A Comparison of Elastic-Plastic and Variable Modulus-Cracking Constitutive Models for Prestressed...

8/13/2019 A Comparison of Elastic-Plastic and Variable Modulus-Cracking Constitutive Models for Prestressed Concrete React

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LA-UR -78-2873

TITLE: A COMPARI SONCONSTI TUTI VEOF ELASTIC-PLASTICANDMODELS FOR PRESTRESSEDVARI ABLECONCRETE MODULUS-CRACKINREACTORVESSELS

AUTHOR S : Charl es A. AndersonPaul D. Smth

SUBMITTEDTO : To be presented at the 2nd J apan-U. S. Semnar onHTGR Safety Technol ogy on November 24-25, 1978 i n.Fuj i , J apan.

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A COMPARI SONOF ELASTIC-PLASTICAND VARI ABLEMODULUS-CRACKINGCONSTI TUTI VEMODELS FOR PRESTRESSEDCONCRETE REACTOR VESSELSbvCharl es P AndersonPaul . . SmthLos Al amos Sci enti f i c LaboratoryLos Al amos, NewMexi co USA

I NTROI ?UCTIONNumeri cal predi cti ons of the behavi or of prestressed concretereactor vessel s (PCRVS) under stati c, dynamc, and l ong- terml oadi ngsare compl i cated by the currentl y i l l - def i ned behavi or of concreteunder stress together w th the three di mensional nature of a PCRV.

Al though much research has recentl y been supported to carry out boththeoreti cal and experi mental i nvesti gati ons of concrete behavi or understress, i t i s cl ear that the questi on of whi ch of the many consti tu-ti ve model s most closel y approxi mates the behavi or of concrete i n aPCRV under l oad has not yet been sett l ed. Furthermore, the l argenumber of equati ons that resul t f romaccurate model i ng of the threedi mensional behavi or of a PCRV, w th even the simpl est consti tuti vel aw can tax the capabi l i ti es of the most up- to-date computi ng systemThe mai n purpose of thi s paper i s to compare the characteri sti cscf two consti tuti ve model s that have been proposed for concrete and tocompare the behavi or of representati ve concrete structures whosemateri al obeys these consti tuti ve l aws. The fi rst concrete mod+l i s avari abl e modul us-cracki ngmodel th t w ~sdescri bed i n the f i rstJ AEB/ USNRC Semnar on HTGR Safety. Thi s model treats nonl i nearcompressi ve concrete behavi or by a vari abl e modul us approach, i taccounts for crack formati on and th~ devel opment of stress-dependentorthotropy, and i t represents the el asti c-pl asti c behavi or of anyreinforcement present i n the concrete. The second model i s anel asti c-pl asti c mode12 whose devel opment cl osel y fol l ows the mathe-mati cal theory of pl asti ci ty w th an arbi trary strai n hardeni ng yi el dsurface. Thi s model takes i nto account the i nfl uence of bi axi alstress on concrete yi el di ng that has been observed experi mental l y.Concrete cracki ng i s not treated expl i ci tl y but i s model ed by thecontracted nature of the yi el d surface whenever at l east one Of thepri nci pal stresses i s i n tensi on. Both of these concrete model s havebeen i ncorporated i n a f i ni te el ement code, cal l ed NONSAP-C, that hasbeen devel oped at the Los Al amos Scienti f c Laboratory for the three-di mensi onal structural anal ysi s of PCRVS.4

Compari sons of the predi cti ons of the NONSAP-C code usi ng theel asti c-pl asti c concrete model and the vari abl e modul us-cracki ngconcrete model and the resul ts of experi ments, where avai l abl e, aregi ven bel ow The concrete structures that are model ed i ncl ude athi ck-wal l ed ri ng under i nternal pressure, a rei nforced concrete pl ateunder transverse pressure and an unrei nforcedPCRV end sl ab modelunder transverse pressure. Thi s l atter structure was one of an exten-si ve seri es of PCRVmodel s that were tested to destructi on at thStructural Engi neeri ng Laboratory of the Uni versi ty of I l l i noi s.8


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PCRV STRUCTURAL RESPONSE TO INi ERNAL PRESSUREExtensi vemodel tests4~5~6 have demonstrated the structural re-sponse of PCRV S to i nternal pressure l oadi ng. General l y, the PCRV heads

act as thi ck pl ates and showsmal l def l ecti ons at i ncreasingpressures.For unrei nforced heads, the fai l ure mode has been observed to be ei ther ashear or fl exure fai l ure w th sudden and catastrophi c rupture of thehead. 4 For the PCRV barrel , the l oad-detecti onbehavi or i s character-i zed by a l i near el asti c response up to a certai n pressure, cal l ed thereference pressure PR, where the prestress i s overcome and cracki ng ofthe concrete starts. Thi s i s fol l uwed by a range of sl i ghtl y nonl i nearbehavi or characteri zed by progressi ve cracki ng and transfer of l oad tothe rei nforcement and the tendons. Fi nal l y, at a pressure Py, a l i mi ti ng regi on of pl asti c behavi or begi ns i n whi ch the concrete no l ongerparti ci pates i n the structural response and the l oad i s carri ed by thePCRV l i ner, the tendons, and the rei nforcement up unti l the ul ti matestrength of the PCRV at an ul ti mate pressure Pu i s reached. Thi sl i mt i ng regi on i s characteri zed by rel ati vel y l arge i ncrementaldef l ecti ons per i ncremental l oad. Fi gure 1 shows schemati cal l y a typi call oad defl ecti on curve for a PCRV l oaded by i nternal pressure. Thedi f ference between the ul ti mate pressure P and the reference (desi gn)Yressure PR i s the reserve pressure capabi i ty of the vessel .I deal l y a fi ni te el ement code shoul d be capabl e of predi cti ng al - ofthe above aspects of PCRV behavi or - compl i cated as they are by thecracki ng phenomena and the heterogenei ty provi ded by the rei nforcement.The esser~i i ali ngredi ents of such a predi cti ve capabi l i ty are model i ng ofthe rei nforcement and an adequate concrete model to represent i ts sti f f -ness, strength, and cracki ng characteri sti cs. Those features of the con-crete model , rei nforcement, and prestress that i nf l uence the three rangesof ?CRV behavi or are i ven i n Tabl e I .

(Degree of contr i buti on i s

i ndi catedbyS (smal l , M moderate), and L (l arge). A bl ank i ndi catesvery l i ttl e contri buti on,A

P ----------- ------PLASTIC BEHAVIOR OF TCNOONS FAILt UE

-------------RWRVEEkI ENSIVC CRAtKINO cArAlllLITvNONLINEAR RtSPONSt, CONCRETECRACKINIJ

R - ------ ------ .-INITIATION OF HINOR CRACKINO

I/ LINtAR RUPONWOWLIcfION

Fi g, 10 Typi cal PCRV response to i nternal pressure l oadi ng,


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TABLE ICONTRI BUTI NG FACTORS TO PCRV STRUCTURAL BEHAVIOR

PF6jEFty - Elasti c - Cracki ng . r.Li mti ngBehavi or Behavi or Behavi orModulus and L L sPoi sson s R. -Concr~i , ~Stren@hYoncrete Compress M L~aF MCornressi ve Stren@h+ - oncrete oppressi ve L M MStress-Strai n Behavi orConcrete Tens~ -r- T . . -Stress-Strai n Behavior . . - -Rei nforcement s M. - L . . -Prestress L L_

CONCRETE CONSTI TUTI VE MODELSThe vari abl e modul us-cracki ng concrete model was presented at thefi rst J AEB/ USNRCSemnar on HTGR Safety and w l l not be di scussed i ndetai l here. As menti oned previ ousl y, i t treats concrete compressi vebshavl or by a vari abl e modul us approach, i t accounts for crack forma-

t. on,and i t handl es the rei nforcement present i n the concrete by asmeared representati on.The second model i s the el asti c-pl asti c consti tuti ve rel ati on forconcrete undar general three-di men i onal stress states that has been3proposed recentl y by Chen and Chen i n whi ch the concrete i s assumedto be acunti nuous, i sotropi c, and el asti c-pl asti c materi al w th astrai n-hardeni ngyi el d surf ace. In thi s theory, an i ni ti al di sconti -nui ty surf ace, subsequent l oadi ng surfaces, and a fai l ure surface forconcrete are def i ned, and el astl c-pl asti c stress- strain i ncrementalrel ati onshi ps are deri ved usi rqjthe classi cal theory of pl asti ci ty.Fi gure 2 i l l ustrates the fai l ure and i ni ti al di sconti nui ty surfaces i npri nci pal stress space for thi s type of materi al . The fa i l uresurfacei s presumed to be dependent on the fi rst stress i nvari ant, 11(proporti onal to the pressure), and the second i nt ari antof the devi a-tori c stress tensor, J 2, of the quadrati c form

1

where A and are materi al constants that can e determned fromthe)concrete tensi l e and compressive strengths. K i s a geometri cparameter and when I C2= 3 a good f i t to experi mental bi axi al concrete


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cr2A ,s 2x

FAILURESU~ /0I NI TI ALOtSCONl l NUOUSSURFACE

I NTERSECTI ONLINECOMPRESSI ONZONE

TENSION-COMPRESSI ON 1

Fi g. 2. Fai l ure and i ni ti al di sconti nuous surfaces i n tri axi alpri nci pal stress space.fai l ure data i s obtai ned. Constl tuti ve rel ati ons for i ncrementalstresses i n terms of i ncremental strai ns are deri ved i n Ref. 2 basedon the normal i ty rul e of pl asti ci ty as appl i ed to the i ni ti aldi sconti nui ty and subsequent l oadi ng surf aces. The three-di mensi onalformgf these rel ati ons has been i ncorporated i n the NONSAP-Ccode.~ The i nput parameters for thi s concrete model are the el asti cpropert i es of the concrete, the ul ti mate compressi ve strai n, thetensi l e, compressi ve, and bi axi al compressi ve yi el d stresses, and thetensi l e, compressi ve, and bi axi al compressi ve ul ti mate strengths.

TEST PROBLEMSOver the past year, a seri es of test probl ems has been anal yzedw th the vari bl e modul us-cracki ng and el asti c-pl astfc model s of the9NONSA -C code and the vari abl e modul us-cracki ngmodel of the ADI NAcode. Di rect compari son of the two model s I s gi ven onl y for thef i rst test probl empresentedbel ow

THI CK-WALLEDCONCRETE RINFi gure 3 i l l ustrates afi nl te el ement mesh of a thi ck-wal l edunrei nforced ci rcul ar ri t] gwhose response to i nternal pressure l oadi ngwas cal cul ated usi ng the NONSAP-C code w th the materi al model s di s-cussed previ ousl y. Fi ve 12-node I soparametr l c el ements were used todescri be the ri ng geometry.


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/= 5

Fi g. 3. Concrete ri ng f i ni te el ement model .Concrete propert i es descri bed i n Tabl e II were used. Based on theformati on of a radi al tensi l e crack i n the el asti c stress fi el d, theul ti mate pressure that the ri ng can w thstand i s 1. 2 MPa (178 psi ),whi ch i s predi cted by the vari abl e modul us theory. For the el asti c-pl asti c concrete model , a pl asti c zone propagates fromthe i nner

el ement through the ri ng unti l the pl asti c zone permeates the ri ng atan i nternal pressure of about 1. 12 ti Pa(160 psi ). From1. 12 MPa up to1. 4 MPa the concrete work hardens correspondi ng to stress statesbetween the i ni ti al di sconti nui ty and fai l ure surfaces. At 1. 4 MPathe stress state throughout the ri ng i s on the fai l ure surface andrupture occurs. Cal cul ated response of the ri ng to the i nternalpressure l oadi ng i s shown i n Fi g. 4, and Fi g. 5 shows the devel opmentof the crack pattern of the vari abl e modul us concrete model and thegrowth of the i nel asti c zones for the el asti c- pl asti c model .TABLE IICONCRETE PROPERTI ES USED IN TEST PROBLEMS

Modul us of El asti ci ty (E) 26000 MPa (3. 8x106psi )Poi sson s Rati o (v) 0.20Compressi ve Stren th (f c)? 46.OMPa (6800 psi )Tensi l e Strength f t) 3. 1 MPa (450 psi )Ul ti mate Strai n (~u) 0.003Bi axi al Compressi veStrength (f bc) 1. 16 f c


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I NNERSURFACELOADVSDEFLECTI ON

P/ PMx1

o

0

0

VARI ABLEMODULUS.

/ELASTI C-PLASTIC.

OF 1 1 1 t -I I 0 0125 0 0250 0 0375 Ot05CI0 0 0625INNER WALL DEFLECTION (MM)

Fi g. 4. Response of the concrete ri ng to i nternal pressure l oadi ng,

PROPAGATI ONFNONLI NEARENAVICIR VS INTERNAL PRESSUREWHwHnvIia. 5 LASTIC-pLWICFi g. 5. Propagati on of nonl i near behavi or for vari abl emodul us-cracki ng and el asti c-pl as. l c concrete model s.


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RECTANGULAR CONCRFTE PLATEA rei nforced concrete slab, simpl y supported on three si des, freeon the fourth, and l oadedw th a uni formtransverse pressure wasanal yzed w th NONSAP-C usi ng the vari abl e modul us cracki ng concretemodel . The f i ni te el ement mesh, consi sti ng of twenty-f our 16-nodei soparametri c el ements, was one el ement thi ck. Equal amounts (0. 5of tensi on and compressi on rei nforcement l ocated as shown i n Fi g. 6were i ncl uded. Usi ng a steel tensi l e strength of 300 MPa and a con-crete compressi ve strength of 30 MP- a l i mt l oad of 28. 3 kPa waspredi cted usi ng yi el d l i ne theory. Fi gure 7 shows t. :~eoad-def l ecti oncurve cal cul ated by NONSAP-C w th the f i ni te el ement model ; a l i mt orul ti mate l oad of 30 kPa i s i ndi cated, whi ch i s i n good agreement w ththe predi cti on of the yi el d l i ne theory.

rl OMF SIMPL E SUPPORT10 M SIMPL E SUPPORT

1SIMPL E SUPPORT

FREE

PLANVI EW

im 1 MT

COMPRESSION SIDE

0 5M -.. .1ss ~ TEhSION SIDE

041 Mf

.

SECTI ONVIEUFi g. 6. Concrete pl ate show ng rei nforcement.


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NONSAP-CCALCLJ LATI O: J

STRUCTURAL ENGINEERING LIMIT LOAD

DEFLECTION INNM

Fi g. 7. Response of the concrete pl ate.ANALYSI S OF PV-26 WTH THE ADI NA CODE

PV-26 i s an axi symmetr i c concrete pressure vessel that was testedto destructi on at the S ructural Engi neeri ng Laboratory of theUni versi ty of I l l i noi s.i Rupture occurred pri mari l y as a shearfai l ure of the head. The vessel was anal yzed usi g the vari abl e9odul us-cracki ng concrete model i n the ADI NA code w th concreteproperti es of Tabl e II The head of PV-26 was unrei nforced.Fi gure 8 shows the two-di mensi onal f i ni te el ement mesh used tomodel PV-26. Seventy-ei ght 8-node i soparametr i c el ements were empl oyed i n the cal cul ati on; prestressi ng and boundary condi ti ons forthe vessel aye as shwn. An i nternal pressure l oadi ng was appl i edi ncremental l y w th the onset of cracki ng of the outer surface of thevessel head occurri ng at sl i ghtl y under 6. 1 MPa (900 psi ). Cracki ngconti nues w th substanti al stress readj ustment up to an i nternalpressure of 11. 5 MPa (1690 psi ) where di vergence of the equi l i bri umi terati on method i ndi cated that head fai l ure had occurred. Si nce thel oad step was 27 psi , a strength of 11. 3 MPa (1663 psi ) was attri butedto the vessel . Fi gure 9 i l l ustrates the l oad-defl ecti onbehavi or ofthe center of the head of PV-26 as cal cul ated by 4DI NA and Fi g. 10shows the devel opi ng cracked regi ons i n the head of PV-26. Dottedregi ons i ndi cate the presence of a si ngl e crack whereas l i ned regi ons


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1300PSI PRESTRESSED

Fi g. 8. PV-26 fi ni te el ement model .i ndi cate the presence of a doubl e or tri pl e crack. The ul ti matepressure sustai ned i n the experi ment was 17. 8MPa (2610 psi ); theul ti mate pressure predi cted by ADI NA was 64 percent of theexperi rmal l y observed val ue.

CONCLUSIONSConclusi ons devel oped f romthi s study are as fol l ows:1. The vari abl e modul us- cracki ngmodel i s capabl e of predi cti ngthe behavi or of rei nforced concrete structures (such as the rei nforcedpl ate under transverse pressure descri bed previ ousl y) wel l i nto therange of nonl i near behavi or i ncl udi ng the predi cti on of the ul ti matel oad.2. For unrei nforced thi ck wal l ed concrete vessel s under i nternalpressure the use of el asti c-pl asti c concrete model s i n f i ni te el ementcodes enhances the apparent ducti l i ty of the vessel s i n contrast tovari abl e modul us-cracki ngmodel s that predi ct nearl y i nstantaneousrupture whenever the tensi l e strength at the i nner wal l i s exceeded.3. For unrei nforced thi ck-wal l ed end sl abs representati ve Uf PCRVheads, the behavi or predi cted by f i ni te el ement codes using vari abl emodul us-cracki ngmodel s i s much sti f fer i n the nonl i near range thanthan that observed experi mental l y. Al thoti ghthe shear type fai l uresand crack patterns that are observed experi mental l y are predi cted bysuch concrete model s, the ul ti mate l oad carryi ng capaci ty and vessel -ducti l i ty are signi f i cantl y underestimated. I t appears that suchmodel s do not adequatel y model such features as aggregate i nterl ockthat coul d l ead to an enhanced vessel reserve strength and ducti l i ty.


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. .

/ -RUpTLSRE

/

13

a/

@ I NI IAL cRACKING

/1 0, 25 0, 50 0, 75

CENTRAL HEAD DEFLECTION IN MM

Fi g. 9. Experi mental and cal cul ated l oad-def l ecti oncurves for PV-26.

L-J=....,......................p= 16n psi

................................:.:.:.:.:.:,:;.:.:.~. . . . . . . . . . . . . . . . . . . . . . . . . .,.


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REFERENCES1.

2.

3.

4.

5.

6.

7.

P, D. SMTHand C. A. ANDERSON, Consti tuti ve Model s for Col reteand Fi ni te El ement Anal ysi s of Prestressed Concrete Reactor~Proceedi ngs of the Japan-UI S. Semnar on HTGR Safety~ogy-Sei smc Research, Vol . I . Brookhaven Nati onalLaboratroy report BNL-NUREG-50689, pp. 282-297, September 1977.A. C. T. CHEN and W F. CHEN, Consti tuti ve Rel ati ons for Concrete,J ournal of the Engi neeri ng ?@chani cs Di vi si on, ASCI L,Vol . 019 PP*465-481, August 1975.P. D. SMTHand C. A. ANDERSON, NONSAP-C: A Nonl i near StressAnal ysi s Programfor Concrete Containment Under Stati c, Di ni mc,Long-TermLoadi ngs, Los Al amos Sci enti f i c Laboratory report7496-MS,November 1978.J D. REI NS, J . L. QUI ROS, W C. SCHNOBF?ICH,and M A. SOZEN,Shear Strengths of End Sl abs or Prestressed Concrete ReactorVessel s, Uni versi ty of I l l i noi s report UI LU-ENG-76-2022, J ul y 1976.FORT ST. VRAI N FI NAL SAFETY ANALYSI S REPORT, Publ i shec by Publ i cServi ce Co. of Col orado, Appendi x E, PCRV Design Data.T. TAKEDA, T. YAMAGUCHI , and K. I MOTO, I nel asti c Anal ysi s of a .Mul ti cavi ty PCRV under I nternal Pressure, 3rd I nternati onalConference on Structural Mechani cs i n Reactor Technol ogy, paperH2/ 5, 1975.K. J . BATHE, ADI NA; A Fi ni te El ement Programfor Automati c DynamcI ncremental Nonl i near Anal ysl~ssachusetts I nsti tute ofTechnol ogy Report 82448-1, September 1975.

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