LEHIGH UNIVERSITY

DEPARTMENT OF CIVIL ENGINEERING AND MECHANICS

BETHLEHEM. PA.

ADDRESS REPLY TO,

FRITZ ENGINEERING LABORATORY

PHONEBETHLEHEM 7·5071

FRITZ E. L, OFFICE EXT. 258

HYDRAULICS LAB. EXT, 279

it··· ; .''Q"~

F.ile: 205A

ERRATA:

to Robert L. Ketter and Lynn S. Beedle, "The

Moment-Curvature Relation for WF Columns", Progress

Re.port P, Fritz Laboratory Report No. 205A.10, Sep­

tember 1952.

page 2 Fig. l(b) the equation should read

t1 =f y cr dAA

page 5 EqU8 tion (3) should read.0',.,

page 14

-J" d"- ,f a 4J""\

A

. Equatiori (5M) should read• p

.' ,....

WELDED CONTINUOUS FRAMES &THEIR COMPONENTS

PROGRESS REPORT P

THE HOHENT - CURVATURE RELATION

FOR WF COLUHNS

..

(Not for Publication)

by

Robert L. Ketter & L~'nn S. Beodle

This work has been carried out as partof an investigation sponsored jointly bythe Welding Research Council, and the NavyDep artment wi th funds furni shed by thefollowing:

American Institute of Steel Construction

American Iron and Steel Institute

Column Research Council (Advisory)

Institute of Research, Lehigh University

Office of Naval Research (Contract No. 39-303)

Bureau of Ships

Bureau of Yards and Docks

Fritz Engineering LaboratoryDepartment of Civil Engineering and Mechanics

Lehigh UniversityBethlehem, Pennsylvania

September, 1952

Fritz Laboratory Report No. 205A.I0

205A. 10

TABLE o F CONTENTS

page

1. Introduction 1

II. Method ·of Solution 4

III. Discussion 9

IV. Summary 11

V. Acknowledgements 11

VI. Nomenclature 12

VII. References 13

APPENDIX

A. Summary of Important Equations 14

B. Outline of Alternate Method of M-¢Deter,1r. inrttion 16

FIGURES

I INTRODUCTION

When extending the methods of analysis used in engineeringmechanics to include the case of elastic-plastic bending, therelation between applied moments, axial thrust and resultingcurvature for the caSe where stresses exceed the elastic limitis basic in importance, since it is the integrated effect ofthis curvature that determines deflections and rotations. Thevalue of moment-deflection data is primarily that it allows thedetermination of true maximum loads. Moment-rotation data isalso valuable since it enables one to evaluate performancerelating to energy absorption.

One assumption that must be made is that the Bernoulli­Navier hypothesis (bending Btrain is proportional to the distancefrom the neutral axis) can be extended to include the case ofplastic deformations.* Using this together with assumptionswith regard to properties of the material in question, thisreport will outline and illustrate a method whereby a relationbetween axial loads, bending moment and curvature can be obtainedeven though stresses exceed the elastic limit.

The curvature, ¢, is a function of the strain developed ata section; the moment, H, is a function of the resultant stress.These ideas are illustrated in Fig. l(a) and l(b).**

*

**

Actually for steel members tris is an idealization. As has beendescribed in previous work(~, it is only required that in the elasticrange , the strain be proportional to the distance from the neutralaxis and in the plastic range the stress be equal to the lower yield­point stress. Thus the curvature at any section is a function of thepart which remains elastic. This is further discussed in section IIof this report.

If the strain-hardenirtg range is considered, it is once againnecessary to make an assumption with regard to strain-distributionin the inelastic range. But strain-hardening has not been consideredin this paper.

Details of the development of these equations are given on paee46of Reference (2).

20M. 10

-r-­d

l

Strain Diagram Stress Diagram

-2-

JJer dA y dyd 11

(a)Fig. 1

(b)

For the case of elasti6 strains illustrated in Fig. 1 (that is,when €1 < 6 y and €2< €y) the distribution of stra~n and dis­tribution of stress are similar, er

1being equal to E€f' etc.

When yielding occurs, however, this linear relation no longerholds. Instead there results a stress distribution composed ofsegments of various straight lines or curves depending on thestress-strain properties of the material in question.

For this report in agreement with those preceeding it, theidealized stress-strain diagram of Fig. 2 has been assumed. Thiscurve closely approximates the experimental stress-straindiagram of mild structural steel in the elastic and p:asticranges. Strain hardening is neglected.

e----Fig. 2

The effect of this choice on the stress and strain distributioncurves, when strains exceed the elastic limit, is shown inFi g. 3.

'205A. 10

1~---·-

L.__

_.._......---- f--7

Strain Diagram( a)

Fig. 3

Stress Diagram(b)

As would be expected, the moment corresponding to a given valueof ¢ would be less than that predicted using the straight linerelation of the elastic solution.

The problem of determining M-¢ relations in the plasticrange has been solved for the condition of pure bending. (0)Since for that problem the neutral axis of strain coincided withthe geometrical center of the section, relatively simple expres­sions were developed relatin~ Mand ¢. If, however, axial loadin addition to the imposed moment is applied, such a conditionwill no longer exist.

The determ.ination of M- ¢ curves including the effect ofaxial load has been developed by Timoshenko for the case ofrectangular sections.* For the case of WF shapes discontinuitiesof the cross-section make direct computation troublesome. In thenext section of this report a method suitable to handling sucha problem is developed.

The purpose of this paper is to present a If'point-by-point"method of computing M- ¢ curves similiar to that used earlier(0)including the influence of axial load and suitable for rolledsections. The method presented is not difficult to set-up andeffectively illustrates the variables involved.

* Pages 51-54, Reference (Z).

'20M. 10 -4 -

II. HETHOD OF SOLUTION

I~I

When a member is loaded within the elastic range by combinedbending and thrust, stress and strain diagrams similar to thoseshown in Fig. 4 result.

Strain DiagramFig. 4

Stress Diagram

If 6 = €1 - €2' using Timoshenko's notation,since the stra.ins are small in comparison tosection, tan ¢ = ¢ or ¢ = 6 Jd. But since

tan ¢ = 6/d.the depth of6. :: lTl-f"2

E

However.the

For members loaded beyond the elastio limit such as thatshown in Fig. 5, a slightly different ex~;ession relating ¢ andtt is obtained.

Mp

T

1:Fig. 5

1~2J

Strain Diagram Stress Diagram

For these distributions of stress and strain

¢ = ::

or ¢ :: . ~ .........•.... (1)

205A. 10 - 5 -

Equation (1) is similar to that obtained for the elastic caseexcept that Yc has replaced d, and ~y replaced~l . Therefore,it is possible to compute ¢ directly from a stress diagramproviding that only that part of the section which remainselastic is considered.

From a stress diagram it is also possible to determineboth the moment acting- on the section and the applied thrust.These quantities are defined by the classic equations: *

P = J cr dA . . . . . . . . . · .. · . . . . • (2)A

M = I I t7" dA y dy • • • • • • • • • • • • • • • • • • • • • • • (3)d A

By using equations (1), (2) and (3) it is then possible todetermine M, P and ¢ for any given stress distribution.

When a member is subjected to both compressive forces andbending, yielding will first occur at the extreme fiber on thecompression side of the specimen. As the load (that is, axialthrust ~nd/or moment) is increased, yielding will penetratethrough the flange on the compression .side. After furtheryielding it will have penetrated to * the depth of the section.Then i the depth, etc. Now consider what effect the tensionside of the specimen has (for a given fixed value of yieldpenetration on the compression side) in determining the valuesof the external loads, D and M.

..

fFig. 6

* Pages 45-54, Reference (r).

205A.IO - 6-

Using Equations (1), (2) and (3) each of the stress diagrams ofFig. 6 define distinct values of M, P and ¢ even though thenet effect of each on the compression side has been tocause yielding to penetrate to a depth of i the section. It ispossible then by selecting various values of u a to determineH, P and ¢* corresponding to each selection of U a ' therebymaking it possible to plot 'curves of P vs ¢ and Mvs ¢ forany given value of yield penetration on the compression side.One such set of curves if shown in Fig. 7 for an 8WF21 sectionwhere yieldting was assumed to have penetrated to a depth of !dfrom one side. These, curves have been made non-dimensional byplotting P/Py vs ¢E/~y and M/M y vs ¢E/ay.(Since My, E and u ywould be constant for any particular section, this is, in fact,an M-¢ curve.) For all points plotted the corresponding stressdistribution diagrams have been shown.

This method of assuming a penetration of yielding on thecompression side of the section and then determining curves ofP vs ¢ and M vs ¢ can be carried out for any value of yieldpenetration. The ones investigated in this study am indicatedby the stress distribution curves of Fig. d.

a. Elastic Limit ;1;,.711I ' 1

. / / I

/~. '/' I/' . I

///

..*

b. Yielding PenetratedThrough Flange

U y

Equation (1), (2) and (3) are not in the most suitable foom fordetermining M, P and ¢ for rolled sections due to the discontinuitiesresulting where the flanges and web join. Therefore, expanded equationsin terms of the section dimensions have been given for WF shapes inAppendixA.

- 7

CT y _t_1---, ~d

c. Yielding Penetrated tok Depth of Section

CT y

T

d. Yielding Penetrated toi Depth of Section

e. Yielding Penetrated toi Depth of Section

Fig. 8

...

Curves similar to Fig. 7 have been computed for each of theseassumed values 0: yield penetration for the 8WF31 section.These are presented in Fig. 9. Since the straight line portionof ~ach of these curves is that corresponding to an elasticstress condition on the tension ;(convex) side of the section,only the limiting cases where CT a = yield point stress in compres­sion and tension need be investigated to define that part ofea~h curve. The limit of this range has been shown as a dash­dot line.

The general method of using these "auxiliary curves".(Fig. 9) is to select a value of P/Py for which an M-¢ curveis desired. For example P= O.2Py. With thi$ value of P/Pydraw a horizontal line cutting the various auxiliary P-¢ curves.At these points of intersection project verticals until theyintersect the corresponding M-¢ auxiliary diagrams. Connectingthese points gives the desired M-¢ curve including the effect

205A.10 -8-

of an axial thrust of O.2P y • (This rrocess is indicated in Fig.9 by dashed lines.) Fig. 10 is the desired curve with theauxiliary construction lines removed.

The curved portions of Fig. 9 have·been plotted to alarger scale in Fig. 11.to facilitate greater accuracy in thisgraphical method. Here M-¢ curves for several different valuesof P/P y have been shown .(dashed lines).

The influence of varying axial load for a greater rangeof P/P y is illustrated in Fig. li. Here M-¢ curves have beenplotted for various values of P/Py ranging from a to 0.8. Asindicated by the dashed line, only when P is relatively smallwill the tension flange be plastic. For that rang~ where itremains elastic, only the straight line portion of eachauxiliary curve need be considered to obtain the desired M-¢curve.

=

205A. 10 - 9 -

III. DISCUSSION

Since it is the integrated effect of curvature thatdetermines the deflection ofa structural member, and sincedesign methods are indirectly based on deflection as the cri­terion of usefulness, it is important to consider the influenceof axial thrust on the moment-curvature relation in the plasticrange.*

The method for determining the M-¢ curves presented inthe previous section, while not the most direct solution tothe problem, appears to be the easiest to use and best illus­trates the variables involved. Possibly a more mathematicalapproach wouid be as follows: Each of equations (4), (5)

and (6), correspond ing to expanded equatione (1), (2) and(3), contain ,5 variables; H, P, cP, YC' and (Ytor O'"a)' where

M = Homen t,P = Concentric axial load,¢ ~ Curvature,Yc = Distance from yield penetration

in compression to extreme fiberin tension,

Depth of yield penetration intension,

0'" a = Stress on tension side of section.The formal procedure wouin be to eliminate the latter two ofthese unknowns thus obtaining an expression relating M and ¢

and including the variable P.** Instead of solving theproblem in this manner, we have chosen a simpler point-by­point method. The procedure used however ~ppears to give aclearer indication of the interrelationship between H, P and¢ as s tresses exceed the 'el asti.c 1imi t.

*

**

Deflection and rotation curves can be obtained by usinC Newmark'smethod for numerically integrating the M-¢ curves defined ip thisreport.

This procedure is outlined in Appendix B.

'20 5A. 10 - 10 -

Only one type of stress-strain cu~ve has been consideredin this paper. This one, which agrees comparatively well withmild structural steel coupon tests in the elastic and plasticranges, is composed of twc straight lines, one at a slope Eup to the yield point and then the other horizontal. One methodof extending the method dnveloped herein to include the case ofcurved stress-strair diagrams woulj be to approximate the curveddiagram with a series of straight lines(5). Any degree ofaccuracy could then be achieved by increasing the number ofstraight-line segments; the resulting expression would be morecomplex.

Residual stresses would cause a deviation from the straightline portion of the I':-¢ curve at a sme.;Ilef val ue of Mthanpredicted by a theory that neglects their effect. The relativedifferences are greater for larger values of axial load. Apartial soluti6n to the problem of the efrect of residualstresses on moment-curvature relations using experimentallymeasured residual stress distribution valUes has been obtained.In general, good agreement was noted between the curvesresulting from that study and full scale column tests.* Solutionof the M-¢ rel~tion including the effects of residual stresseswill be presented in a future report.

More detailed studies of the problem of beam bending basedof -the various "theories of plasticity" have been presented byAris Phillips in his various reports relating to ONR TASK ORDER11. (6) These mathematical derivations frequently result in thesolution of complex partial differential equations;however, the method gives an "exact theory of bending" providedall of the implied conditions of material behavior are fulfilled.

* See for exarr~le, Column Research CounciJ Progress Report,Vol. IV,No.2, April 1952, page 6.

205A.IO

IV. S U H H A R Y

- ·11 -

A method has been developed for determining the influenceof axial load on the moment-curvature relation in the plasticrange. The method used employs a point-by-point procedure ofdetermining "auxiliary curves"(Fig. 7 or Fig. 9) prior toobtaining the final M-¢ relation (Fig. 10). Howev~r, oncethese "auxil i ary curves II have been P1.ll.ot ted for a particul arsection, the M-¢ relation can be obtained conveniently forany desired value of axial load (Fig. 12'.

V. ACKNOW LED. GEM E NTS

The authors wish to express their appreciation to Mr.Don C. McCullough for the detailed numerical work of thisreport. The work is being carried out under the direction ofthe Lehigh Project Subcommittee of the Structural SteelCommittee, Welding Research Council, at Fritz EngineeringLaboratory of which Profe.ssor William J. Eney is Directo·r.

'205A. 10 - 12 -

VI • NOM E N C L A T U R E

A Area of cross-sectionb Flange widthd Depth of sectionE Young's modulus of elasticityM Moment

My Moment at which yield point is reached in flexureP Concentric axial load on a column

Py Axial load corresponding to yield point stressacross entire section

S Section modu~us

t Flange thicknessw Web thicknessYo Distance 0nm yield penetration in compression

to extreme fiber in tensionYt Depth of yield penetration in tension6 The difference in strains in the outermost fibers

on the convex and the concave sides of amember

€ Strain

e y Strain corresponding to yield point stress~ Normal stress&y Lower yield point stress¢ Curvature at a section (the reciprocal of the

radius of curvature)

205A.l:C

VII. REF ERE NC E S

- 1-3 -

(1) Yang, C. H., liThe Plastic Behavior of Steel Beams II ,Ph.D.Disser~~tiontLehigh Univ~rsity) ~m.lg51.

(·2) Timoshenko, S., "Theory of Elastic Stability",First Edition, MoGraw-Hill Book Co.,Inc.,New York, 19 36 •

( 3) Luxian, W. W., and J0 hnston, B. G., II PI as t i 0

Behavior of Wide F'lange Beams ll, The Welding

Journal Supplement A.W.S., vol. ·27, No. 11,November 1948.

(4) Ketter, Robert L., Beedle, Lynn S.; and Johnston,B. G., IIStrength of Columns Under CembinedBendin~ and Thrust ll

, (scheduled for publica­tion in the Welding Journal of the AmerioanWelding Society);

(5) Philli.ps, AriS, rtBending with Axial Force ofCurved Bars in Plasticity", Technical ReportNo. 10, Division of Engineering Mechanics,Stanford Univer~t3ity, June, 1951.

(6) Phillips, Aris, "Pure Bending with Axial Forcein the Theory of Plastic Deformations",Technical Report '0. 3, Division of Engineer­ing Mechanics, Stanford University, Jan.,1949.

205A. 1'0 - 14 -

APPENDIX A

Sm1MARY OF UiPQRTM;T EQUATIONS FOR DETERHINING H, P anu

_I.__~LASTIC LIMIT CASE

O"a

Stress Diagram

P = (O"y;'0".:i)[2bt + ~).(d-2t)J

H = {(]" -0" ) fbt (3d L 6dt.+4t 2}y a _ 6d .

¢ = (~-(]" a)

Ed

................. ~

!

!i

W (d rt ). "'J [+ - -2;" i12d !

_.J

•••••••••••••• ( .4)

II. ONE FLANGE PLASTIC, THE OTHER ELASTIC

Limits:

Yc <: (d- t)

P = (CTyfCTal[bt- wtJ f ICTy-CTal[~~:

M = (O" t"U )[~(y -t)2(3d-:2y -4t) +y 8' 12Ye e c

¢ = {(j Y":'(7 a ~Ey

e

WV wt 2] i~ e ;- -2 - -r'- + wdo- y ~, ·i,Y c ~

b. t( 6'Jr d --2" t-3dt+2t2~ I... (5)l'iy, e ole j

~ , Ii!...........:

'205A. 10

III. BOTH FLANGES PLASTLC

Limits:

Yc < (d-t)

Yc>Yt>t

- 15 -

p = Wor [d - Yt - Yc]

...........................,i:

~i

M= "y~ t (d - t) + W( Yt - t)( d - Yt '- t) + ~ (Yc - Yt ) ( 3d- 2yc - 4Yt )J i.... (6)

I!:i

........., , ,~.

Note: The effect of the fillet has been neglected in theseequations.

'205A. 10 - 16 -

A P PEN D I X B

OUTLINE OF ALTERNATE METHOD OF M-¢ SOLUTION

I. From equation (5), page 14, (tension flange elastic)

(a) ¢ =(oy -~3)

YcE

(b) Mw 2 etc.= (oy - O:J,) U (Yc - t) ( ...... ..Ye

• • • ! •• ' etc.

1) Substitute the value of u obtained from eq. (a) into eq. (b)a .and simplify. The resulting expression is of the form

M = E¢ [ai + b;l~ + c~c + d1]

where ~, ~, ~ and ~ are various' geometrical oonst~nts

of the section.

2) Make the same sUbstitution in equation (c) and solve for y~

V2 bcr: Ut riP] Iy.= (a) + .2..;X.+ J. C + _"2_ + e

c . Z EcIJ E¢ 2 oy a

Here again Q1' b2

, C2

' d2 an.d e2

are geometr ical constants.

3) Replace ~ in the moment equation with the expression determinedin step 2). This will be the desired M-cIJ relation.

II.

1)

2)

For the case where. both flanges are plastic (Eq. (6), page15, the above outlined reasoning will result in the followingequations:

M = u rl1. y 2+ b Y + C.3cr

Yy, + ~ + eJY[-:1 c .3 c E¢ c E¢ .3

[cr Pcryb4]y= .:.L.+a-

C E ¢ -4

Therefore,

3)

205A. 10 '

}-_.l--_...l..-_..L--_-1-_--'-_..:...-l L-_._.L.-.::::=====._-Q-__l 0o

'0 ~ 1.1n l-I

I. I 1.0l \ I

~i

->+. i\ /c;;>/ '\ \ \\ 0.90.9 "\ <0 I

\i

~o- i !\ I

\ ee- I \0.8 \

'l. I I _. 0.8~ i

\ \

\\0.7 \ 0.7

\ \

\ \

I\0.6

I \ iI 0.6I I J1 jP

···I~··· L Z- M-Py I "I ::::-",,/ '=7 "-

0.5 - .......... 0.5 L~,- ) '-

..... ' -. ._- ' ._- -_. ._.

I I 1 II iI !

0,4 I It

! I 0.4I I

; iI ,

! /0.3 j ~. l \,

0.3'I I

/ I CJ':> i

I/ I0.2 I / I

0.2i

1o. I ! I . Ir I~

II!

O. , 0.2 0.3 0.4 0.5

¢E

Fig. 7

'205A. 10

1.2

CI) Elastic Limit

(?) Yield Penetrated thru Flange

CD Yi e Id Penet rated to <t Depth of Sect ion!

® Yield Penetrated to ~. Depth of Section!

® Yi e Id Penet rated to ~ Depth of Sect i on1,

a

0.2

I

I1.0

0.8

_E-DI y

0.41

0.2 0.4 0.6 0.8 1.0 1,2

¢E

Fig. 9

~1-'"

GD.

f--.J.o

«q!;;;

oI\)

o~

?0\

oOJ

o

:::1-'- 1\:>« -0(Jl

0 0 0 0 0 0 0 0 0 - - :t:-O . . .

t·) \..N ~ \•.'1 0\ -J OJ l,c, 0 - I--'0

Initial Yield ..

" 1A " /'Yle "ln9 Penetrated thru Flange-----------/

Yi el d i n9 Penet rated to It Depth of Sec t ion _

Yielding Penetrated to ~ Depth of Section - _

\J

II

ol'\)

\J

Yielding Penetrated to n Depth of Section ~

~~

205A. 10

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ,. I 1.2I

1.3

I. I

1.0

0.9

0.8

------------------------_.-- --(;:1- 4> Curve for P=O. IPy)

--_ .... ­.. --------

/

0.3 --

0.2

0.1

a I02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I. I 1.2 1.3

¢Ecry

Fi g. 11

205A.10

1.21.0---

0.80.6

-_...-

0.4

\-'--

¢E

F· ~19. 12

DIS1'. aFebruary 18, 1952

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SuperintendentNaval Gun Factory',--jashinvton 25. D. C.

--' " (1)

O'fHER GOVERNHT!~NT 1'.GENCIJi.:~

u. S. Atonic Enor2,y COlnnissionDivision of Research';cJashinGton, D. C. (1)

DiroctorNation2l Bureau of Standards'!ashinstou, D. C.Attn: 1::1' •.:/. H. RarJborg (1)

u. S. Coc-st (~uard

1300 E. Strc')t,· N. 1';.~Vashington D. C.Attn~ Chief, Testinc &

DCvo1opnont Division (1)

{]lJi af of Buroau of Yal' ds t~ Docks:'bvy Department~~3hington 25, D. C.t~ :; tn: Code P -314 (1 )

: Code C-313 (1)P_.~~~ /1 i_. _1lI/\I \ .,_ .'

National .i\dvlsory Comroi ttoe forAoroniluticsLanGley Field, Virginiai\ttn: 1':11'. E. Lundquist (1)

u. S. ~farit inc Comai ssl onTecrolicnl BureauWashinGton, D. C.Attn: Mr. V. Russo (1)

Supplci':1cnt ..fl.To Distri"!)ution List for Technical and Final Roports

IssuoC: Undor O:NR Pl'OjOC-C NR -Q64-C,45

Contractors a~1c1 r)t~1.or Invostir:ators J',ctively J:nra,:.ad in Relatod Rcooareh

Dr. N. M. N01.'Jl;lO..rlcDopnrtrnont of Civil ~:~n[,ineoring

Univorsity of IllinoisUrbona, 111inoi s (1)

Dr. N. J. Hoff, HondDepartment of tGron~utical

EnLino oring t: j~ppl5.0 0, 1'~O cha nic sPolytochnic lnst. of Brooklyn99 LivinLston StrootBrooklyn 2, New York ,(1)

Dr. \"j. H. HoppmnnnDo;xlrtmont of 1'.rmliod ~.IIochanics

Johns Hopkins UnivorsityB['.ltir~or·:), r.:oryland (1)

Professor Jesso Orncndrflyc1Uni vorsi ty of TUchJ<:;o.nAnn j..l.rbor, ~!Iichisan

Dr. J. N. GoodierSchool of EngineeringStunford UniversityStanforcl, Cal ifornla

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Professor D. K. KrcfcldCol1Q[',0 of Encinc',ringColunbia UnivorsityNow York, Naw York

Profossor R. M. HermesUniversity of Santa ClaraSanto. 01 ara, Cnlifornia

Dr. TI. P. Pcto~son

Direct orAppliod Physics DivisionSandie. I,"l.b r'ra.toryAl buqucrqu 0, New l'1Gx:i.c 0

(1)

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Dr. ~. OSGood, IllinoisInstitute of ToclmoloLYTochnoloCY CenterChicago, Illinois

Professor B. Fried'::Llshingt on StatG r} 011 osePUllman, "!ashington

Dr'. L. Phill ipsSchool of EngineorinGStanford Univorsi t',fStanford, California

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Dr. R. J. Hnl1sonrf[assachusctts lnst. of' 11ochnolo~;y

Ca~0ridGo 39, Mass. (1)

Professor Vito Sc.larnoLopartmcnt of ~cronautical Engr.0.11.6 Lpplior'l EcchcmicsPolytechnic Inst. of Brool::1J'Ll85 LivinGston St.Brooklyn 2, N. Y. (1)

LibriJ.ry, EnG5.nc :;ring Ii'ounc1ati on29 ~. 39th StreetNow York City, Now York (1)

:i):,:,. '.-;-. ?ragerC":'CtC'1uato Division of AppliodJ\1Jthona tl cs13r0\'m Uni versi tyProvidence, Rhode Island (1)

Professor L. S. J~cobsen

.::: L8.nfol"'d Uni vor sityStanford, California (1)

r'ICi:,lbcrs, Project Staff

Projc ct File

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