The Stress Determination by the Photo Elastic Method of the Shrinkage Stresses in a Gravity Dam Citation Smits, Howard Gardner (1933) The Stress Determination by the Photo Elastic Method of the Shrinkage Stresses in a Gravity Dam. Master's thesis, California Institute of Technology. doi:10.7907/g3vr-5d04. https://resolver.caltech.edu/CaltechTHESIS:09232024-220037743 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): von Kármán, Theodore Thesis Committee: Unknown, Unknown Defense Date: 1 January 1933 Record Number: CaltechTHESIS:09232024-220037743 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09232024-220037743 DOI: 10.7907/g3vr-5d04 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16754 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 24 Sep 2024 21:35 Last Modified: 24 Sep 2024 21:56 Thesis Files PDF - Final Version See Usage Policy. 6MB Repository Staff Only: item control page

The author wishes to express hie appreeiation tor the valuable suggestions made and the continual interest shewn by Dr. von K!rm!n,, Dr. Blot, Professor Martel• and Dr. Brahtz .. stl:'$ase$ in a theoretical gravity dam oo:etian on an elastic ba.se -due to ME'lJiOD T.ba photo;-elasti,e method is particularly adiipte~ to this problem,,, (}lUY a two-dim:enaional stl'e.as analysis bai11g 1•~quired,. A e~ie'te de:se:ription of the ·theo,retieal ba-okgr-0und of this phot-o elaet:f.e Inethod is given by Fi.lon &nd Coker in their 'book, "Photo Elastieity"f'- published by the Cambridge Press., A working summery @f' the theel"Y' together w1 th some laboratory technique is given in a ser ies 0-r five articl-e-s publi.shed every t wo months i n the General .Eleetri~ Review: by Fil.o n s-tarting with the I'lo:vember issue ot l.920. Tire only differe•n ce between the app.aratus used he'I'e and that common throughout the other :pllotG-elastie laoo,1 •ato:ries is that eight inch mirro;rs we'l'e used in obtaiuing pai~allal light instead o:r the more usual four or six inch lenses. A greater flexibility of' appantus and a larger working area on the model is permitted 'by this substitution o>f> :m:ir1-ors fe-r lenses. Foth mono,• ehrornatie and whitce light wel"e available but: whi ta light was used throughout this test by using a: blue eolor filter in photographing the isoohromati es. To obtain the ef':!'eet experlmentally of ha:Ving the dam shrink as e. unit upon an el.astie f oundation, Dr. Biot suggested elongating the base by me,an..~ .o f t.ansion directly a pplied. relative to the dam. relative to the base .• '!hu.a the base t ends to elongate Aetually the rever$e takes :ple.ee; t:he dam shrinks T110 mst desirable pro,eedu:re would be to stret,ah the dam ODE by 1Je~urel along the base to iihw st.r ~as free elas'tie gr@Ulld An. T-ti,$ is m t .m w 1~ep,resentatiott of unifoJ.im a."l~i'llkage i n a thooretioat. g~av!cy dam E ground in t he region near t h@ eenter t ground was, stretched wit h a dir.ee,t tensional load"' r:t'his gave, a high tension of aay P l'bs./sq,.tn., 1-n the regi ons A and B, cM.u1t~ary to the ab0.ve desc:ri ptior.i. ot: tbe pr(ltblent.. This nmst be earreeted., Therefore.., a.:tt-er the eompl.ete stre.ss 'a rutly~i$' had been detetmned a blanket ho:d.zontal eomp·r ~sion ean be arb.itl"e.ril.y la.id dGwn on the grollnd ef' P lbs.Jsq,-.in., gi;~ing zero stress in t he regions A a.no l3;: tension in the re'8triet-ed regions G an<a H; and e0r«-prooaian in the ragie,n :F.. This: proouees a mos t decided discontinuity alone; the base line CD,. This is as it should be .~, «am pl'opet",; 'I'hu-s it is .seen that the analogy chosen holds rer the but not at all .for t he ground •. Aft.er the lines of eonstant shear and d.irec.tj;on lines of prineitml stress had been determined it wa.s found desirable t0 find the prtneipal stress indtvidually,.,. (a} Three mathod,s were open:- Graphiea:l integrat.ion.• ( b J An eleett•i cal anal<tlgy .. {e) A membrane anale-gy. Graphical integrat:ton is in rather eo:ruon us:age ,; but is· long a111d tedi.ous.,,. 1~o record was f'ound of the electrical analogy lle.ving ·b een ooed in this w,or.k.. Th@• Prandtl ·membrane analogy,, also -s,u.ggested by Dr,,. .Biot,. has been. used in t he solution of La Pla:ce·•·s aquat'ion i,n probl.ems of eha:f1;s Cal'I'Ying torque. Here ·s oap mm_l'tbranes have been used.. A rubber membrane analogy was used en this problem,. Too analogy is based on the f'aet that both the sum of the prin·e ipal stcr:e,sses mnd tlte dis·p la.eement e-f the Jt,embrane obey La Place*s equation. 'l:b.e model.:: Let P mid ei be the p:dnetpal. atre.s ses and Airy.'a stress function . Then: - /7-t-(j?~ff ~ ;;;z ,;: t i ;;i? I/ 8y2 -f-~2 p-=pr t? d x..z Therefore T ELEME~T OF THE MEMBRANE Y- -----+------f----.Y T T ,x T T T ~ ,x L The ioombrane: - X _y_ z Let the tension in the membrane be a eonst.a nt T. Then it 1s in equilibrium as is e.vide.n t from the vi ew s howing the ,.x-y plane. In the X ~ plane the angle t .h at T makes w1 th the 1/ plane is ..J--_ /CIIJ -I 'c)~ ;;;x • If the angle be small the tangent may be t aken eq_u.el to the .a ngle. The OOn:l]i)enent of each T aeting i.11 the Cons-f ds:ring the ele!lll?,nt z dh~eetion th.en. is T a.x -;}z • o'.'.'.r' ,, t he resultaut ~ompanent - of f"oree in the Z airee'ti{).l'l. then 'ts: ~r:lz)r cJX olX -Si.'Itilar re,a.so:ni :ng in the / z plane leads to 2-(~)T ~y ;;Jy '!'o have eqUili bri U.'ll the foll owi ng must hold true: ,,~ z ~ I f - -;;;z=r=o S>,x~ !}J'z or' {7,ec= O It is seen that ~ can be used a s P + Q,.,_ P - Q is determined 1'1'0r!! ·the is.o,ehroruat ie lines, and at the boundary o.f' t he i:rod,el one of t he P and Q a:re loiown along ~he boundary giving immediately P + ~~ i .f tbe membrane be deformed in. the i: direetio:n proportional 'fllus to P + Q. al.011g t he. geoi:oo-trie shape of the model bolm.dax-;1.esli 1c ean be measured at every point within t he model thtta dete·riui:ni,ng P + Q. Thia prooo-dure was followed so that a oomplete stress analysis resulted f-or t he model,. The moe_el was made up -so that tension eould be put into the ground by means of pins as shown in Figure 3o To be sure of axial loading the tension strap was made 0 ( ayrmnetrical a'bou.t its ans., thus t he optical :pattern was a1,1ay.s a gaug~ o:t the symmetry of loading.. Th-e sf.z·a of t 1e model was limited. to the eight ine...'11 working eirele· of lj_gh:t-,, the thickneea being fixed by the standard 1/2ff plates of bakelite available which we.re milled to 0.. 000 ineb.es and then polished. APP.ARA It was necessary to construc-t to pieces of apparatus for the solution or th1s p:roblem. a loading mechanism and the mbrane mechanism. LoatU,ng 48.chine The loading fran'C is shown •1th th& model in place In the accompanying :pb.otogre.ph. Figure 4. Plans of 1ts eonstruction are provided in the appendix of original data. A load ot 1220 lbs. per sq. in .. was used 1 the strap ot tbe model . embrane APP!E!tue Several general vie e ot the membrane With the deforming walls or boundaries are shown on the f'ollow.lng page. The component parts, con- THE MEMBRANE stretching remained ei.rcles on str,.ateh:'i.n g ,. b:0undar ies are shovm. The deforming walls ,o:r Bo·lih a. pos:ltive and a negative WBre made so as t-o be S'l.U'e the rubbet- wa.s at al l time.a in contact with the boundary.. The latter were mde of l/8tr aluminum plate -serewed to wood bl0,eks to insure their rata.i:ning the cor!'ect shape. The nega.tiv,e and positive bloeks were een.tt1red through t he rubber by meane of a ball and soeket. The maximum difference in height in the boundary, tw'htch was twice the model size) was o•.4 inches g1V1.ng rise to -a :maximum slope in the membrane of approximately 45f> . This steep slope was limited, hOVJever,, to very restricted regionst nal!lely, the comers , o:f the dam base. 'rhe elevation of the defonned membrane was obtained by means o:r a micrometer .screw mounted. upon a two way slide carriage provided with .scales for pur-poses of ol'ientation., Readings in elevation of' the robber sheet oould be reproduced to .1i thin 0.,003 inehes, giving an aeeuracy of greater· than on~ percent o Conte.et between the :pointer and the rubber was noted by using a light fo.c:used to serui light praetically :parallel to the rubber. TJ:ms J when the p0inter and its shadow met, oonta.et was made .., After l!lllc.h experimental work, it \'1aa found •chat f'or- best 1-eaul ta the bakelite available shoul..d be altll;eal,ed be~ore null ing .. T'ilie was done i n an el.eetrieall.y operated oven , loaned. by the Che1nistry Department -. Best r·esults iere obtained by placing -the bakelite on a: gl:ass plat.e horizontally in the ove.n, allowing one hour for the temperature to rise to 80° F.. This temperature was held constant for one hour when the plate was t'U:rned. It was held then f or anothe r hour at thi.s temper- aturt\. af'ter whieh it was cooled at the rate of 6° per hour. All OUl"Vat~ o.f.' the bakeli te having been t~ken out• , the plate was milled to o._.zoo inehes t hiQ'knesa tn rou:r. euts take.n alternately 011 each Mr. San.dale tlien machined out t he model ,. side . A small t est beam was w..ade out of the same pie.es f0r optical cali b:c•atfon,. From this the A st-rass-strain diagram. was plotted to 1250 lbs. per sq.• inch, giving a beautiful etrai~ht line relationship leadi ng to a modulus of elastieity of 740.000 lbs., per sq., inch. A complete set ot 1.ooelini cs was taken o-n Eastm.~n s:u11er,,-s~si ti ve pa.achromatic film, a few samples of which are shown on t he following pages. I.sechromatics were ta.ken on co:mme!!:ieal or'tho with t he aid of a blue filter, {see Fi gure .5 .) White light was use.a throughout .. ISOCLINIC PICTURES ISOCLINIC PICTURES 10. ISOCHROMATIC PICTURE FIG. 5 o~ the principa1 botm ry of tlle model trom d to d brane f r the comp Choe t:reDse a round for all points on the ich the ooun r:, ls were ns then sune~ d, the data thus de. The btai.n:ed being us d • ation of the principal tensions and compressions. agai t quili br d • by obtain g t e foree perpendicul r to and the shears along various eeotione through t model to check on the experiment accuracy. The resul ta are tbe rollo ng age. They uow:• (a) A 15% error ill the shear alo (b) AB. An 8% ettor in the tension along AB. 0 (e) A 5% orror i n t h e t ens i on alo.Iig CCo (d) A 5% erro:r i n t he t ensi.o-n p.J..ong DD,, (e} A Z% e!'r or in the t;ension along EE. .All results were drawn up and are presented in the following prints. I DI RE CTI ON OF LIN ES PRINCIPAL STRESS 111■■■■■■■■111 111■■■■■■■■111 111■■■••·••111 111■■■••··••11 ........... , ~ ,,11111111.~ -~•········•·\;,~ 1 11 ~",,..........,,~~ ~,,..........,,,, 111■■••···••11 ~l11■llllllllll111~1 11■■••····••11 1111••········••11 .,,• • • • • • • • • • • •,,j r&\\\l■■■•••■■■■r,,~ 8,••••••••••,,~~ 11•■■■•■■■■IH 111■■■■■■■■111 111■■■■■■■■111 111■■■••··••11 111■■■■■■■■111 are. only of :philosophice.1 tnteree.t . Pointa (tf high ~rtre-ss am indicated,, however~ and -a gereral feeling for the behavi0r o:f darl'lS undel: shrinkage. s-tresse:a ea.n be obtained from the results. Tho aeel:l:raey of the method is :remarkably good , espeeial.l:y as the ioomb:mne boundaries were in t his ease rather 01."tlde . It can be said Results shown hold only in the dam pr.eper• not 1n the :foundation material.. It an 1D$:1ght is desire@ i nto th.ts stress 0011oiti-on ,, a rou,~ pietu:re may be bad by supe,r imposing a ooii'orm horizontal. compression on di tio.n is given on page .Z •O f' this paper. . v1here the photo ...elasti-e method. of stress determination Ls used ...