FINITE ELEMENT ANALYSIS OF CONSOLIDATION PROBLEM IN SEVERAL TYPES OF COHESIVE SOILS USING THE BOUNDING SURFACE MODEL

Transcription

1 ARPN Journl of Enginring nd Alid Sins Asin Rsrh Publishing Ntwork (ARPN). All rights rsrvd. FINITE ELEMENT ANALYSIS OF CONSOLIDATION PROBLEM IN SEVERAL TYPES OF COHESIVE SOILS USING THE BOUNDING SURFACE MODEL Qssun S. Mohmmd Shfiqu Drtmnt of Civil Enginring, Nhrin Univrsity, Irq E-Mil: ABSTRACT Finit lmnt nlyss of onsolidtion roblm in svrl tys of sturtd ohsiv soils wr rformd using th lstolsti bounding surf modl. In this r, th modl nd th finit lmnt formultion wr dsribd nd xmls of modl rdition nd ury of th finit lmnt formultion wr givn. Th trnsint rsons of th sturtd orous mdi is bsd on Biot s thory of onsolidtion. Trnsint nlysis of two-dimnsionl onsolidtion roblm involving flxibl stri footing on ly lyr of finit thiknss is thn rrid out whih dmonstrt th ffts of onsolidtion ross nd modl rmtrs on th or rssur rsons nd ground movmnts undr th stri footing. Kywords: modl, finit lmnt, onsolidtion, bounding surf, stri footing. INTRODUCTION Th dformtion nd or-wtr rssur rsonss of lyy soils r of grt intrst to ivil nginrs. Th stbility of foundtions nd rthworks in sturtd fin-grind soils is tim-dndnt ross. This is bus ny hng in totl norml strss is initilly rsistd by or rssurs, whih thn dissits ovr riod of tim. In gnrl, it is diffiult to urtly rdit or bk-omut th or-wtr rssur rsonss in lys in filds, silly in som ss in whih, ftr th omltion of strutur onstrution, th or-wtr rssur in th foundtion soils ontinuously inrsd for rtin riod of tim (Kbbj l l., 1988). Consolidtion of soils hs bn n imortnt subjt studid for mor thn 5 dds. Th thory of D onsolidtion ws first formultd by Biot (1941). In sil ss, suh s stri, xisymmtri, or squr footings with uniform lod intnsity rsting on linr lsti orous mtril, nlyti solutions hv bn found (Shiffmn t l., 1969). Howvr, if th mtril is onsidrd to b nonlinr lsti or lsti lsti or if th boundris r omlitd, numril mthods must b mloyd to find th solutions (Chng nd Dunn, 198). Th most widly usd mthod my b th finit-lmnt (FE) mthod. So fr lrg numbr of studis using th FE mthod with ithr linr lsti, nonlinr lsti, or lstolsti modls hv bn rformd for onsolidtion of soils. Th FE mthod is rltivly mturd mthod. Th onsolidtion nlysis using lstolsti modls hs bn n tiv rsrh r in rnt yrs (Adhi t l., 1996 nd Tibt nd Crtr, 001). In this r, th onsolidtion bhvior of soft soil undr stri footing is nlyzd using th FE mthod with bounding surf modl whih is on of th most sohistitd soil onstitutiv rltions to modl th bhvior of soil. Th modl rdits mongst othrs th strss-strin rltions nd or rssur hngs whn th soil is subjtd to xtrnl lods s it hs rominnt ftur tht inlsti dformtions n our for strss oints within th surf. In th following stions, th roosd modl nd th finit lmnt formultion r dsribd nd xmls of modl rdition nd ury of th finit lmnt formultion r givn. Th onsolidtion bhvior of soft soil undr flxibl stri footing in diffrnt ohsiv soils is thn studid using th modl in ordr to show th influn of th onsolidtion ross nd modl rmtrs on th bhvior. THE ELASTOPLASTIC BOUNDING SURFACE MODEL Dtils of th lstolsti formultion, th numril imlmnttion of th modl nd th rmtrs ssoitd with th modl r vilbl lswhr (Dflis nd Hrrmnn, 1986 nd Hrrmnn t l., 1987). Thrfor, only th lstolsti rt rltions r givn hr. Th totl strin rt is onsisting of two rts: lsti strin nd lsti strin: ε & = ε& + ε& (1) Th invrs form of th onstitutiv rltions is obtind s: σ & = Dklε& kl () D kl = G KF δ +, I ( δ δ + δ δ ) + K ( / ) ki G J kj ii ( G) δδ kl G F, α sins F δ +,J osα bj J nj S s 4 J δ h L G G F, α sinsnj S s δ KFδ + δ + F (), I,J B 4 J osα bj J J Whr 8

2 ARPN Journl of Enginring nd Alid Sins Asin Rsrh Publishing Ntwork (ARPN). All rights rsrvd. 1 G L = KF ε& kk + F s ε&, I,J B J G F α + os α bj s iks J S s 4 J, kj & kk ( ) + G( F ) + G( F / bj),i,j ε ε& (4) B = K + 9K F (4b) nd whr K nd G rrsnt th lsti bulk nd shr moduli, rstivly, δ is th Kronkr dlt, K th lsti modulus, I nd J r th strss invrints, 1 b nd F rrsnts th nlytil xrssion of th bounding surf. Rquird modl rmtrs Th rmtrs in this tgory r dtrmind from rsults of stndrd lbortory tsts of short nough durtion to nsur tht visolsti ffts r ngligibl. Th mtril rmtrs usd to ort th lstolsti bounding surf modl r (Klikin, 005): λ = slo of onsolidtion lin, κ = slo of swlling lin, N(α) = slo of ritil stt lin, N = N in omrssion, N = N in xtnsion, υ= Poisson s rtio R(α) = R > 1dfins th oint I = 1 Io / R (Figur-1), whih togthr with oint J dfin th oordints of oint H whih is 1, α th intrstion of F = 0 nd CSL, omrssion, R = R in xtnsion, R = R in A(α) = rmtr dfins th distn D = AIo of x H of th hyrbol from its ntr G intrstion of th two symtots nd thus rtins only to th omosit form of th surf, A = A in omrssion, A = A in xtnsion, T = I t / Io rmtr whih dtrmins th urly tnsil strngth of th mtril, nd T is lso rtins to th omosit form of th surf, C = 0 C < 1 rmtr whih dtrmins th ntr of th bounding surf I = CI o. s rmtr whih dtrmins indirtly lsti = nulus. For s =1th lsti nulus dgnrts to oint surf nd s s ntr of bounding th lsti nulus xnd towrds th bounding surf. h = slo-hrdning ftor, whih is funtion of lod ngl (α), = for omrssion ( h h( π / 6) ) ( h( π / 6) ) h I h h =, = for xtnsion =. Figur-1. Th bounding surf in th strss invrints (ftr Dflis nd Hrrmnn, 1986). FINITE ELEMENT FORMULATION Th lstolsti bounding surf modl dsribd bov is inorortd in finit lmnt rogrm, whih hs th ftur of modling twodimnsionl (ln strin nd xisymmtri) gothnil roblms suh s onsolidtion, writtn by FORTRAN90 lngug. This rogrm is rimrily bsd on th rogrms rsntd by Smith nd Griffiths (004) for th nlysis of on nd two-dimnsionl solid by finit lmnt mthod utilizing lsti onstitutiv rltionshi nd whih is modifid for th uros of this study. So in ddition to th lstolsti bounding surf modl, th rogrm llows on to ssign linr lsti bhvior to ny rt of th roblm gomtry. Dsrition of ll of th 8

3 ARPN Journl of Enginring nd Alid Sins Asin Rsrh Publishing Ntwork (ARPN). All rights rsrvd. rogrm fturs is byond th so of this r, nd brif summry of th ftur rlvnt to this study is givn blow. Trnsint formultion In th s of stri footing on sturtd orous mdium, th loding is tim-dndnt, so n inrmntl formultion ws usd in th following work roduing th mtrix vrsion of th Biot qution t th lmnt lvl rsntd blow (Lwis nd Shrflr, 1987). K T L L u K L u df/dt + C = T + S+ αh t L S ( 1 α) H t F k k (5) whr: K = lmnt solid stiffnss mtrix, L = lmnt ouling mtrix, H = lmnt fluid stiffnss mtrix, u = hng in nodl dislmnts, = hng in nodl xss or-rssurs, S = th omrssibility mtrix, F = lod vtor, t = lultion tim st, α = tim sting rmtr ( = 1 in this work), df / dt = hng in nodl fors. VERIFICATION PROBLEMS Consolidtd undrind trixil omrssion for normlly onsolidtd ly This roblm hs bn drwn from Hrrmnn t l., (1981), s rortd by Dflis nd Hrrmnn (1986) nd Klikin nd Dflis (1989). A lbortory rrd Kolin ly ws tstd nd th msurd dt r tkn from th lttr uthors, whrs th omosit bounding surf modl rmtrs r tkn from th formr. Th vlus of rmtrs r listd in Tbl-1. Th modl bhvior ginst th xrimntl dt is illustrtd in Figurs nd b.th rsults of th bov roblm suort th vrifition ross of th usd rogrm, nd indit tht th modl sussfully rdits rsults for soils undr omrssion lodings. Tbl-1. Bounding surf rmtrs vlu. Prmtrs Vlu Prmtrs Vlu λ 0.15 A κ C 0.7 υ 0. s 1.0 M 1.5 h 50.0 M h R.5 T -0.1 R b A 0.0 w b Mtril rsons in xtnsion ws not simultd b A bounding surf onsisting of two lliss Dvitori strss, q (kp) undrind Undrind trixil trixil tst Exrimntl Prsnt study CSL rogrm Exrimntl Mn fftiv strss, (kp) Figur-. Shr strss-strin urv for undrind trixil omrssion of normlly onsolidtd ly. Por wtr rssur (kp) undrind trixil Axil strin (%) Figur-b. Por rssur-xil strin urv for undrind trixil omrssion of normlly onsolidtd ly. Elstolsti nlysis of two-dimnsionl onsolidtion roblm Figur- shows th finit lmnt msh usd; th width of th lodd r, b, is ssumd qul to (.05m). Th roblm is solvd using th inut mtril rmtrs shown in Tbl-. 1.m b=.05m 1 o = kp Dring 5 0 Prmbl 18.m Figur-. Finit lmnt msh for th two-dimnsionl onsolidtion roblm. Th lssil mtril rmtrs, whih hv bn usd by Siriwrdn nd Dsi (1981), r tkn s th sm s rortd by thm. Th othr rmtrs r tkn s

4 ARPN Journl of Enginring nd Alid Sins Asin Rsrh Publishing Ntwork (ARPN). All rights rsrvd. tyil vlus in th mor limitd rngs for rtil litions (Klikin, 005). Th lod lid t 5dys or tim ftor of T v = Tbl-. Bounding surf rmtrs (ftr Klikin nd Dflis, 1991 nd Siriwdn nd Dsi, 1981). Prmtrs Vlu Prmtrs Vlu λ 0.14 A 0.08 κ 0.05 C 0.4 υ 0.4 s 1.0 M 1.05 h 4.0 M 0.89 h 4.0 R.7 h o 4.0 R.18 m 0.0 A 0.1 v k h 5 k = m / dy Figur-4 shows tim wis vrition of surf sttlmnts, using th modifid Cm-ly nd bounding surf modls. It n b sn tht sttlmnt vlus obtind by th two modls do not diffr signifintly t th rly stg of tim lvls. Howvr, t ltr tims th bounding surf lstiity rsults show highr sttlmnts but smllr finl sttlmnt. Dislmnt 100/b Two-dimnsionl onsolidtion roblm Modifid Cm-ly modl (Siriwrdn nd Dsi, 1981) Bounding surf modl (Prsnt study) T = T = 0.14 T = Horizontl distn (x/b) Figur-4. Surf sttlmnts vrsus horizontl distn. TIME-DEPENDENT BEHAVIOR OF A SOFT SOIL UNDER A FLEXIBLE STRIP FOOTING Trnsint nlysis of two-dimnsionl onsolidtion roblm involving flxibl stri footing on ly lyr of finit thiknss is studid in this stion using th bounding surf modl. Th finit lmnt msh rrsnting th roblm is illustrtd in Figur-. This is th sm roblm rviously onsidrd to show th bility of th bounding surf modl to solv th onsolidtion roblms but with diffrnt rmtrs of th modl ording to th ty of ohsiv soil (Klikin v v v 6 nd Dflis, 1991). Th rmtrs r tbultd in Tbl-, whr th rmtrs S, nd w r fixd for ll th tys of soils s 1, 1. nd 5, rstivly. Th loding will b lid in 5dys or tim ftor of T v = Th rsults wr obtind t th nd of loding whn th tim ftor T v = 0.07 nd ftr 100 dys or tim ftor T v = 0.78.In th following stions, nlyss r rrid out in ordr to study th ffts of th onsolidtion ross nd th bounding surf modl rmtrs on th or rssur rsons nd ground movmnts undr th stri footing. In gnrl, ll tys showd th sm bhvior but with rltiv hngs. This my b du to th vrition of th rmtrs ording to th ty of ohsiv soil. Th dislmnts during onsolidtion undr th stri lod 47.9 kn / m r shown in Figurs 5 nd 7, xggrtd by ftor of 5 nd lottd with th originl finit lmnt rry. At ll tims th sttlmnts r bowlshd nd th initil dislmnts involv downwrd motion undr th lod nd gnrl horizontl dislmnt wy from th lodd r. Th uwrd motion t th surf just outsid th lodd r is inrsd somwht by th rigid ltrl boundris. During onsolidtion th mtril sttls furthr undr th lod nd movs horizontlly towrd th lod s th xss or rssurs dissit with tim nd s shown in Figurs-6 nd 8, whih drws th ontours of xss or rssur t th bginning nd during th onsolidtion ross. Figur-6 shows th xss or wtr rssur ontours t th nd of lying stri loding for th fiv ohsiv soils. It should b obsrvd tht ll of or rssurs r ositiv rflting th loding usd by stri footing nd th lrgst or wtr rssur lis dirtly blow th bottom of footing whr th loding is onntrtd. Ths xss or rssurs dissitd with tim s du to wtr flow wy from th loding nd s shown in Figur-8, whih drws th xss or rssur ontours for th fiv ohsiv soils nd t tim ftor T v = 0.78 using lrgr fftiv strsss nd thus inrsing th dislmnts. Also from Figurs 5 nd 7, in gnrl it ws obsrvd tht for ohsiv soils tht hv highr vlus of th modl rmtrs λ nd κ nd lowr vlus of M, ν,r,h,a nd C, highr surf sttlmnts wr rditd silly downwrd motion undr th lod nd gnrl horizontl dislmnt wy from th lodd r. Also, lowr dislmnts undr th stri lod wr rditd for Kolin Mix tht hs th lowst vlus of λ nd κ, nd th highst vlus of M, ν,r,h nd A with lrgr mount of th rojtion ntr C. 85

5 ARPN Journl of Enginring nd Alid Sins Asin Rsrh Publishing Ntwork (ARPN). All rights rsrvd. Soil ty Tbl-. Bounding surf rmtrs for th fiv ohsiv soils (ftr Klikin nd Dflis, 1991). Pror Kolin mix Kolin Mrin silty Grnobl Umd λ κ υ M M R R A A C h h Mtril rsons in xtnsion ws not simultd. Kolin Mix Kolin Mrin Grnobl Umd Figur-5. Dformd msh t th nd of loding in th fiv ohsiv soils. 86

6 ARPN Journl of Enginring nd Alid Sins Asin Rsrh Publishing Ntwork (ARPN). All rights rsrvd. Kolin Mix Kolin Mrin Exss or rssur(k) Grnobl Umd Figur-6. Contours of xss or rssur t th nd of loding in th fiv ohsiv soils. Kolin Mix Kolin Mrin Grnobl Umd Figur-7. Dformd msh t tim ftor ( =0.78) in th fiv ohsiv soils. Kolin Mix Kolin Mrin Exss or rssur(k) Grnobl Umd Figur-8. Contours of xss or rssur t tim ftor ( = 0.78) in th fiv ohsiv soils. 87

7 ARPN Journl of Enginring nd Alid Sins Asin Rsrh Publishing Ntwork (ARPN). All rights rsrvd. CONCLUSIONS A fully trnsint nlysis of onsolidtion roblms in sturtd orous mdi is rrid out. This ws to llow th trnsition btwn th stts of drind nd undrind bhvior to b invstigtd. An lgorithm for rrying out suh n nlysis hs bn rsntd. Th trnsint rsons of th sturtd orous mdi ws bsd on th thory of onsolidtion (Biot, 1941). Also it should b mhsizd tht th rsults rsntd hrin wr bsd on lstolsti bounding surf soil modl whih hs th ftur tht th inlsti dformtions our for strss oints within th surf. Th lstolsti bounding surf modl imlmnttion ws vrifid using xrimntl nd numril rsults. Thn th rsults of th lstolsti nlyss of onsolidtion roblm involving flxibl stri footing on svrl ohsiv soils r rsntd. Th following onlusions wr obsrvd: At ll tims th sttlmnts r bowl-shd nd th initil dislmnts involv downwrd motion undr th lod nd gnrl horizontl dislmnt wy from th lodd r. Th uwrd motion t th surf just outsid th lodd r is inrsd somwht by th rigid ltrl boundris; During onsolidtion th mtril sttls furthr undr th lod nd movs horizontlly towrd th lod s th xss or rssurs dissit with tim; For ohsiv soils tht hv highr vlus of th modl rmtrs λ nd κ nd lowr vlus of M, ν,r,h,a nd C, highr surf sttlmnts wr rditd silly downwrd motion undr th lod nd gnrl horizontl dislmnt wy from th lodd r; nd Lowr dislmnts undr th stri lod wr rditd for Kolin Mix tht hs th lowst vlus of λ nd κ, nd th highst vlus of M, ν,r,h nd A with lrgr mount of th rojtion ntr C. REFERENCES Adhi, T., Ok F. nd Mimur M Modling Asts Assoitd With Tim Dndnt Bhvior of Soils. Msuring nd modling tim dndnt soil bhvior, T. C. Shhn nd V. N. Klikin, ds., Nw York Biot M.A Gnrl Thory of Thr-Dimnsionl Consolidtion. Journl of Alid Physis. 1: Chng, C.S. nd Dunn J.M Consolidtion Anlysis for Prtly Sturtd Cly by Using An Elsti- Plsti Efftiv Strss-Strin Modl. Intrntionl Journl for Numril nd Anlytil Mthods in Gomhnis. 7: Soils. Journl of Enginring Mhnis. 11(1): Hrrmnn L. R., Shn C. K., Jfroudi S., DNtl J. S. nd Dflis Y. F A Vrifition Study for th Bounding Surf Plstiity Modl for Cohsiv Soils. A Rort. Drtmnt of Civil Enginring, Univrsity of Cliforni, Dvis, USA. Hrrmnn L. R., Klikin V. N., Shn C. K., Mish K. D. nd Zhu Z Numril Imlmnttion of Plstiity Modl for Cohsiv Soils. Journl of Enginring Mhnis. 11(4): Kbbj M., Tvns F. nd Lrouil S In Situ nd Lbortory Strss-Strin Rltionshi. G othniqu, London. 8: Klikin V. N Prmtr Estimtion For Tim- Dndnt Bounding Surf Modls. Go-Frontirs Confrn (005); Soil Constitutiv Modls: Evlution, Sltion, nd Clibrtion. Gothnil sil ublition, No.18, Klikin V. N. nd Dflis Y.F Simlifitions to th Bounding Surf Modl for Cohsiv Soils. Intrntionl Journl for Numril nd Anlytil Mthods in Gomhnis. 1: Klikin V. N. nd Dflis Y.F Dtils Rgrding th Elstolsti-Visolsti Bounding Surf Modl for Isotroi Cohsiv Soils. Civil Enginring Rort No.91-1, Univrsity of Dlwr, Nwrk. Lwis R. W. nd Shrflr B.A Th Finit Elmnt Mthod in th Dformtion nd Consolidtion of Porous Mdi. John Wily nd Sons Ltd., London. Shiffmn R. L., Chn T. F. nd Jordn J. C An Anlysis of Consolidtion Thory. Journl of Soil Mhnis nd Foundtion Division. 95(1): Siriwdn H. J. nd Dsi C.S Two Numril Shms for Nonlinr Consolidtion. Intrntionl Journl for Numril Mthods in Enginring. 17: Smith, I. M. nd Griffiths D.V Progrmming Finit Elmnt Mthod. 4 th Ed., John Wily nd Sons. Tibt H.T nd Crtr J.P A Smi-Anlytil Finit Elmnt Mthod for Thr-Dimnsionl Consolidtion Anlysis. Comutrs nd Gothnis. 8: Dflis Y. F. nd Hrrmnn L. R Bounding Surf Plstiity II: Alition to Isotroi Cohsiv 88