Engineering and Research Center

Bureau of Reclamation

April 1981

Water and Power TFCHP

A System for Measuring Steady-State Torque on a Rotating Shaft

I E. Campbell I 9. P E R F O R M I N G O R G A N I Z A T I O N N A M E A N D ADDRESS

Bureau of Reclamation Engineering and Research Center Denver, Colorado 80225

12. SPONSORING A G E N C Y N A M E A N D ADDRESS

I S . S U P P L E M E N T A R Y N O T E S

:PORT STANDARD T ITLE PAGE 3. R E C I P I E N T ' S C A T A L O G NO.

5. R E P O R T D A T E

April 1 98 1 6. P E R F O R M I N G O R G A N I Z A T I O N C O D E

8. P E R F O R M I N G O R G A N I Z A T I O N R E P O R T NO.

REC-ERC-8 1 -2

10. WORK U N I T N O .

I ' W w t I. CONTRACT^ ;t NO.

A 4 13. TYPE^ PERIOD

C O V E R E D

Bureau of R e c l a m a ~ Denver, Colo&

14. SPONSORING A G E N C Y C O D E

DlBR

Microfiche and/or hard copy available at the Engineering and Research Center, Denver, Colorado Editor RNW

16. A B S T R A C T

The report describes a system to transfer torque signals from a rotating shaft to stationary equipment using a transformer concept. The system has error correcting capabilities to eliminate drift. The system has been successfully used on large generators and pumps. Includes 8 figs, 2 refs., 15 pp.

17. K E Y WORDS A N D D O C U M E N T A N A L Y S I S

o. DESCRIPTORS-- / *torque/ "steady-state/ rotation/ drift1 compensation1 *measurement/ strain

c COSATI F ~ e l d i G r o u p 14 COWRR. 1400 21 N O O F P A G E

15 22. P R I C E

18. D I S T R I B U T I O N S T A T E M E N T

A v a r l o b l e from the N o t r o n o l T e c h n r c o l In formatron Servrce, Operatrons Drvrsron, Sprrngfreld, Vrrgrnro 22 10 1 .

(Microfiche and/or hard copy available from NTIS) G P O 854 - 6 12

19 S E C U R I T Y C L A S S ( T H I S R E P O R T )

U N C L A S S I F I E D 20. S E C U R I T Y C L A S S

I T H I S PAGE) UNCLASSIFIED

A SYSTEM FOR MEASURING STEADY-STATE TORQUE ON A ROTATING SHAFT

by E. Campbell

April 1981

Power and Instrumentation Branch Division of Research Engineering and Research Center Denver, Colorado

UNITED STATES DEPARTMENT OF THE INTERIOR * BUREAU OF RECLAMATION

As the Nation's principal conservation agency, the Department of theInterior has responsibility for most of our nationally owned publiclands and natural resources. This includes fostering the wisest use ofour land and water resources, protecting our fish and wildlife, preser-ving the environmental and cultural values of our national parks andhistorical places, and providing for the enjoyment of life through out-door recreation. The Department assesses our energy and mineralresources and works to assure that their development is in the bestinterests of all our people. The Department also has a major respon-sibility for American Indian reservation communities and for peoplewho live in Island Territories under U.S. administration.

In May of 1981, the Secretary of the Interi~proved changing the Water and Power Reso~rc~:

IService back to its former name, the Bureau ofReclamation. All references in this publication tothe Water and Power Resources Service should beconsidered synonymous with the Bureau

~Of

Reclamation.IL--~--- ~~-_._-

CONTENTS

PageIntroduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

General description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Circuit description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Strain gage bridge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Bridge amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Pulse generator... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Bridge excitation.............................................Synchronizing pulse generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Pulse receiver...............................................Synchronizer card. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Frequency transducer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Decoder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

FIGURES

12345678

Block diagram of complete system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Torque sensor, encoder, transmitter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Waveforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Receiver-decoder block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Receiver schematic... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Synchronizer schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Frequency transducer schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Decoder schematic.........................................

Hi

114444449

12

12

12

235678

1011

1111'1.

INTRODUCTION

A system for measuring steady-state torque on arotating shaft has been developed using atransformer technique rather than conventionalslip rings to couple the signal off the shaft. Thesystem comprises a small battery-operated sens-ing unit riding on the shaft and a stationary signalprocessing unit for recording and indicating theresults. The equipment is easy to install and iscompensated for drift in the signal processingcircuits.

CONCLUSIONS

1. Satisfactory drift compensation has beenachieved allowing steady-state torquemeasurements to be made on a rotating shaft.

2. The system is relatively easy to use becauseslip rings are not required. It has been used suc-cessfully on shafts from 100 to 2500 millimeters( 4 inches to 8 feet) in diameter and with as littleas 50 mm (2 in) of exposed shaft available.

APPLICATION

The torque measurement system was developedfor the purpose of determining the feasibility ofmaking a real-time mechanical input powermeasurement by sensing shaft torque as ameasure of prime mover mechanical input power.The device evolved from earlier use of the sameprinciple to sense shaft torsional oscillations [1 ]

* ;however, it is useful as a strain indicator for anytype of strain measurement including static orrotational.

GENERAL DESCRIPTION

As shown on figure 1, the system is divided intotwo parts. One part (fig. 1A) rides on the shaftand the other (fig. 1B) is stationary.

S~aft torque !s. measured using a strain gagebridge comprising four active strain gages

* Numbers refer to the bibliography.

oriented to sense torque[2]. The bridge output,after amplification by an instrumentationamplifier, frequency modulates a voltage controll-ed oscillator VCO making the signal medium avariable frequency. The VCO output is convertedto low duty cycle relatively high current pulseswhich are easily transformer coupled off the shaftto the stationary processing unit. This signalcoupling method obviates slip rings, resulting ineasier and more universal application with lowbattery drain.

Two major sources of error in a system of thistype are d-c amplifier drift and VCO frequencydrift. Compensation for both error sources is in-herent in the encoding-decoding scheme usedhere. Drift, due to the temperature coefficient ofthe strain gages, is compensated by the sym-metry of the full bridge configuration.

The effect of amplifier zero drift and VCO fre-quency drift is avoided by interrupting the bridged-c excitation at a 500-Hz rate. The 500-Hzmodulation frequency is derived by dividing the10-kHz VCO frequency by 20, thus preventingundesirable beat frequencies between the VCOand bridge excitation signals. The slight frequencyvariation of the bridge excitation when the VCO ismodulated by the strain signal does not affect themeasurement system output. The strain informa-tion is contained in the frequency difference oc-curing when the bridge is alternately excited thenunexcited. This frequency difference is deter-mined only by the bridge unbalance multiplied bysystem gain if the drift rates are slow compared tothe modulation period. The receiver demodulatorconverts the variable frequency pulses back to asquare wave while the synchronizing circuits keepthe decoder square wave in phase with the cor-responding 500-Hz bridge power waveform. Twosample-hold circuits continuously monitor the ex-cited and unexcited system outputs respectivelywhile a difference amplifier responds only to theresulting difference in the two levels. The dif-ference amplifier output just described is thus ad.c. level proportional to shaft torque. The errorsources associated with strain gage torquetransducers are well documented in the literatureand will not be discussed here.

CIRCUIT DESCRIPTION

Strain Gage BridgeThe strain gage arrangement is a full bridge con-figuration (fig. 2). In other words, all four straingages are active, resulting in increased sensitivity

10 kHz

Strain VCO Pulsegage

10 kHz gen.bridge Coupl ing

transformerprimary

500 Hz Sync.Bridge

-:- 20 pulsepower gen.

A

Couplingtransformer

secondary

Pul sereceive Sync.

-;-2

5 kHz

Freq.demod.

-;-20

500 Hz

Decode

Output Samplehold

B

Figure1. - Block diagram of complete system.

2

11'11'1.

+9V

-5V

CD

8

76

5

5.lk 1l10kHz

+9V

+9V

UJ~

@ 10 kHz

I 20 CYCLES jCD JLS ~L.r4JLJ0~~~@Jl n JL@ I n I

@~~L

0

10kHz

500Hz

SYNCDELAYSYNCPULSE

TXOUTPUT

Tr+

I

2 VCO

3 566

420k Il.

I. 2V d..c.47pF

10k Il.

2.67k.1L

STRAIIoIGAGES

+9V

10k.!1...

470pF+9V

20 k11.

@

@

Figure 2. - Torque sensor, encoder, transmitter.

+5V

10k 11.O. I JJ.F

Ik.!1...

2000pF

+9V

10k.!1...

10k 11.. I kA

10Il. (OUTPUT COUPLING TRANSFORMER

~

~Ok11. 1@

I PRIMARY GONTO RECEIVER (FIG.4-5!

SHAFT33k11.. SECONDARY IS STATIONARY

2N2222 I IOOO5.aFL-

!DISC.

-9V10kA 10kA

-5V

2N2222

SYNCPULSEINJECTOR

+

- 'T' +12V

~10.J.lF

to torque. This configuration produces optimumtemperature compensation of the bridge includingeliminating the effects of all strains other thantorsion.

Bridge Amplifier(refer to fig. 2)The bridge difference amplifier A 1 is a 521 in-tegrated circuit instrumentation amplifier having adifferential input impedance of a 3 x 109 ohmsand a common mode input impedance of 6 x 1010ohms making the effect of amplifier loading of thebridge negligible. The amplifier gain is 100 or1000 and its output is the control voltage input ofa 566 voltage controlled oscillator VCO operatingat nominally 10kHz. The frequency modulationsensitivity of the VCO is approximately 4.6 kHz/Vand is linear.

Pulse Generator(refer to fig. 2)The square wave output of the VCO is differen-tiated via the 47 -pF capacitor and the negativegoing transition momentarily turns on a com-plimentary transistor pair, discharging a 0.005-~Fcapacitor through the primary of the outputcoupling transformer resulting in waveform 3A(fig. 3).

Bridge Excitation(refer to fig. 2)The 500-Hz bridge excitation signal (fig. 3B) isgenerated by dividing the 1O-kHz VCO signal by20. The resultant 500-Hz square wave drives atransistor switch, coupling the strain gage bridgeto a 1.2-V d.c. regulated voltage source. Whenthe bridge is balanced, equal 500 Hz squarewaves are coupled to the two inputs of the in-strumentation amplifier, resulting in zero volts d.c.out of the amplifier. Any unbalance of the bridgecauses the amplifier output to alternate between ad.c. level proportional to the bridge unbalance,and a zero reference output when the bridge isunexcited. Of course, this amplifier output is a500-Hz square wave causing the VCO to alter-nate between a zero reference frequency fl and ashifted frequency f2, proportional to bridge un-balance. Thus the signal information is containedin the frequency difference f2 - fl which is not af-fected by zero drift of the amplifier or frequencydrift of the VCO.

Synchronizing (sync.) Pulse Generation(refer to fig. 2)A synchronizing pulse (fig. 3D) is generated by adual one-shot multivibrator (OSI. fig. 2). Thepositive going transition of the 500-Hz square

11'11'1.

wave described above triggers a delay one-shotmultivibrator (OSI, fig. 2) which in turn triggersthe second half of the dual one-shot. The secondone-shot output (fig. 3D) constitutes a synchro,-n izing pulse which is sandwiched into the pulsetrain (fig. 3E) delivered to the output couplingtransformer primary. The sync. pulse ensures thatthe measurement system output polarity willalways correspond to the sense of the applied tor-que that is positive voltage out for positive appliedtorque.

Pulse Receiver(refer to fig. 4)Figure 4 is a composite block diagram of thereceiver decoder. The pulse receiver (figs. 4 and5) is a discreet transistor, FET input amplifier. Thetransistors are hooked as a complimentary pairwith negative d.c. feedback for bias stability;however, the bias resistors are bypassed at thesignal frequency to allow adequate gain. An NPNtransistor interfaces the receiver with the 74121TTL one-shot multivibrator. The one-shot outputdelivers constant amplitude, constant widthpulses (fig. 3F) to the sync. card.

Synchronizer Card(refer to figs. 4 and 6)The purpose of the synchronizer card is togenerate a 500-Hz square wave and synchronizeit to the 500-Hz bridge excitation signal in thetransmitter. The 500-Hz bridge excitation signal(fig. 3B) is generated by dividing the 10kHz VCOoutput by 20. The required 500-Hz signal for thedecoder (fig. 3H) is generated in a similar way.The 1O-kHz pulse train out of the receiver(waveform F, fig. 3) is divided by 20 also.However, the sync. pulse must be removed fromF before applying this pulse train to the divider D1(figs. 4 and 6). The sync. pulse is removed byOSI, a one-shot with its pulse length adjusted a lit-tle wider than the sync. delay (fig. 3C). Thisresults in constant amplitude, constant widthpulses (fig. 3G) to the divider D1 (fig. 4 and 6)with the sync. pulses removed. The divider D1has two outputs. One is a divide-by-2 output (5kHz) which goes to the demodulator and a divide-by-20 output (500 Hz) for the decoder. An addi-tional requirement remains that the 500-Hzdecoder square wave must be synchronized inphase with the 500-Hz bridge excitation in thetransmitter. Synchronization is accomplished asfollows (figs. 3 and 4): The sync. pulse (fig. 3D)follows a positive transition of the bridge excita-tion signal (fig. 3B) by a fixed delay determined bya delay one-shot (OSI, figs. 2 and 3C). A similar

4

A

B Jc Jl

0 _IE

Encoder - transmitter waveforms

fo=10 kHz

r-- Bridge excitation

L Sync. delay

L Sync. pulse

Receiver-decoder waveforms, sync. pUlse

F

G UUUUUUUlJ

I

fL

L

H ~

J

~,- sync. delay

J'--sync. gate

K JL Jr- sync. delay

r~ J~sync. gate

N Torque signal+ drift (excited)

Zero ref. (drift only)(unexcited)

0

p

J I,--Torque signal S/H delay

nTorque signal S/H gate

Q Zero ref. S/H delaY~1

10 kHz with sync. pulses

I

n

I

Synchronizerwaveformsout of sync.

Synchronized

rLzero referencesample - hold gateR

S Sample point~I I I I

Figure 3.- Waveforms.

5

Output signa I

" I I 5 kHz

L

OSIV SHII

I

52 I

~~~OSV SH2

I

0> Retriggerable -.Jone-shot

8~-

Decoderfigure

L ~Synchronizer figure 6 ~

Figure 4. - Receiver-decoder block diagram.

-..J

NOTE

* Pulse transformerPCA 101-0.15

or equivalent

Pickup

cr=p *

~Chassisground

+15V

~.~ 12

>---=2N2222 To

sync.card

+5Vpin 2

Sync. pulsefa = 10 kHz

+ 15V

1. 211.n.

1k.o.50..1l.

I00 Jl.

2N3251

0.01 .u.F

IOk..1l.

IOkA

0.01 J.l. f

Figure 5. - Receiver schematic.

+5V

lkJ\.

2N2222

10k.n.

741211 14

~ 133 124 115 106 97 8

20kJ\.

500pF

Sync. searchosc.

O.OI..lLF100k.lL

IOk...l1. 10k.lL

Note: The sync. search OSC periodically removes the incomingsignal causing the 500-Hz synchronizer signal to assume a newphase relation to the 500-Hz signal of the transmitter. The properphase relation is detected by a nand gate when the incomingsync. pulse is coincident with the 500-Hz sync. gate. When thetwo 500-Hz signals are synchronized properly, the nand gateoutput triggers a retriggerable one-shot and bypasses the searchOSC interrupt switch.

+15V

Ik.lL

10kJl. 10k.lL

~ 10kHz in

500 Hz/sync. pulse sync. gate

r ~t-ll c;-=JL~r-IL OS11

.00111F

21

-sGTo sync. indicator ~

LED

+15V

+15V

45181 162 153 144 13

5 -I- 126 117 108 9

+20

500Hz

+15V

12.lk.1L +15V3300pf4528

1 12>, 1

+15V 3~ 14~ ~ 13 +15V5 . ~ 12

u6 ~ 117VJ 10

9

ex>

+15V

+15V

+15V

OS111 out goes + with sync.-';'25000 Hzout to freq. Xd

pin 2115 Fig. 7

500 Hzout to decode pin 8

120Fig. 8

Figure 6.- Synchronizer schematic.

pulse (fig. 3J) is generated by 0511 (fig. 4) in thesynchronizer circuit and is delayed an equalamount after the positive transition of the 500-Hzsquare wave H of the synchronizer. This seconddelayed pulse J is called the sync. gate. The500-Hz bridge excitation signal (fig. 3B) and the500-Hz square wave in the synchronizer (fig. 3K)are synchronized properly when the sync. pulse Fand sync. gate M are coincident. Coincidence isdetected by a nand gate N1 (figs. 4 and 6) and thenand gate output triggers a retriggerable one-shot05111 which in turn closes switch 52 (figs. 4 and6). If the pulses described above are not coinci-dent, there are no pulses out of the nand gate N1;therefore, the retriggerable one-shot 05111outputstays low and 52 remains open. With 52 open, thefree running oscillator driving 5, alternately opensand closes the signal path causing the relativephase of the two previously described 500 Hzsquares waves to vary. When proper synchroniza-tion occurs (the pulses D and M, fig. 3, becomecoincident). the retriggerable one-shot 05111(figs.4 and 6) output goes high, closing 52 which main-tains the signal path and the two 500-Hz squarewaves are synchronized. The high output of 05111lights an LED indicating proper synchronization.The 500-Hz square wave (figs. 3K and 6) out ofthe synchronizer is used to time the two sample-hold gates of the decoder properly. The 5-kHzoutput of the synchronizer is routed to the fre-quency demodulator, a linear frequencytransducer (figs. 4 and 7). The frequencytransducer output alternates between zero voltswhile the strain gage bridge is unexcited (cor-responding to the zero reference frequency of theVeO) and a voltage proportional to the bridge un-balance when the bridge is excited (correspondingto the shifted frequency of the VeO). The signalof interest is the difference in voltage between theexcited and unexcited levels out of the frequencytransducer (waveform N, fig. 3).

Frequency Transducer(refer to figs. 4 and 7)The frequency transducer receives the 5-kHzvariable frequency signal (fig. 4) from the syn-chronizer card and converts it to a d.c. level pro-portional to frequency. The conversion is accom-plished by using the 5-kHz square wave(waveform A) as the gate signal via A51-b of agated integrator 11 (fig. 7). The output of the in-tegrator is a linear ramp (fig. 7 waveform B) dur-ing the half period while the gate switch is closed.The final value of the ramp at the end of the half

period is the desired level that is sampled and re-tained by a sample-hold circuit 5H1. The in-tegrator then is reset via A51-a to be ready for thenext half period. Waveform e and D (fig. 7) arethe sample and reset timing signals, respectively.The d.c. output voltage of the frequencytransducer sample-hold circuit is a function ofveo frequency; however, a frequency transducerof this type inherently is nonlinear. The frequencytransducer is linearized using an analog divider D1(fig. 7) as shown below. An expression for therate of change of voltage across the integrator 11capacitor is

dVldt = ilC, volts per second (1 )

where i is the charging current and C is thecapacitance.

Since i and C are constant in this circuit, dVldt isconstant, making the integrator output voltageproportional to the time T/2 which is a half periodof the 5-kHz input signal. Therefore, the capacitorvoltage at the end of a half period is

Vc = ilC (TI2) = KT, volts (2)

where K = il(2C).

The 5-kHz waveform period T can be expressedas

T = 1If, seconds (3)

where f is the frequency (nominally 5 kHz).

Substituting 3 into 2 gives

Vc = Kif, volts (4)

indicating the nonlinear characteristic of this typeof frequency transducer. Voltage Vc then isdelivered to an analog divider which has an outputvoltage of

Va = 1Oxly, y > 0 (5)

where x is a constant and y = KIf.

Therefore. the linearized frequency transduceroutput is

Va = K'f, volts (6)

where K' = 10xlK.

9

;;;;

0

I TI2 IA ~ 5 kHz (7510 pin 4 and 4528 pin 5)

B /L/l Integrator out

c ~ S/H (7510 pin 5-6)

~ Reset (7510 pin 3)0

-15V

3.01 k.n..

2k.JL ASI7510

5 kH z in ~( no sync)

Reset

S/H

Tr- + 15V

Figure 7. ~ Frequency transducer schematic.

Offset10klt

-15V

499k..!L

q-><0......

] k.n..

r4700pF

O.0015ll.F

49.9k1l.Freq. Xdout

To decode

pin 20

Figure 8

~21

SHI

01Eo

6 + 15VZ

Numerator lOkJ!.Z magnitude

Eo = 10 ~X

Note: X always will be >0with finite frequency.

+15V +15V +15V +15V

10ksL as 1l as IT

10kfl..... 500 Hz....

In8 3 +15V 3

-<Tr + 4 13 4 13

15 12 15 12From

6 II . IISynchronizer

7 107 10

. 9 8 9

From freq. Xd

120

Figure 8.- Decoder schematic.

Decoder(refer to figs. 4 & 8)The frequency transducer output (fig. 3N) and the500-Hz square wave (fig. 3K) are routed to twodifferent sample-hold circuits (SH1 and SH2).Sample-hold SH1 samples during the excited por-tion of the bridge power waveform and the SH2during the unexcited portion. Proper timing for thesample gates is derived by triggering delay one-shots (OSIV and OSV) at the beginning of therespective excited and unexcited portions of the500-Hz decoder square wave using the positivetransition to trigger SH1 and the negative transi-tion for SH2 (fig. 3, waveforms 0, P, Q, R).

The output of SH1 is a d.c. level proportional tosignal + drift and SH2' s output is proportional todrift alone. Amplifier A 1 responds to SH1 - SH2which corresponds to (signal + drift - drift) Gwhere G is the gain of A1 (figs. 4 and 8). The out-put of A 1 is a d.c. level proportional to torque.

CALIBRATION

There are no calibration adjustments once theoverall system gain is decided upon. The gain, ofthe bridge amplifier in the sensor section, isshown having two switch selectable values. Thisfeature is needed only if the maximum torque ex-

pected would saturate the system. The overallsystem gain in the high sensitivity position of theswitch mentioned above is on the order of10 000. Calibration can be affected by applying aknown torque - if possible - or by unbalancingthe bridge a known amount by applying high valueknown precision resistors across appropriate pairsof the strain gages and noting the output shift.Then the output can be measured while the inputis calculated. Various commercial calibrationschemes are available also.

The procedure for adjusting the analog dividershould follow the recommendation of themanufacturer. This would be a one-time "factoryadjustment" and is not difficult.

BIBLIOGRAPHY

[1] Eilts, L. E., and Eugene Campbell, Shaft Tor-sional Oscillations of Hydrogenerators,USDI, Bur Reclam, res. rep. REC-ERC-79-6,25 pp., August 1979.

[2] Perry, C. C., and Lissner, H.R., Strain GaQeprimer, McGraw-Hili New York, p. 185,1955.

12

A free pamphlet is available from the Bureau of Reclamation entitled, "Publications for Sale". It describes some of the technical publications currently available, their cost, and how to order them. The pamphlet can be obtained upon request to the Bureau of Reclamation, E&R Center, PO Box 25007, Denver Federal Center, Bldg. 67, Denver, CO 80225, Attn: 922.