier (1 ) gives a solution based o n th e method o f least squares , but it scomplexity i s rathe r a disadvantage . Th e followin g simpl e solutio n i stherefore presented, sinc e its reduction o f observed strains into terms of stresswill be ver y much easier.

Since th e averag e o f any tw o strain s measure d a t righ t angles give s theposition o f th e cente r o f Mohr' s circle , w e therefor e hav e fo r th e T-delt a

The


rosette th e quantit y

From Eq . (8.33a ) for the delt a rosette ,

Therefore, fo r the T-delt a rosette ,

Again, fro m Eq . (8.33b ) fo r th e delt a rosette ,

If is substituted fo re 3 from Eq . (b) , the n

The expressio n fo r B can no w b e writte n as

Again, from th e delt a rosette , th e valu e of tan 2 9 is given by Eq . (8.36) .Thus,

The valu e o f e give n b y Eq . (c ) ca n b e substitute d int o


Eq. (d) to produc e

The valu e o f tan 2 # given by Eq . (8.47 ) i s th e rati o o f th e quantitie s unde rthe radical in Eq. (8.46). The rule s for assigning the two value s of 9, obtainedthrough th e use of Eq. (8.47), to the correct principa l axe s is exactly the sam eas in the cas e o f the equiangula r (delta ) rosette .

For th e T-delt a rosette , th e value s o f th e principa l strain s ma y b eexpressed a s

Substituting th e value s o f A an d B , give n b y Eqs . (8.45 ) an d (8.46) ,respectively, into Eq . (8.48 ) yields

Insertion o f the value s o f A an d B , Eqs . (8.45 ) an d (8.46 ) respectively,into Eqs. (8.4) and (8.5 ) gives the expressions for the principal stresses. Thus,

Summary of equations

Three-element rectangular rosette:

When e t > (e a + e c)/2, 0 lies between 0° and +90° .Three-element delta rosette:

STRAIN GAG E ROSETTE S AND DAT A ANALYSI S 28 1

When e lies between 0° and +90° .Four-element rectangular rosette:

or

When e lies between 0 ° and +90° .Four-element delta rosette:

When E lies between 0 ° and +90° .

Directions of principal axes for all the summary equations are given by

8.7. Graphical solutions

If a numbe r o f rosett e observation s ar e t o b e analyzed , th e tas k ca n b etime-consuming and tedious . The data, however, can be reduced rapidl y andeasily with the use of a programmable calculato r or a small computer. Ther eare times , though, when graphical solution s may b e desirable, eithe r fo r thepurpose o f severa l peopl e checkin g eac h other , o r i f a compute r i s no tavailable.

For ou r purpose , th e discussio n o f graphica l method s o f solvin g th erosette equation s wil l b e confine d to th e genera l case . Thi s method , whichhas been put forward by McClintock (2) , applies to the general case in whichthe rosett e axe s may have any arbitraril y chose n axes , 9 ab, and Q bc, betweenthem. A rosette can always be represented diagrammatically so that 9 ab + 9bcis always less than 180° , as indicate d in Fig . 8.22 .

The objectiv e i s t o establis h Mohr' s circl e fo r strai n b y a ver y simpleprocedure. Th e followin g step s ar e employe d for finding the strai n circle :

1. Th e rosett e axe s are rearranged, b y extending them if necessary, so thatthey are arranged in sequence in order of ascending or descending strai nmagnitudes (algebrai c order) . Th e include d angl e betwee n th e axe s of


FIG. 8.22. Arbitrar y rosett e axes.

minimum an d maximu m strai n mus t b e les s tha n 180° . Fo r th erearranged rosette , the angl e betwee n th e maximu m an d intermediat estrain axes i s designated a s a , while the angl e between the intermediat eand minimu m strain axes is designated a s /?. For th e rearranged rosette ,compute a an d /? , then plac e th e intermediat e axi s i n th e vertica lposition an d la y off , on eithe r side , th e maximu m and minimu m axes.The possible arrangement s an d the values of a and ft are shown in Figs.8.23 and 8.24 . Note, in Fig. 8.24 , that th e maximum and minimu m axeshave bee n extende d an d th e angle s a an d / ? are als o show n belo w th ecrossover point . The reaso n fo r thi s wil l b e explaine d i n a subsequen tstep.

2. La y ou t a strai n scal e paralle l t o th e directio n o f the absciss a (whic hwill b e establishe d later) . Next , dra w i n ordinate s a t location s corres -ponding t o zer o strain , t: a, eh, an d E C. Thi s procedur e i s show n i n Fi8.25. While the strain values shown in Fig . 8.26 are positive , they mightall b e negative or som e positiv e and som e negative . Furthermore , th emeasured strains , Ea, eb, and e c may hav e any relatio n wit h each other .The strain s i n Fig . 8.2 5 hav e bee n plotte d i n sequenc e accordin g t omagnitude.

3. Whe n th e diagra m correspondin g t o Fig . 8.2 5 has bee n drawn, choos eany point , D, on th e ordinate correspondin g t o th e intermediate strai nvalue. From poin t D draw straigh t lines D E an d EF , makin g angles aand /? , respectively, with th e ordinat e o f intermediat e strain , t o mee tthe ordinates o f emax and £ rain a t £ an d F , respectively. Notice tha t ther eare tw o possibilitie s for drawing the line s DE an d DF , since th e anglesof a an d f t ca n b e measured fro m eithe r th e upwar d o r th e downwar ddirection o f the ordinat e o f intermediate strain , as show n i n Fig . 8.26 .The choic e i s governed a s follows :

(a) I n Fig . 8.23 , th e right-han d diagram s sho w th e strai n axe s i nsequence. Her e i t ca n b e see n tha t the y g o i n a counterclockwis edirection fro m e max to e int to £ min. In thi s case, the axi s of maximumstrain fall s t o th e right of the intermediate strain axis , and s o a an d/? are measure d fro m th e upwar d direction .


FIG. 8.23 .

(b) I n Fig . 8.24 , th e right-han d diagram s sho w th e strai n axe s i nsequence. Her e i t ca n b e see n tha t the y g o i n a counterclockwis edirection fro m e min to e int to e max. In thi s case, the axis of maximumstrain fall s t o th e lef t o f the intermediat e strai n axis , and s o a an d/? are measured from th e downward direction. This i s shown by th eextended line s in Fig . 8.24 .

4. Th e final step is to draw a circle through point s D , E, and F . This wil lbe Mohr' s circl e fo r strain. The abscissa , whic h ca n no w b e drawn in ,will pas s throug h th e cente r o f the circle , an d th e extrem e right-han dand left-han d positions o f the circumference will represent th e principalstrains e x an d e 2. Case (a ) fro m Fig . 8.2 3 and Cas e (b ) fro m Fig . 8.2 4are plotte d a s Figs . 8.2 7 and 8.28 , respectively.

The point s A , B , and C , which represen t th e strain s alon g th e rosett eaxes, can now be located o n th e circumference of the circl e according t o th efollowing tw o requirements :


FIG. 8.24 .

1. Th e magnitude s of the strain s e , an d 2. Th e sequenc e as we go along th e circumference of the circle . This must

correspond t o th e sequenc e i n th e physica l layou t o f the rosette . Fo rexample, i f the rosett e axes follow th e sequenc e A , B , and C when oneproceeds i n the counterclockwise direction, the same order mus t prevailas one goe s around Mohr' s circl e in the sam e sense .

Although ther e are tw o possibl e position s for each o f points A , B , an dC tha t wil l satisf y th e firs t requirement , the secon d requiremen t eliminateshalf o f them. This mean s tha t ther e i s only on e arrangemen t fo r th e point sA, B, and C on th e circumference of the circle.


FIG. 8.25.

FIG. 8.26.

Angle of reference, 9^

As soon as poin t A ha s been locate d o n th e circumferenc e of the circle , th eangle betwee n th e radia l line s t o poin t A an d t o E J will establis h th e angl e20l5 as shown in Figs . 8.2 7 and 8.28 . From thi s we can determin e th e angl e$! and locat e th e axis of el5 the algebraically large r principa l strain , relativeto th e A axi s of the rosette .

Principal stress determination

Once the magnitudes of the principal strains, e: an d e2, have been determined,then th e principa l stres s value s can b e computed fro m Eqs . (8.2 ) and (8.3) .


FIG. 8.29. Mohr' s circle for Example 8.3.

Thus,

Example 8.3. Three strain gages are arranged int o a rosette as shown in Fig.8.22. Th e followin g data ar e given : £ „ = -32 5 ustrain ; e b = 130 0 ustrain; e c =250 ustrain; 9 ab = 55° 75°. Construct a Mohr' s circle an d determine'^£2, and 9 1.

Solution. Sinc e s b > ec > ea, a rearrangemen t o f the rosett e axe s wil l produc ethe configuratio n show n i n Fig. 8.23b. The angles , a and , are


FIG. 8.30 . Computer-based data-aquisitio n system . (Courtes y o f Measurement s Group , Inc. )

Lay ou t a horizonta l strai n axis . O n thi s axis , erec t vertica l line s representin g£„, s b, an d B C. Since th e maximu m strai n axi s fall s t o th e righ t o f th e vertica lline representin g th e intermediat e axis , th e angles , a an d fj , wil l b e measure dfrom th e upwar d vertical , a s show n i n Fig . 8.23b . Th e constructio n o f Mohr' scircle i s shown in Fig. 8.29 . From the circle, the following values are obtained:

«, = 1550ustrain , i: 2 = -44 0 tistrain , 20 l = 151

Machine solutionsIn situation s involving the solutio n of large numbers of rosette equations , th eemployment o f machine s ca n b e ver y advantageou s fo r econom y o f bot htime an d cost . A number o f special-purpose computer s hav e bee n develope dover the years in order t o evaluate rosette data (3-7). Today, however , manyhand-held programmabl e calculators , som e wit h graphics display , are avail -able a t smal l cost . Fo r reductio n o f data fo r a fe w rosettes a t a time , thes eare quit e convenient . Smal l desk-to p computer s ar e als o no w availabl e a treasonable price s an d ar e foun d i n nearl y ever y organization . Thes e ca nreduce an d prin t ou t larg e quantitie s of data i n a shor t perio d o f time onc ethe ra w dat a hav e bee n entered .

The ultimat e aim, however, has bee n t o develop a combined computer -plotter-tabulator fo r direct connectio n t o strai n gages . Suc h system s (8) arenow availabl e tha t are dedicated solel y to th e acquisition of strain gage dat a(also transducers , thermocouples , etc.) . N o programmin g i s necessary ; th eoperator enters th e required constants an d th e machine automaticall y scan sthe tes t point s an d reduce s th e data . Suc h a syste m i s shown i n Fig . 8.30 .

Problems

8.1. A tensil e specime n ha s tw o gage s bonde d t o it s surface , on e aligne d alon g th elongitudinal axi s an d th e othe r perpendicula r t o it . Sho w tha t onl y th e


longitudinal gag e i s require d i n orde r t o determin e th e longitudina lstress.

8.2. I n a long, thin-walled pressure vessel , the hoo p stress is twice the longitudinalstress. I f the vesse l is made o f steel, determine the rati o o f the hoo p strai n t othe longitudina l strain .

The followin g rectangula r rosettes , illustrate d i n Fig . 8.2 , ar e bonde d t o stee l withgage A aligne d alon g th e x axis . Fo r th e reading s shown , i n uin/in , comput e th eprincipal strains , th e principa l stresses , thei r orientatio n relativ e to th e x axis , an dthe maximu m shear stres s a t th e point . Sketc h th e principa l stress elemen t an d it srelation t o th e x y coordinat e system . Chec k th e analytica l result s b y usin g Mohr' scircle.

a e, b s c8.3. 122 5 11 5 90 58.4. 39 5 -76 0 98 58.5. 100 0 100 0 100 08.6. -72 5 -28 5 53 08.7. -94 0 54 5 21 0

The followin g delt a rosettes , illustrate d i n Fig . 8.3 , are bonde d t o stee l with gage Aaligned alon g th e x axis . For th e reading s shown , in uin/in , compute th e principa lstrains, the principal stresses, their orientation relative to the x axis, and the maximumshear stres s a t th e point . Sketc h th e principa l stres s elemen t an d it s relation t o th exy coordinat e system . Check th e analytica l result s by using Mohr's circle .

8.8. -88 0 0 -88 08.9. 45 5 -20 5 11 08.10. -61 0 23 5 -10 58.11. 97 5 43 5 43 58.12. -72 0 -61 0 -18 58.13. A four-elemen t rectangula r rosette , illustrate d i n Fig . 8.4 , i s bonde d t o

aluminium wit h gage A aligned along the x axis . The strain observations, givenin uin/in , are th e following : e. a = -29 5 , e.h = -350 , e c = 550 , s d = 605 . UsinE = 10. 5 x 10 6 psi an d v = 0.33 , determine <T , and a 2-

8.14. I f a T-delt a rosette , illustrate d in Fig . 8.5 , i s applie d a t a poin t whos e strai nfield is identical to tha t o f Problem 8.13 , determine th e rosett e readings .

8.15. Solv e Proble m 8. 3 by graphica l methods .8.16. Solv e Proble m 8. 7 by graphica l methods .8.17. Solv e Proble m 8. 9 by graphical methods .8.18. Solv e Problem 8.1 1 by graphica l methods.

REFERENCES

1. Meier , J . H. , "Improvements i n Rosett e Computer, " SESA Proceedings, Vol. Ill ,No. 2 , 1946 , pp . 1-3 .

2. McClintock , F . A. , "On Determinin g Principa l Strain s from Strai n Rosette s withArbitrary Angles, " Lette r t o th e Editor , SESA Proceedings, Vol. IX, No . 1 , 1951,pp. 209-210.

0

55

0

0

5

0

0 0


3. Hoskins , E . E . and R . C. Olesen , "A n Electrica l Compute r fo r th e Evaluatio n ofStrain Rosett e Data, " SESA Proceedings, Vol . II , No . 1 , 1944, pp. 67-77 .

4. Meier , J. H. and W. R. Mehaffey, "Electronic Computin g Apparatus for Rectangularand Equiangula r Strai n Rosettes, " SESA Proceedings, Vol . II , No . 1 , 1944 , pp .78-101.

5. Murray , W . M. , "Machin e Solutio n o f th e Strai n Rosett e Equations, " SESAProceedings, Vol. II , No . 1 , 1944, pp . 106-112 .

6. Bassett , W . V., Helen Cromwell , an d W . E . Wooster, "Improve d Technique s an dDevices fo r Stress Analysi s with Resistanc e Wire Gages," SESA Proceedings, Vol.Ill, No . 2 , 1946 , pp. 76-88.

7. Williams , S . B. , "Geometr y i n th e Desig n o f Stres s Measuremen t Circuits ;Improved Method s Throug h Simple r Concepts, " SESA Proceedings, Vol . XVII,No. 2 , 1960 , pp. 161-178 .

8. "Syste m 4080, " Bulleti n 235-B , Measurement s Group , Inc. , P.O . Bo x 27777 ,Raleigh, N C 27611 , 1985 .

STRAIN GAGE ROSETTES AND TRANSVERSESENSITIVITY EFFECT

9.1. Introduction

In Chapter 7 the effect o f transverse sensitivity on a strain gage measurementwas considered . I t wa s pointed ou t tha t th e tota l uni t resistanc e chang e i na gag e was made u p o f two parts : namely, (1) the uni t resistanc e chang e i nthe gage' s axia l direction , an d (2 ) th e uni t resistanc e chang e norma l(transverse) to the gage axis. Furthermore, th e axial strain sensitivity , Fa, andthe norma l strai n sensitivity , F n, ar e define d b y Eqs . (7.3 ) an d (7.4) ,respectively. The transvers e sensitivity of the gage is then taken a s the rati oof th e norma l sensitivit y t o th e axia l sensitivity , or K = F n/Fa.

It was also stated tha t i f a strain gage is used under conditions differin gfrom thos e o f calibration, a n erro r wil l exis t in th e indicate d valu e o f axialstrain. Thus , i f th e strai n i s measure d b y a singl e gag e unde r biaxia lconditions, th e erro r wil l depen d o n bot h th e valu e o f th e transvers esensitivity factor , K , an d th e rati o o f the norma l strai n t o th e axia l strain ,£„/£„. Fortunately , thi s erro r i s usuall y rathe r smal l an d ca n b e neglected .For instance , i f the norma l strai n doe s no t excee d th e axia l strai n an d th evalue o f K i s 3 percen t o r less , the n th e maximu m erro r wil l no t excee d4 percent. Thi s i s easily verified b y usin g Eq. (7.29) to comput e th e error .

When strai n gag e rosette s wer e examine d i n Chapte r 8 , th e effec t o ftransverse sensitivit y was no t take n int o account . I n general , though , theeffect o f transverse sensitivity should b e considered when using strain gage sin a biaxia l stres s field (1-4). I f it ca n b e demonstrate d tha t th e transvers eeffect i s negligible, then th e expressions i n Chapter 8 may b e used ; if, on th eother hand , th e effec t i s not negligible , then th e expressions fo r determiningthe actua l strai n tha t wil l b e developed her e should b e used.

9.2. Two identical orthogonal gages

Figure 9. 1 show s tw o identica l gage s mounte d a t 90 ° to eac h other . Th elongitudinal axi s o f gage a is aligned alon g axi s OX , whil e the longitudina laxis of gage b is aligned along axis O Y. The strain in the axial, or longitudinal ,direction o f a gage i s represented b y ea, while the strai n norma l (transverse )to th e gag e axi s i s represented b y £„ . In orde r t o identif y th e gag e tha t i s

9


FIG. 9.1. Tw o identica l strai n gage s aligned alon g th e O X an d O Y axes .

subjected t o strai n and th e strai n direction, a doubl e subscrip t wil l b e used .The firs t subscrip t denote s th e strai n directio n whil e the secon d subscrip tidentifies th e gage . Fo r instance , i f a strai n i s designated £ aa, th e firs t sub -script show s the strai n i s in th e axia l direction o f th e gage , an d th e secon dsubscript identifie s th e gag e a s gag e a . The strai n E nb is the transvers e strai non gag e b .

Since th e gage s ar e identical , they hav e equa l axia l strai n sensitivities ,Fa, equal manufacturer' s gag e factor , G f, an d equa l transvers e sensitivities,K. When th e gages are subjected t o an unknow n biaxial stress field, the unitchange i n resistanc e fo r eac h gag e i s

The right-han d side s o f Eqs . (9.1 ) and (9.3 ) may b e equated , an d als o th e

where c' aa = indicated strai n fo r gag e ae'ab = indicate d strai n for gag e b

Using Eq . (7.13) , we can als o writ e for each gag e

STRAIN GAG E ROSETTE S AN D TRANSVERS E SENSITIVIT Y EFFEC T 29 3

right-hand side s o f Eqs. (9.2 ) and (9.4) . This give s

Dividing both side s o f each equatio n b y G F produces

Substituting th e valu e o f FJG F give n b y Eq . (9.7 ) into Eqs . (9.5) and (9.6)yields

Since th e gage s ar e orthogonal , w e know tha t

Substituting th e value s o f th e transvers e strain s give n b y Eqs . (a) an d (b )into Eqs . (9.8) and (9.9) , respectively , results in

From Eq. (7.21)mit is seen that


Equations (9.10 ) an d (9.11 ) ar e no w expresse d i n term s o f th e strain sin th e axia l directio n o f eac h gage , an d s o th e firs t subscript , a , ca n b edropped. Th e apparent , o r indicated , strain s are no w expresse d i n term s ofthe actua l strains i n the axia l directions of the gages . Thus,

If Eqs . (9.12) and (9.13 ) are solve d simultaneously, the actua l strains , eaand £;, , will b e determined i n term s o f the apparen t (indicated ) strains. Thisoperation gives

Equations (9.14 ) and (9.15 ) show that, in order t o determine the actua lstrain i n a desire d direction , tw o gage s mus t b e used . On e gag e i s aligne din th e desire d direction ; th e othe r gag e i s mounted norma l t o th e directionof th e desire d strain . I f on e choose s t o ignor e th e transvers e sensitivit y(K = 0) , then Eqs . (9.14 ) and (9.15 ) reduce t o e a = e'a and e h = e'h.

9.3. Two different orthogonal gages

The cas e ca n no w b e considered i n whic h there ar e tw o orthogona l gages ,each wit h a differen t F a, GF, and K . Agai n the gage s ar e arrange d a s shownin Fig . 9.1 . Fo r gag e a w e hav e th e axia l strai n sensitivity , F aa, th emanufacturer's gage factor, GFa, and the transverse sensitivity factor, Ka. Th ecorresponding value s for gage b are F ab, GFb, and K b. W e can us e Eqs . (9.5)and (9.6 ) t o writ e the apparen t strains , s' aa and e' ab, in term s o f th e actua lstrains, £ aa and e ah, and th e individua l gage factors . Thus,

STRAIN GAG E ROSETTE S AN D TRANSVERS E SENSITIVITY EFFEC T 29 5

Equation (7.21 ) also show s that , for each gage ,

Substituting the values of Faa/GFa an d F ab/GFb give n by Eqs. (9.18) and (9.19 )into Eqs . (9.16) an d (9.17 ) result s in

From Eqs . (a ) an d (b ) i n Sectio n 9.2 , the norma l strain s ca n b e expressedin term s o f the axia l strains ; tha t is , ena = eab and E nb = saa. Replacin g th enormal strain s with axia l strains i n Eqs . (9.20) and (9.21) , we obtain

Again th e firs t subscript , a , for eac h strai n ca n b e droppe d sinc e th estrains ar e i n th e axia l directio n o f each gage . Th e apparen t strains , e' a ande'b, in term s o f the actua l strains , e a and e fc, wil l now b e

If Eqs . (9.24 ) an d (9.25 ) ar e solve d simultaneously , th e tru e (actual )strains wil l b e expresse d i n term s o f the apparen t (indicated ) strains . Thi s


operation gives

If K a = Kb = K, the n Eqs . (9.26 ) and (9.27 ) reduce t o Eqs . (9.14 ) an d(9.15). Furthermore , i f the transvers e sensitivit y factor , K , i s take n a s zero ,then £ „ = £' „ an d e b = r,' h.

9.4. Three-element rectangular rosette

A three-element rectangular rosette, with all gages different , wil l be examinednext. Th e rosett e i s show n i n Fig . 9.2 . The apparen t strain s fo r eac h gag e

FIG. 9.2. Three-elemen t rectangula r rosette .

can be developed and expressed in the same manner a s those leading to Eqs .(9.20) an d (9.21) . The thre e equations ar e

STRAIN GAG E ROSETTE S AND TRANSVERS E SENSITIVIT Y EFFEC T 29 7

The norma l strain s fo r eac h gag e mus t b e expresse d i n term s o f the axia lstrains. Fo r gage s a and c we have s na = sac and s nc = eaa, since thes e gage sare 90 ° t o eac h other . Fo r gag e b , however , a Mohr' s circl e o r th etransformation equatio n give n b y Eq . (2.32 ) mus t b e use d i n orde r t odetermine the normal strain . Th e transformation equatio n i s

Before proceedin g t o determin e e nb, which is 90° from gag e b and 135 °from gag e a , the shearin g strai n in th e plan e mus t b e determined. I n orde rto d o this , Eq . (9.31 ) i s use d wit h s and — 45° .Using thes e values,

Equation (c ) reduces t o

The norma l strain s ar e no w i n terms o f the axia l strains.Equation (9.32 ) could als o have been obtaine d b y considering the first

strain invariant . That is,

constan tor

The value s o f th e norma l strain s ca n no w b e substitute d int o thei r

From Eq . (a),

Equation (9.31) is once again used with sx, = s nb, 9 = 135° , and the valueof y xy/2 give n b y Eq . (b) . Thus ,


respective equations ; tha t is , Eqs. (9.28) , (9.29) , and (9.30) . Thi s gives

The firs t subscript , a , for each strai n ca n no w b e dropped, an d s o Eqs . (d) ,(e), and (f ) becom e

Equations (9.33) , (9.34) , an d (9.35 ) ma y b e solve d simultaneousl y fo rthe actua l strains , s a, e fc, and e c. This operatio n result s i n

When th e transvers e sensitivitie s of gage s a an d c ar e th e same , the n


Ka = Kc = X ac. Fo r thi s condition Eqs . (9.36) , (9.37) , and (9.38 ) becom e

The actua l strains , e a, eb, and e,. , have bee n determine d b y takin g intoaccount th e transverse sensitivitie s of the gages making up the three-elementrectangular rosette . I n orde r t o determine the principal strains , the principalstresses, an d th e direction s o f th e principa l stres s (o r strain ) axes , th eequations develope d i n Section 8. 4 can b e used . The appropriat e equation sfrom tha t sectio n wil l b e identified and renumbere d here .

The principa l strain s in terms of the gage values are given by Eq. (8.19).The expression i s

The principa l stresse s are give n by Eqs. (8.21) and (8.22) . These areare

If K a = Kb = Kc = K , the n Eqs. (9.39), (9.40), and (9.41 ) reduc e toto


In orde r t o establis h the angula r relationship , 0lf betwee n the 0 1 axi sand th e O X axis , Eqs . (8.23), (8.24), and (8.25 ) were given. An y tw o o f th ethree equations are needed in order to establish 0l. The three equations are

Graphical method s could , of course, also b e used .

Example 9.1. Th e followin g dat a ar e give n fo r a three-elemen t rectangula rrosette:

f.' = 145 0 uin/in, K a = — 6.0 percen t

s' — —96 0 uin/in, K b = 2. 5 percen t

r/ = 87 0 uin/in , K c = — 5.0 per cen t

(a) Determin e th e actua l strains , t. b, and (b) Determin e th e principa l strains , E I an d (c) Wha t erro r exist s i f th e principa l strain s ar e compute d usin g apparen t

strains rathe r tha n actua l strains ?

Solution, (a ) Th e actua l strain s may b e computed usin g Eqs . (9.36), (9.37), and(9.38).


(c) Equatio n (9.45 ) i s agai n used , bu t th e strain s wil l b e th e apparen tstrains.

The principa l strains computed by usin g the apparen t strain readings areslightly more than 7 percent lower than the actual principal strains.

9.5. The equiangular or delta rosette

The equiangula r o r delt a rosett e i s shown i n Fig . 9.3 . The apparen t strain sfor eac h gag e are given by Eqs. (9.28) , (9.29), and (9.30) . The transformatio nequation, Eq . (9.31) , must b e used t o determine e na, e nb, and e nc, the strain snormal t o gage s a , b, and c .

The X axi s is established alon g gage a, and since these two axes coincide, In orde r t o determine th e strain s normal t o th e gages , th e values of

and xy/2 mus t firs t be computed . Tw o expression s ar e obtaine d throug hthe us e o f the transformatio n equation ; thes e ar e solve d simultaneousl y for

E nd The firs t equatio n use s gag e b . Here s' x = £ab, 0 = 120° , an d

Substituting these value s into Eq . (9.31 ) gives

(b) Th e principa l strains are give n b y Eq . (9.45) .


FIG. 9.3. Th e equiangula r or delt a rosette .

The secon d equatio n use s gag e c . Her e E' 240°, an d Substituting thes e value s int o Eq . (9.31 ) give s th e secon d independen tequation tha t i s needed .

Solving Eqs . (a ) and (b ) simultaneously for e v and y xy/2 produce s

Since the values of £x, ey, and 7^/ 2 ar e how known in terms of the actua lstrains alon g eac h gag e axis , th e strain s norma l t o eac h gag e ma y no w b edetermined throug h th e us e o f th e transformatio n equation , Eq . (9.31) .Because & na i s along th e Y axis, its value is the same a s e y. Thus, s na is written


as

The transvers e strain , e nb, ha s a n angl e o f 210° relativ e to th e X axis .Letting £ x. = £nb and 6 = 210°, the transformatio n equatio n i s

If £ x = £ aa and th e value s o f e ^ and y xy/2, give n b y Eqs . (9.51 ) an d (9.52) ,respectively, are substituted int o Eq . (d) , the n

The require d norma l strain s are given by Eqs. (9.53), (9.54), an d (9.55) .As pointe d ou t i n Sectio n 9.4 , onc e e na wa s establishe d th e firs t strai ninvariant coul d b e used t o determin e e nb and e nc. Thus,

If E X = £ aa and th e value s o f £ y an d y xy/2, give n b y Eqs . (9.51 ) an d (9.52) ,respectively, are substitute d int o Eq . (c) , the n

Finally, the transverse strain, £nc, has an angl e of 330° relative to th e Xaxis. Letting e x. = e nc and 6 = 330°, the transformation equation i s

or

Also,

or

The expression s fo r th e norma l strain s ar e no w give n i n term s o f th eaxial strains. Substitutin g the value s of eno, £nb, and e nc, given by Eqs. (9.53) ,(9.54), an d (9.55) , respectively , into Eqs . (9.28) , (9.29) , an d (9.30 ) wil l giv ethe indicated strain s in terms o f the axial strains a t each gag e location. Also ,


since only axial strains are involved, the first subscript, a, for each strain ca nnow b e dropped . Carryin g ou t thes e substitution s gives

If tw o gage s hav e th e sam e transvers e sensitivit y factor , the n thes eexpressions ca n b e simplifie d accordingly. I f gages a an d c hav e th e sam etransverse sensitivit y factor, the n K a = K c = Kac. Unde r thes e condition sEqs. (9.59) , (9.60) , an d (9.61 ) reduce t o

Equations (9.56) , (9.57) , and (9.58 ) may no w b e solve d simultaneouslyfor £„ , eb, an d e c. This yields

STRAIN GAG E ROSETTE S AN D TRANSVERS E SENSITIVIT Y EFFECT 30 5

The denominator s o f Eqs. (9.62) , (9.63), an d (9.64 ) ar e alike . However , Eq.(9.63) can b e simplified furthe r b y finding common factor s i n the numerato rand denominato r (1) .

If al l gage s hav e th e sam e transvers e sensitivit y factor , the n th eexpressions simplif y further . Thus , fo r Ka = K b = Kc = K, w e have

The actua l strains , sa, z b, an d e c, have been determine d b y accountin gfor th e transvers e sensitivities of the gages making up th e delta rosette . Th eequations develope d i n Sectio n 8. 5 can no w b e use d t o determin e (1 ) th eprincipal strains , (2 ) th e principa l stresses , an d (3 ) th e orientatio n o f th eprincipal axe s relativ e t o th e origina l coordinat e system . Fo r eas e o f use ,the pertinen t equation s wil l be repeated an d renumbere d here.

The principa l strains , give n by Eq . (8.32) , ar e

T h e p r i n c i p l e s t r e s s e s , g i v n b y E q s . ( 8 . 3 4 ) a n d ( 8 . 3 5 ) , a r e


Example 9.2. I f the strai n gage s use d i n Exampl e 9. 1 ar e arrange d i n a delt arosette, a s shown in Fig . 9.3 , determine the apparen t strain s indicate d b y eac hgage whe n subjected t o th e stres s field of Example 9.1.

Solution. Th e followin g actua l strains have been determine d i n Exampl e 9.1:

«„= 153 2 uin/in, O = 0°

ec = 959 uin/in, 9 C = 90°

Since th e gage s i n th e delt a rosett e ar e arrange d a t 9 a = 0°, 9 b = 120° , an d9C = 240°, the actual strain s in these directions are required before the apparen tstrains ca n b e computed. I n orde r t o comput e th e actua l strains , th e shearin gstrain must first be determined through the use of Eq. (9.31), the transformatio nequation. Fo r thi s purpos e E X = 153 2 uin/in, e, y = 959 uin/in, e x. = «45 =-1041 uin/in , and 0 = 45°. Equation (9.31 ) is

Thus,

From this ,

In orde r t o establis h th e angle , # 15 between th e 0 1 axi s an d th e O Xaxis, Eqs . (8.36) , (8.37) , and (8.38 ) are used . Any two o f the thre e equationsare needed i n order t o establish 9 l.

45- 1041 uin/in


The actua l strai n i n th e axia l directio n o f gage b can b e determine d b y usingthe transformatio n equatio n wit h 9 = 120°.

= 308 3 uin/in

The actua l strai n i n th e axia l directio n o f gage c may b e determine d b y usingthe transformatio n equatio n wit h 9 = 240°.

= - 878 uin/in

The apparen t strains , e' a, 4 , an d e.' c ar e give n b y Eqs . (9.56), (9.57) , an d(9.58), respectively. The value of e'a, however, must be the same for both rosettes ,since bot h ar e aligne d alon g th e sam e axis . Thus ,

s'a = 145 0 uin/in

The apparen t strai n reading s o n gage s b and c for thi s rosett e ar e quit edifferent fro m thos e for the rectangular rosette . Whethe r o r no t thes e values ar ecorrect ca n b e verifie d b y usin g Eqs . (9.59), (9.60) , an d (9.61 ) t o comput e th eactual strain s a t eac h gage , whic h are alread y known .

Problems

9.1. Tw o identica l strai n gage s ar e arrange d a s show n in Fig . 9.1. The transvers esensitivity factor is K = —0.026 . If the indicated strain s are e' a = 76 5 uin/in an de'b = 255 uin/in, determine (a ) the true strain in each direction , an d (b ) the erro rif the transvers e sensitivit y factor i s ignored .


9.2. Tw o differen t gage s ar e bonde d t o a thin-walle d pressure vessel . Gage a withKa = 3.0 percen t i s aligne d i n th e longitudina l direction , whil e gag e b wit hKb = —3. 9 percent i s aligne d i n th e hoo p direction . Th e followin g dat a ar eavailable fo r the vessel : internal pressure = 80 0 psi, diamete r = 6 0 in, and wal lthickness = 1.2 5 in. Determine th e indicate d strains.

9.3. Tw o lik e gages wit h K = — 1.7 percent ar e bonde d alon g th e principa l stres saxes o f a roun d shaf t subjecte d t o pur e torsion . Determin e th e percen t erro r ifthe transvers e sensitivit y facto r i s ignored .

The give n data fo r Problem s 9. 4 through 9.8 , wit h al l strain s i n uin/i n ar e fo rthree-element rectangula r rosettes . Determin e th e tru e strain s fo r eac h rostt e an dthen comput e th e erro r i f the transvers e sensitivity facto r ha d bee n ignored .

9.4.9.5.9.6.9.7.9.8.

F,'a

960-565

135-355

800

Ka, percen t1.31.52.0

-2.01.8

l-h150

-760-820

460800

Kh, percen t0.7

-0.51.01.51.8

e,[.445315865

-715800

Kc , percent1.31.52.02.01.8

The give n data for Problem s 9. 9 through 9.13 , with al l strains i n uin/in , ar e fo r threeelement-delta rosettes . Determine th e true strain s fo r each rosett e an d the n comput ethe erro r i f the transvers e sensitivit y facto r ha s bee n ignored .

f,'a K a, percen t e.' b K h, percen t e' c K c, percen t9.9. 44 5 3. 0 -22 5 1. 0 -56 5 -3. 09.10. 81 0 3. 0 40 5 1. 0 -19 5 3. 09.11. 1000 1.8 1000 1.8 1000 1.89.12. 800 -1.3 0 0.7 800 -1.39.13. -565 2.0 260 2.0 695 2.0

9.14. A three-element rectangula r rosette i s bonded t o a steel specimen, a gage factorof 2. 0 is set o n th e strai n indicator , an d th e recorde d dat a ar e a s follows :

Gage factorK, percentStrain, fiin/in

Gage a2.151.8

200

Gage b2.051.0

1608

Gage c2.151.8

850

(a) Correc t fo r th e gag e facto r setting.(b) Determin e th e tru e strains .(c) Comput e th e principa l strains .(d) Comput e th e principa l stresse s an d thei r orientation relativ e to th e axi s of

gage a . Sketch th e element .(e) Comput e th e maximu m shearin g stres s a t th e point .

REFERENCES

"Errors Du e t o Transvers e Sensitivit y in Strai n Gages, " TN-509 , Measurement sGroup, Inc. , P.O . Bo x 27777 , Raleigh, NC 27611 , 1982 .

1.


2. Dove , Richard C . and Pau l H . Adams , Experimental Stress Analysis an d MotionMeasurement, Columbus , OH , Charle s E . Merrill Books , Inc. , 1964 , pp. 243-251 .From Experimental Stress Analysis and Motion Measurement b y Richard C. Dov eand Pau l H . Adams . Copyrigh t © 1964 . Reprinte d by permission o f Merrill, animprint o f Macmillan Publishin g Company .

3. Dally , James W . and Willia m F. Riley , Experimental Stress Analysis, 2n d edition ,New York , McGraw-Hill , 1978 , pp . 328-329 . Materia l i s reproduce d wit hpermission o f McGraw-Hill, Inc .

4. Handbook o n Experimental Mechanics, edited by Albert S. Kobayashi, Englewoo dCliffs, Prentice-Hall , 1987 , pp. 52-54 .

10STRESS GAGES

10.1. Introduction

There ar e a numbe r o f situations i n whic h on e wishe s to determin e eitherthe norma l o r shearin g stres s i n som e particula r directio n withou t bein grequired t o establish the complete stat e of stress at an y particular point . Fo rexample, i f it i s desired t o evaluat e th e radia l forc e a t a give n cross sectio nof a n aircraf t propelle r blade , thi s ca n b e accomplishe d b y multiplyin g theaverage radia l stres s b y th e are a o f cross sectio n o f the blade . Thi s sound ssimple, bu t i t ma y involv e the us e o f a grea t dea l o f equipment , especiallyunder dynami c conditions whe n all strain observations, at al l gage locations ,may hav e t o b e made simultaneously .

The standar d procedur e fo r approachin g thi s proble m woul d b e t omount rosett e gage s a t eac h o f the desire d station s aroun d th e blade , an dthen calculate , fro m th e thre e strain s indicated b y each rosette , th e corres -ponding stres s i n th e radia l direction , an d henc e th e radia l forc e a t thi ssection. Thi s involve s considerable computation , and s o one can appreciat ethat a gage whose response i s directly proportiona l t o normal stres s wil l no tonly reduc e th e amoun t o f instrumentatio n required , bu t i n additio n wil lreduce th e amoun t o f calculation involve d i n determinin g th e fina l result .Thus, usin g a stress gag e rathe r tha n a three-elemen t rosette a t each statio nreduces the numbe r o f channels from thre e t o one .

10.2. The normal stress gage (1)

A much simple r method , however , involves the us e of the stres s gage, whichhas the capacity to measure tw o strains at righ t angles and to combine the min th e prope r proportion s s o tha t it s indication , whe n multiplie d b y th eproper constant , gives the valu e of the stres s in the give n direction. Th e us eof a stres s gag e reduce s th e amoun t o f instrumentatio n require d b y two -thirds, and th e time involved in data reductio n b y even mor e tha n that .

Theory of the normal stress gage

Let u s conside r th e reference s axes, O A an d ON , whic h are a t righ t angle son a free surface in a two-dimensional stress system (Fig. 10.1) . The followin g

STRESS GAGES 311

FIG. 10.1. Reference s axe s OA an d ON .

relations exist between normal stress an d linea r strain:

Simultaneous solution of Eqs. (10.1) an d (10.2 ) for a a in terms of the strainsgives

In passing , on e should observ e that th e directions of the axes , OA an dON, althoug h 90° apart, have no particula r inclination s with respect t o th edirections of the principa l axes .

Let us now examine the expression for the indication from a strain gage,which wil l be a dimensionles s quantit y in term s o f R/R. Sinc e Eq. (10.3 )involves strains in two perpendicula r directions , on e can refe r t o Chapte r 7on latera l effect s i n strain gages for a general expression for the uni t changein resistanc e o f a strain gage . From Eq . (7.13) we have the genera l relationthat


where F a and K ar e constant s fo r th e gage . Furthermore , from Eq . (7.21) ,

where v0 is the Poisson ratio o f the materia l upon whic h the gage facto r wasdetermined. Substitutin g the valu e of F a into Eq . (7.13 ) produces

Equations (7.13 ) an d (10.3 ) indicat e tha t R/ R i s proportiona l t o(£„ + Ke n) an d a a i s proportiona l t o (r, a + ve n). Therefore, i f K = v , then

This means that , in orde r fo r th e gag e t o respon d directl y in proportion t othe norma l stres s i n th e directio n of OA , K mus t b e equa l t o v.

From Eq. (10.4) , we can find the valu e of (£ fl + Ks n) an d the n substitutethis valu e into Eq . (10.3 ) for (e fl + vej . Thi s gives

We wil l no w conside r certai n grid configurations , for bot h wir e gage sand foi l gages , whic h posses s characteristic s suitabl e fo r stres s gages .Fortunately, strai n gages wit h meta l sensing elements lend themselves ratherwell t o fulfillin g th e requirement s fo r stress gages.

Single round wire in an L configuration

The L is the simples t configuration , as show n i n Fig . 10.2 . It consist s o f tw ostraight part s o f roun d wire , a t righ t angles , s o proportione d tha t th efollowing rati o exists :

where v is the Poisso n rati o of the material upon whic h the gage i s to b e usedas a stres s indicator .

The following assumptions will be made in the analysis for the transvers esensitivity factor :

1. Th e chang e i n directio n fro m th e longe r piec e o f wir e t o th e shorte rpiece o f wire takes place ver y abruptly.

STRESS GAGE S 313

FIG. 10.2. Singl e round wir e i n a n L configuration.

2. Th e latera l effec t o f the wir e i s zero, due t o th e lac k o f efficiency o f th ebonding agen t i n thi s direction . (Th e reade r shoul d appreciat e tha t thi smay not b e true for a slender stri p of foil, in which the width of the elementmay b e severa l time s th e thickness. )

If a gage of this configuration is subjected to strains e fl and e n, the chang ein gag e resistance , R, i s

where k = resistance pe r uni t length o f the wireS, = strain sensitivit y of the wire

The valu e of R/R i s then

The transvers e sensitivit y factor, K , fo r a gag e o f the L configurationmay b e computed b y usin g Eq . (7.14) , which is

The valu e o f R/R give n b y Eq . (b) , bu t subjec t t o th e restriction s o n th e


strains o f Eq. (7.14) , may b e substitute d int o Eq . (7.14) . Therefore,

Equation (c ) show s tha t R/ R wil l b e proportiona l t o a a whe n thi sconfiguration is used as a stress gage for the normal stress in the OA direction .However, due t o th e amount o f wire required t o make a practica l gage , thi sform wil l usuall y occupy to o muc h space . O n thi s account , i t i s customar yto arrang e th e wire , or foil , i n a mor e compac t gri d form.

Two orthogonal gages of different resistances

Let u s imagine tha t tw o strai n gages wit h resistances R a an d R n hav e bee ninstalled i n directions paralle l t o th e referenc e axes, O A an d ON , whic h ar eat righ t angles . Thes e tw o orthogona l gages , connecte d i n series , hav e acombined outpu t expresse d b y Eq . (7.34) . I t i s

where the subscript s a and n refer t o th e gage s which are paralle l t o th e O Aand O N axes , respectively, and / ? = R n/Ra.

It wil l b e assume d tha t th e gag e factor s fo r bot h gage s hav e bee ndetermined o n the sam e calibratin g device so tha t v 0 is the sam e for the tw ogages. Equatio n (7.34 ) can b e simplied t o

where

Rearrangement o f Eq . (10.7 ) gives

STRESS GAGES 31 5

or

It i s seen tha t

so

Furthermore, i f

then

This concep t cover s al l values of gage facto r an d transvers e sensitivityfactor, whic h may b e differen t fo r both gages , fo r an y particula r valu e of .However, since commercially available gages may not be obtainable t o satisf ythe require d value s of v and , it ma y b e necessar y t o see k a compromise ,or, possibly , some othe r method .

The above relations ar e somewhat complicated, so a first approximationmay b e examined . I f th e tw o gag e factor s ar e nearl y equa l an d th e tw otransverse sensitivit y factor s ar e als o nearl y equal , on e ca n mak e th eapproximation o f equality withou t causin g ver y muc h erro r (possibl y les sthan th e erro r i n the valu e of the modulu s o f elasticity) by using th e valuesof th e gag e facto r an d th e transvers e sensitivit y factor fo r th e gag e i n th edirection o f th e O A axis , especiall y i f th e strain , £ a, i s somewha t large rnumerically tha n £„ . According to thi s approximation ,

Consequently,

Using thes e values , Eq . (7.34) becomes

Two specia l case s o f interest can no w b e examined. The firs t case take sK = 0 fo r bot h gage s an d (G F)a = (G F)n = G F. Usin g thes e value s i n Eq .(10.12) produce s

where f t = R n/Ra ca n hav e an y valu e betwee n 0 an d 1 . However, i f fl = v,the tw o gage s i n serie s represen t a singl e stres s gage . Fo r th e secon d cas ewhen (G F)„ = (G F)n = G F and K a = K n = K = v , the gage in the direction ofthe O A axi s is a stress gag e b y itsel f an d th e secon d gag e i s not required . I nthis case / ? = 0 . Consequently, usin g these value s in Eq . (10.12 ) gives

The proble m i s t o selec t gage s wit h appropriat e gag e factor s an dtransverse sensitivit y factors and the n t o establis h a suitabl e valu e o f th eratio /? = R n/Ra. I t wil l be best to commence by choosing a pair of gages withequal gage factors and equa l transverse sensitivity factors. If gages are chosensuch tha t th e transvers e sensitivit y factor s ar e no t quit e equal , a n averag evalue migh t b e use d withou t causin g seriou s error , sinc e thes e factor srepresent a secondar y effect .

10.3. The SR-4 stress-strain gage

The stress-strain gag e show n in Fig . (10.3 ) was produced a s a specia l item,but ha s sinc e bee n discontinue d (2) . I t is , however, an interestin g concep tand worth examining. The gage consists of a pair of foil strai n gages mounted


Rearranging gives

From Eq . (10.12 ) one ca n se e that if

then

STRESS GAGES 317

FIG. 10.3. Stress-strai n gage. (From ref. 2.)

at righ t angle s o n a commo n carrie r an d possessin g a rati o o f resistancessuch tha t

where v is the Poisso n rati o of the materia l upon whic h the stres s gage is tobe used . Th e tw o gage s ar e arrange d wit h three-lea d connection s s o tha teither gri d ca n b e use d independentl y t o measur e th e strain s i n th e tw operpendicular directions , a- a an d n-n. I f th e tw o gage s ar e connecte d i nseries, however, they can be used together to indicate stress in the a-a direction .

Since the Poisson ratio of the material on which the gage might be usedcould hav e man y differen t values , the gage s wer e limited t o tw o particula rvalues, namely , 0.28 fo r stee l an d 0.3 3 fo r aluminu m an d it s alloys . Thesegages wer e als o furnishe d i n temperatur e compensation s fo r us e o n mil dsteel, stainles s steel , and aluminum . The resistance s o f al l larg e grid s wer efixed at 35 0 ohms, whil e the resistance s o f the smalle r grid s wer e either 98or 11 5 ohms, to correspon d wit h the differen t Poisso n rati o value s for stee land aluminum.

The use r was supplied wit h the followin g thre e factors:

GFa = factor for sensing strain alon g th e a- a axi s

GFn = facto r for sensing strai n alon g th e n- n axi s

GFs = factor for sensing stress along th e a-a axi s


The resultin g strain s ar e give n by

The resultin g stres s alon g th e a a axi s i s given b y

In discussing the stress-strain gage, the description b y Hines (3) outlinesits essentia l characteristics , an d hi s argumen t wil l b e followe d here . Fro mprevious work , w e know tha t th e relationshi p betwee n stres s an d strai n i s

where a = stres s alon g th e n-n axi s

From Chapte r 7 , the uni t change i n resistance o f a gage i s given b y Eq .(7.11). Fo r th e stress-strai n gage , thi s is

where F a = strai n sensitivit y o f th e gag e element s fo r uniaxia l strai nalong axi s a-a wit h zero strai n alon g axi s n- n

Fn = strai n sensitivit y o f th e gag e element s fo r uniaxia l strai nalong axi s n- n wit h zer o strai n alon g axi s a- a

If th e values o f K a and e n given b y Eqs . (10.18 ) an d (10.19) , respectively , ar esubstituted int o Eq . (10.20) , th e resul t i s

where

stress along the a-a axis

STRESS GAGES 31 9

Rearranging,

Since F n = KF a, Eq . (10.21 ) ca n b e rewritten as

When th e gag e i s being calibrated , th e followin g two condition s mus tbe met :

1. Whe n R/R = 0 , <ra = 0 . This means the change in gage resistance mustbe independent o f the transvers e stress, a n.

2. Whe n a n = 0, then R/R = (aaGFs)jE. Thi s mean s th e chang e i n gag eresistance must be proportional t o the stress, a a, applied alon g th e a- aaxis.

From th e first condition, Eq . (10.22) gives

where v 0 = Poisso n rati o o f the tes t material . Thus , Eq . (10.23) shows tha t

K = v0

For th e secon d condition , Eq . (10.22) gives

where

Equation (10.24 ) shows that in order for the gage to have a unit chang ein resistanc e proportiona l t o a a, th e stres s gag e mus t hav e a transvers esensitivity facto r equa l t o th e Poisso n rati o o f th e materia l t o whic h i t i sbonded. Equatio n (10.26 ) show s how th e stress-gag e factor , G Fs, i s relate dto th e other gag e constants .

In Fig . 10.3 , elemen t 1 i s th e principa l strain-measurin g grid , whil eelement 2 provides th e necessary transvers e sensitivit y when th e tw o grid sare connected in series. Therefore, the transverse sensitivity of the entire gagemay be controlled by the ratio o f the resistance o f element 2 to the resistanceof element 1 ; that is , R2/R1. Th e ratio R 2/Ri = v0 = K i s only approximate ,since each elemen t ha s a smal l bu t measurabl e transvers e sensitivit y factor.


This coul d b e corrected , however , throug h calibratio n o r b y computatio nfrom th e know n characteristic s of each element .

10.4. Electrical circuit for two ordinary gages to indicate normal stress

The circui t show n i n Fig . 10. 4 wa s develope d i n 194 5 by S . B . Williams i norder t o produce indications whic h are directl y proportiona l to th e normalstresses i n th e direction s o f the gag e axes . Th e circui t wa s firs t reporte d b yKern (4 ) an d the n appeare d i n late r paper s (5 , 6) . I f th e direction s o f th eprincipal axe s coincid e wit h th e gag e axes , thi s provide s a metho d fo robserving th e principa l stresses directly.

The valu e of the resistor , R c, i s given b y th e expressio n

where K i s the transverse sensitivity factor of the strain gages. If K = 0 , then

If K = v , then R f become s infinite , an d i n thi s case th e tw o gages , aligne din th e X an d Y directions , ar e themselve s stress indicators , a s eac h wil lrespond i n direc t proportio n o f th e norma l stresses , a x an d <r v, respectively.

For furthe r details on the circuit, the reader should consult the referencescited.

FIG. 10.4. Circui t fo r conversio n o f T-strai n gag e rosett e int o tw o equivalen t stres s gag ecircuits. (Fro m ref . 4.)

STRESS GAGE S 321

10.5. The V-type stress gage (4 , 7)

Schematic diagram s o f two form s o f wir e grid s o f a V-typ e stres s gag e ar eshown in Fig. 10.5 . Although these are depicte d a s wir e gages, foi l gage s ar ealso manufacture d a s a singl e uni t wit h thi s configuration . Th e grea tadvantage o f this shape , whic h i s formed b y tw o lik e gages , i s that i t ca neasily b e mad e u p b y connectin g tw o ordinar y strai n gage s i n series ,providing, however , tha t th e angl e betwee n thei r axe s correspond s t o th evalue o f the Poisso n rati o o f the materia l upo n whic h they wil l b e use d t oindicate stress .

For commerciall y manufactured gages o f this nature, the angle betweenthe axe s o f th e tw o grid s ca n b e determine d unde r controlle d factor yconditions. I f two separat e gage s are to b e installed i n the field, the enginee rin charg e wil l nee d t o b e particularl y careful t o se e that th e tw o gage s ar emounted with the correct relativ e inclination, 20, betwee n their center lines.The directio n i n whic h the stres s i s to b e determined wil l b e established b ythe directio n o f the bisecto r o f the angl e betwee n th e gri d axes . Ther e are ,therefore, tw o consideration s abou t whic h the installe r o f the gage s shoul dbe meticulous . They ar e

1. Th e angl e betwee n the gag e axes .2. Th e directio n o f the bisecto r o f the angl e betwee n th e gag e axes .

The followin g tw o consideration s shoul d als o be kept i n mind:

FIG. 10.5. Schemati c diagrams of two form s o f V gage.


1. Foi l and wire strain gages respond essentiall y to linear strains, primarilyin th e axia l direction , bu t frequentl y an d t o a lesse r extent , t o th etransverse strai n a s well .

2. Fo r tw o lik e grids i n series , th e combine d outpu t wil l correspon d t othe averag e outpu t fro m eac h o f the tw o gages .

Mathematical analysis of the V-type configuration

The developmen t tha t Ker n (4 ) used wil l b e followe d bu t extende d t o tak ein th e latera l respons e o f th e strai n gage s t o strain s i n th e directio nperpendicular t o th e gag e axes . Le t u s consider a stres s gag e o f the V typeas consistin g o f two lik e strai n gage s connecte d i n series , a s show n i n Fig .10.5. Fo r eac h gage , Eq . (7.13 ) can b e used t o giv e

where F a i s th e axia l strai n sensitivit y and K i s th e transvers e sensitivit yfactor.

The stress , a a, i n th e directio n o f th e axi s O A ma y no w b e state d i nterms of the principal strains. Note, in Fig. 10.5 , that th e principal stres s axis,1, makes a n angl e o f 8 1 wit h respect t o th e O A axis . Again, aa is

where £a and £ „ are th e strain s alon g an d transverse , respectively, to th e O Aaxis. Th e tw o strains , £ „ and £„ , are t o b e writte n i n term s o f th e principa lstrains, ex and e 2. In order to do this , the transformation equation, Eq . (2.32),can b e writte n i n term s o f th e principa l strain s b y takin g eand } > = 0 . Thus ,

The strain , £„ , normal t o O A become s

The strain , e a, in the O A directio n become s

STRESS GAGE S 32 3

Substituting th e value s o f e a an d e tt given by Eqs . (b ) an d (c) , respectively,into Eq . (a ) result s in

Equation (10.31 ) contains tw o terms . The firs t term is

which represent s th e hydrostati c component o f the principa l strain s an d i sthe cente r of a Mohr' s strai n circle . The secon d ter m is

which represent s th e pur e shea r componen t o f the principa l strain s an d i sthe radiu s o f a Mohr' s strain circle . Since the hydrostati c strain i s the samein al l directions, the axial an d norma l strain s acting on th e gage due to thiscomponent are the same. Using Eq. (10.29), the hydrostatic component givesa uni t resistance change of

In term s of principal strains , R/R i s

In a simila r manner , sinc e th e pur e shea r componen t correspond s t o tw oequal strain s o f unlik e sign , th e uni t resistanc e chang e du e t o thi s com -ponent i s

In term s of principal strains , R/R i s

The averag e uni t resistanc e chang e fo r th e V-typ e gag e (tw o gage s i nseries), is R/(2Rg), wher e Rg i s the resistanc e o f one hal f o f the tw o gages ,


or grids . This ca n b e written as

where e fll +0 an d are the strains the grids are subjected to . These strain sare

If the right-hand sides of Eqs. (e) and (f ) are multiplie d and divide d by ( 1 — v )and thes e value s ar e the n substituted int o Eq . (d) , w e have

the quantitie s i n th e squar e bracke t o f each equatio n wil l b e identical , s othat R/ R wil l b e directl y proportional t o a a. Thi s mean s tha t whe n on ecomputes th e correc t angl e betwee n th e tw o gages , o r grids , one wil l have astress gage . Fro m Eq . (i) , the relativ e angl e o f inclination between th e tw o

Examination o f Eqs. (10.31 ) and (10.34 ) tells us tha t whe n

Substituting th e valu e of the su m o f the cosin e term s give n by Eq . (h ) int oEq. (g) , the n multiplyin g and dividin g the coefficien t o f 2 cos 29 1 co s 2<p b y(1 + K), th e final expression fo r R/R i s

where

STRESS GAGE S 32 5

gages, o r grids , can b e established. Thus ,

This expressio n ca n als o b e written as

Since v will be between 0 and 0.5 , and becaus e K ca n b e expected t o b eless than 0.04 , th e produc t v K wil l be very smal l wit h respec t t o unity , andso b y neglectin g th e product , vK , th e valu e of cos 2</ > become s

which is a very close approximation, especially when K tends towards zero.The term , co s 2(/> , can als o be writte n a s

Using th e valu e of co s 2</ > given b y Eq . (10.36) , th e valu e of ta n </ > becomes

If th e produc t v K i s ignored, the n

Taking th e transverse sensitivit y factor, K, equa l t o zer o gives

Stress gages ma y b e made usin g either wir e or foil . A typical foi l stres sgage i s shown i n Fig . 10.6 .

326 TH E BONDE D ELECTRICA L RESISTANCE STRAI N GAGE

FIG. 10.6 . Foi l stress gage in V configuration. (Courtes y o f Measurements Group, Inc.)

10.6. Application of a single strain gage to indicate principal stress (8)

There ar e certai n situation s in whic h th e direction s o f the principa l stresse sare known . This may b e through the conditions o f symmetry, a preliminarystudy wit h a brittl e lacque r coat , o r b y som e othe r method . Unde r thes econditions, i f tw o strai n gage s ar e installed , on e i n th e directio n o f eac hprincipal axis , the tw o strai n reading s thu s obtaine d wil l provid e sufficien tinformation t o enable the computation o f the two principal stresses . At eachpoint t o b e investigated, this saves the us e of at leas t one se t of instrumenta-tion whe n al l reading s ar e require d t o b e mad e simultaneously . Also, thetime involve d wit h calculation s wil l b e muc h reduce d i n compariso n wit hthat require d for standard rosett e analysis , which might employ equipmentfor thre e or fou r observation s at eac h statio n fo r each loa d level .

If on e i s only intereste d i n on e o f the tw o principa l stress magnitudes,however, an d know s t o whic h axi s thi s corresponds , a singl e strai n gag eindication for each loa d a t eac h statio n is all that i s required t o provid e thenecessary information . Such a gag e i s shown in Fig . 10.7.

The amoun t o f require d instrumentatio n is reduce d b y two-third s i ncomparison wit h th e general rosette method . The corresponding calculatio ntime ca n als o b e reduce d b y a n eve n greate r proportion , especiall y i f th eequipment ca n b e calibrated t o giv e a direct readou t i n terms of the desire ddata.

Due t o th e fac t tha t th e strai n condition s are symmetrica l with respectto a principa l axis , i f a V-typ e stress gag e i s t o b e use d an d line d u p wit hthe principa l axis , bot h halve s o f th e gri d wil l b e subjecte d t o exactl y th esame strai n condition s an d wil l sho w th e sam e uni t chang e i n resistance .This means tha t on e hal f of the V-typ e gage wil l b e redundant . Becaus e ofthis only one hal f o f the gag e needs t o b e used , which means a singl e straingage incline d a t th e correc t angl e </ > wit h respec t t o th e principa l axi s wil

STRESS GAGES 327

FIG. 10.7. Singl e strain gage inclined a t a n angle 0 wit h respec t to the principa l axis.

yield al l the necessary information to enabl e th e evaluation of the principa lstress magnitude. As shown in Section 10.5 , tan 0 = ^/ v — K. I f K = 0, thentan

10.7. Determination of plane shearing stress

Wire and foi l gages have little response to shearing strains so, if this quantityis t o b e determined , i t i s necessar y t o mak e th e approac h throug h th emeasurement o f linear strains , whic h can b e converte d int o th e equivalentof plane shear strain , and the n into terms o f shearing stress by means of theshearing modulus of elasticity. In Chapter 8 on rosette analysis, it was shownthat linea r strain s i n certai n give n directions ca n b e converte d int o thei requivalent value s in term s of the hydrostati c component, S H, and th e pur eshear component , s s. Thes e tw o values are written in terms of the principalstrains a s follows :

(position o f the cente r o f Mohr's circle)

(radius o f Mohr's circle )

This is shown in Fig . 10.8 .As shearin g stresse s ar e currentl y bein g considered , th e hydrostati c

component, which is equivalent to tw o principal strains of equal magnitudeand lik e sign , wil l no t concer n us . Th e pur e shea r component , whic h i sequivalent to tw o principal strain s of equal magnitud e bu t o f opposite sign ,will be examined rather carefully . I t i s from th e pure shear componen t tha tthe shearin g conditio n i n an y particula r directio n ca n b e determined , a s


FIG. 10.8 . Shea r strain i n terms of pure shear component.

shown i n Fig . 10.8 . Sinc e th e data produce d wit h any rosett e arrangemen tcan b e converted into it s hydrostatic and pur e shea r components , i t can b eseen that rosette s might, in general, be used to find the shearing strain in anyparticular direction . However, except in two special cases, the data reductio ninvolved i n thi s procedure i s too complicate d an d time-consuming .

Let us now look into the simpler procedures which can be accomplishedwith just tw o strai n indications . Consider th e rectangula r rosett e wit h fou rstrain observations , which , fo r thi s specia l case , ca n b e reduce d t o tw omeasurements. Thi s rosett e consist s o f fou r strai n gage s mounte d i n th edirections OA, OB, OC, and OD , as indicated i n Fig. 10.9 . For thi s particula rrosette, th e value s o f % , e s, an d ta n 20 ar e give n b y Eqs . (8.41a) , (8.41b) ,and (8.42) . These are

Furthermore, i f gages A an d C an d gage s B an d D ca n b e connecte d i nadjacent arm s o f th e Wheatston e bridge , th e differences , (c a — ec) an d

STRESS GAGE S 329

FIG. 10.9. Arrangemen t of gage axes for four-elemen t rectangula r rosette.

(sb — Ed), ca n b e read ou t directl y so that th e determinatio n of the maximumshear strai n and th e directions o f the principa l axe s may b e found fro m jus ttwo strain observations . Thi s save s both time and equipment . It shoul d als obe note d tha t wit h thi s form o f rosette i t i s also possibl e t o determin e th emagnitudes of the tw o principa l stresse s a s wel l a s th e directions o f the tw oprincipal axes .

When K ^ 0 , the gage s respon d t o latera l strai n i n additio n t o axia lstrain. Thus ,

There i s no chang e i n the direction s o f the principa l axes , since the valu e oftan 2 9 i s independen t o f K . Figur e 10.1 0 show s a n availabl e stacke dfour-element rectangula r rosett e o f the configuratio n shown in Fig . 10.9 .

If stres s gage s rathe r tha n strai n gage s wer e used , th e chang e i n thes erelations woul d b e du e essentiall y t o th e differenc e i n th e numerica l valu eof K , whic h would the n tak e o n th e particula r valu e o f K tha t woul d b e

FIG. 10.10. Four-elemen t rectangula r rosette. (Courtes y o f Measurements Group, Inc. )


equal t o th e valu e o f Poisson's rati o o f the materia l upo n whic h th e gage swere installed . Hence ,

When K = v,

This ca n b e rewritten a s

or

where G , the shea r modulus , is

The shear gage

In orde r t o determin e th e linea r strai n equivalen t o f shearin g strain , tw ogages ca n b e use d tha t ar e mounte d wit h thei r axe s havin g an y arbitrar yangle between them. In orde r to sho w this , the argumen t given by Perry (9)will be followed. The two gages are bonded t o the test material as illustratedby Fig . 10.11 .

The expression s fo r the strains , ea and £ b, can b e written as

Solving Eqs . (10.46 ) an d (10.47 ) simultaneously for y xjl produce s

STRESS GAGES 331

FIG. 10.11. Arbitrar y gag e arrangement .

If gage s a an d b ar e bisecte d b y th e x axis , the n 6 A = — 9B an d s ocos 9A = cos 9B. In tha t cas e Eq . (10.48) i s rewritten as

Furthermore, si n 29A = —si n 26B, whic h further reduce s Eq. (10.49) t o

Perry (9 ) generalizes these result s as follows : Th e difference i n normal strainsensed by any two arbitrarily oriented strain gages in a uniform field isproportional to the shear strain along an axis bisecting the strain gage axes,irrespective of the included angle between the gages.

An examination of Eq. (10.50) shows that if the two gages are 90° apartthe denominato r become s unity , since 9A = 45 ° (or 9 B = —45°) . Thus ,

Equation (10.51 ) tell s u s tha t th e shearin g strai n alon g th e bisecto r o f thegages' axe s i s equal t o th e differenc e i n th e norma l strains . I t ca n b e seen ,then, that a two-element rectangular rosette makes an ideal shear gage whenthe two gages are arranged i n adjacent arms of a Wheatstone bridge . Figure10.12 shows the gag e and bridg e arrangement .

Unless th e tw o gage s happe n t o b e line d u p wit h th e principa l axe s(when th e individua l strai n indication s wil l correspon d t o eac h o f th eprincipal strains) , no information about the principal strai n magnitudes , o rthe directions of the principal axes, is available from th e two gages. However,


FIG. 10.12. Two-elemen t rectangula r rosette arrange d t o measure

if on e i s onl y intereste d i n shea r i n a give n direction , th e tw o gage s wil lprovide the necessar y data .

Since two strai n gages onl y occup y hal f o f the Wheatston e bridge, th ebridge outpu t ca n b e double d b y addin g tw o mor e gage s wit h thei r axe sparallel an d perpendicula r t o th e firs t two . I f the bisecto r o f the gag e axe shappens t o lin e u p wit h the principa l axes , the n twic e th e maximu m shea ris indicate d b y th e outpu t fro m th e complet e Wheatston e bridge . I nparticular, i t shoul d b e note d that , since th e fou r gage s i n th e Wheatston ebridge onl y measur e strain s i n tw o direction s (at righ t angles) , the y wil l d onothing to determine th e directions of the principal strai n axes . Figure 10.1 3shows severa l configuration s of commerciall y availabl e four-elemen t gage sfor determinin g shear strain.

Care mus t b e take n i n orde r t o avoi d confusio n betwee n th e four -element shea r gag e an d th e four-elemen t rectangular rosette . Bot h emplo yfour activ e strai n gages . Th e shea r gage , whic h measure s strai n i n tw operpendicular directions , involve s a simpl e procedur e fo r establishin g th eshearing strain , and thu s the shearin g stress , but i t wil l onl y permi t on e t ofind this i n a particula r direction . A rectangular rosette , o n th e othe r hand ,is muc h mor e genera l i n nature , permittin g strai n observation s i n fou rdifferent direction s space d successivel y at 45° . Wit h th e rosette , on e ca ndetermine th e tw o principa l strains , th e direction s o f th e tw o principa laxes, an d th e shearin g strai n i n an y direction , includin g th e maximu mvalue. However , th e correspondin g computation s ar e somewha t mor eelaborate.

Equations (10.50 ) an d (10.51 ) wer e develope d b y considerin g th etransverse sensitivity factor, K, t o b e zero. I f K i s to b e accounte d for , thensa an d E b in Eqs . (10.50 ) an d (10.51 ) wil l hav e t o b e modified . Thi s ca n b eaccomplished b y returnin g to Eqs . (9.14) and (9.15) . Thes e ar e

STRESS GAGE S 333

FIG. 10.13. Ful l bridge s fo r shea r measurement . (Courtes y of Measurements Group, Inc.)

These equation s represen t strain s for a pai r o f orthogonal gages . Sincethe gage s use d t o indicat e shea r strai n ar e arbitraril y oriented , thes e tw oexpressions wil l be rewritten . Thus ,

where £„' and ej, ' are strain s perpendicular t o e' a and e' b, respectively. The value sof e' ^ and ej, ' can b e determine d b y usin g th e firs t strai n invariant , s o tha t


Substituting th e value s o f e^ given b y Eq . (a ) int o Eq . (10.52) ,

Substituting th e valu e o f 4' give n b y Eq . (b ) int o Eq . (10.53) ,

Since th e shearin g strai n i s proportiona l t o (e a — eb), we hav e fro m Eqs . (c )and (d )

This reduce s t o

Note tha t £' a and e' b are indicate d strains .Equation (10.54 ) show s tha t Eqs . (10.50 ) an d (10.51 ) can b e correcte d

for transvers e sensitivit y b y multiplyin g th e shearin g strai n b y ( 1 — v 0K)/(1 - X) . Therefore, Eq . (10.50) become s

Likewise, Eq . (10.51 ) fo r the two-elemen t rectangula r rosett e become s

Problems

10.1. Tw o strai n gages with thei r axes perpendicular to each other are to be used asa stres s gage . Th e followin g data ar e availabl e for the gages : R a = 350 ohms,(Gr)a = 2.15 , K a = 0.007, R n = 12 0 ohms, (G f )„ = 2.05 , K n = 0.009. Wil l thi sarrangement b e suitabl e fo r a stres s gage? I f so, specif y th e materia l on whichit ma y b e used.

STRESS GAGE S 335

10.2. Fo r th e V stress gag e show n i n Fig . 10.6 , determin e th e include d angle, usingK = 0, when designed for use on materials having the following Poisson ratios:

(a) v = 0.25 ; (b ) v = 0.30 ; (c ) v = 0.34 .

10.3. A single strain gage i s used t o measure the longitudina l stress at a point on a naluminum cantilever beam. Using v = 0.3 3 and E = 10 x 10 6 psi, determine thefollowing:

(a) Th e angl e (f > between th e bea m axi s and th e gag e axi s(b) Th e longitudina l stres s fo r a recorde d strai n o f 884 microstrain.

10.4. Thin-walle d pressur e vessel s are t o b e mad e fro m severa l differen t materials .They have an internal pressure p, a diameter d , a wall thickness t, Poisson rati ov, and a modulus of elasticity E. A single strain gag e i s to b e bonded t o eac hvessel s o the hoo p stress , a H, ma y b e monitored an d th e vesse l automaticallyshut dow n i f a specifie d stres s i s exceeded . Develo p a n expressio n fo r <J H interms o f the vesse l dimensions, the materia l propertie s an d th e strain .

10.5. Chec k th e expression s develope d i n Proble m 10. 4 b y usin g p = 900 psi,d = 40 in, t = 1. 5 in, v = 0.29 , and E = 2 8 x 10 6 psi.

10.6. O n th e vesse l i n Proble m 10.5 , a lin e 45° from th e longitudina l axis i s drawnfrom the origin of the longitudinal and circumferential axes. Two identical straingages ar e bonde d t o th e vesse l a t 15 ° on eithe r sid e o f thi s line . Fo r th econditions in Problem 10.5 , determine the strain s a t eac h gag e an d sho w theymeasure the maximum shearing strain .

10.7. A two-element rectangular rosette is to be bonded t o the web of a beam, whosecross sectio n is shown in Fig. 10.14 , in order to determin e the maximu m shearstrain a t tha t sectio n an d thereb y th e maximu m shea r stress . Th e materia lproperties o f the bea m ar e v = 0. 3 and E = 3 0 x 10 6 psi.

(a) Determin e th e poin t wher e the maximum shear stres s occurs .(b) Sketc h the gag e arrangement .(c) Determin e th e strain a t eac h gag e when the tota l vertica l shearing force i s

480001b.

FIG. 10.14 .


REFERENCES

1. Williams , Sidney B. , "The Dyadi c Gage, " SESA Proceedings, Vol. I , No. 2 , 1944 ,pp. 43-55.

2. "SR- 4 Stress-Strain-Gage, " Produc t Dat a 4323, BLH Electronics , Inc., 75 ShawmutRd., Canton, M A 02021 , May , 1961 . (Now ou t o f print.)

3. Hines , Fran k F. , "Th e Stress-Strai n Gage, " Proc. 1s t International Congress o nExperimental Mechanics, 1963 , pp. 237-253 .

4. Kern , Rober t E. , "Th e Stres s Gage, " SESA Proceedings, Vol . IV , No . 1 , 1946 ,pp. 124-129 .

5. Kern , Rober t E . an d Sidne y B . Williams , "Stres s Measuremen t b y Electrica lMeans," Electrical Engineering, Transactions, Vol . 65, March 1946 , pp. 100-107 .

6. Williams , Sidne y B. , "Geometry i n th e Desig n o f Stres s Measuremen t Circuits ;Improved Method s Throug h Simple r Concepts, " SESA Proceedings, Vol . XVII,No. 2 , 1960 , pp . 161-178 .

7. Sevenhuijsen , Pieter J. , "Stres s Gages, " Experimental Techniques, Vol . 8 , No . 3 ,March 1984 , pp. 26-27.

8. Lissner , H . R . an d C . C . Perry , "Conventiona l Wir e Strai n Gag e Use d a s aPrincipal Stres s Gage," SESA Proceedings, Vol. XIII , No . 1 , 1955, pp. 25-34 .

9. Perry , C. C., "Plane-shear Measuremen t wit h Strai n Gages," Experimental Mech-anics, Vol. 9, No. 1 , Jan. 1969 , pp. 19N-22N .

11TEMPERATURE EFFECTS ON STRAIN GAGES

11.1. Introduction

When usin g strai n gages , th e enginee r want s t o measur e strain s produce donly b y the loadin g o n th e structur e and t o eliminat e strains produce d b yother variables, particularly temperature. Since metals change their resistancewith temperature as well as with strain, the purpose of the electrical resistancestrain gage is to measure the strain-induced resistance change independentlyof the temperature-induced resistance change. Therefore, we want to account ,or t o compensat e automatically , for the effect s o f temperature o n th e strai nobservations.

The physica l phenomen a occurrin g i n a strai n gag e bonde d t o a tes tspecimen are complex when a change of temperature takes place (1) . Amongthem ar e th e following :

1. Th e bas e material expand s o r contracts .2. Th e strain-sensitiv e filamen t o f the gag e expands o r contracts .3. Th e resistanc e o f the filamen t changes .4. Th e gag e factor o f the gag e i s subject to variation .5. Th e bon d betwee n the gage and th e bas e materia l ma y b e affected .6. Du e t o the transverse sensitivity, dimensional changes which take place

in th e latera l direction , eithe r i n th e gag e o r i n th e bas e material , wil lshow a n indicate d chang e i n resistance .

7. Th e carrie r o n which the sensitiv e filament is mounted ma y chang e it sproperties.

11.2. Basic considerations of temperature-induced strain (2-4 )

In Chapte r 1 , in th e discussio n o f the strai n sensitivit y of a wire , Eq. (1.18 )was developed . From this , th e expressio n fo r th e uni t chang e i n resistanc ecan b e written as

where v = Poisson' s ratioL = conducto r lengt hp = resistivity of the conducto r materia l


Equation (11.1 ) show s that th e uni t change i n resistanc e i s dependen ton th e uni t change s i n lengt h an d resistivit y o f th e conductor . I f n omechanical strai n take s place, a uni t resistance change ca n stil l occur whenthe conducto r i s subjected t o a temperatur e change.

Consider a strai n gag e bonde d t o a bas e materia l an d connecte d t o astrain indicator . I f the bas e materia l i s unrestraine d and the n undergoe s atemperature change , th e strai n indicato r wil l sho w a n indicate d strai nconsisting o f the algebrai c su m o f three components.

1 . Th e base material to which the gage is bonded expands or contracts inthe directio n o f the gag e axis . This uni t change i n length , or strain , is

where a m = coefficien t o f thermal expansion of the bas e materialAT = temperature change fro m a referenc e temperature .

2. Th e strain gage grid material expands or contracts du e to the tempera -ture change . This uni t change in length , or strain , is

where <x g = coefficien t o f thermal expansio n o f the gri d material .3. Sinc e th e resistivit y o f th e strai n gag e gri d materia l change s wit h

temperature, the gage resistance will change. The uni t resistance changeof th e gag e is

where f t = resistance-temperature coefficien t o f th e strai n gag e gri dmaterial. Equation (11.4 ) can be expressed in terms of strain by dividingboth side s o f the equatio n b y G F, the gag e factor . Thus ,

The strain , called therma l outpu t (sometime s referre d t o a s apparen tstrain), tha t wil l b e registere d o n th e strai n indicato r ma y b e expresse d a sthe algebrai c su m o f the thre e strains . Hence,

TEMPERATURE EFFECT S O N STRAI N GAGES 339

FIG. 11.1. Apparen t strain vs. temperature for strain gage alloys bonded to steel. (From ref. 2.)

or

We are not t o assume that th e strain given by Eq. (11.6) is a linear functionof temperature. I t is not, since the coefficients a m, ag, and ft are also functionsof temperature. We must know, therefore, the temperatur e characteristic s ofeach gage used as wel l as the temperatur e characteristic s o f the materia l o nwhich th e gag e is bonded .

For illustratio n only , Fig . 11. 1 show s th e variatio n o f strai n wit htemperature fo r severa l strai n gag e material s bonde d t o steel . Th e figur eshows tha t larg e error s ca n occu r whe n the strain gag e and th e material towhich it is bonded ar e subjected to temperatures differin g fro m th e reference ,or bonding , temperature . Thi s illustrate s the nee d fo r correctio n whe n th estrain gage system is subjected t o temperature fluctuations. Corrections ma y


be accomplishe d b y computatio n usin g a temperature-strai n calibratio ncurve, o r b y usin g a compensating , o r dummy , gag e i n a n adjacen t ar m o fthe bridg e that i s subject to certai n restrictions . If the correctio n i s made b ycomputation, the n a temperatur e recor d mus t b e kep t durin g th e test .

Gage factor variation with temperature

The gag e factor , GF, also varie s with temperature. I f the temperatur e rang eis smal l an d th e variatio n in G F is slight, then a correctio n ma y b e ignored .If, however , the tes t temperatur e rang e i s large an d th e variatio n in G F withtemperature cannot b e disregarded, then, depending on the required accurac yof th e strai n measurement , a gag e facto r correction ma y b e necessary . Thi sis illustrate d i n Fig . 11.2 , which show s th e variatio n i n gag e facto r wit htemperature fo r severa l strai n gag e alloys . Severa l o f th e alloy s ar e linea rover a considerable temperatur e range and sho w quite a variation i n slopes .Constantan show s a n increas e o f les s tha n 1 percen t pe r 100° F wit hincreasing temperature, while Nichrome V shows a decrease o f over 2 percentper 100° F wit h increasin g temperature . Isoelasti c ha s a ver y sligh t chang ein gag e facto r betwee n roo m temperatur e an d 200 CF, bu t change s quit eperceptibly outsid e o f thi s region. This latte r material , however , is use d fo rdynamic measurement s rathe r tha n stati c measurements . Unde r dynami cconditions, other error s may be considerably greater than the change in gagefactor, an d s o correctin g the gag e facto r may b e inessential.

In orde r t o correc t th e gag e facto r fro m it s valu e a t th e referenc etemperature t o it s valu e a t th e tes t temperature , a simpl e procedur e i sfollowed. I f Fig . 11. 2 is examined, i t i s seen tha t th e percen t change in gag efactor i s plotted versu s temperature. The gag e factor , G F T, a t som e tempera -ture differen t fro m th e referenc e temperature i s

where G FR = gage facto r a t the reference temperature, generall y at roo mtemperature wher e G fR = G f

Gf(%) = percentag e chang e i n th e gag e facto r fro m th e referenc etemperature t o th e tes t temperatur e

The prope r sig n o f AGF(%) must, o f course, b e use d i n Eq . ( 1 1.7).

Method of determining gage factor variation with temperature

The method o f determining th e variation in the gage factor with temperatur efor resistanc e strai n gage s i s give n b y ASTM(5) . Tw o methods , stati c an ddynamic, are discussed, bu t onl y the static method wil l be outlined here . Thetest apparatus, shown in Fig. 1 1.3, consists of a beam havin g a uniform stress

FIG. 11.2 . Gag e facto r variatio n wit h temperature fo r severa l strai n gag e alloys . (From ref . 3.)


FIG. 11.3. Apparatu s for stati c determination o f gage-factor variatio n vs . temperature. (Fromref. 5 with permission . © ASTM. )

area tha t i s directl y proportiona l t o th e deflectio n o f th e en d poin tcontaining th e rider , whic h is located a t th e apex o f the angl e formed by thebeam sides .

The fixture holding th e bea m mus t b e much more rigi d tha n th e bea min orde r t o preven t error s du e t o it s own deformation . Th e slidin g steppe dblock ha s fou r surface s that mus t b e o f nearly equal steps , with the surfacesparallel t o eac h othe r an d t o th e slidin g surfaces. The step s ar e suc h tha t amaximum surface strain of 1000 + 10 0 uin/in is produced o n the beam. Whenthe ride r i s resting o n th e lowes t surface of the slidin g block, the en d o f th ebeam shoul d b e deflecte d about 2 percent o f it s tota l planne d deflectio n inorder t o insur e positiv e contact .

The gages to be tested ar e symmetrically mounted i n the constant-stressarea o f the beam an d aligne d wit h th e longitudinal axis . Thermocouples ar emounted a s near the gages as possible an d a t each en d o f the constant stres sarea. The entire tes t uni t is then placed i n a temperature chamber , th e gage sare connected t o the instrumentation, and th e fixture and bea m ar e allowe dto com e t o equilibriu m a t th e referenc e temperature, whic h is usually roo mtemperature. Wit h th e ride r restin g o n th e lowes t ste p o f the slidin g block ,the instrumentatio n is balanced, the n gage outpu t i s recorded a s th e ride r i sdisplaced t o subsequen t step s o n th e slidin g block . Readings , take n thre etimes, are recorde d fo r both increasin g and decreasin g deflections .

The tes t chambe r i s brough t t o previousl y selected temperature s an d

TEMPERATURE EFFECT S O N STRAI N GAGE S 34 3

the process i s repeated afte r th e temperature has stabilized. Th e temperaturedifference over the constant stres s area shall not exceed 5°F (3°C) or 1 percentof the temperatur e o f the gag e area , whicheve r is greater . Neithe r shal l th etemperature change more than 5°F (3°C) during a test at any temperature.

The chang e i n gag e facto r i s compute d a s th e differenc e betwee n th egage output due to the strain for a given temperature and that at the referencetemperature. Thi s i s expresse d a s a percentag e change . Thus , th e percen tchange i n gag e facto r i s

where E t = gage outpu t a t tes t temperatur eEr = gage outpu t a t referenc e temperatur e

If mor e accurac y i s desired , correction s ca n b e mad e fo r th e therma lexpansion o f the bea m an d th e steppe d block . Thi s gives

where a b = coefficient o f thermal expansion o f the bea mas = coefficien t of thermal expansion of the slidin g bloc k

AT = difference betwee n the tes t and referenc e temperatur e

11.3. Self-temperature-compensated strain gages (2 , 4)

The manufacture r of strain gag e alloy s can contro l temperature-resistanc ecoefficients withi n reasonabl e limits . Wit h carefu l selectio n o f particula rmelts, followed by judicious process control, the alloy will exhibit a minimumtemperature response ove r a given temperature rang e when bonded to a testspecimen whos e coefficien t o f thermal expansio n matche s tha t o f the strai ngage alloy . B y choosin g a gag e tha t i s temperatur e compensate d fo r th ematerial being tested, a three-wire, quarter-bridge circui t may be used rathe rthan usin g a half-bridg e circui t with a matchin g dummy, or compensating ,gage. In the case o f the quarter-bridge circuit , the Wheatstone bridg e can becompleted b y usin g a stabl e precisio n resisto r i n th e adjacen t ar m a t th einstrument, o r b y usin g a n instrumen t tha t accommodate s th e three-wire ,quarter-bridge circuit . This circui t (wit h lead-lin e resistance ) was discusse din Chapte r 5 . Figur e 11. 4 show s th e three-wir e circui t (withou t lead-lin eresistance) wit h the bridge-completio n resistor .

Self-temperature-compensated gage s ar e readil y availabl e fro m strai ngage manufacturers. In the gage designation code, a number usually appearsthat indicate s th e materia l fo r which the gag e i s temperature-compensated .For instance , 6 or 06 indicates a gage compensated fo r mild steel , where the


FIG. 11.4. Quarte r bridg e wit h thre e lea d wire s an d bridg e completio n resistor.

thermal expansion coefficient is 6 (uin/in)/°F. This is also designated as partsper million pe r degree Fahrenhei t an d abbreviate d t o ppm/°F . A graph ma ybe include d i n th e gag e packe t showin g th e variatio n wit h temperatur e o fboth th e thermal output , expressed i n microstrain, and the gage factor . Ther emay also be a polynomial expression givin g the thermal output as a functio nof temperature . Figur e 11. 5 shows a typica l grap h o f bot h therma l outpu tand gag e facto r variatio n vs . temperature.

In developin g th e therma l outpu t vs . temperature curve, the data givenare for a foi l lo t rathe r tha n a gage designation. A test gage made fro m a foi llot i s bonde d t o a tes t specime n an d th e procedur e give n b y AST M i sfollowed (5) . The tes t specime n i s placed i n a temperatur e chambe r an d th egage i s connecte d t o a strai n indicator , th e gag e facto r i s se t (usuall y 2.0 )and th e instrumen t i s then balance d a t th e referenc e temperatur e o f 75°F .The tes t specime n i s unrestrained an d allowe d t o expan d o r contrac t freel yas th e temperatur e i s varied . Sinc e n o mechanica l o r therma l stresse s ar epresent a t th e equilibriu m temperatures , th e recorde d strai n a t thes etemperatures i s du e onl y t o th e therma l effects , thu s enablin g on e t o plo tthe therma l outpu t vs . temperature . Th e therma l outpu t ma y als o b eexpressed a s a polynomia l give n as

where e TO = therma l outpu t i n microstrai nT = temperatur e

The coefficients , A , B , C , D, an d E ma y b e give n fo r bot h th e Fahrenhei tand Celsiu s temperatur e scales .

If greater accurac y i s required i n determining the therma l outpu t whentesting an actual structure , the gage, or gages, may be bonded t o the structurealong wit h th e adjacentl y place d temperatur e senso r fo r eac h gage . Th estrain-measuring instrumen t i s balance d a t th e referenc e temperature , an dthe structur e i n th e unrestraine d stat e (n o mechanica l o r therma l stresse s

TEMPERATURE EFFECT S O N STRAI N GAGES

Foil Lo t No . A38AD31 5

345

FIG. 11.5 . Therma l output and gag e factor variatio n vs. temperature.

present) is subjected to th e tes t temperature , o r temperatures . The thermaloutput (microstrain ) a t eac h equilibriu m temperatur e i s recorded , whic hallows subsequent correctio n in order to obtain the strains due to the loading.

Table 11. 1 is a listin g of the coefficient s fo r Eq . (11.10 ) fo r severa l foi llot numbers . The temperatur e coefficien t o f the gag e facto r fo r eac h foi l lo tis also included.

Thermal output correction

When th e structure carryin g th e bonded strai n gag e i s loaded and teste d a tsome subsequen t temperature , the strai n indicato r wil l sho w a n indicate dstrain, e| , tha t i s made u p no t onl y of the thermal outpu t (apparen t strain )but als o the strain produce d b y the mechanical an d therma l stresse s due tothe loading . Settin g the actua l valu e of th e gag e factor , G F, give n o n th estrain gag e dat a sheet , on th e strai n indicator , a firs t approximation o f thestrain a t th e test temperature is obtained b y subtracting the thermal outpu tfrom th e indicated strain . Whe n doin g this , car e mus t be taken i n using th eproper sig n of the strain . This gives

Table 11. 1

Foil lo t no . Tes t materia l

A11BP11 10 1 8 Steel

A

2.31 x1.37 x

103

103

B

-2.79 x-5.51 x

10'10'

Temperature

A12BJ01 101 8 Steel

2024-T4 Al

7.03 x4.21 x2.98 x1.88 x

102

102

102

102

-8.31-1.67 x-2.94-7.00

101

Temperature

A12BJ03 101 8 Steel

2024-T4 A l

7.47 x4.47 x2.40 x1.54 x

102

102

102

102

-8.80-1.78 x-2.26-5.58

10'

Temperature

A38AD315 101 8 Steel -9.59 x-2.64 x

101

10'3.022.53

Temperature

-4.59-1.23

coefficient

-1.78-4.22-1.76-3.89

coefficient

-1.86-4.75-1.44-3.84

coefficient

-2.89-6.96

coefficient

C

x 10- 2

x 10 -1

of gag e facto r

x 10 -2

x 10 - 2

x 10 - 2

x 10 - 2

of gag e factor

x 10 -2

x 10-2

x 10 -2

x 10 - 2

of gag e facto r

x 10 -2

x 10 -2

of gag e factor

8.604.79

= (0. 7

5.152.756.063.19

= (0. 8

4.192.362.541.58

= (0. 8

8.134.29

= (0. 8

D

xX

+

X

X

X

X

+

X

X

X

X

+

X

X

+

10-5

10-4

0.2)/100°C

10-5

10-4

10-5

10-4

0.2)/100 C

10-510-4

10-510-40.2)/100°C

10-5

10-4

0.2)/100°C

E

-2.9 x 10 -8

-3.11 x 10 -7

-3.37 x 10 -8

-3.53 x 10 - 7

-4.59 x 10 -8

-4.82 x 10 -7

-1.09 x 10 -8

-1.14 x 10 -7

1.26 x 10 -8

1.32 x 10 -7

-5.99 x 10 -8

-6.29 x 10 - 7

°F°C

=F'C3FJC

=F"C°F!C

°F"C

Source: Courtes y o f Measurements Group, Inc .

TEMPERATURE EFFECT S O N STRAI N GAGES 34 7

where e, ' = strai n indicato r readin g unde r tes t condition sETO = thermal outpu t fro m th e data shee t (microstrain )

The valu e of e, - give n b y Eq . (11.11 ) ma y b e o f sufficien t accuracy , bu te'i was obtaine d wit h the actua l valu e of GF set on th e strai n indicator , whilethe therma l output , S TO, wa s determine d wit h a differen t valu e o f G f(generally 2.0) . Therefore , if further accurac y i s desired, th e therma l outpu tshould b e correcte d fo r th e differenc e i n gag e factors . Thi s i s a simpl eprocedure, a s one ma y recal l fro m Chapte r 5 . The correctio n i s

where G F1 = gag e facto r set on th e strain indicato rGF2 = gage factor of the strai n gag e

£i = indicated strai n o n the strain indicato r£2 = correcte d strai n

Since eTO was obtained b y setting a gage facto r o n th e strai n indicato rother tha n th e actua l gag e factor , a correcte d valu e o f the therma l outpu tmay b e calculate d b y usin g Eq . (11.12) . B y takin g G F1 = G$, G F2 = G F,ei = £ ro> an<3 £2 = e ro> we hav e the corrected therma l outpu t as

where G f i s th e gag e facto r used i n determinin g the therma l outpu t curve .Using Eq . (11.13) , Eq . (11.11) ca n b e rewritten as

If desired , th e entir e therma l outpu t vs . temperatur e curv e coul d b ecorrected beforehan d fo r th e actua l gag e facto r b y usin g Eq . (11.13 ) an dplotting a ne w curve . Note , however , tha t thi s correctio n woul d b e fo rreference temperatur e gag e factor s an d woul d no t accoun t fo r gag e factorvariation wit h temperature.

Example 11.1. Tw o identica l gage s ar e attache d t o a structur e a t differen tlocations. Th e gag e data an d therma l outpu t curv e are show n i n Fig . 11.5 . Agage facto r o f 2.05 i s se t on th e strai n indicator , th e instrumen t i s balanced a t75°F, an d th e structur e i s loaded and brough t t o it s test temperatur e o f 300°F.The followin g readings ar e obtained :

Gage 1 E I = 1180|iin/i nGage 2 s 2 = -2060uin/i n


Use Eq . (11.14 ) t o determin e th e corrected strai n fo r each gage.

Solution. Fro m Fig . (11.5) , K ro = — 80 ^in/in (G f = 2. 0 fo r thi s curve) . Usin gEq. (11.14) ,

Note tha t the gage factor correctio n amounte d to onl y 2 uin/in.

Correcting for thermal output and gage factor variation

In th e precedin g example , th e therma l outpu t wa s correcte d fo r th e actua lgage factor at the reference temperature, since that value was set on the strainindicator whe n makin g th e strai n measurements . I f th e gag e facto r i s als oto be corrected a s the temperature changes, then the indicated strai n as takenfrom th e strai n indicator , a s wel l a s th e therma l output , mus t b e correcte dfor th e chang e i n gag e factor . I f th e therma l outpu t curv e wa s develope dusing a gag e facto r o f G* = 2.0 , bu t a differen t valu e o f G F was se t o n th estrain indicato r whe n th e strain s wer e measured , the n eac h strai n valu e(indicated an d therma l output ) woul d hav e t o b e correcte d individuall ybefore makin g th e fina l correction .

As lon g a s a few identical gage s wer e used i n a test , thi s process, whileinconvenient, coul d b e used . Generally , however , ther e woul d likel y b e amixture of gages with different gag e factors and therma l output curves. Underthese conditions i t i s simpler t o us e a metho d correctin g bot h th e indicate dstrain an d therma l outpu t i n one operation .

A simpl e metho d o f correctin g fo r th e therma l outpu t an d th e gag efactor variatio n i s available. Since the therma l output curve s ar e develope dusing a particular gage factor o f Gf (usuall y 2.0) at the reference temperature,set the sam e gag e facto r on th e strai n indicato r when conducting tests . Th estrain readin g ca n the n b e corrected fo r th e effec t o f the therma l outpu t b yusing Eq . (11.11) . Th e nex t ste p i s t o correc t th e actua l gag e facto r t o it sproper valu e at th e tes t temperature . This ca n b e done b y usin g Eq . (11.7).Combining thes e tw o equation s give s th e actua l strain , e , a t th e tes ttemperature. Thus ,


where E TO = thermal outpu t a t th e tes t temperatur e£• = strai n indicato r readin g unde r tes t condition s

Gj£ = gag e factor at whic h the therma l outpu t wa s recorde dGFR = actual gag e factor a t th e reference temperature

AGF(%) = percen t variatio n i n gag e facto r a t tes t temperature , wit hthe prope r sig n

Example 11.2. A strain gag e havin g a gag e facto r o f 2.15 i s bonded t o a stee lstructure an d a gage factor o f 2.0 is set on the strain indicator, which is balancedat roo m temperature . A t the tes t temperatur e th e followin g data ar e recorded :

Indicated strai n = 267 5 uin/i n

Thermal outpu t = —85 0 |iin/in

AGF(%) = 0.7 5 percen t

Solution. Usin g Eq . (11.15) ,

= 325 5 uin/i n

Note that the reference temperatur e gage factor. G FR, wil l be the manufacturer' sgage factor , G T, providing the referenc e temperatur e fo r thi s tes t i s 75°F.

The precedin g metho d i s the easies t t o use , since the correctio n ca n bemade i n one step . Othe r way s may b e employed.

1. Th e manufacturer' s gage facto r ca n b e se t o n th e strai n indicato r an dthe instrumen t balanced . Thi s woul d requir e th e indicate d strai n an dthe thermal output to be corrected separately for gage factor variation.

2. Se t any arbitrar y gag e factor , or th e tes t temperatur e gag e factor , o nthe instrumen t an d balanc e i t a t th e reference temperature.

3. Se t th e tes t temperatur e gag e facto r o n th e strai n indicator . Brin gthe tes t structur e t o th e tes t temperatur e an d balanc e th e indicato rbefore applyin g th e load . Th e drawbac k here , o f course , i s bein gsure th e structur e is stress fre e whe n balancing th e instrument .

11.4. Strain gage-test material mismatch (2)

Self-temperature-compensated strai n gage s ar e manufacture d fo r material sthat hav e coefficient s o f thermal expansio n rangin g fro m 0 t o 1 8 parts pe rmillion pe r degre e Fahrenhei t (ppm/°F) . Thes e value s cove r a rang e o fcommonly used engineering materials. Gages used o n plastics , however, are


Table T1.2. Thermal expansio n coefficient s o f commo nmaterials

Expansion coefficient

Material (per F ) (per C )

Aluminum, 2024-T4 , 7075-T 6Beryllium coppe r 2 5Brass, 30-7 0Bronze, phospho r (10%)CopperIron, gra y cas tMagnesium, AZ-31 BMolybdenumMonelSteel, 1008 , 101 8Steel, 434 0Steel, 30 4 stainles sSteel, 31 6 stainles sTin, pur eTitanium, pur e

12.99.3

11.110.29.36.0

14.52.27.56.76.39.68.9

13.04.8

23.216.720.018.416.710.826.14.0

13.512.111.317.316.023.48.6

Source: referenc e 2 .

manufactured wit h coefficient s o f 30 , 40 , an d 5 0 ppm/°F. I f a strai n gag ecompensated fo r steel , fo r instance , i s use d o n a secon d materia l wit h adifferent coefficien t o f therma l expansion , the n th e therma l outpu t curv efurnished wit h the gage wil l no longe r b e directly applicable. Th e amoun t o fdeviation wil l depend o n th e differenc e i n the therma l expansion coefficient sof th e tw o materials . Tabl e 11. 2 i s a partia l listin g o f therma l expansio ncoefficients fo r some common engineerin g materials .

If a strai n gag e i s use d o n a materia l fo r whic h i t i s no t compensated ,and i f the differenc e i n therma l expansio n coefficient s i s not to o large , the nover a limite d temperatur e rang e nea r th e referenc e temperatur e th e erro rproduced i n using the given thermal output curve may b e acceptable. A s thedifference betwee n th e therma l expansio n coefficient s become s larger , som esteps shoul d b e taken to determin e th e therma l outpu t fo r th e gag e whe n i tis used o n materia l fo r which it i s not compensated . Thi s i s particularly tru efor plastics , no t onl y becaus e o f th e wid e variety , but als o becaus e o f th edifference betwee n manufacturer s fo r supposedl y th e sam e typ e o f plastic .When suc h gage s ar e t o b e use d o n a specifi c application , i t migh t b eadvisable t o determin e th e therma l outpu t curv e fo r tha t particula r strai ngage-plastic combination. Thi s can be done in accordance wit h the procedur ediscussed earlier .

When a strai n gag e i s applied t o a material fo r whic h i t i s mismatched ,an approximat e correctio n ca n b e mad e b y usin g Eq . (11.6) , whic h i s th eexpression fo r therma l output . I f th e strai n gag e i s applie d t o th e firs t

TEMPERATURE EFFECT S ON STRAI N GAGE S 35 1

material, then the therma l output is

where th e subscrip t 1 refers t o th e firs t materia l tested . I f a simila r gage isapplied t o a second material , then the thermal output is

where the subscrip t 2 refer s t o th e secon d material .In Eqs . (a ) an d (b) , th e las t ter m o n th e right-han d sid e i s the same ,

since i t refer s t o th e strai n gage . Thus ,

If the thermal output, £T01, i s known, then the thermal output, er02, can beapproximated by usin g Eq. (c) . This gives

Equation (11.16 ) give s a firs t approximatio n fo r th e therma l outpu twhen the strain gage is applied to a second material. This amounts to rotatingthe give n therma l outpu t curv e about th e referenc e temperature . I f a m2 i slarger than <x ml, the rotatio n wil l be counterclockwise ; if am2 i s less than <x ml,the rotatio n wil l b e clockwise.

Figure 11. 6 shows the therma l outpu t an d gag e facto r variation curvesfor a strai n gag e manufacture d fro m foi l lo t numbe r A12BJ03 , Table 11.1 .Tests gage s o f thi s foi l lo t wer e bonde d t o bot h 101 8 stee l an d 2024-T 4aluminum to produc e th e curves shown. The thermal output curve for steelshows large changes with temperature, bot h abov e and belo w the referenc etemperature. The thermal outpu t curve for aluminum, on th e other hand , ismuch flatter . I t i s eviden t tha t gage s mad e o f thi s foi l an d bonde d t oaluminum woul d give good result s in the low-temperatur e region .

Example 11.3. Usin g Fig . 11.6 , assum e tha t onl y th e therma l outpu t dat aavailable i s fo r 101 8 steel . Mak e a firs t approximation , usin g Eq . (11.16) , fo rthe therma l outpu t fo r 2024-T 4 aluminum . Compar e i t wit h th e actua l curv efor aluminu m in Fig . 11.6 .

Solution. Fro m Tabl e 11.2 , the therma l expansio n coefficien t fo r 101 8 stee l i s6.7 ppm/°F and 12. 9 ppm/°F for 2024-T4 aluminum. Using Eq . (11.16) ,


FIG. 11.6. Therma l outpu t an d gag e facto r variatio n vs . temperature.

where e,Tol i s the therma l output fo r 101 8 steel and e. T02 i s the calculate d valu eof th e therma l outpu t fo r 2024-T 4 aluminum . Determin e e rol a t variou stemperatures fro m th e curv e fo r 101 8 steel, o r comput e i t b y usin g th epolynomial coefficient s give n i n Tabl e 11.1 . Carrying ou t th e calculations , th eresults ove r a temperatur e rang e fro m — 100°F to SOO T are tabulated .

r, °F-100-50

050

100150200250300350400450500

£TOl, fan/in

13981135747266

-278-856

-1439-2003-2524-2979-3364-3608-3747

ETm, fan/in

313360282111

-123-391-664-918

-1129-1274-1349-1283-1112

TEMPERATURE EFFECT S O N STRAI N GAGES 353

FIG. 11.7. Therma l outpu t an d gag e facto r variatio n vs . temperature.

The results are plotted i n Fig. 11.7 . For th e values of the thermal expansio ncoefficients used , a first approximation o f the thermal output , sT02, shows tha tthe approximated value s are positive and slightl y higher tha n th e tes t value s attemperatures below 75°F, while the approximated value s are negative and belowthe test values for temperatures above 75°F. The error i n the approximate value sranges betwee n 1 5 and 2 0 percen t u p t o 300°F , the n increase s considerabl yabove tha t temperature . I n lie u o f other information , however , thi s correctio nfor th e therma l outpu t woul d no t b e unreasonable , particularl y i f the strain simposed b y mechanica l an d therma l stresse s wer e large . I t doe s illustrate ,though, the need for an actual test if more exact values of the strain are required .

11.5. Compensating gage

It was pointed out i n Chapter 5 that two identica l gages placed in adjacentarms o f a half-bridg e circuit an d bonde d t o th e sam e materia l woul d givetemperature compensation if both gages were subjected to the same tempera-ture. This also applies to a full-bridge circui t if all four gages , bonded t o th esame material, were always at the same temperature during the test. Further-more, in either of these circuits the lea d wire s must be routed together an dbe at th e same temperature.


FIG. 11.8 . Temperature-compensate d circui t usin g a dumm y gage.

A commo n arrangemen t fo r temperatur e compensation i s the us e o f adummy gage, identical to the active gage, in a half-bridge circuit. The dummygage must be bonded to a stress-free piece of material identical to the materialon whic h the activ e gage i s bonded an d place d a s clos e t o th e activ e gageas possibl e s o tha t i t experience s th e sam e temperature . Th e lea d wire s ofboth the active gage and dummy gage should be routed together. This circuitis shown in Fig . 11.8.

If the bridg e i s initially balanced, then , fro m Eq . (5.1),

where R ! = activ e stain gag eR2 = dummy strain gag e

If ther e is a chang e i n temperatur e only , AR 1 = AR2, sinc e the activ e an ddummy gage s ar e identical , are mounte d o n th e sam e typ e of material, an dare maintained a t identica l temperatures . Therefore ,

This shows that th e bridge remains balanced, irrespectiv e of the temperaturechange, sinc e th e influenc e o f temperatur e ha s bee n eliminated , an d an yunbalance o f the bridge wil l be due solely to mechanica l strai n o n the activegage.

An alternative method woul d employ the dummy gage and activ e gagein quarter-bridge circuits . The dummy gage would be placed adjacen t to th eactive gag e an d record s kep t o f bot h gag e outputs . Th e therma l outpu trecorded fro m th e dumm y gage woul d b e subtracte d fro m th e activ e gag estrain. This would , of course, double the require d strain gag e channels .

If th e ful l bridg e i s considered, an d i t i s assumed fo r simplicit y that al lresistances ar e identica l strai n gage s o f resistance R an d mounte d o n th e


same material , the n th e circui t outpu t give n by Eq . (5.6 ) is, taking n = 0,

If eac h of the resistanc e change s is composed of load-induced chang e plu stemperature-induced change , and all gages have undergone the same tempera-ture change, then Eq . (11.19) can b e written as

where subscrip t L = load-induced resistanc e changesubscript T = temperature-induced resistanc e chang e

Since

This shows that we have temperature compensation fo r an initially balancedbridge a s lon g a s adjacan t arm s ar e mad e u p o f strain gage s o f the sam etype, bonded t o th e sam e material , an d kep t a t th e same temperature .

Although the basic idea for temperature compensation is simple enough,nevertheless, like many other aspect s of strain gage work , attention to detai lis essential if optimum results are to be achieved. One must always rememberthat th e observation s fro m what , in othe r respects , i s a practicall y perfec ttest can b e made quite valueless by faulty temperatur e compensation. Som epoints t o b e kept i n mind ar e th e following :

1. Th e magnitud e o f th e erro r include d i n th e indicate d observatio ndepends upo n(a) Change s i n temperature betwee n active and dumm y gages.(b) Th e gages and materia l upo n whic h the y are mounted .(c) Th e operatin g temperatur e level .

2. Th e piec e o f materia l upo n whic h th e dumm y i s mounte d ma y b eunintentionally subjecte d t o mechanica l strain.

3. Th e therma l connectio n betwee n the bloc k carryin g th e dumm y gageand the material upon which the active gage is mounted may not be verygood s o tha t a temperatur e differentia l i s set up.

4. Ther e wil l b e a differenc e betwee n gage s o f the sam e lot , particularlyat th e higher temperatures .

There ar e condition s unde r whic h temperatur e compensatio n ca n b eattained b y havin g th e dumm y gag e pla y a n active , rather tha n a passive,


FIG. 11.9. Cantileve r beam s wit h strai n gage s aligne d paralle l t o th e longitudina l axi s an dtemperature-compensated.

role i n th e measuremen t o f stress-induce d strains , an d a t th e sam e tim eincrease th e outpu t signal . On e suc h arrangemen t i s show n i n Fig . 11.9 ,which consists o f a thin cantilever beam. Since top and botto m surface strainsat a given section ar e equal in magnitude but o f opposite sig n when the loa dF i s applied, either a half-bridge circuit consisting of gages 1 and 2 (or 3 and4) i n adjacen t arms , o r a full-bridg e circuit , a s shown , ca n b e used . Thi srequires, o f course, tha t n o therma l gradien t exist s in the bea m an d tha t al lgages ar e a t th e sam e temperature . Th e outpu t signa l i s either tw o o r fou rtimes tha t o f a singl e gage , dependin g o n th e circuit , and wil l giv e strain sdue onl y t o th e bendin g cause d b y load F .

Figure 11.1 0 show s a tensio n membe r wit h fou r gages . Unde r th erequirement tha t all gages are at the same temperature , the full-bridge circuitshown wil l b e temperature-compensated , an d th e outpu t signa l wil l b e2(1 + v ) times the average longitudina l strain. This circui t wil l read only th e

FIG. 11.10. Tensio n membe r wit h strain gages .

TEMPERATURE EFFECT S O N STRAI N GAGE S 357

FIG. 11.11. Strai n gage s arranged fo r measurin g torque.

effect o f the axia l load F and wil l cancel bending strains . On th e other hand ,if only gages 1 and 2 were used in adjacent arm s o f a half-bridge circuit, thecircuit would be temperature-compensated bu t th e relationship betwee n thestrains, g j and e 2, would no t b e known unles s the load , F , was a pur e axialload.

A thir d exampl e i s illustrated i n Fig . 11.11 . I f gages a , b , c , and d ar earranged i n a full-bridg e circui t wit h gages a an d b i n adjacen t arm s an dgages c an d d i n arm s opposit e a an d b , respectively , then th e bridg e (al lgages mus t b e a t th e sam e temperature) , wil l b e temperature-compensate dand th e outpu t wil l b e a function of the torqu e only.

As a final example, an instrument was used in which a full-bridge circuithad t o b e wire d t o externa l bindin g post s tha t wer e arrange d i n a fairl ycompact rectangl e o n th e instrument' s to p surfac e an d adjacen t t o th ebalancing control . Tw o activ e gage s wer e arrange d i n opposit e arms , an dsince testin g took plac e a t roo m temperature , tw o precisio n resistor s wer eplaced i n th e othe r tw o arm s a t th e bindin g posts . Whe n balancin g th einstrument, driftin g was observed an d balanc e coul d no t b e obtained . Thi scontinued fo r som e tim e unti l th e operato r realize d tha t hi s arm , whe nadjusting th e instrument , wa s directl y above on e o f the precisio n resistors ,causing its temperature t o change. Replacing the two precision resistor s withtwo dumm y gages bonde d t o a piec e o f the tes t materia l an d movin g themaway fro m th e instrument solve d th e problem .


Problems

11.1. A strain gage , whos e thermal outpu t curve is shown in Fig . 11.5 , is bonded t oa machine element made o f mild steel. The strai n indicato r i s balanced a t 75° Fusing a gag e facto r o f 2.0 , the n th e machin e elemen t i s heate d t o th e tes ttemperature of 200°F and a load i s applied. If GF = 2.05 from th e gage packag edata an d th e strai n indicato r shows , afte r loading , a n indicate d strai n o fK', = —246 5 u,in/in , determine the actua l strain, correcting fo r both th e therma loutput an d th e gag e facto r variation.

11.2. Th e machin e element in Proble m 11. 1 has it s temperature raised t o 250° F an da ne w loa d i s applied. Afte r a readin g i s taken th e temperatur e i s then raise dto 400°F and th e loading i s again changed . For th e followin g data , correc t th estrains a s i n Proble m 11. 1 and determin e th e differenc e i n strai n between th etwo temperatures :

r=250°F K - = -187 5 uin/i nT = 400°F f, \ = - 362 8 uln/in

11.3. A strai n gag e o f th e sam e typ e a s use d i n Proble m 11. 1 is bonde d t o a tes tspecimen an d th e indicato r i s balanced a t roo m temperature . A gage facto r of3.0 was inadvertently set on th e strai n indicato r rathe r tha n 2.0 . When th e tes tspecimen was brought to its test temperature of 300°F and loaded , the indicatedstrain wa s e.' t = 193 6 uin/in. Determine th e actua l strain .

11.4. Plo t th e therma l outpu t curve s for th e foil s give n in Tabl e 11.1 .11.5. A strai n gag e wit h foi l lo t numbe r A12BJ0 1 is t o b e use d o n a magnesiu m

member. Determin e th e approximat e therma l outpu t curve , £7-02 , usin g th ethermal outpu t curv e for steel for values of erol.

11.6. A strain gag e wit h foi l lo t numbe r A11BP1 1 is to b e use d o n a plasti c whosethermal expansion coefficien t i s 40 ppm/°F. Determine the approximate thermaloutput curve.

11.7. A rectangula r rosett e ha s a nomina l gag e facto r o f 2.1 2 for al l sections . Th ethermal outpu t curv e associate d wit h thi s rosett e wa s obtaine d o n 2024-T 4aluminum. Th e rosett e i s bonde d t o a stee l tes t membe r an d a gag e facto r of2.0 is set o n th e instrument , which is then balance d a t 75°C . The tes t membe ris loade d an d brough t t o a temperatur e o f 300°F . A t thi s temperatur e th ethermal outpu t (fo r aluminum ) i s t. TO = — 950 uin/in an d th e gag e facto rvariation i s 1 percent. Gag e a i s aligned alon g a chose n coordinat e axi s an dall angles ar e measure d fro m thi s axis . The followin g strains were recorded:

< = 87 5 uin/in a t 0 = 0°4 = - 196 0 uin/in at 0 = 45°s'c = - 157 5 uin/in at 9 = 90°

(a) Determin e th e principa l strain s fo r the uncorrecte d readings .(b) Determin e th e principa l strain s for th e correcte d readings .

11.8. A rectangula r rosett e ha s a nomina l gag e facto r o f 2.145 for al l sections . Th ethermal outpu t curv e associated wit h this rosett e wa s obtaine d o n 101 8 steel .The rosett e i s applied t o a stee l test member , a gag e facto r o f 2.0 is set o n th estrain indicator , an d th e instrumen t i s balance d a t 75°F . Th e tes t membe r i sloaded an d brough t t o a temperatur e o f — 50°F, wher e the therma l outpu t i s— 500 uin/in and th e gag e variatio n facto r is —0. 5 percent . Gage a i s aligned


along a chosen coordinat e axi s and al l angles are measured fro m thi s axis. Thefollowing strain s were recorded:

s'a= -68 5 nin/i n at 9 = 0°4= -182 5 uin/i n at 6 = 45°e'c= 133 5 uin/in a t 0 = 90°

(a) Determin e th e principa l strain s fo r the uncorrected readings .(b) Determin e th e principa l strain s for the corrected readings .

11.9. A delta rosette, bonded t o aluminu m and havin g a nominal gage factor of 2.08for al l sections, is loaded t o it s tes t temperatur e of 350°F. A t thi s temperaturethe therma l outpu t i s S TO = — 90 uin/in an d th e gag e variatio n facto r i s 1. 2percent. The strain indicator was initially balanced with GF = 2.0 at 75°F. Gag ea i s aligne d alon g a chose n referenc e axi s an d al l angle s ar e measure d fro mthis axis . The followin g strain s were recorded :

£'„= -53 5 uin/i n at 0 = 0°4 = -84 5 uin/i n at 9 = 120°s'c= 18 0 uin/in at 0 = 240°

(a) Determin e th e principa l strain s for the uncorrecte d readings .(b) Determin e the principal strains for the correcte d readings .

REFERENCES

1. Murray , Willia m M . an d Pete r K . Stein , Strain Gage Techniques, Lectures an dlaboratory exercises presented at MIT, Cambridge, MA: July 8-19,1963, pp. 95-96.

2. "Temperature-Induce d Apparen t Strai n an d Gag e Facto r Variatio n i n Strai nGages," TN-504, Measurements Group, Inc., P.O. 27777 , Raleigh, NC 27611,1983.

3. "Catalo g 500: Part B—Strain Gage Technica l Data, " Measurements Group, Inc.,P.O. Bo x 27777, Raleigh, NC 27611 , 1988 .

4. "SR- 4 Strain Gage Handbook," BLH Electronics, Inc., 75 Shawmut Road, Canton,MA 02021 , 1980 .

5. 798 6 Annual Book a t ASTM Standards, 191 6 Race St. , Philadelphia , P A 19103 ,"Performance Characteristic s o f Bonde d Resistanc e Strai n Gages, " Vol . 03.01 ,Designation: E251-86 , pp. 413-428. Copyright ASTM. Reprinted with permission.

12TRANSDUCERS

12.1. Introduction

When on e o r mor e strai n gage s ar e use d t o measur e some quantit y whos emagnitude ca n b e determine d b y th e indicatio n o f strai n o n som e load -bearing member, the whole unit i s frequently describe d a s a transducer . Th eload-bearing membe r ma y hav e one , two , three , o r mor e strai n gage smounted o n it , dependin g o n th e quantit y t o b e measured , th e precisio ndesired, an d th e influenc e o f extraneou s effects , som e o f whic h ca n b eeliminated o r reduce d t o negligibl e proportions .

In general , th e load-bearin g element s fo r transducer s ma y b e divide dinto a few categories whic h depend upo n wha t i s to b e measured, a s wel l asbeing dependent o n spac e requirements . These includ e direct stres s (tensio nor compression ) fo r th e measuremen t o f large forces , member s i n bendin gfor determinin g mediu m o r smal l forces , th e indicatio n o f torsion , th emeasurement o f fluid pressure, etc. Ther e ar e man y variation s of apparatu sto accomplis h thes e end s an d considerabl e overlappin g o f th e differen tprocedures, an d som e o f th e device s tha t hav e bee n develope d fo r specia lconditions ar e exceptionall y ingenious. A review o f some o f the mor e usua ltypes o f transducer i s presented i n thi s chapter .

Let u s now examin e a simpl e case involvin g fou r strai n gage s (on e fo reach ar m o f th e Wheatston e bridge ) a s indicate d i n Fig . 12.1 . Her e/?! = R 2 = R3 = R4 = Ra and , i f idealized bridg e condition s ar e assume dand the bridge is initially balanced , E = 0 and /^^ = R 2R4- Fo r thi s case ,then, the bridg e rati o is

From Eq . (5.39) , th e bridg e output , A£ 0, ca n b e writte n as,

Since th e uni t change s i n resistanc e wil l b e smal l compare d t o unity , th e

TRANSDUCERS 361

FIG. 12.1. Wheatston e bridge .

nonlinearity ter m i s determined b y usin g Eq . (5.42) . Thus ,

Equation (12.1) , the general expressio n fo r the output fro m th e initiallybalanced Wheatston e bridge , tell s on e tha t th e bridg e outpu t i s directl yproportional t o th e applie d voltage , V , an d fo r smal l uni t change s i nresistance i s nearly proportional t o th e following :

1. Th e algebrai c differenc e betwee n th e uni t change s o f resistanc e i nadjacent arm s o f the bridge .

2. Th e algebrai c su m o f th e uni t resistanc e change s i n opposit e arm sof th e bridge .

In additio n

3. I f two o r mor e gage s happe n t o b e connecte d i n serie s i n on e ar m o fthe bridge , th e averag e valu e o f th e correspondin g strain s wil l b ereflected i n the bridg e output .

This mean s tha t i f the gage s ar e appropriatel y located , a bridg e outpu t wil lbe produced representin g the addition, subtraction, or the average of strainsat certai n particula r locations .

The full bridge

Since Eq . (12.1 ) fo r A£ 0, th e outpu t o f a n initiall y balance d Wheatston ebridge, contains a n equa l numbe r o f terms wit h positiv e and negativ e signs,this suggests that if one were designing a transducer fo r full-bridge operation ,


strain gage s wit h positiv e and negativ e gage factors might be considered i norder t o achiev e the maximu m output , o r indication , pe r uni t load . A s itmay no t b e possible, or desirable , to us e gages wit h gag e factor s of oppositesign, i t i s fortunate that th e sam e result can b e achieve d b y using like gagesand mountin g them alternately in regions o f tension and compressio n o f theload-carrying elemen t o f th e transducer . This i s commo n practice , whic hworks bes t whe n th e strain s i n tensio n an d compressio n ar e o f equa lmagnitude.

Equation (12.1 ) also tell s u s tha t i f all gages ar e alik e and th e gage s i nadjacent arm s o f th e bridg e ar e subjecte d t o strain s o f opposit e sign , th eindication A£ 0 wil l b e large r tha n tha t fro m a hal f bridge . I n th e sam emanner, i f the strain s on th e gage s ar e o f the sam e sign , th e bridg e outpu twill b e less tha n tha t fro m a hal f bridge whos e gages i n opposit e arm s ar esubjected t o the two largest strains. In the worst case, there may be no bridg eoutput al l (AE0 = 0).

The half bridge

There are certain situations in which i t wil l be more convenient to us e a hal fbridge instea d o f a ful l bridge . I n thi s case tw o activ e gage s ar e employe dinstead o f four. Th e tw o gages represente d b y R3 an d R 4 i n Fig . 12. 1 can b ereplaced b y any tw o equal fixed resistors (for initial bridge balance) , o r theymay b e left out . In tha t event one must be sure that th e applied voltag e doesnot sen d a curren t throug h th e gage s i n exces s o f th e norma l carryin gcapacity, which is usually about 3 0 milliamperes.

With tw o fixed resistor s fo r R 3 an d R 4, AR 3 = A.R 4 = 0 . In thi s cas eEq. (12.1 ) reduces to

Equation (12.3 ) is the same a s Eq . (4.26) for the potentiometri c circui t wheno = l . Th e nonlinearity factor i s given by

The hal f bridg e i s particularl y usefu l fo r bendin g member s wit h asymmetrical cros s sectio n i n whic h th e tensil e an d compressiv e strain s o nopposite surface s ar e o f equa l magnitude . Fo r thi s cas e n = 0 an d th ehalf-bridge outpu t become s

TRANSDUCERS 36 3

The quarter bridge

When on e want s t o measur e strai n a t a singl e point , o r i n rar e cases , t oproduce a transduce r wit h a singl e active gage , th e singl e activ e gage an dthree fixe d resistor s ca n b e use d i n th e Wheatston e bridge . I n orde r t oaccomplish bridg e balanc e a t zer o load , on e o f the thre e resistor s mus t b eequal to that o f the gage. The other two, then, can have any resistance values,but the y must be equal t o eac h othe r i f a four-ar m bridg e is being used .

A convenien t way to provid e the thre e fixed resistors, although not th eonly one , i s to moun t thre e strai n gages , identica l t o th e activ e one, o n apiece o f materia l simila r t o tha t upo n whic h th e activ e gag e ha s bee nmounted. This arrangemen t give s an equal-arm Wheatston e bridg e suitablefor bot h static and dynami c measurements. If the material carrying the threeinactive, o r dummy , gage s ha s th e sam e therma l characteristic s a s th ematerial carrying the active gage, the system will be temperature-compensated .

If dynami c measurement s onl y ar e t o b e mad e an d temperatur ecompensation i s of no concern , i t woul d b e preferabl e to us e the potentio -metric circuit and change the ballast rati o fro m 1 to about 10 . This increasesthe circui t efficienc y fro m 5 0 percent t o abou t 9 0 percent.

12.2. Axial-force transducers

Tension-compression load cell

This type of transducer, generally called a load cell , is one o f the earlies t t obe used . By proper en d connection s either tensil e o r compressiv e loads , o rboth, ma y b e measured . Th e centra l sectio n wher e th e strai n gage s ar ebonded i s made long enough s o that the strains at the gage location are no taffected b y th e en d conditions . Thi s sectio n i s designed s o tha t maximu mpossible strains are reached, yet the member remains within the elastic regionand wel l below the yiel d point o f the materia l in orde r t o reduc e hysteresis.The cross section of the load cell at the strain gage location can have differen tgeometries, with cylindrical, square, or tubular cross sections being common.

Figure 12.2 shows a cylindrical load cel l for both tensile and compressiveloads. Fou r gage s ar e shown ; gages 1 and 3 are 180 ° apar t an d aligne d i nthe longitudina l direction , whil e gage s 2 and 4 are 180 ° apar t an d aligne din th e transvers e direction . Th e gage s ar e arrange d int o a ful l bridg e a sillustrated, wit h E J = e 3 = E and e 2 = e 4 = — vs. Thi s bridg e arrangemen tcancels bendin g strain s an d i s temperature-compensate d a s lon g a s n otemperature gradient s exis t in th e member .

Instead o f using Eqs. (12.1) and (12.2 ) in determining the bridge output,AE0, an d th e nonlinearit y factor , (1 — n), we wil l retur n t o Eqs . (5.39 ) an d(5.40) for each case. The bridge output, A£0, for this bridge arrangement is


FIG. 12.2 . Loa d cel l fo r tensil e an d compressiv e loads .

The nonlinearit y factor , ( 1 — n), i s

These result s correspon d t o th e result s obtaine d i n Exampl e 5.2 , where thisbridge arrangemen t wa s examined .

Equation (12.6) shows tha t th e bridge gives an outpu t o f 2(1 + v ) timesas grea t a s tha t o f a singl e longitudina l gage , considerin g th e nonlinearit yfactor t o b e unity . Substituting the nonlinearit y factor, (1 — n), given b y Eq .(12.7) into Eq . (12.6) , the resultin g bridg e outpu t i s

The indicate d strain , £ t, in term s o f th e actua l strain , E , is

TRANSDUCERS 36 5

Solving Eq . (12.9 ) for e gives

The ratio , £/e ;, is

When usin g thes e equations , not e tha t th e strain s mus t b e entere d a s£ x 10~ 6 in/in.

These equations sho w that the bridge output is nonlinear, since all gagesdo no t se e the sam e strai n magnitudes . Th e transvers e strains , becaus e ofthe Poisson effect, are about 30 percent of the longitudinal strains. Equation(12.10) show s tha t th e actua l strai n wil l b e larger tha n th e indicate d strai nfor a tensil e force , whil e th e convers e i s tru e fo r a compressiv e force . Th ebridge nonlinearit y a t a strai n leve l o f 100 0 uin/in , fo r eithe r tensio n o rcompression strains , i s about 0.0 7 percent .

Another nonlinearit y facto r i s present i n th e geometr y o f the loa d cel lin tha t th e area changes unde r load . Thi s can b e approximated fo r a roun dcross section by considering the change in diameter due to the Poisson effect .The diameter , d , at an y load withi n the elasti c region , is

where d0 is the diameter a t no load. The bridge an d geometry nonlinearities ,however, are offsetting , wit h th e bridge nonlinearit y being th e highe r o f thetwo.

It may be desirable, a t times , to have only gages 1 and 3 active. In tha tcase, precisio n resistor s coul d b e used i n arm s 2 and 4 to complet e th e ful lbridge; however, a more convenien t wa y would be to bon d tw o strain gagesto materia l simila r t o th e transduce r an d us e them a s dumm y gages . Thi swould giv e temperatur e compensatio n provide d al l gage s remaine d a t th esame temperature . The bridge output, A£0, woul d be double tha t o f a singlegage an d woul d be nonlinear .

Ring-type load cell

The proving ring has been i n use for years as a standard fo r the calibratio nof tensile-testing machines . The diametra l deflectio n of the ring is a measureof the applied load, where the deflection is measured by means of a precisio nmicrometer. Th e thicknes s o f th e cros s section , whic h i s th e differenc ebetween th e inne r an d oute r radii , i s small compare d t o th e mean radius .

Rather than measur e diametra l deflection , strain gages may be bonde d


FIG. 12.3. Ring-typ e loa d cell .

to the ring , as shown in Fig. 12.3 , arranged int o a ful l bridge , and th e bridg eoutput use d a s a measur e o f the applie d load . A n axia l forc e an d bendin gmoment ac t a t th e sectio n containin g th e strai n gages , a s illustrate d in th efree-body diagra m o f th e uppe r hal f o f th e loa d cell . Sinc e eac h gag e i ssubjected t o the same axial strain due to the axial force, F/2, these strains arecanceled an d th e bridge responds onl y to the strains induced b y the bendingmoment, M 0. Fo r th e tensil e load, F , gage s 1 and 3 wil l b e i n compressio ndue t o bending , an d gage s 2 and 4 wil l b e i n tension . The convers e wil l b etrue for a compressive load. Furthermore, thi s bridge arrangemen t give s ful ltemperature compensation .

The strain at the gages may be estimated from th e bending stresses. Themoment, M 0, is

Since th e cros s sectio n i s rectangular, the strai n du e t o M 0 i s

TRANSDUCERS 367

FIG. 12.4. Curve d beam cross section.

This reduce s to

where h « R .If th e differenc e betwee n th e inne r an d oute r radi i increases , th e loa d

cell wil l n o longe r b e considered a thi n ring . In thi s case , a bette r estimat eof the bending strains due to M0 may be obtained fro m curve d beam theor y(1). In the curved beam, the centroidal axis and neutra l axis do not coincide ,with th e neutra l axi s shifte d inwar d toward s th e cente r o f curvature . Th egeometry, shown in Fig . 12.4 , has th e followin g notation :

Rj = radius o f inner fiber

Rn = radius of neutral axi s

R = radiu s of centroidal axi s

R0 = radius o f outer fiber

e = distance betwee n the centroidal an d neutra l axes

h = section thickness , R 0 — Rt

w = section width

yt = distanc e fro m th e neutra l axi s to th e inner fiber

y0 = distanc e fro m th e neutral axi s to th e oute r fiber

The bending moment , M 0, now become s


Since e = R — Rn, th e radius , /?„, must be computed . I t i s

The stresse s a t th e inner an d oute r fibers are

where A i s the cross-sectional area. For th e sens e of M0 show n in Fig . 12.3 ,o-; will be a tensile stress and a 0 wil l be a compressive stress. The correspond -ing strain s ar e

For a tensile force, F , acting o n the load cell , we see that £ t = e 3 = — £0 ands2 = e 4 = E ; and s o the bridg e outpu t i s nonlinear. Fo r th e developmen t ofthese equations , th e reade r i s referred to Referenc e 1 .

The expressions for the strains at the gage locations ar e estimates, sincethe bosses where the load i s applied have a stiffening effect . The y can b e usedfor design , but calibratio n i s essential.

12.3. Simple cantilever beam

A devic e ofte n use d a s a transduce r i s th e cantileve r beam . Amon g it sapplications, i t may b e used t o measur e force , t o serv e as a comparator , o rto determine deflections in areas not readil y accessible to othe r instruments.

Single active gage

The most basic application uses a single active gage and thre e fixed resistorsin th e Wheatston e bridge . Th e bea m ca n b e made fro m a piec e o f unifor mbar stoc k o f rectangula r cros s section , wit h th e strai n gag e mounte d nea rthe fixed end o n th e longitudina l center lin e of the uppe r surface . Th e force ,F, ca n the n b e measure d afte r a suitabl e calibration o f the bea m ha s bee nperformed. Thi s devic e is subject to th e followin g limitations:

TRANSDUCERS 36 9

1. Th e outpu t will be low because only one arm o f the Wheatstone bridgeis active.

2. Th e lin e of action o f the applie d force , F , must alway s remain paralle lto itsel f (includin g calibration ) an d a t th e sam e distance , L , fro m th ecenter o f the gage .

3. Unles s a self-temperature-compensated strain gage is used to match thethermal properties o f the beam, the apparatus can only be used preciselyat th e temperatur e o f calibration, otherwise serious error s ma y occur .

4. N o compensatio n i s provide d fo r force s (i f any) , othe r tha n F ,which ma y produce latera l bending , torsion, o r direc t axia l thrust.

The bridge outpu t i s given as

The nonlinearit y facto r is

The indicate d strain , e t, in term s o f the actua l strain , £ , is

Solving Eq. (12.24 ) for e produce s

The ratio of the actua l strai n t o th e indicated strai n is

Longitudinal and transverse gages on the same side

There may be some cases for which two gages mus t be mounted on the sam eside of the beam . Her e on e ca n tak e advantag e o f the Poisso n effect , whichproduces a latera l strai n o f opposit e sig n fro m th e axia l strain . Thi sarrangement use s one gage mounted in a longitudinal direction and a second


gage bonded in the transvers e direction. Thi s arrangemen t ca n b e used a s ahalf bridg e whe n th e gage s ar e connecte d i n adjacen t arms . Wit h th e tw ogages a t th e sam e temperature , th e bridg e outpu t wil l b e automaticall ytemperature-compensated. However , al l othe r characteristic s o f th e singl egage applicatio n als o appl y t o this . Depending upo n th e valu e o f Poisson'sratio for the ba r material , the output for this bridge wil l be about 3 0 percentgreater tha n tha t o f a singl e gage.

For thi s bridge, the longitudinal strain is £a = s and the transverse strainis £„ = — ve. With the gage s placed i n bridge arms 1 and 2 , the uni t resistancechanges ar e

From Eq . (7.21),

Using th e value s of R l/R1, R 2/R2, an d F a from Eqs . (a), (b), and (7.21) ,the bridg e output , E 0, i s

The nonlinearit y factor , (1 — n), can b e writte n as

TRANSDUCERS 37 1

If K i s ignored ( K = 0), then A£ 0 an d ( 1 — ri) revert t o

Two longitudinal gages on opposite surfaces

Provided tha t th e tw o side s o f the bea m ar e fre e fro m an y obstruction , aconsiderable advantag e i n outpu t ca n b e obtaine d b y mountin g the gage sback t o bac k o n opposit e surfaces . Becaus e the y are subjecte d t o strain s ofequal magnitude but o f opposite sign , the y ca n be placed in adjacen t arm sof a hal f bridge. Sinc e s i = s an d & 2= — e, the bridg e output , A£ 0, wil l b elinear an d i s

This arrangemen t i s compensate d fo r temperatur e change s provide dboth gage s are maintaine d a t lik e temperatures . I t i s also compensate d fo rdirect axia l thrust , which wil l produc e th e sam e resistance changes i n bot hgages, althoug h axia l thrus t wil l produc e bridg e nonlinearity . For metalli cgages, the variatio n o f this nature in ( 1 — n) will usually be smal l enough t obe neglected .

Full bridge, two gages back to back on opposite surfaces

Four gage s mounted back t o back i n pairs on opposite surface s of the bea mand arrange d a s a ful l bridg e wil l giv e th e larges t bridg e output . Sinc eei = e 3 = £ and E 2 = £ 4 = ~ £> the bridge output , A£ 0, wil l be linear and is

Provided al l gage s ar e maintaine d a t th e sam e temperature , thi sarrangement gives temperature compensation. A s in the two-arm bridge, thestrains cause d b y axia l thrus t wil l b e nullified , althoug h thes e strain s wil lproduce bridg e nonlinearity .

There ar e severa l comments i n orde r concernin g th e cantileve r beam .

1. I f the line of action o f the force remains paralle l t o itself , the moment a tthe gage section decreases because o f the shortening of the moment ar mdue t o th e curvatur e of the beam .


2. Th e strai n alon g th e lengt h o f the strain gag e i s not constant . Thi s ca nbe alleviated b y designing a constant-stres s bea m o f uniform thicknessand a triangularly shaped width , or unifor m widt h and a parabolicall yshaped thickness . Th e loa d i n eac h cas e i s applie d a t th e narrowes tpoint o f the beam . Fo r a tapered widt h beam, se e Fig. 7.5 .

12.4. Bending beam load cells

A variet y of load cell s can b e constructed b y usin g different configuration sof beams. Whethe r or not al l make satisfactory load-measuring devices mustbe determine d b y a combinatio n o f analysi s and testing . Severa l differen ttypes, among th e man y available , wil l b e discussed here .

Fixed-end beam

One may consider a beam wit h fixed ends an d cente r loading , show n in Fig .12.5, for use as a load cell . One placemen t o f the strain gage s and th e bridg earrangement ar e shown , with gage s 1 and 3 being in compression an d gage s2 an d 4 bein g i n tensio n fo r th e loadin g illustrated . Becaus e th e bea m i ssymmetrical, th e reaction s a t eac h built-i n end ar e identical . Furthermore ,the support s ar e ver y stif f compare d t o th e beam . Expression s fo r th emoments an d reactiv e forces a t th e fixed ends ma y b e developed o r foun din a tex t on mechanic s o f materials.

Since th e end s o f the bea m ar e constraine d fro m movin g laterally, thisinfluence wil l no t b e accounted fo r i n th e expression s fo r th e en d reactions .Because of this constraint, a horizontal force is produced tha t affect s bendin g

FIG. 12.5. Bea m with fixed ends.

TRANSDUCERS 373

moments i n th e bea m an d therefor e the deflectio n curve . While the strain sproduced b y this force are canceled by the bridge arrangement, nevertheless,they wil l produc e nonlinearit y in the bridge output .

Two fixed-end beams

A beam-typ e loa d cel l tha t overcome s th e lac k o f lateral movemen t o f th efixed end s i s show n i n Fig . 12.6 . Durin g loading , eithe r i n tensio n o rcompression, the ends are free to move laterally and thus eliminate horizontalforces o n th e beams . Th e centra l sectio n wher e the loa d i s applied an d th etwo end support s ar e ver y stif f compare d t o th e thinned bea m sections , an dso practically all of the deflectio n is produced i n the thi n sections . Thi s loa dcell, however, has twice the deflection of the single beam shown in Fig. 12.5.

FIG. 12.6 . Dual-bea m load cell .

One arrangemen t o f four strai n gage s for a ful l bridg e i s shown. Gages1 and 4 are subjected to strain s of like sign, and gage s 2 and 3 are subjectedto strain s o f like sign. If the loa d i s compressive, for instance , gages 1 and 4will have tensile strain s an d gage s 2 and 3 will have compressive strains.

When designin g this load cell , one wants to estimat e th e strain leve l atthe gag e locations . I n orde r t o accomplis h this , a free-body diagram o f theupper bea m i s show n i n Fig . 12.7 . Section s A- B an d C- D hav e th e sam emoments o f inertia, while section B-C ha s a much larger momen t o f inertiain orde r t o reduc e the deflectio n i n thi s section. Althoug h section B- C wil ldeflect slightl y (dependen t o n th e valu e o f its momen t o f inertia compare dto section A-B), mos t of the deflection wil l occur in sections A-B an d C-D.Since the beam is symmetrical, the reactions a t both end s are equal; however,the beam i s statically indeterminate t o the first degree, since M0 is unknown.Knowing th e slop e o f the deflectio n curv e i s horizontal a t poin t A an d a tthe center under the load, M 0 in terms of the beam dimension s and th e load,F, can b e computed. The moment-are a method , fo r instance, ma y easil y beused.

If th e gages can b e located s o the strain s a t gage s 1 and 4 are equa l i n


FIG. 12.7 . Free-bod y diagram o f the uppe r beam.

magnitude but opposite in sign to gages 2 and 3 , the bridge output, A£0, wil lbe linear . Becaus e o f the latera l movemen t o f the ends , though , ther e ma ybe som e nonlinearit y effec t becaus e o f the sligh t change i n geometry . Also,if al l gage s ar e subjecte d t o th e sam e temperature , th e bridg e wil l b etemperature-compensated.

S-shaped dual beam

The S-shaped, dual beam load cell uses two beams attached t o sections whosestiffness i s much large r tha n tha t o f th e beams . I t i s used fo r direc t tensil eor compressiv e loads , a s show n i n Fig . 12.8 . For bes t results , the loa d cel lshould b e machined fro m a solid bloc k o f material. Eccentri c loading error sare minimize d an d th e gage s ar e easily protected .

Figure 12. 9 shows the loadin g (fo r a tensil e force) acting o n on e o f thebeams, along with the deflection curve, for estimating the strains in the beam.

FIG. 12.8. S-shaped , dual-bea m load cell .

TRANSDUCERS 375

FIG. 12.9. Fixed-en d beam wit h en d displacement.

From symmetry, the reactions a t each end of the beam ar e identical. Durin gloading, th e right-han d sid e o f th e beam , relativ e t o th e left , move s u pthrough a distance , y , thereb y producin g th e reaction s shown . Th e shea rforce, R 0, an d th e movement , M 0, ca n b e determine d i n term s o f th edeflection, y , an d th e bea m dimensions . Onc e again , th e moment-are amethod lend s itsel f to th e determinatio n o f the reactions . Th e value s of R 0and M 0 i n terms of the bea m deflectio n and bea m dimension s ar e

For th e gage placements shown , gages 1 and 3 are in tension an d gages2 and 4 are i n compression fo r a tensil e load, whil e the converse applies fo ra compressiv e load. Furthermore , as long as all gages are subjecte d to thesame temperature , th e bridg e wil l b e temperature-compensated . Th e non -linearity of the bridge will depend on the values of the strains at each gage.

There are a number of other beam-bending load cells in use or that couldbe constructe d fo r laborator y use . A n examinatio n o f a manufacturer' scatalog wil l show beam-bending loa d cells are used for applications involvin gloads fro m les s than 1-l b a t th e lo w end t o abou t 100 0 Ib at th e uppe r end .For load s i n exces s of 100 0 Ib, other design s ar e generall y utilized . For a nexcellent discussio n o f strain-gage-based transducers , se e Reference 2 .

12.5. Shear beam load cell

The shea r beam loa d cell , usually designed fo r hig h loads , i s in th e for m o fa cantilever beam wit h a cross section larg e enough tha t the beam deflection


FIG. 12.10. Shea r bea m loa d cell .

is kep t t o a minimum . Sinc e th e bendin g stresse s o n th e oute r surfac ewould b e quit e lo w unde r thes e conditions , a reces s i s machine d o n eac hside s o th e cros s sectio n forme d resemble s a n I-beam . Here , mos t o f th ebending moment i s resisted b y the flanges , whil e the vertica l shear i s carriedby th e web . Th e shearin g stres s i s maximu m a t th e neutra l axis , an d s othe dimension s o f th e I-bea m sectio n ca n b e chose n s o tha t th e strain swill produc e a desire d bridg e output . Suc h a loa d cel l i s show n i n Fig .12.10.

Because ther e i s pur e shea r a t th e neutra l axis , th e principa l stresses ,and therefor e th e principal strains , are a t ±45 ° fro m th e neutra l axis . Twopairs o f strain gages, bonded bac k to bac k on opposite surface s o f the recess,can b e centere d acros s th e neutra l axi s a t +45° . Althoug h th e gage s ar esubjected t o a slight amount o f bending strain because the y extend o n eitherside o f th e neutra l axis , thi s effec t tend s t o b e self-canceling . A bette rarrangement, for instance, would use a two-element 90° gage, generally usedfor torqu e measurements , o n eac h surface . Choosin g a torqu e gag e wit helectrically independen t element s whos e grid s ar e +45 ° t o th e gag e longi -tudinal axi s allow s th e gag e t o b e bonde d s o tha t it s longitudina l axi scoincides wit h the neutra l axi s of the beam . I n thi s manner, th e element s ofthe gag e wil l experienc e bendin g strain s o f th e sam e magnitud e bu t o fopposite sign . When th e gage s ar e arrange d int o a ful l bridge , th e bendin gstrains wil l cancel . Furthermore , thi s arrangemen t wil l als o cance l an ybending due t o sid e loading .

As lon g a s th e loa d i s to th e righ t sid e o f the recess , a s show n i n Fig .12.10, th e bridg e outpu t i s relativel y insensitiv e t o th e poin t o f loa dapplication. Althoug h i t i s desirable to kee p th e bea m a s shor t a s possible ,

TRANSDUCERS 377

FIG. 12.11. Folde d shea r beam.

the load mus t be far enough fro m th e recess that its localized effect s wil l no tinfluence th e bridg e output .

The shearing stress at th e neutral axis, and thereb y the shearing strain ,must be determined in order t o estimate the bridge output , A£0, fo r a givenload. Th e equatio n fo r shea r stres s in th e web , which can b e found in textson mechanic s o f materials, is

where V = vertical shear forc e o n the sectio n/ = momen t o f inertia abou t the neutral axist = web thickness

Q = first moment o f the area above th e neutra l axis

The principa l stresses , a t 45 ° on eithe r sid e o f the neutra l axis , are equa l i nmagnitude bu t opposit e i n sign , giving a1 = — cr2 = T . The principa l strain sare

The shea r bea m loa d cel l may als o b e constructed s o that it s profil e i sS-shaped, as shown in Fig. 12.11 . This configuratio n is also referre d to a s afolded shear bea m b y some manufacturers . The lin e of action o f the applie dforce goe s throug h th e cente r o f th e strai n gag e bridge , thu s eliminatin gbending a t tha t section .


12.6. The torque meter

Although many different type s of torque meters have been devised , probabl ythe mos t common consist s o f a shaf t o f circular cross sectio n wit h four lik estrain gage s mounte d a t 45 ° to th e axi s of the shaft . Car e must b e taken i nassuring the gages ar e mounted a t precisel y 45°, and tha t companion gage ssubjected t o tensio n (o r compression ) ar e bonde d exactl y opposit e eac hother. A typica l torque meter i s shown i n Fig . 12.12 .

In constructing a torque meter, one should b e aware of its characteristics ,which ar e state d a s follows :

1. Th e uni t is automatically compensated fo r changes in temperature. Thisis due to the fact tha t a uniform temperatur e change will produce equa lresistance change s in al l four arm s o f the bridge , thereby producing n ochange i n th e conditio n o f balance.

2. Theoretically , th e instrumen t wil l no t respon d t o th e effect s o f axialthrust, i f suc h shoul d exist . Thi s i s becaus e axia l thrus t wil l produc eequal resistanc e change s i n al l fou r arm s o f th e Wheatston e bridge ;therefore, ther e wil l b e n o chang e i n th e conditio n of balance .

3. Ther e wil l be no respons e t o bending , if such should occur , because theresistance change in the two front gage s wil l be equal i n magnitude bu topposite in sign to the resistance changes in the two gages at the back.

4. Th e outpu t o f the bridg e wil l b e linea r wit h respec t t o th e torque , T ,because th e nonlinearit y factor i s (1 — n) = 1 ; that is , n = 0 .

Due t o the location o f the gages , torque produces resistance changes in

FIG. 12.12. Torqu e meter .

TRANSDUCERS 37 9

each bridge arm. Since the gages are alike , the bridge ratio i s a = 1 , and th eresistance changes ar e

This reduces t

Also, substitutin g thes e resistanc e change s int o Eq . (12.2 ) show s tha t th enonlinearity term , n, is zero, giving a linear bridg e output .

The case ca n b e examined where there i s not onl y torque bu t a n axia lload actin g o n th e meter . Unde r thes e conditions , th e resistanc e chang e i neach gag e i s

The subscript s T an d A refe r t o torqu e an d axia l thrust , respectively . Thebridge output no w becomes

Equation (12.38) shows that the bridge output does not change becauseof th e axia l load , providin g th e nonlinearlit y factor, ( 1 — n), i s unity . Th enonlinearity factor, however, must be examined to se e if it affect s th e bridg e

Simplifying, thi s reduces t o

Using these resistance changes, the bridge output, A£0, given by Eq. (12.1) is


output. Rathe r than us e the approximat e expressio n for the nonlinea r ter mgiven b y Eq . (12.2) , Eq. (5.40 ) wil l b e used . I f th e resistanc e changes give nby Eqs . (a) , (b) , (c) , and (d ) ar e substitute d into Eq . (5.40) , it wil l b e foundthe nonlinearit y factor reduces t o

Equation (12.39 ) show s tha t th e nonlinearit y term wil l hav e an effect ,although minor , whe n a n axia l force is combined wit h torsion . Thi s mean sthat th e nonlinearit y ter m wil l hav e a differen t valu e fo r eac h differen tcombination o f axial thrust and torque .

If th e torqu e mete r i s use d i n a stationar y application , th e lea d wire sfrom th e strain gage bridge may be readily connected t o a suitable indicator.For limite d angular motion a t a low rate o f rotation, the lea d wire s may beof suc h a lengt h tha t windu p i s permitted . If , however , the torqu e mete rrotates, the n som e arrangemen t mus t b e mad e t o brin g th e signa l t o th einstrumentation, eithe r throug h sli p rings , radiotelemetry , o r som e othe rmethod.

12.7. The strain gage torque wrench

Mechanical torqu e wrenche s have been in use for many years, with the mos tcommon havin g a pointe r attache d t o th e hea d en d an d extendin g ove r ascale, calibrate d t o rea d torque , attache d a t th e handl e end . Th e handl e i spin-connected t o th e wrenc h bod y s o tha t th e forc e i s transmitte d t o th ebody through the pin, thereby keeping the force at a fixed point. Rather thanuse a pointe r an d scale , strai n gage s coul d b e bonde d t o th e wrenc h bod ynear th e head end , the system calibrated, and th e torque rea d o n a suitabl estrain indicator . Th e force , however , would stil l hav e t o b e applie d a t afixed point . Ca n one , then , arrang e strai n gage s s o tha t th e indicato rreading i s a measur e o f th e torqu e an d independen t o f th e poin t o f forc eapplication?

Meier (3) investigated this problem and arrived at a bridge arrangementso tha t th e bridg e outpu t wa s linearl y relate d t o th e torqu e a t th ewrench-head cente r line , yet wa s independen t o f the poin t o f force applica -tion. Figur e 12.1 3 shows th e wrench , th e strai n gag e placement , an d th ebending momen t diagram .

The bending moment is maximum at section 3 where the torque is beingapplied. Since it is impractical to measure bending at this section, the bendingmoment, M3, can be related to the bending moments, M1 an d M 2, at section s1 an d 2 , respectively. Any force s an d moment s applie d t o th e wrenc h mustbe t o th e lef t o f section 1 , with non e applie d betwee n sections 1 and 2 . Th e

TRANSDUCERS 381

FIG. 12.13. Gag e arrangement on torque wrench for direct torque measurement. (From ref. 3.)

Since the bendin g moments , M i an d M 2, ca n b e expressed i n term s of

moments o f the thre e sections ar e

If Eqs . (b ) an d (c ) are solve d for th e force , F , then

From Eq . (d) , M3 is

Taking L 2 = 2L1(


strain,

where Z is the section modulus for bending and E is the modulus of elasticity.Substituting the value s of Mj an d M 2, given by Eqs . (e ) and (f) , respectively,into Eq . (12.41 ) produces

The eight strain gages bonded to the wrench can be arranged int o a ful lbridge to produc e th e operation indicate d i n parentheses i n Eq. (12.42). Forthe bendin g momen t diagra m shown , gage s a , b , an d c wil l experienc e acompressive strai n o f — e2, whil e gage d wil l hav e a compressiv e strai n o f— Ej. Gage s e , f , an d g wil l hav e tensile strains o f e 2, whil e gage h wil l hav ea tensil e strain o f el. Th e bridg e output, A£0, fo r the give n strains is

Comparing Eqs . (12.42 ) and (12.43) , it can b e seen tha t th e bridg e outpu t i sproportional t o the torque , M 3.

The torque wrenc h can easily be calibrated using known weights. Meierfound th e calibratio n curv e o f indicator readin g agains t th e torque , M 3, t obe ver y consisten t an d straigh t ove r a wid e range o f level arms and applie dweights. Whil e th e uni t ha s bee n describe d a s a torqu e wrench , i t ca n b eapplied t o othe r situation s requirin g a torqu e arm . On e application , fo rinstance, would be the determination of reaction torque for a cradle-mountedpiece of equipment, using the device described a s the arm fo r the measurementof torque abou t th e cradl e axis .

12.8. Pressure measurement

The measurement of pressure is often require d during the course of a project .There ar e man y device s available usin g pressur e forc e t o ac t o n a n elasti cmechanical element , thereby causing i t to deflect . Among these elements ar ethe Bourdon tube with different configurations , diaphragms, bellows, straighttubes, and flattened tubes . These elements are used in conjunction with somesort o f measuring system , so thei r deflection is an indicatio n o f pressure. I fthe pressure-measurin g devic e i s t o b e constructe d rathe r tha n purchased ,there ar e severa l options , depending o n projec t requirements .

TRANSDUCERS 383

FIG. 12.14. Thin-walle d pressur e transducer.

Thin-walled cylindrical tube

For stati c o r slowl y varying pressures, a simpl e an d effectiv e metho d i s t oconstruct a thin-walle d cylindrica l tube , wit h tw o gage s mounte d i n th ecircumferential (hoop ) direction . A ful l bridg e ca n b e use d b y placing thesetwo activ e gages i n opposit e bridg e arms , the n completin g th e bridg e b ybonding two dummy gages on an unstrained piece of similar material place dadjacent t o th e cylinder, or b y extending the soli d uppe r en d o f the cylinderand bondin g the dumm y gage s to thi s unstrained portion . If al l gage s ar emaintained a t th e sam e temperature , th e bridg e wil l b e temperature -compensated. A typical transducer o f this type is shown in Fig . 12.14 .

The circumferentia l strain, E H, and th e longitudina l strain , eL, are

where p = internal pressur ed = inner diamete rt = wall thicknes s

E = modulus of elasticityv = Poisso n rati o


With only the circumferential strain gages activ e and i n opposite bridg e arms,the bridg e output , A£ 0, i s

The nonlinearit y factor ,

Although th e circumferentia l stres s i s twice the longitudina l stress , th esame i s not tru e fo r th e strains . Usin g Eqs. (12.44 ) and (12.45) , the rati o ofstrains i s

For stee l with v = 0.3 , E H = 4.25s L. If all gages were bonded t o th e cylinder ,two circumferentia l and tw o longitudinal , and arrange d int o a full y activ ebridge, th e bridg e outpu t woul d b e reduce d b y approximatel y 2 4 percent .

This type of pressure transduce r i s best use d a t relativel y high pressur efor a compac t design . A s Eq . (12.44 ) indicates , th e diameter , d , mus t b eincreased and/o r the wall thickness, t, decreased i n order to obtain reasonabl estrain reading s fo r lowe r pressures . Onc e th e transduce r dimension s hav ebeen chosen , however , i t ca n b e constructe d an d calibrate d b y usin g adeadweight tester , fo r instance . Th e frequenc y respons e ca n b e improved b yreducing th e interna l volum e throug h th e insertio n o f a soli d plug , thu sreducing th e flow caused b y pressure variation .

Diaphragm pressure transducer

A second type o f pressure transducer uses a diaphragm. The diaphragm maybe made fro m a thin sheet o f flat material clampe d betwee n tw o element s ofthe transduce r body , o r i t ca n b e machine d a s a n integra l par t o f th etransducer body . The informatio n outlined her e ma y b e used t o arriv e a t apreliminary design , bu t th e fina l outpu t o f th e instrumen t wil l hav e t o b eobtained b y calibration. I n determining the characteristics of the diaphragm ,the followin g restrictions apply:

1. Th e diaphrag m i s rigidly clamped a t it s oute r edge .2. Th e diaphrag m i s fla t an d o f uniform thickness.3. Th e deflectio n of the cente r wil l no t excee d one-hal f o f the diaphrag m

thickness.4. Th e natura l frequenc y o f th e diaphrag m mus t b e hig h enoug h t o

respond adequatel y to fluctuating pressure .

TRANSDUCERS 385

FIG. 12.15. Clampe d circula r plat e with strai n distribution .

In determinin g th e characteristic s o f the diaphragm , th e analysi s fo r auniformly loade d thi n circula r plat e clampe d a t th e edg e ca n b e use d (4) .The plate an d it s loading ar e show n in Fig . 12.15 . The pressur e act s o n th eupper surfac e and th e strain gage s ar e bonded to the unde r surface .

The tangentia l bendin g moment , M t, an d th e radia l bendin g moment ,Mr, a t an y radiu s ar e

The corresponding stresse s ar e

The strain s follo w a s


The strain s give n by Eqs . (12.53 ) and (12.54 ) are als o plotte d i n Fig . 12.15 .At r = 0 the tangentia l and radia l strain s ar e identica l an d expresse d a s

At r = R the tangentia l strain is zero and th e radial strain becomes

Equations (12.55 ) and (12.56 ) show where the gage s shoul d b e placed .A pai r o f stacked orthogona l gage s coul d b e use d a t th e center , whil e tw oradial gage s coul d b e place d a s clos e t o th e boundar y a s possible , the narranged int o a ful l bridge . Althoug h th e bridg e woul d b e temperatur ecompensated, a n examinatio n o f Eq. (5.40) , usin g thes e strains , show s tha tthe nonlinearit y factor i s not zero .

Special gages , Fig . 12.16 , have bee n designe d fo r us e wit h diaphragm s(5). This gage takes advantage o f the strai n distribution show n in Fig. 12.15 .Since the tangential strain decreases mor e slowly with increasing radius thandoes the radia l strain , the centra l elemen t i s designed t o measur e tangentialstrain. Th e oute r element s ar e the n arrange d i n a radia l directio n t o tak eadvantage o f the radia l strai n a t th e boundary , where it i s maximum. If thestrain i s average d ove r th e regio n covere d b y eac h element , an d usin gGr = 2.0 , the bridg e outpu t i s approximately

The maximu m deflection , at th e cente r o f the plate , is

In order to have the transducer respond satisfactoril y to pressure pulses ,the natura l frequenc y o f the diaphrag m mus t b e a t leas t thre e t o five timeshigher tha n th e forcin g frequenc y (5) . The undampe d natura l frequenc y o f

The deflectio n a t an y radiu s is

TRANSDUCERS 38 7

FIG. 12.16. Diaphrag m strai n gag e fo r a pressur e transducer . (Courtes y o f Measurement sGroup, Inc. )

the diaphrag m i s

where g = gravitational constant, 386. 4 in/sec2

y = specifi c weight o f diaphragm material , lb/in 3

Comments

The transducer s describe d i n thi s chapter hav e th e inten t o f do-it-yourself ,where suc h a n instrumen t wil l b e use d wit h existin g strain-measurin ginstrumentation. They are , therefore, not designe d t o stan d alone . Wit h th edo-it-yourself transducer , desirable adjustments ca n be made a t th e instru -ment (suc h a s gag e facto r adjustment) t o brin g i t withi n the desire d limits.For mor e precis e compensatio n procedures , th e reade r i s referre d t o th epaper b y Dorsey (6 ) or t o Referenc e 2.


Problems

In all problems use steel with v = 0.3 and E = 30 x 106 psi.12.1. Th e load cel l shown in Fig. 12. 2 is used t o measur e loads between + 750001b .

The loa d cel l ha s a diamete r o f 1.5 0 in, GF = 2.15 , and R g = 12 0 ohms. Wit hthe load a t bot h extremes , determine the following :

(a) Th e bridg e nonlinearity.(b) Th e geometri c nonlinearity.

12.2. I n Proble m 12. 1 the bridg e is rearranged so that gages 1 and 3 are active gages ,with R2 = R4 = Rg being dummy gages bonded to a similar piece of unstrainedmaterial. Fo r th e sam e loadin g conditions , determine th e following :

(a) Th e nonlinearit y factor.(b) Th e bridg e output , A£ 0, i f the suppl y voltage i s 1 0 volts.

12.3. A compressive force , F , acts on a ring-type load cell . By considering the strain sbased o n a curved beam, <•: , = c 3 = E O and r, 2 = c 4 = — f,,., writ e th e expressionfor th e nonlinearit y factor , 1 — n.

12.4. A ring-typ e loa d cel l i s subjecte d t o a tensil e loa d o f F = lOOOOlb . It sdimensions are R0 = 3.0 in, K, = 1. 5 in, and w = 0.7 5 in. Determine the following:

(a) Th e percentag e differenc e i n M 0 betwee n Eqs . (12.13 ) an d (12.16) .(b) Th e strain s a t eac h gag e usin g thin-ring equations.(c) Th e strain s at eac h gag e base d o n curved-bea m equations.

12.5. Th e cantileve r bea m i n Fig . 12.1 7 is to hav e a constan t strai n o f I200uin/i nalong its tapered length when the load i s applied at th e verte x of the equilateraltriangle formed b y it s tw o sides . Determin e w.

12.6. Desig n a loa d cel l havin g th e geometr y show n i n Fig . 12.6 . Th e widt his limite d t o 0.62 5 in, th e rati o I BC/IAB i s 25 , th e tota l deflectio n i s no tto excee d 0.01 5 in , an d th e maximu m loa d i s 50 0 Ib. Fo r th e fina l gag elocation, compute th e bridg e nonlinearit y at th e maximum load .

FIG. 12.17 .

TRANSDUCERS 38 9

12.7. Desig n a load cel l having the configuratio n shown in Fig . 12.8 . The widt h islimited t o 1 in, and th e maximu m deflection must no t excee d 0.012 in a t th erated loa d o f 700 Ib. Determine the gage locations and thei r strains . Calculat ethe bridg e nonlinearit y at th e maximum load.

12.8. Desig n a shea r bea m loa d cel l havin g th e configuratio n show n i n Fig .12.10. Th e maximu m loa d o f 5000 0 Ib i s t o produc e A£ 0/K o f approxim -ately 2 mV/V.

12.9. Usin g Eq . (5.40) , derive Eq . (12.39).12.10. A torqu e mete r ha s a diamete r o f 1.2 5 in an d use s fou r 350-oh m gages ,

with G F = 2.10, t o for m a ful l bridge . I f A£ 0 /K=2mV/V a t maximu mtorque, determin e the valu e of the torque .

12.11. I f the transvers e sensitivity of the gage s i n Proble m 12.1 0 is 0.9 percent, whatwill b e th e percentag e chang e i n torqu e i f th e transvers e sensitivit y i sconsidered?

12.12. Desig n a torqu e wrench, shown in Fig . 12.13 , to th e followin g specifications:

(a) Th e maximu m torque i s 200 ft-lb.(b) Th e overal l length of the wrenc h must no t excee d 1 8 in.(c) Th e maximu m strain a t sectio n 2 shall be 100 0 uin/in at ful l torque .

12.13. A thin-walle d cylindrica l pressur e transduce r ha s a n interna l diamete r o f1.25 i n an d a wal l thicknes s o f 0.0 5 in. Tw o circumferentia l gage s wit hRg = 35 0 ohms an d G F = 2.10 ar e bonded t o the cylinder . If the hoop strain,EH, i s limite d t o 100 0 uin/in , determine the maximu m interna l pressur e an dthe corresponding valu e of A£0/K

12.14. Mak e a preliminary design o f a diaphragm pressur e transduce r t o measur e apeak pressur e o f 7 5 psi a t a frequenc y o f 10 0 cycles/sec. Th e desire d bridg eoutput, A£ 0/K, is 1.5 mV/V at the peak pressure . Use GF - 2.0 .

REFERENCES

1. Cook , Rober t D . an d Warre n C . Young, Advanced Mechanics o f Materials, Ne wYork, Macmillan , 1985 , Chap. 10.

2. Strain Gage Based Transducers, Their Design an d Construction. Measurement sGroup, Inc. , P.O . Bo x 27777, Raleigh, NC 27611 , 1988.

3. Meier , J. H. , "Some Phases o f the Technique of Recording Performanc e Data o nLarge Machines, " SESA Proceedings, Vol. X, No. 1 , 1952, pp. 35-52 .

4. Timoshenko , S. , Strength o f Materials, 3 d edition , Par t II , Ne w York , Va nNostrand Reinhold , 1958 , Chap. 4.

5. "Desig n Consideration s for Diaphragm Pressur e Transducers," TN-510 , Measure -ments Group , Inc. , P.O. Bo x 27777, Raleigh, NC 27611 , 1982 .

6. Dorsey , James, "Homegrow n Strain-gag e Transducers, " Experimental Mechanics,Vol. 17 , No. 7 , July 1977 , pp. 255-260 .

13STRAIN GAGE SELECT10N AND APPLICATION

13.1. General considerations

On first observation, the strain gage appears to be a rather simple instrumentthat ca n b e applie d wit h minimu m effort . Thi s ma y b e so , perhaps , i f th egage is to b e bonded t o a fairl y larg e plan e are a wit h ample workin g roomand using a quick-setting cement. The novice soon learns , however, that eventhe supposedly simpl e tas k o f satisfactorily soldering lead wires to the gage'ssolder tab s i s not easy . Whe n h e moves on t o bondin g gage s i n a confinedspace and the n attaching lead wires , his appreciation o f the talent of a skilledtechnician rise s rapidly .

When selectin g a strai n gage , o r gages , fo r a project , th e condition sunder whic h th e gag e wil l operat e mus t b e considered . Whe n al l o f th evariables that go into gage construction are examined (backing material, foil ,gage length , configuration , etc.) , thousands o f type s ar e available . A studyof manufacturer's catalogs shows that gage s ar e divided into related groups ,or series , o f the sam e basi c construction . Sinc e gage s belongin g t o a serie shave simila r characteristic s an d capabilities , th e tas k o f choosing a gag e i stherefore reduced .

The firs t ste p i n choosin g a strai n gag e installatio n i s to lis t a s man yconditions a s possible affectin g th e system. Some of these are the following:

1. I s the strai n t o b e measured i n elastic or plasti c region? I f the strai n i sin the plastic region, for instance, then a post-yield gage will be chosen.

2. I f th e stres s fiel d i s uniaxial , a single-elemen t gag e aligne d alon g th eprincipal stres s directio n wil l suffice . I f th e fiel d i s biaxia l an d th eprincipal strain directions are known, a two-element rectangular rosett ecan b e used . I f the principa l strai n direction s ar e unknown , a three -element rosett e wil l b e required .

3. Wha t is the duration o f the test ? Will it be measured i n minutes, hours,or years ? The concer n her e i s the shiftin g o f the zer o referenc e point .

4. Ho w difficul t wil l th e installatio n o f gages be ?5. Wil l th e test s b e stati c o r dynamic ? I f the y ar e purel y dynamic , the n

consideration ha s t o b e give n t o a foi l tha t exhibit s goo d fatigu eproperties.

6. Th e temperatur e rang e ove r which the gage wil l operate an d th e choiceof it s self-temperature-compensation numbe r mus t b e considered .

STRAIN GAG E SELECTIO N AND APPLICATIO N 39 1

7. Ar e strain gradients perpendicular t o th e tes t surface or i n the plane ofthe tes t surface ?

8. Th e choice of an adhesive is important and canno t be over emphasized.

Depending o n the specia l requirements of a given test, other condition s ca nbe added t o thi s list . Th e cos t o f the strai n gages , however , may hav e lowpriority, since the gag e cos t i s generally smal l whe n compared t o th e tota lcost o f a test .

After th e condition s tha t affec t th e syste m are listed , a manufacturer'scatalog ca n b e consulte d i n orde r t o choos e a specifi c gage. Here wil l b efound a designation code giving the features of the gage. They are as follows:

1. Gag e series an d typ e of strain-sensing allo y2. Backin g o r carrie r materia l o f the strain-sensin g alloy3. Self-temperatur e compensation numbe r4. Th e activ e gage length5. Gri d an d ta b geometr y6. Gag e resistance7. Options , i f desired

13.2. Strain gage alloys (1, 2)

Constantan

One o f the most common strain gage foils i s a copper-nickel alloy generallyknown as constantan. I t finds wide use in static strain measurements as wellas bein g employe d i n transducers . I t als o show s goo d fatigu e lif e whe napplied t o alternatin g strains , providin g th e strai n level s ar e kep t belo w±1500 uin/in. I t ha s a lo w an d controllabl e temperatur e coefficien t o fresistance as well as good strain sensitivity, which gives a nominal gage factorof 2.0 . Furthermore , th e gag e facto r i s relativel y insensitive to strai n leve land temperature .

Constantan can be processed fo r self-temperature compensation so thatit matche s th e therma l expansio n characteristic s o f a numbe r o f commo nengineering materials wit h thermal expansion coefficient s rangin g from zer oto 5 0 ppm/°F. W e have seen how mismatching of the coefficient s of thermalexpansion o f the gag e an d tes t materia l rotate s th e therma l outpu t curv earound th e referenc e temperature i n orde r t o obtai n th e mos t favorabl eresults in a desired temperature range. An examination of strain gage catalogsshows that satisfactory gage resistance is obtainable even for very small gagesmade o f this material .

If very large strains, on the order o f 5 percent, are to be measured, thenan anneale d constanta n foi l is used . If gage lengths of | inc h or larger arused, strains in excess of 20 percent ca n b e measured. Annealed constantan ,however, i s not recommende d fo r cyclic strains, sinc e permanen t resistanc echange occur s a s a functio n o f number o f strain cycles.

392 TH E BONDE D ELECTRICA L RESISTANCE STRAI N GAGE

Constantan ha s severa l disadvantages . I f the tes t temperatur e i s abov e150°F (66°C) , i t show s referenc e point drift , whic h i s undesirabl e fo r test sconducted ove r a lon g perio d o f time. The therma l output i s also ver y highat temperature s belo w -SO T (-45°C ) and abov e 400T (205°C) .

Isoelastic

Isoelastic, mos t generall y use d fo r dynamic strain measurement , i s a nickel -chromium-iron allo y wit h molybdenu m added . It s hig h gag e facto r o fapproximately 3.2 improves the signal-to-noise ratio in dynamic testing. This,coupled wit h superio r fatigu e life , make s i t particularl y usefu l fo r dynami cstrain measurement .

The therma l output o f isoelastic is about 8 0 uin/in/T and i t cannot b eself-temperature compensated , thereb y makin g i t generall y unsuitabl e fo rmeasuring static strains. This feature makes i t undesirable for any long-rang emeasurements i f a stabl e referenc e poin t mus t b e maintained. Furthermore ,its respons e become s nonlinea r at strain s on th e orde r o f 0.5 percent; hence ,it i s confine d t o strai n measuremen t i n th e elasti c region . I n specia l cases ,where a high-outpu t response i s desired, i t ma y b e use d wit h a full-bridg ecircuit, thereby obtainin g circui t temperatur e compensation .

Karma

Karma, a nickel-chromiu m allo y wit h smal l percentage s o f iro n an daluminum, i s anothe r desirabl e material , sinc e gage s mad e o f thi s materia lshow minima l referenc e point drif t wit h tim e an d temperature . Becaus e o fthis stability , it is a fine choice fo r long-time static measurement s at o r nea rroom temperature . I t i s recommended fo r stati c strain measurement s fro m-452T (-270°C ) t o SOO T (260°C) , bu t encapsulate d gage s ca n b e use dto 750 T (400°C ) fo r shor t tim e periods . Th e materia l als o exhibit s goo dfatigue lif e wit h minimum reference point drif t eve n after bein g cycled a largenumber o f times . Becaus e o f it s hig h resistivity , smalle r gage s fo r a give nresistance can b e manufactured.

Karma ca n be self-temperature compensated ove r a broad temperatur erange, bu t i t i s mor e limite d tha n constanta n i n th e numbe r o f therma lexpansion coefficient s fo r whic h i t ma y b e compensated . A n advantage ,however, i s a flatte r therma l outpu t curve . Another feature is a gag e facto rthat goe s negativ e wit h increasin g temperature , thu s compensatin g fo rthe temperature-induce d chang e i n th e modulu s o f elasticit y o f th e tes tmaterial.

Karma ha s severa l disadvantages. I t i s difficul t t o solder , an d fo r thi sreason gage s wit h copper-clad tab s are available . Gages o f this material ar ealso mor e difficul t t o manufacture , making them mor e expensiv e than gage susing constantan .

STRAIN GAG E SELECTIO N AN D APPLICATIO N 39 3

Platinum-tungsten

A platinum-tungste n allo y ha s bee n develope d fo r high-temperature use . I thas unusua l stability and fatigu e lif e a t temperature s abov e 750° F (400°C) ,does no t underg o an y metallurgica l change s t o abou t 1650° F (900°C) , an dso it s resistanc e remain s essentiall y unchange d wit h time . I t ha s a high -temperature coefficien t o f resistance that i s not adjustable , although repeat -able; thus , it cannot b e self-temperature compensated. I f temperature comp -ensation i s desired, i t should b e done throug h circui t compensation .

This materia l i s use d fo r dynami c strai n measurement s t o 1500° F(815°C) an d fo r stati c strai n measurement s t o 1200° F (650°C) . I t ha s ahigher strai n sensitivit y tha n copper-nicke l o r nickel-chromiu m alloys ,but i t i s nonlinear . Th e strai n rang e i s generall y limited t o approximatel y+ 0.3 percent.

13.3. Grid backing materials (1-4 )

The strain-sensin g elemen t (eithe r foi l o r wire ) of a strai n gag e i s mounte don a backing (carrier) material. The backing material serves several purposes.

1. I t protect s th e strain-sensin g gri d fro m damag e durin g handlin g an dinstallation.

2. I t provide s a bondin g surfac e t o th e tes t piece .3. I t transmit s strai n fro m th e tes t piec e t o th e strain-sensin g alloy . It s

stiffness mus t be low enough s o it can follow the strains in the test piecewithout affectin g it . On th e other hand, i t must be stif f whe n comparedto th e strain-sensin g allo y s o tha t th e conducto r materia l follow s th estrains without irregular distortion .

4. I t provide s electrical insulation between the strain-sensing element andthe tes t piece .

Paper carriers

One o f th e firs t backin g materials , an d on e stil l used , i s a nitrocellulos epaper. Strai n gage s usin g thi s readil y availabl e backin g materia l easil yconform t o th e surfac e o f a tes t specimen . Th e gage s ar e usuall y bonde dto a tes t piec e wit h a nitrocellulos e cemen t tha t impregnate s th e paper' spores an d cure s b y evaporation. Gage s bonde d i n thi s manne r ca n operat ebetween - 100° F and 180°F (-7 3 t o 82°C), although the y can be used forshort periods of time beyond th e upper temperatur e limit. At room tempera -ture, this combination o f paper an d adhesive , when properly applied , can besubjected t o strain s in exces s of 1 0 percent befor e breaking down .

Polyimide resins

Polyimide resin s ca n b e provide d i n bot h cas t fil m an d glass-reinforce dlaminated construction . I t i s a general-purpos e materia l use d fo r bot h


static an d dynami c strains . The cas t fil m type s are tough , flexible , an d ca nbe elongate d u p t o 2 0 percent . Becaus e o f thei r flexibility , the y ca n b econtoured t o fi t smal l radii . Thi s materia l ca n b e use d a t temperature sranging from cryogenic to 400°F (205°C). For highe r temperatures, however,the resi n ca n b e reinforce d with glass and th e gag e encapsulate d fo r us e t o700°F (370°C), although th e temperature can b e increased to 750° F (400°C )for short-duratio n tests.

Epoxy resins

Epoxy resins reinforced with glass fibers were developed i n order t o improvetemperature capabilities . Thi s material has an operatin g temperatur e range ,for bot h stati c and dynami c strain measurements , fro m cryogeni c t o abou t550°F (290°C), with an uppe r limit of 750°F (400°C) fo r short-duration tests.This backin g als o ha s improve d dimensiona l stabilit y for us e i n precisio ntransducers. The glass reinforcement, however , reduces the maximum strai nto abou t 1 percent bu t result s in a n extremel y thin carrier . Sinc e i t i s morebrittle than polyimide , it requires more care i n handling in order t o preventdamage.

Metallic carriers

Metallic carrier s hav e bee n discusse d i n Chapte r 1 under weldabl e gages .Weldable wir e gages ar e covere d i n Sectio n 1.5 , while weldable foi l gage sare discusse d i n Sectio n 1.6 .

13.4. Gage length, geometry, and resistance (1, 2)

Gage length

When referrin g t o gag e length , it i s the activ e o r strain-measurin g portio nthat is referred to, not th e overall or matrix length. A major purpose of usingstrain gages i s to determin e strains at critica l points on a structure . Becausethese points are ofte n wher e stress concentrations exist , thereby resulting instrain gradient s whic h may b e quit e steep , consideratio n mus t b e give n t othe strai n gradien t alon g th e gag e length . Stee p strai n gradient s ma y als ooccur i n dynami c measurements , suc h a s occu r whe n th e propagatio n o fstress wave s i n a materia l i s being studied . Sinc e a strain gag e average s th estrain alon g it s activ e length , choosin g a gag e lengt h considerabl y longe rthan th e pea k strai n regio n result s i n a strai n readin g o n th e lo w side ;therefore, a gag e lengt h consisten t wit h th e pea k strai n regio n shoul d b echosen. For nonhomogeneous materials , however, a gage length long enoughto spa n th e representativ e structure of the materia l should b e used i n orderto averag e the strai n ove r voids , etc.

When possible , gage s wit h lengths from ^ in to ^ in are preferable , sincethey are easie r to apply , offe r th e larges t numbe r o f geometries and options ,


and are less expensive. For gage s o f identical resistance an d applie d voltage ,the large r gag e wil l dissipat e hea t mor e easil y becaus e o f th e lowe r hea tgenerated pe r uni t area . Thi s i s particularl y importan t whe n th e gag e i sbonded t o a materia l wit h poor heat-transfe r qualities.

Gage geometry

When choosing a strain gage for a particular test, several elements ente r intothe decision . Amon g thes e ar e th e shap e o f th e strain-sensin g grid , th enumber o f grid s an d thei r orientatio n relativ e t o eac h other , solde r ta barrangement, an d spac e availabl e fo r mounting . I f th e principa l stres s i sknown to be uniaxial and it s direction is also known, then a single grid gagemay be used. This condition generall y does not exis t and singl e gages shouldbe use d only when one i s absolutely sur e one ha s a uniaxia l stress state .

For th e biaxial stress state a three-element rosette is used if the principalstress direction s ar e unknown . Th e grid s o f th e rosett e ma y hav e an yorientation relativ e to eac h other , bu t rosette s hav e bee n standardize d o nthe delt a an d th e rectangula r configuration . Thi s make s dat a reductio nsimpler, particularly for the rectangula r rosette . Whe n bondin g a rosette t oa tes t specimen , any orientatio n ca n b e used , but usuall y one rosett e le g isaligned alon g som e chose n axi s of the specimen .

If mounting space is confined, stacked rosette s are preferred, particularlywhen ther e is a high strain gradien t in the plane o f the mountin g surface. I nthis case they give a closer approximation of the strain at the point, but hea tdissipation ma y b e a problem . The y ar e stiffe r tha n th e plan e rosett e an dconform les s easily to curved surfaces. O n th e othe r hand , plane rosettes ar epreferred whe n th e strai n gradien t i s normal t o th e surface , sinc e al l grid sare a s close t o th e specime n surfac e a s possible .

When th e principa l stres s direction s ar e known , the n a two-elemen trosette ma y b e used . The principa l stres s direction s ma y b e apparen t fro mthe geometr y o f the tes t specimen , suc h a s a thin-walle d tube wit h interna lpressure, fo r instance . Generally , th e principa l direction s ar e determine dthrough th e us e o f a brittl e lacque r coa t o r a photoelasti c coating . I t i sobvious tha t i f the principa l axe s ar e known , considerable saving s in tim eand labo r ca n b e attained i n wirin g a numbe r o f two-element gage s rathe rthan three-elemen t gages .

Special-purpose gages , suc h a s gage s fo r residua l stres s measurement ,crack detection, or diaphragm gages for pressure transducers, are available.

When a high strain gradient transverse to the gage axis exists, a gage witha narrow grid width should be chosen i n order t o giv e a bette r strai n average .The reduce d gage area , though , will reduce the abilit y to dissipat e heat.

Gage resistance

An examination o f manufacturer's catalogs show s that strai n gage s may b eobtained wit h resistances up to 100 0 ohms. The two most common resistanc e


values, however , are 12 0 and 35 0 ohms. A s we sa w i n Chapte r 5 , lead-lineresistance desensitize s th e circuit . I f lead-lin e resistanc e o r othe r parasiti cresistances ar e present , the n choosin g a higher-resistanc e gag e wil l reduc ethe circui t desensitization . This i s illustrated in Exampl e 5.4 . For th e sam eapplied voltage , a higher-resistanc e gag e reduce s th e hea t generated . A120-ohm gage , fo r instance , generate s nearl y thre e time s th e wattag e o f a350-ohm gage. Conversely, if the wattage remains the same, a higher voltagemay b e used o n th e highe r resistance gag e i n order t o increas e th e output.

Self-temperature compensation

Choosing a gage fo r the prope r self-temperatur e compensation numbe r i s amatter of examining a strain gage catalog in order to determine the availablethermal expansio n coefficients . Whe n th e desire d numbe r i s chosen , i t i s amatter o f addin g i t i n th e prope r plac e i n a manufacturer' s strai n gag edesignation code . Self-temperatur e compensatio n an d it s us e hav e bee ncovered i n Chapte r 11 .

Options

Both standar d option s an d specia l option s ar e available . Amon g thes e ar eattached lea d wires , gag e encapsulation , solde r dots , an d etche d integra lterminals, t o nam e a few . For a complet e descriptio n o f options , bot hstandard an d special , consul t a manufacturer' s catalog .

13.5. Adhesive s (1, 2 , 4)

Successful us e o f strai n gage s i s ver y muc h dependen t upo n satisfactoril ybonding th e gag e t o th e tes t specimen . Th e chose n adhesiv e mus t hav esufficient shea r strengt h i n order t o transmi t strain s i n the tes t specime n t othe strain-sensing grid, yet it must be compatible with both the gage backin gmaterial an d th e tes t materia l s o tha t neithe r i s damaged . Further , th eadhesive shoul d hav e long-term stabilit y s o tha t i t doe s no t decompos e o rshow appreciabl e cree p ove r th e test' s lifetime .

The manufacturer' s instructions i n th e us e o f th e adhesiv e mus t b efollowed carefully , particularl y i f the adhesiv e call s fo r mixin g a resi n an dhardener. Th e adhesiv e mus t b e capabl e o f forming a thi n glu e line fre e o fvoids, with minimum curing time being a desirabl e feature . I t als o help s t oelectrically isolat e th e gri d fro m th e tes t material . Whe n checkin g th eresisance betwee n th e gri d an d tes t specimen , th e recommende d resistanc eis 1000 0 megohms minimum , but preferabl y higher . An adhesiv e should b ecapable o f high elongation a s wel l as have the abilit y t o operat e ove r a widetemperature range .

A large number of adhesives are available, each with detailed techniquesfor it s application . Manufacturer s wil l suppl y th e use r wit h instructions .One thin g that i s crucial to satisfactor y strai n gage performance , regardless


of th e adhesiv e used , i s cleanliness. Th e tes t surfac e mus t b e fre e o f grease,rust, or other contaminants s o that the bare base material i s exposed. Duringthe preparation proces s the hands must be kept clean and care taken no t t otouch th e surface . Afte r th e surfac e i s prepared th e gage s should b e applie dwithout undu e delay.

Although ther e ar e numerou s adhesive s available , onl y th e mor ecommonly use d one s wil l b e discussed . Detaile d informatio n o n specifi cadhesives and their methods of application can be obtained fro m informatio nbulletins supplied by manufacturers.

Nitrocellulose

Nitrocellulose adhesive s (suc h a s Duco ) wer e onc e widel y use d whe npaper-backed gage s wer e prevalent . Thi s typ e o f adhesiv e set s b y solven tevaporation; thus , it s us e today i s limited t o paper-backe d gage s o r gage swith a porou s backing . A minimum pressur e ha s t o b e applied durin g th ecuring process, which is usually in excess of 24 hours, depending on humidityand temperarure . Applicatio n o f hea t wil l accelerat e th e curin g process ,however. The curing process may be monitored b y periodically checking thegage resistance to ground, since the resistance increases as the adhesive sets.

Gages bonde d wit h nitrocellulose adhesive s may b e used u p t o 180° F(82°C). The y ar e hygroscopi c (i.e. , they absorb moistur e fro m th e air ) an dmust be protected wit h a moisture-resistant coating once the adhesive is fullycured to ensure electrical and dimensiona l stability. Because adhesives of thistype ar e vulnerabl e t o ketoni c solutions , the y ar e easil y remove d withou tsurface damag e b y usin g a ketonic solution.

Cyanoacrylate

Cyanoacrylate adhesive s ar e widel y use d a s general-purpos e cement s tha tare fas t curin g and simpl e to use , since no mixin g is required. While the lif eof an unopened containe r o f Cyanoacrylate is approximately 9 months whenstored a t roo m temperature , th e lif e ca n b e extende d b y refrigeratio n a t40°F (4°C) . Whe n remove d fro m th e refrigerator , the adhesiv e shoul d b eallowed t o com e t o roo m temperatur e befor e openin g i n orde r t o preven tcondensation an d possibl e damag e t o th e material . Onc e th e container ha sbeen opene d i t shoul d b e stored i n a cool , dar k are a rathe r tha n returningit t o th e refrigerator.

In preparin g a gag e fo r bonding , th e gag e backin g materia l i s treate dwith a catalyst , sparingl y used , an d allowe d t o dr y fo r approximatel y 1minute. A thi n coa t o f adhesiv e i s pu t on , th e gag e i s place d o n th e tes tspecimen, and thum b pressure i s applied t o th e gage . Polymerizatio n take splace i n th e adhesiv e film in approximatel y 1 minute. The bonde d gag e i sready fo r use by the tim e lead wire s are attached .

The glue line is sensitive to moisture and must be protected by a coating.A properl y protecte d gage , however , can b e use d i n we t atmosphere s fo r


short-duration tests . Thes e adhesive s ar e excellen t fo r short-ter m test s bu tare seldo m use d fo r test s extending ove r lon g tim e periods , sinc e th e bon dis subjec t t o embrittlemen t wit h age. Gages bonde d wit h these cement s ca nbe use d t o measur e strain s o f the orde r o f 1 5 percent an d ca n operat e ove ra temperatur e rang e o f -25 ° t o 150° F (-3 2 t o 65°C).

Epoxies

Epoxy adhesives, in use for many years, come in a wide selection o f two types;namely, on e wher e polymerizatio n take s plac e a t roo m temperatur e an danother tha t require s the application o f external heat fo r correct polymeriza -tion. Wit h epoxies , ther e i s n o solven t evaporatio n involved , ver y littl eshrinkage, an d a goo d permanen t bon d i s forme d wit h a wid e variet y ofmaterials. Epoxie s als o exhibi t excellent moistur e an d chemica l resistance ,and can be used over a temperature range from cryogeni c to 600°F (315°C).

One typ e o f epoxy, using an amin e catalyst , cures a t roo m temperatur ethrough th e exothermi c reactio n produce d whe n th e adhesiv e components ,hardener an d resin , are mixe d together. Anothe r type o f epoxy, activated b yan acid anhydride catalyst, requires external heat fo r polymerization t o occurproperly. A temperatur e o f a t leas t 250° F (120°C ) mus t b e maintaine d fo rseveral hours . Bot h type s requir e a clampin g pressur e durin g th e curin gprocess. Furthermore , i f either typ e i s t o b e use d a t a temperatur e highe rthan th e curin g temperature , the n a post-cur e temperatur e abov e th eexpected maximum test temperature should b e maintained for several hours .For th e room-temperature-curin g epoxy , the post-cur e temperatur e shoul dbe 70 degF to 85 degF (40 to 47 degC) above the maximum test temperature.For th e hot-cur e epoxy , th e post-cur e temperatur e shoul d b e 8 5 degF t o115 degF (4 7 to 6 4 degC) abov e th e maximu m tes t temperature .

Other adhesives

Other availabl e adhesive s ar e generall y use d fo r mor e specialize d applica -tions. Among these are phenolic, polyimide, and cerami c adhesives . Phenoli cadhesives are littl e used because the y require complicated, lon g curin g cyclesand hig h clampin g pressure . Polyimid e adhesive s ar e difficul t t o wor k wit hand the solvents in them are not easily removed. Remaining solvents degradethe adhesive properties. Cerami c adhesive s are applied t o free-filament gage sand thermocouple s fo r temperature s tha t excee d th e limit s o f organi cmaterials. Again, for special applications, consul t the manufacturers and theirapplication departments .

13.6. Bonding a strain gage to a specimen

Bonding a strai n gag e appear s t o b e a simpl e process , bu t clos e attentio nmust b e pai d t o eac h step . Thi s involve s surfac e preparatio n o f th e tes tspecimen, cementing the gage to that surface, soldering lead wires , and finall y


applying a protectiv e coa t t o th e installation . Cleanlines s canno t b e over -emphasized; th e hand s shoul d b e washe d frequentl y durin g th e proces s o rcleaned wit h neutralizer , n o materia l shoul d b e reused , an d th e wor k are amust b e kept clean .

Surface preparation

1. Usin g a degreasin g agent , suc h a s trichloroethylen e o r carbo n tetra -chloride, clean the test surface, being sure to have adequate ventilation.

2. San d th e degrease d surfac e i n orde r t o remov e al l scale , dirt , o r dus tparticles.

3. Clea n th e surfac e wit h a spong e o r tissu e saturate d wit h th e cleanin gsolvent.

4. Usin g a metal conditioner, wet lap the area with silicon-carbide paper.5. Usin g a clea n tissue , wipe the are a dr y wit h one stroke . D o no t reus e

the tissue . Repeat severa l times.6. Usin g a ballpoin t pe n o r 4- H pencil , locat e an d mar k referenc e line s

for gag e alignment. Do not use a scribe: make certain you d o not scratchthe surface.

1. Usin g a cotton-ti p swab , di p i t int o meta l conditione r an d scru b th esurface. Wip e dr y wit h one strok e usin g a clea n tissue . Usin g a clea nswab an d tissue , repea t severa l time s unti l th e cotton-ti p show s n oforeign material .

8. Di p a cotto n swa b into neutralize r an d scru b th e surface . Wip e clea nwith one stroke using a clean tissue . Repeat severa l times to ensure thesurface i s neutralized.

9. Instal l th e gag e a s soo n a s possible .

Bonding the gage

Since bonding technique s wil l differ dependin g o n the adhesive , th e metho dfor a cyanoacrylat e adhesiv e wil l b e described , sinc e i t i s a widel y use dcement.

1. Remov e the gag e fro m it s packe t an d plac e it , bondin g sid e down , o na clea n surface . Position a separat e termina l stri p relativ e t o th e gag etabs.

2. Usin g a piec e o f cellophane tape , plac e i t ove r th e gag e an d termina lstrip. Pul l th e tap e fro m th e surfac e a t a shallo w angle , bein g certai nthe gage an d termina l stri p ar e firmly attached .

3. Plac e the gage o n th e tes t specimen , alignin g the referenc e tabs o n th egage wit h th e marke d referenc e system o n th e tes t surface . The tap eand gag e ar e no w in the desire d position .

4. Lif t on e end of the tape from th e test surface unti l the gage and termina lstrip ar e just clear . Th e remainde r o f the tap e i s stil l attache d t o th especimen.

400 TH E BONDE D ELECTRICAL RESISTANC E STRAI N GAGE

5. Pul l the fre e en d o f the tape back unti l the bonding surfaces o f the gag eand termina l stri p ar e exposed . Brus h catalys t sparingl y ont o th ebonding surfaces . Allow t o dr y fo r 1 minute.

6. Appl y one o r tw o drop s o f adhesive at th e boundar y lin e o f the tap eand tes t surface . Pull the fre e en d o f the tap e tau t and toward s th e tes tspecimen, makin g a shallo w angle . A t th e sam e time , usin g a clea ntissue, wipe over the tap e fro m th e boundar y line towards the fre e en dso that th e cement spreads unde r the gage and termina l strip , bondingthem t o th e surface . Appl y thum b pressur e t o th e gag e an d termina lstrip fo r approximatel y 1 minute.

7. Afte r severa l minutes, grasp one end of the tape and slowl y and carefull ypull back on itsel f until it is removed. The gage is now ready for soldering.

Completing the installation

Now tha t th e gage is successfully bonde d t o th e test specimen, there remainsthe tas k o f attaching th e lea d wire s and the n applyin g a protectiv e coat t othe entir e installation . The procedur e i s outlined i n the following:

1. I f the gage has an open grid, cover the grid area with a piece of maskingtape, leaving the solde r tab s exposed .

2. A 30-40-watt solderin g iron wit h a smooth , tinne d ti p i s required.3. Us e a fine rosin-core solde r whose melting temperature is compatibl e

with th e tes t environment.4. Wit h the soldering iron at the proper temperature, lay the solder across

the gage tab an d appl y the iron firmly for a second. Lif t th e solder an diron a t th e same time, leaving behind a shiny mound o f solder o n th etab.

5. Lightl y tin th e termina l strips .6. Separat e th e individua l leads o f the composit e lea d wir e and remov e

about inc h to f inc h insulation from each . O n eac h individua l lead,separate on e strand, twist the remaining strands togethe r and ti n for ashort distanc e a t th e insulation . Snip of f the remainin g end, leavingabout inc h of the tinne d bundle. The singl e strand wil l b e use d a s ajumper wir e from th e termina l stri p to th e gage tab .

7. Solde r th e tinne d lead wire s to th e termina l strip.8. Usin g the singl e strand o f each lead , solde r t o th e gag e soldering tab ,

arranging it so there is some slack between the terminal strip and gag etab. (Fin e insulate d wir e may b e use d i n place o f the singl e strand. )

9. Clea n al l solde r joints wit h rosin solvent , remove th e maskin g tape ,and clea n th e gag e wit h rosin solvent .

10. Secur e the lea d wire s so the y cannot accidentl y be pulle d loose .11. Chec k th e resistance between the gage an d th e specimen . It should be

at leas t 1000 0 megohms.12. Appl y a protectiv e coatin g t o th e gage , termina l strips , an d a shor t

distance ont o th e lea d wir e insulation.


REFERENCES

1. "Catalo g 500: Par t B—Strai n Gage Technical Data, " Measurements Group, Inc. ,P.O. Bo x 27777, Raleigh, NC 27611 , 1988 .

2. "SR- 4 Strain Gage Handbook," BL H Electronics, Inc., 75 Shawmut Road, Canton,MA 02021 , 1980 .

3. "Weldabl e and Embedable Integral Lead Strain Gages," Applications and Installa-tion Manual , Eato n Corp. , Ailtec h Strain Gag e Products , 172 8 Maplelawn Rd.,Troy, MI 48084 , 1985 .

4. Vaughn , John, Application o f B & K Equipment to Strain Measurements, Brue l &Kjaer, Naerum , Denmark , 1975 , Chaps. 3 and 4 .

ANSWERS TO SELECTED PROBLEMS

2.2. a , = 2 0 500 psi; a2 = 450 0 psi; 6 = 45°.2.4. = <j 2 = 950 0 psi ; Mohr' s circl e i s a point .2.6. a i = -<r 2 = 750 0 psi; 0 = 45°.2.7. = -700 0 psi; = -2300 0 psi; 0 = 135°.2.9. (a ) Poin t A: = -<r 2 = 625 8 psi; Point B : =2651 0 psi; a2 = - 147 7 psi.

(b) Poin t X: ma x = 625 8 psi; Point B : ma x =13944 psi.2.11. Lef t bearin g reactions : F y = 30 6 Ib; Fz = 217 Ib.

Right bearin g reactions : F y = 31 0 Ib; Fz = -569 Ib ; Tmax = 861 4 psi.2.12. ma x = 3309 6 psi; raa x =1679 9 psi.2.14. = 1 1 701 psi; 2 = -470 1 psi; 0 = 18.8° .2.16. K ! = 181 8 nin/in; £ 2 = 19 7 nin/in; 0 = 161.8° .2.18. £ , = 75 3 nin/in; £2 = -18 3 nin/in ; 0 = 102.3° .2.20. £ , = -£ 2 = 250 nin/in; 0 = -45° .2.22. fi j = 40 0 nin/in; £2 = -80 0 nin/in ; (9 = 90°.2.24. £ , = 125 8 nin/in: £2 = -18 8 nin/in ; 0 = 139.2° .2.25. (a ) E I = 2654 nin/in; s2 = - 115 4 nin/in; y max = 190 4 nradians .

(b) Rectangula r rosette : e a = 2500 nin/in; e b = 150 0 nin/in; E C = —100 0 nin/in.Delta rosette : e 0 = 250 0 nin/in; £b = 52 5 nin/in; EC = —77 5 nin/in.

2.26. fi , = 124 4 nin/in; «2 = -84 4 nin/in ; f = 101.7° .2.28. £ j = 102 7 nin/in; K, = -202 7 nin/in ; 0 = 65.5° ccw from gag e b .2.30. <T ! = 3 2 664 psi; <2 = 1552 1 psi .2.32. a i = 1381 0 psi; a2 = -5 6 66 7 psi.2.34. <7 3 = 1 0 385 psi.2.36. (b ) E , = 179 1 nin/in ; £ 2 = —69 3 nin/in; £3 = —47 1 nin/in .

(d) CT, = 5220 3 psi; <r2 = -514 5 psi; 0 = 17.4° .2.38. s a = 0 nin/in; £6 = 6 8 nin/in; e c = 95 1 nin/in.3.2. / = 0.02 0 15 amps; n = 0.0074 .3.4. / = 0.028 amps; n = -0.12 .3.5. R p = 5880 ohms .4.5. r \ = 9 0 percent ; e . = 0.10204 1 in/in ; No .4.8. (a ) £ „ = -84 4 nin/in ; £ b = 195 0 nin/in; e. c = 384 4 nin/in; e. d = 105 0 nin/in.

(b) A £ = 0.0312 5 volts.(c) « = 0.000 625.

4.9. (a ) £ 9 = 185 5 nin/in; sb = -75 7 nin/in .(b) AR g = 0.4630 ohms; AR b = -0.1889 ohms.(c) E = 0.033 96 volts.

ANSWERS T O SELECTE D PROBLEM S 40 3

5.2. A£ 0/K = (GFe)/(4 + 2G Fe); E/E, = 2/(2 - G F£;).5.4. A£ 0/K=(GF£)/2;£/£;= 1.5.6. A£ 0/F = G F(1 + v)e/2 ; e/g; = 1.5.8. <7 t = 3 2 490 psi; <r 2 = - 1 9 560 psi; F = 340 923 Ib.5.11. P, = 1 5 000 psi.5.12. F x = 220 9 Ib ( + x); F , = 11 5 Ib ( + y); F z = 104. 7 Ib (-z) .

EI = —18 0 nin/in; s 2 = 24 0 |iin/in; e3 = 30 0 nin/in; £4 = — 120 nin/in.5.14. (a ) F ! = 5301 4 Ib; F2 = 6202 7 Ib.

(b) M I = 1590 4 in-lb (cw); M2 = 1491 0 in-lb (ccw).5.16. Increas e = 11.7 5 percent.5.18. (a ) A£ m0 = 3.8 3 mV; (b) A£m0 = 52.2 2 mV.5.21. (b ) A£m0 = 14.8 5 mV; A£m0 = 14.2 8 mV; A£m0 = 11.7 2 mV.5.23. W=2.94\b.5.25. E ! = 2000 nin/in; e2 = 120 0 nin/in; £3 = -500 nin/in ; £4 = 93 7 nin/in.6.2. K p = 468.7 6 ohms.6.3. R p = 1114. 1 ohms; K s = 14.4 8 ohms.6.5. R s = 40 ohms.6.7. K p = 327. 8 ohms; R s = 69.3 ohms.6.9. K s = 77. 4 ohms.6.11. R p = 690 ohms; R s = 30.96 ohms.7.1. (a ) e H — 680 nin/in ; «L = 16 0 nin/in.

(b) E' H = 69 3 nin/in ; E' L = 18 6 nin/in.(c) r\ H = 1.91 percent ; ?; L = 16.2 5 percent.

7.3. e' a = 101 5 nin/in.7.5. G F = 0.939 .8.2. e. H/e.L = 4.25.8.4. ff l = 6 3 718 psi; a2 = -4576 psi; tmax = 3 4 147 psi; 9 = 129.3°.8.6. a , = 1 0 890 psi; <r2 = - 1 9 290 psi; T max = 1 5 090 psi; 9 = 98.3°.8.8. CT! = -1 1 61 1 psi; a2 = -3 8 70 3 psi; T mal = 1 9 352 psi; 0= 120° .8.10. a l = 447 4 psi; <r2 = -1 8 18 8 psi; Tmax = 1 1 331 psi; 9 = 101.8°.8.12. <T ! = -1 4 12 0 psi; a2 = -2 9 16 6 psi; imax = 1 4 583 psi; 0 = 65.6° .8.14. B a = —29 5 |iin/in; Eb = 75 2 nin/in; e c = —7 5 nin/in; £,, = 55 0 nin/in.9.2. e ^ = 14 6 nin/in; 4 = 53 3 nin/in.9.4. £ „ = 95 1 nin/in; e fc = 14 1 nin/in; £c = 43 1 nin/in.9.6. e, a = 11 7 nin/in; s b = —87 0 nin/in; ec = 858 nin/in.9.8. £ „ = e b = ec = 782 nin/in.9.10. £ „ = 80 7 nin/in; £6 = 40 1 nin/in; sc = -219 nin/in .9.12. £ „ = £, . = 80 7 nin/in; sb = - 8 nin/in .9.14. (b ) £ „ = 17 1 nin/in; eb = 157 1 nin/in; ec = 78 4 nin/in.

(c) £ j = 161 2 nin/in; £ 2 = —65 6 nin/in.(d) <T J = 46655 psi; a2 = -568 4 psi; 6 = 52.8° .(e) T max = 26170psi .

10.2. (a ) <t > = 26.6° ; (b) 0 = 28.7° ; (c ) 0 = 30.2° .

10.4. CTH = -^ = ——— [3(1 - v ) + (1 + v) cos 21 — v 8t( l — v)


10.6. y xy = 276 (iradians .11.1. £ = -230 0 nin/in.11.3. E = 288 5 nin/in.11.7. (a ) E ! = 167 3 |j.in/in ; £ 2 = —237 3 nin/in.

(b) K ! = 375 4 nin/in; e2 = ~2 6 )j.in/in .11.9. (a ) ei = 207 nin/in ; «2 = - 100 7 jiin/in.

(b) E I = 28 3 nin/in; e, 2 = —87 1 |j.in/in .12.1. (a ) Tension : nonlinearity factor = 0.9989 . Compression : nonlinearity factor

1.0011.(b) Tension : nonlinearit y facto r = 1.008 . Compression : nonlinearit y factor

0.9992.

12.3.

12.5. w = 1 . 6 i n .12.10. r=2106in-lb .12.11. Percen t differenc e = 0.86.12.13. A£ 0 /K= 1.05mV/V .

INDEX

Adhesives, 396-8cyanoacrylate, 397- 9epoxies, 398nitrocellulose, 397

Axial strai n sensitivity , 236

Backing material , 393- 4Baker, M . A. , 9Ballast circuit , see Potentiometric circui tBallast resistor , 100- 1Biaxial stress , 45Biermasz, A. J. , 1 6Bonded wir e strain gage, 24- 7Bridge input resistance, 151-2, 173- 5Bridge outpu t resistance , 151-2 , 176- 7Bridge ratio , 155 , 159-61Brittle lacquer coatings , 3 , 36- 8

Calibrationpotentiometric circuit , 141- 4strain gag e us e under othe r conditions ,

246-8Wheatstone bridge , 193- 5

Circuits, elementaryconstant current , 94-6

advantages, 96- 7constant voltage , 91- 4nonlinearity, 93- 4

Circuits, potentiometri cadvantages an d limitations , 105- 6applications, 104- 5ballast resistor , 100- 1calibration, 141- 4characteristics, 102- 3circuit analysis , 106- 9circuit efficiency , 111-1 2circuit equations , 101- 2components, 10 0

dynamic strains , 118-1 9gages i n series , 112-1 4linearity consideration s

fixed ballast resistance , 123- 4variable ballast resistance, 121- 3

measurements, stati c vs. dynamic,114-15

nonlinearity, 102 , 108-9signal measurement , 147- 9static strain , 118temperature effects , 129-4 1

ballast an d gag e leads, 133- 5voltage limitation , 110

Coatingsbrittle lacquer , 3 , 36-8photoelastic, 38

Compensating strai n gage, 27-8, 353-7 ,363

Constantan, 14 , 339-41, 391- 2Crack measurin g gage, 34- 5Cyanoacrylate cement, 397-8

Data analysis , 253-6Delta rosett e

analysis, 267-9Mohr's circle , 269-73principal stres s directions , 269T-delta, 278-81transverse sensitivity , 301- 6

Desensitization o f circuitsfull bridge , 227-31half bridge , 218-2 5kinds, 207meter resistance , 175- 9power suppl y resistance , 173- 5reasons for varying, 205-6single gage , 207-17

combination, serie s an dparallel, 211-16

406 INDEX

Desensitization o f circuits (contd.)single gag e (contd.)

resistance i n parallel , 209-11resistance i n series , 207-9

temperature effects, 21 6Dorsey, J. , 8 , 387Dummy gage, see Compensating strai n gage

Embedment gage , 3 6Epoxy cement , 398Equiangular rosette, see Delta rosette

Four-element rosett erectangular, 275-8T-delta, 278-8 1

Friction gage , 35- 6

Gage factordetermination, 242-3manufacturers, 26 , 236

relation wit h axia l an d norma l strains ,240-2

variation wit h temperature , 340-3Gages

crack measuring , 34- 5embedment, 3 6friction, 35- 6semiconductor, 32- 3temperature, 33- 4

Mines, F . F. , 318Hydrostatic strai n component , 6 9Hydrostatic stres s component , 5 6

Lateral effect , 234-5 1basic equations , 236-42transverse sensitivit y factor , K, 238-40

Lead-line resistance , 180-9 1full bridge , 180- 1half bridge—fou r wire, 181- 4half bridge—thre e wire , 184- 6quarter bridge—thre e wire , 187- 8quarter bridge—tw o wire, 188-90

Load cellaxial force , 363- 5bending beam , 372- 5ring type , 365-8shear beam , 375- 7

Maslen, K . R. , 10Material, backing, 393- 4

epoxy resins , 394metallic, 39 4paper carrier , 393polyimide resins , 393-4

Material, strai n gag econstantan, 14 , 339-41, 391- 2isoelastic, 392karma, 392platinum tungsten , 39 3properties desired , 10

McClintock, F . A. , 281Measurements, fundamenta l laws, 97- 8Meier, J . H. , 10 , 278, 380, 38 2Meter resistance , 175- 9Mohr's circl e

delta rosette . 269-7 3rectangular rosette , 261- 5strain, 68-70stress, 54- 7

Multiple circuits . 195- 7

Indicated vs . actual strain, 165-8 , 206Invariants

strain, 81stress, 8 1

Isoelastic, 240 , 392

Nitrocellulose cement , 397Nonlinearity o f circuits

elementary. 93- 4potentiometric, 102 , 108- 9Wheatstone bridge , 150-1 , 163- 4

Normal strai n sensitivity , 236

Jones, E. , 10

Karma, 39 2Kelvin, Lord , 5Kern, R . E. , 320 , 32 2

Perry, C . C., 330- 1Photoelastic coating , 38Plane shearing stress, determination, 327-30Plane strain , 62- 5Plane stress , se e Biaxial stres sPlatinum tungsten , 393

INDEX 407

Poisson's ratio , 17 , 73Potentiometric circui t

advantages an d limitations , 105- 6applications, 104- 5ballast resistor , 100- 1calibration, 141- 4characteristics, 102- 3circuit analysis , 106- 9circuit efficiency , 111-1 2circuit equations , 101-2components, 10 0dynamic strains, 118-1 9gages i n series , 112-1 4linearity consideration s

fixed ballast resistance , 123- 4variable ballast resistance , 121- 3

measurements, static vs. dynamic, 114-18nonlinearity, 102 , 108-9signal measurement , 147-9static strains , 11 8temperature effects , 129-4 1

ballast an d gag e leads , 133- 5voltage limitation, 11 0

Pressure transduce rdiaphragm, 384- 7thin-walled cylinder , 383- 4

Principal strains , 64-5Principal stresses , 48-53, 260-5, 269-73

Rectangular rosett eanalysis, 258-61four element , 275-8Mohr's circle , 261-5principal stres s directions , 260- 1transverse sensitivity , 296—300

Resistance, basic equations fo r uni t change ,236-8

Resistor, ballast , 100-1 , 119-26Rosettes

deltaanalysis, 267-9Mohr's circle , 269-73principal stres s directions , 26 9transverse sensitivity , 301-6

geometry, 256-8graphical solutions , 281- 7rectangular

analysis, 258-61four element , 275-8Mohr's circle , 261-5principal stress directions, 261transverse sensitivity , 296-300

stress equations , summary , 280-8 1

T-delta, 278-81transverse sensitivity

delta, 301- 6rectangular, three-element , 296-300two differen t orthogona l gages , 294-6two identica l orthogona l gages , 291-4

Sanchez, J . C, 126Semiconductor gages , 32-3Semiconductor materials , 8- 9Sensitivity variation

full bridge , 227-3 1half bridge , 218-25reasons, 205- 6single gage , 207-16

Shear gage , 330-4Shear strain , Mohr' s circl e sign convention,

68-9Shear stres s

biaxial stres s state , 51- 2determination o f plane, 327-30Mohr's circl e sign convention, 54-6

Shoub, H. , 14 , 22Stein, P . K. , 157Strain

apparent, se e Thermal outpu tbasic concepts , 61- 2correcting fo r thermal outpu t an d gag e

factor variation , 348-9elastic, in metals , 7- 8indicated vs . actual, 165-8 , 206invariants, 81Mohr's circle , 68-70nonlinearity, 102 , 108-9, 163- 4plastic, i n metals, 8principal, 64-5shear, sign , 63- 4small vs . large, 20- 4temperature-induced, 337-4 0thermal outpu t correction , 344- 7transformation equations , 63- 5

Strain gag ealloys, 391-3basic principle , 5bonding, 398-40 0characteristics, 4- 5compensating, 27-8, 353-7, 363foil, 29-31gage length , 394-5general considerations , 390- 1geometry, 395lateral effect , 234-51orthogonally crossed pair , 248-51

408 INDEX

Strain gag e (contd.)properties desired , 1 0resistance, 395- 6self-temperature compensated , 131-2 ,

343-5self-temperature compensation , 39 6temperature effects , 33 7test materia l mismatch , 349 5 1use under conditions differin g fro m

calibration, 246-8weldable

foil, 3 1wire, 27- 9

wire, 2 4 9Strain sensitivity

analysis, 14-2 4general case , 14-1 7small vs . large strain , 17-2 4uniform straigh t wire , 17-2 0

axial an d normal , 236- 8gage facto r relation , 240-2

definition, 5-8 , 23 6material properties , desired , 1 0numerical values , 11-13reasons fo r varying , 205-6

Strain transformatio n equations, 63- 5Stress

basic concepts , 43- 4biaxial, 4 5circuit, indicatio n o f normal stress , 32 0fields, 253-6invariants, 81Mohr's circle , 54- 7principal, 48-53 , 260-5 , 269-73

using a singl e gage, 326- 7sign convention , 45transformation equations , 45-5 3

Stress gagenormal, 310-1 2single roun d wire , L configuration,

312-14two orthogona l gages , 314-1 6V-type, 321- 5

Stress-strain gage , 316-2 0Stress-strain relations , 72- 7Stress transformatio n equations , 45-5 3

Temperature gages , 33- 4Temperature-induced strain , 337-4 0Thermal expansio n coefficients , 350Thermal output , 338 , 34 4

correction, 344- 7Thevenin's theorem , 17 3

Torque meter , 378-8 0Torque wrench , 380-2Transducers

axial force , 363- 5bending beam , 372- 5cantilever beam , 368-7 2full bridge , 361- 2half bridge , 362pressure measurement

diaphragm. 384-7thin-walled cylinder , 383-4

quarter bridge . 36 3ring-type, 365- 8shear beam . 375- 7torque meter , 378-8 0torque wrench , 380-2

Transformation equation sstrain, 63- 5stress. 45-53summary, strain , 65summary, stress , 52- 3

Transverse sensitivitydefinition, 238-4 0delta rosette , 301- 6determination. 244-6rectangular rosette , 296-30 0two differen t orthogona l gages , 294-6two identica l orthogona l gages , 291- 4typical values , 239-40

Unbonded wir e strai n gage , 2 4

Weibull, W. , 12 , 14 , 20, 2 2Weldable strai n gag e

foil, 3 1wire, 27- 9

Weymouth, L . J. . 141Wheatstone bridg e

bridge inpu t resistance , 151-2 , 173- 5bridge outpu t resistance , 151-2 , 176- 7bridge ratio , 155 , 159-6 1calibration, 193- 5derivation o f elementary bridg e

equations, 157-65 , 169-7 2elementary bridg e equations , 149-5 2general bridg e equations , 172- 9lead-line resistance , 180-9 1meter current , 152-3 , 179meter resistance , 175- 9nonlinearity. 150-1 , 163- 4null balanc e referenc e bridge, 15 4null balanc e svstem , 153

INDEX 40 9

reference system , 153- 4 unbalanc e system , 153resistance i n serie s wit h bridge , 172- 5 Williams , S. B., 320, 33 5summary o f properties, 155- 7 Wnuk , S. P., Jr., 141unbalance referenc e bridge, 15 4 Wright , W. V. , 126

the bonded electrical resistance strain gage an introduction 019507209x

Documents