A PLASTIC DAMAGE MODEL WITH STRESS TRIAXIALITY-DEPENDENT HARDENING

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1 8th Intrnational Confrnc on Structural Mchanic in Ractor Tchnology (SMiRT 8) Bing, China, Augut 7-, 5 SMiRT8-B- A PLASTIC DAMAGE MODEL WITH STRESS TRIAXIALITY-DEPENDENT HARDENING Xinpu SHEN Collg of Architctural Enginring, Shnyang Univrity of Tchnology Fax: xinpuhn@vip.ina.com Guoxiao SHEN Collg of Architctural Enginring, Shnyang Univrity of Tchnology Fax: guoxiaohn_ut@ina.com Lin ZHOU Collg of Architctural Enginring, Shnyang Univrity of Tchnology Fax: linzhou_ut@ina.com ABSTRACT Empha of thi tudy wr placd on th modlling of platic damag bhaviour of prtrd tructural concrt, with pcial attntion bing paid to th tr-triaxiality dpndnt platic hardning law and th corrponding damag volution law. A dfinition of tr triaxiality wa propod and introducd in th modl prntd hr. Druckr-Pragr -typ platicity wa adoptd in th formulation of th platic damag contitutiv quation. Numrical validation wr prformd for th propod platicity-bad damag modl with a drivr ubroutin dvlopd in thi tudy. Th prdictd tr-train bhaviour m raonably accurat for th uniaxial tnion and uniaxial comprion compard with th xprimntal data rportd in rfrnc. Numrical calculation of comprion undr variou hydrotatic tr confinmnt wr carrid out in ordr to validat th tr triaxiality dpndnt proprti of th modl. Kyword: Damag; platicity; contitutiv modl.. INTRODUCTION Th inlatic failur of concrt-lik matrial and tructur i charactrizd by th initiation and volution of crack and th frictional liding on th clod crack urfac. Platic damag modl ar th major maur to dal with cracking-rlatd failur analyi, and ar widly ud by variou rarchr,.g. Lmaitr (99); Chaboch (99); Swryn and Mroz (998); d Bort, Pamin, Gr (999). Owing to it implicity and a raonabl capacity of problm rprntation, th iotropic damag modl i th mot popular damag modl in th imulation of th failur phnomna of concrt tructur and i thrfor th choic of thi tudy. Anothr important raon for th choic of an iotropic damag modl in thi tudy i du to that th aim of thi invtigation i to analy th cracking proc of concrt matrial of a prtrd concrt tructur. Conquntly th modlling of pr-pak nonlinarity of tr-train curv undr comprion and th tr-triaxiality dpndnt platic hardning law and th corrponding damag volution law ar th mot important concrn of thi tudy. Copyright 5 by SMiRT8

2 On of th platicity-bad iotropic damag modl for concrt i th o-calld Barclona modl which i rportd by Lublinr, Olivr, Ollr, Onat (989), and ha rcntly bn adoptd by L and Fnv (998) and Nchnch, Mftah and Rynouard (). In thi modl, a holonomic rlationhip btwn damag and quivalnt platic train ha bn prntd, and two damag variabl hav bn dignd for tnil damag and compriv damag rpctivly. It hould b notd that mot of th xiting platic damag modl hav not mphaizd on th tr-triaxiality dpndnt platic hardning law. Th framwork of th contitutiv modl in thi tudy i contructd on th bai of th platicity-bad damag modl rportd by Saanouni, Fortr and Bn Hatira (994). Although thi modl wa propod for th imulation of th platic damag phnomna of mtal, it i till an attractiv modl for th purpo of thi tudy bcau of it important advantag ovr th othr availabl damag modl: firtly, in thi modl, th damag volution i not only cloly connctd to th incra of platic train, but it i alo influncd xplicitly by th latic train; condly, th damag volution i coupld with an incra of platic train; finally, thi modl i rlativly ay to modify to mak it uitabl for th imulation of platic damag phnomna of concrt-lik matrial. Th gnralizd Druckr-Pragr critrion introducd in Lib and Willam () for platic loading, togthr with it platic potntial for non-aociatd platic flow rul, i rfrrd to hr. Th contxt of th articl i organid in th following ordr: in ction, th quation of th contitutiv modl ar givn for lato-platicity coupld with damag in gnral; a dfinition of tr triaxiality i propod and latr introducd in th platic hardning and damag volution law. A drivr ubroutin for validation of th contitutiv modl i dvlopd with rfrnc to th principl propod by Hahah, Wotring, Yao, L and Fu (). In ction, rult of numrical tt of th propod modl ar givn for om typical loading ca. Som concluion ar givn in ction 4.. FORMULATION OF THE PROPOSED MODEL. Fundamntal Equation of th Platicity-bad Damag Modl With th Enrgy Equivalnc Principl, th fundamntal rlationhip of th platicity-bad damag modl propod by Saanouni, Fortr and Bn Hatira (994) ar litd in Eqn.() a: σ E σ =, =, E = D I I = σ J = = ii,, σ δ Y = E () p σ = E ( ) p Q F & = & λ, D& = & λ Y whr σ and ar total tr and train tnor rpctivly, uprcript p tand for platic and rprnt latic quantiti, ovrhad tilt rprnt th quantiti for fictitiou nt matrial, i th dviatoric tr tnor, & λ i th inlatic multiplir, D rprnt th iotropic damag variabl, Y i th damag conjugat forc, E i th laticity tnor of th intact matrial, δ i th nd ordr unit tnor, I i th um of th ffctiv principal tr, J i th cond invariant of th dviatoric ffctiv tr tnor, F i a platic damag potntial function, Q i th platic part of th potntial in th ffctiv tr pac. It i n from th abov quation that th damag volution i dignd to b accompanid by platic train incra, and it quantity of incrmnt i alo dpndnt on latic train tnor via damag conjugat forc. Copyright 5 by SMiRT8

3 . Spcification for Druckr-Pragr Typ Platicity Coupld With Damag With rfrnc to th gnralizd Druckr - Pragr critrion introducd in Mnétry and Willam (995), togthr with th hardning modl introducd in Bon (), th platic damag loading condition i primarily dfind in th ffctiv tr pac in th following form (tr triaxiality will b introducd latr): / bλ f = α F I + J [ k + k )] = () whr k i initial har trngth contant, k i th train hardning limit of th fictitiou nt matrial, which corrpond to infinit quivalnt platic train, i.. λ, and α F i a matrial contant dignd for prur-nitivity proprti, paramtr b i a modl contant which can b dtrmind by fitting xprimntal phnomna. Th platic part of th potntial, i.. Q, i givn in th ffctiv tr pac a: / b Q = α [ ( )] λ Q I + J k + k () whr α i th dilatancy contant for non-aociatd flow rul if α α. Q Q F Th following form of platic damag potntial function F i adoptd in ordr to hav non-aociat platic flow in th ffctiv tr pac: + S Y φ F = Q + (4) + S whr, S, φ ar matrial paramtr, D i damag variabl, Y i damag conjugat forc. It i obrvd that th xprimntal rult of tr-train curv of concrt-lik matrial ar highly dpndnt on th tr triaxiality. In fact, phnomna of tr triaxiality dpndncy of th tr-train curv xit in nginring for a wid rang of matrial uch a gomtrial, cramic, compoit and mtal. In ordr to imulat thi kind of phnomna, th tr triaxiality wa ud by vral rfrnc ( Alv and Jon, 999; Hortmyr, Lathrop, Gokhal and Digh, ; Borvik, Hopprtad and Brtad, ; Li, Zhang and Anari, ). Th form of xprion of tr triaxiality ar diffrnt from on to anothr: Alv and Jon (999), Borvik t al () and Hortmyr t al () dfin thir tr triaxiality xplicitly, whil Li, Zhang and Anari () implicitly account for tr triaxiality influnc in it platic hardning law by uing th tr invariant I and J. Hr, for th convninc of modl formulation, togthr with a rfrnc to th convntional xprion adoptd in vral rfrnc (Sfr, Carol, Gttu and Et ; Et and Willam 999), th tr triaxiality i dfind a: I γ =, J (5) J Th tr triaxiality i introducd into th damag platic loading condition and th damag platic potntial function in th following form: / bγλ f = α I + J k + γk = (6) [ ( )] F + Sγ Y F = Q + Sγ + ( φ (7) whr / b Q = α [ γ ( )] QI + J k + k (8) Conquntly platic train incrmnt i obtaind a: p F Q & = & λ = & λ (9) with Q = α δ + It i obtaind with Eqn.(5) that: k ( ) Q D J γ γ bγλ + γbλ bγλ γ () Copyright 5 by SMiRT8

4 γ η δ I, if I < = ( ) ( J ), η = (), 6J 6 if I Th damag volution law can b drivd a: Y D& & φ = λ ( & = λy Sγ () with E Y = φ () Sγ Th paramtr ud in thi modl ar: E, ν, k, α, α for platicity, and, S, φ for damag. Str triaxiality γ i a pcial variabl introducd in thi modl.. Contitutiv Bhaviour for a Finit Diplacmnt Incrmnt F Q With abov contitutiv modl, th contitutiv bhaviour can b drivd for a known initial tr tat p ( σ,, and a givn train incrmnt. Th tr incrmnt can b obtaind by making total diffrntial opration ovr total tr tnor in abov Eqn.() and a ubqunt linarization ovr th tim incrmnt t, thu p p ( ( ) E ( ) σ = E Q = λ E With th abov quation, th following quation i obtaind: σ = σ + σ ( + E Y + E D p = E D ) ( ) + E D ) Q p D ) + E D )( ) (5) λ E Y Th platic damag multiplir λ can b dtrmind xplicitly with th following conitncy condition: f = f = (6) Conquntly it i obtaind th following quation: f = σ + D + λ D λ Q = E λ E + E Y Y (7) D λ = Thu, th following xprion for platic multiplir can b drivd: E λ = Q E ( E ( (8) + Y Y D λ For th ak of brvity, Eqn.(8) can b r-writtn in anothr form a: A λ = (9) B with (4) Copyright 5 by SMiRT8

5 A = E () Q B = E + E Y Y D λ For a givn train incrmnt, th tr tnor incrmnt can b obtaind by ubtituting Eqn. () into Eqn. (4) uch that: A Q σ = Er ( + Er ry + E () B r Thrfor th algorithmic tangntial tiffn tnor can b dducd a σ Q p A E = = E Er + Er ry () r B Th latoplatic damag loading condition for a givn train incrmnt can b xprd concptually in th ffctiv tr pac a f = f + λ () ( λ) whr f i th valu of yilding function at th tarting ffctiv tr tat σ. With Eqn. (), th following rlationhip i obtaind: ( αi + J ) D λ = + + (4) ( λ) ( λ) ( λ) λ ( λ) With Eqn. () and (), th tnor and vctor on th right hand id of Eqn.(4) ar obtaind a ( ) γ γ bγλ bγλ γ = αδ + k + γbλ (5) D J σ σ σ Q Y φ = E + E ( ) ( ) D (6) λ Sγ ) Y = D E (7) f bγλ = bγ k λ (8) D ( ) ( λ) Y = Sγ φ ( D ) (9) Th formulation of Nwton-Raphon itration quation btwn λ and f i formd a: λ = λ f () λ whr f i th valu of yilding function at th tarting ffctiv tr tat σ.. NUMERICAL VALIDATION AT LOCAL LEVEL Hr a drivr ubroutin i dignd for th purpo of validation of -dimnional contitutiv modl at local lvl, i.. for a matrial point only, with rfrnc to th algorithm propod in Hahah, Wotring, Yao, L, and Fu (). It principl can b illutratd with Figur a: a mixd loading condition i applid with = () t, and σ = σ = cont, which man that a train loading will b applid incrmntally undr a contant tr confinmnt in th othr two dirction. Th train loading i applid latically in dirction, whil th lf-quilibrium mchanim at thi matrial point will rult in variation of latral train nonlinarly 4 Copyright 5 by SMiRT8

6 (for th ak of damag) in ordr to kp th latral confinmnt contant. Th dtail of th numrical calculation will b givn in th following contxt. σ σ σ σ Figur. Illutration of th mixd loading condition.. Itration Procdur for th Modl Validation: Extrnal Equilibrium Itration. Th function of th xtrnal quilibrium itration i: for a known tr and train tat and a t of intrnal p variabl ( σ,,,, apply a load incrmnt (, σ, σ ), to find out quai-latically th rpon of ( σ,,, by an itrativ procdur. Th following quai-latic quation (i.. latic rlationhip for a finit tim incrmnt t) ar adoptd in th calculation: E E σ E = () E E σ E whr E ar componnt of th laticity tnor of damagd matrial which i xprd in Eqn.(). Th principl of th xtrnal quilibrium itration i illutratd in Figur. In Figur, UMAT i th contitutiv modul which chck latoplatic loading tat and mak latic and/or lato-platic damag calculation. Subroutin CONSTITUERE will b calld in th UMAT. Th function of ubroutin CONSTITUERE i to carry on th contitutiv intgration and will b introducd in dtail in th following ub-ction. Th latoplatic-damag loading tiffn, which alo known a algorithmic tangntial tiffn, will b updatd aftr vry itration, and will b ud in th quai-latic calculation of and at vry firt itration tp at ach of th loading incrmnt. pd E 5 Copyright 5 by SMiRT8

7 σ, iload=, Load incrmnt loop: iload=,n iload=iload+ Input, σ, σ a loading data; Calculat n σ = σ, σ = σ, kitr= iload = iload = n Calculat ; kitr=kitr+, ; with th givn Itration loop; If kitr=mitr Stop (divrgnc) UMAT Loading loop; If iload=nload, top (Fin) p σ, p σ p r p = σ σ = σ σ,, σ = σ σ η = r r ( σ ) + ( σ ) k + k + r k + =, σ = σ, σ = r Fal If >Tolr Tru σ r p Figur. Flow chart of global quilibrium itration.. Itration Procdur for th Modl Validation: Intrnal Elato-platic Damag Itration By uing th fixd-point mthod dcribd in Chaboch and Cailltaud (996), for a givn finit train, th only unknown in th lato-platic damag calculation at local lvl i th platic-damag multiplir λ. Th olution tp adoptd in th procdur of CONSTITUERE ubroutin ar: p Stp : Initiat th tr tat and tat of all th intrnal variabl: σ,,, D ; Stp : Apply train incrmnt, with = for i j obtaind from th outr global quilibrium itration; Stp : Calculat λ with givn initial tr tat, train incrmnt and linarizd Eqn.(8); Stp 4: With thi λ obtaind in tp, calculat conquntly th following quantiti: 6 Copyright 5 by SMiRT8

8 D = D () = () = Q + λ p () + = () D ) E + λy, with Y = D Sγ σ = E () ) φ Stp 5: With Eqn.(4) and (), calculat itrativly th platic-damag multiplir with th following quation: ( ) λ = f λ λ = λ + λ ( ) Stp 6: Chck convrgnc: if ( ) Tolranc λ, ca th itration and continu to th nxt load incrmnt; Othrwi, mak λ = λ Rturn to tp to carry on th nxt itrativ calculation up to th maximum itration limit.. Numrical Exampl In thi ub-ction, numrical validation of th contitutiv modl at local lvl ar carrid out with th drivr ubroutin dvlopd hr for kind of typical loading ca, i.., () uniaxial tnion; () uniaxial comprion; and () uniaxial comprion undr variou hydrotatic tr confinmnt. With rfrnc to th xiting litratur ( L and Fnv, 998; Ghavamian and Carol, ; Et and Willam, 999), th following valu of matrial paramtr ar adoptd in th calculation. Thy ar: E=4MPa, ν=., α F =α Q =.5, k=.mpa, =, S= - for tnion and 4-5 for comprion, φ=-., b=88 for tnion and 5 for comprion, k =MPa for tnion and 48MPa for comprion. Tolranc= -5 for intrnal itration (i.. for ( λ) ) and -4 for xtrnal itration (i.. for contant latral tr confinmnt)... Uniaxial Tnion Th tr-tain bhaviour undr uniaxial tnion of th modl i hown in Figur. No platic hardning bhaviour i obrvd in thi ca. Comparion btwn th xprimntal data (Gopalaratnam, Shah, 985) and th numrical rult of tr-train rpon undr uniaxial tnion indicat that th prpak tr-train bhaviour can b prdicatd vry wll, whil th potpak bhaviour can only b prdictd with a raonabl accuracy by th propod modl. Figur 4 how th numrical rult of th rpon of th latral train and volumtric train. Th damag rpon in Figur 5 how that damag valu aymptotically tnd to maximum. with th incra of train loading. 7 Copyright 5 by SMiRT8

9 4.5 Str_(MPa) Exprimntal Numrical Strain_ Figur. Str-train bhaviour undr uniaxial tnion: comparion btwn numrical and xprimntal rult. 4.5 tr_(mpa) σ v σ σ train_ train_ Figur 4. Str-train bhaviour undr uniaxial tnion: latral and volumtric rpon..8 Damag train_(%) Figur 5. Damag volution undr uniaxial tnion. 8 Copyright 5 by SMiRT8

10 .. Uniaxial Comprion Th tr-tain bhaviour and damag volution rpon of th propod modl undr uniaxial comprion ar hown in th following Figur. Comparion btwn th xprimntal data givn by Karan, Jira (969) and th numrical rult of tr-train rpon undr uniaxial comprion in Figur 6 indicat that th prpak bhaviour can b prdicatd vry wll, and th potpak bhaviour can b prdictd with a raonabl accuracy. Th numrical rult of th rpon of th latral train and volumtric train hown in th am Figur 6 indicat th dilatancy proprty of th modl: thr i a aturation of dilatancy at which th volumtric train ca to incra. Th corrponding damag volution bhaviour i prntd in Figur. tr_(mpa) σ σ : numrical : xprimantal train_ train_ σ Figur 6. Str-train bhaviour undr uniaxial comprion... Comprion with Confinmnt Th tr-train bhaviour of a modl for concrt undr hydro-tatic tr confinmnt i an important apct: it indicat th prur-nitivity bhaviour of th modl. In th numrical tt prformd hr, th procdur of loading i givn a: th hydro-tatic confinmnt, i.. σ I m, i applid bfor train loading in dirction bing carrid on. Figur 7 how th variation of th tr-train rpon caud by th confinmnt of th tr-tain bhaviour and damag volution rpon, with th othr paramtr wr kpt unchangd: with th incrmnt of th tr confinmnt, th oftning phnomna bcom wakr and wakr. It m that th tr-triaxiality dpndnt platic hardning phnomna ar proprly imulatd by th modl propod hr. In Figur 8 and 9, th tr-train curv undr confinmnt of -MPa and -MPa ar givn rpctivly, togthr with th rpon of σ and v σ. Th dilatancy phnomna bcom wakr with th incra of confinmnt prur. Th prpak nonlinarity of th tr-train curv m raonably imulatd. In ordr to illutrat th propod modl furthr in-dpth, in Figur and, th pak trngth nvlop obtaind numrically with th propod modl ar givn out. Bcau of th diffrnt paramtr valu adoptd in th calculation for tnion and comprion, it i n in Figur that th hap of tnil trngth nvlop i quit diffrnt from th hap of th compriv trngth nvlop. Th axial point in Figur ar obtaind analytically by uing th Drackr-Pragr condition dirctly, bcau thr i no prpak nonlinarity for tnil ca, which i judgd by critrion I. In Figur, th nvlop of pak-trngth of comprion undr vry high confinmnt up to -MPa i hown in ordr to how th validity of th modl for a wid rang of hydrotatic tr confinmnt. 9 Copyright 5 by SMiRT8

11 tr_(mpa) MPa - MPa MPa - train_ Figur 7 Influnc of hydrotatic tr confinmnt: tr-train bhaviour ( σ =, -, - MPa). m tr_(mpa) σ v σ train_ train_ σ Figur 8. Str-train bhaviour undr comprion with -MPa confinmnt tr_(mpa) σ v σ σ train_, train_ Figur 9. Str-train bhaviour undr comprion with -MPa confinmnt. Copyright 5 by SMiRT8

12 Damag - MPa - MPa MPa train_(%) Figur. Damag-train bhaviour undr comprion with variou tr confinmnt. a) qrtj_ (MP I_/qr I t (MPa) J (MPa) Figur. Pak trngth nvlop in th I - J pac (with confinmnt up to -MPa). a) qrtj_ (MP J 7 (MPa) I_/qr I t (MPa) Figur. Pak trngth nvlop in th I - J pac (with confinmnt up to -MPa). Copyright 5 by SMiRT8

13 4. CONCLUSIONS In thi articl, a platicity-bad iotropic damag modl ha bn propod with pcial concrn bing givn to th modlling of th tr-triaxiality dpndnt platic hardning law and corrponding damag volution. Numrical validation hav bn mad for th propod modl with a drivr ubroutin dvlopd in thi tudy. Th numrical rult of tr-train bhaviour m raonably accurat for th uniaxial tnion and uniaxial comprion ca compard with th xprimntal data of th rfrnc. Th phnomna of tr-triaxiality- dpndnt platic hardning hav bn imulatd. Owing to th highly nonlinarity xiting among th modl paramtr and th xprimntal phnomna, it m ncary to choo valu of contitutiv paramtr by om kind of tchniqu of invr analyi. ACKNOWLEDGEMENTS Thank ar du to th NNSF of China for th financial upport via contract no. 477, and to th Dpartmnt of Education, Liaoning Provinc, China, for th financial upport via contract no. 5C5. REFERENCES: Alv, M., N. Jon, (999). Influnc of hydrotatic tr on failur of axiymmtric notchd pcimn. J. Mch. Phy. Solid, 47, Bon, J., (). Mcaniqu t Ingniri d Matriaux - Eai mcaniqu, Eprouvtt axiymtriqu ntaill, Hrm (in Frnch). Borvik, T., O.S. Hopprtad, T. Brtad, (). On th influnc of tr triaxiality and train rat on th bhaviour of a tructural tll. Part II. Numrical tudy. Eur. J. Mch. A/Solid,, 5-. Chaboch, J.L, (99). Damag inducd aniotropy: on th difficulti aociatd with th activ/paiv unilatrial condition. Int. J. Dama. Mch.,, Chaboch, J.L., G. Cailltaud, (996). Intgration mthod for complx platic contitutiv quation. Comp. Mthod. Appl. Mch. Engng.,, d Bort, J. Pamin, M.G.D. Gr, (979). On coupld gradint-dpndnt platicity and damag thori with a viw to localization analyi. Eur. J. Mch. A/Solid, 999, 8, Dragon, A., Z. Mroz, (979). A continuum modl for platic-brittl bhaviour of rock and concrt. Int. J. Mch. Sci., 7, -7. Et, G. and K. Willam, (999). Failur analyi of latovicoplatic matrial modl. ASCE J. Engng. Mch., 5(), Ghavamian, S., I. Carol, (). Bnchmarking of concrt cracking contitutiv law: MECA projct. In: Computational Modlling of Concrt Structur. N. Bicanic, R. d Bort, H. Mang, G. Mchk (Ed.), Swt & Zitingr, Li, Gopalaratnam, V.S., S.P. Shah, (985). Softning rpon of plain concrt in dirct tnion. ACI J., 8(), -. Hahah, Y.M.A., D.C. Wotring, J.I.C. Yao, J.S. L, Q. Fu, (). Viual framwork for dvlopmnt and u of contitutiv modl. Int. J. Numr. Anal. Mth. Gomch., 6, Hortmyr, M.F., J. Lathrop, A.M. Gokhal, M. Digh, (). Modling tr tat dpndnt damag volution in a cat Al-Si-Mg aluminium alloy. Tho. Appl. Frac. Mch.,, -47. Karan, I.D., J.O. Jira, (969). Bhaviour of concrt undr compriv loading. ASCE J. Engng Mch, 95(), L J., G.L. Fnv, (998). Platic-damag modl for cyclic loading of concrt tructur. ASCE J. Engng Mch., 4(8): Lmaitr, J, (99). A Cour on Damag Mchanic, nd d., Brlin: Springr. Li, Q., L. Zhang and F. Anari, (). Damag contitutiv for high trngth concrt in triaxial cyclic comprion. Int. J. Solid Struct., 9(5), Copyright 5 by SMiRT8

14 Lib, T., K. Willam, (). Localization propriti of gnralizd Druckr-Pragr latoplaticity. ASCE J. Engng Mch., 7(6): Lublinr, J., J. Olivr, S. Ollr, E. Onat, (989). A platic damag modl for concrt. Int. J. Solid & Struct., 5(), Nchnch, W., F. Mftah and J. M. Rynouard, (). An lato-platic damag modl for plain concrt ubjctd to high tmpratur. Engng. Strut., 4(5), Saanouni, K., C. Fortr, F.B. Hatira, (994). On th anlatic flow with damag. Int. J. Dama. Mch.,, Swryn, A., Mroz, Z., (998). On th critrion of damag volution for variabl multiaxial tr tat. Int. J. Solid & Struct., 5(4): Sfr, D., I. Carol, M., R. Gttu; and G. Et, (). Study of th bhavior of concrt undr triaxial comprion. ASCE J. Engng Mch, 8(), Copyright 5 by SMiRT8