Evaluation of Finite Element Formulation for One-Dimensional Consolidation

Transcription

1 Inernionl Jornl of Scienific & Engineering Reserch,Vole 3, Isse 5, y- ISSN Elion of Finie Eleen Forlion for One-Diensionl Consolidion Wn NrFirds Wn ssn, ishohd Absrc Consolidion process is defined s he progress of hich excess pore er pressre genered by exernl loding dissiped o of he soil pores nd sbseqenly csing he soil o copress. The biliy o predic he dissipion of he excess por e er pressre is iporn in he ssessing he perfornce of fondions. In his pper, forlion of Finie Eleen (FE) ehod is deeloped for soling ncopled consolidion proble nd is lidiy is exined. The bsic rix eqions for ncopling nlysis sin g finie eleens re deried bsed on he Glerin s eighed residl ehod. Spil ribles re discreied sing he se shpe fncion nd flly iplici schee is sed o discreie he ie doin. Using he ATAB progr for riing he FE code, he resls re copred ih h obined by clssic Terghi s one diensionl consolidion heory. The FE nlysis shoed good greeen ih he closed for solion. Index Ters Consolidion, ncopled pproch,sred soil, Terghi s heory, Finie Eleen ehod (FE), iplici. INTRODUCTION S oil de p essenilly of soil pricle ih oids in beeen. The oid generlly occpied prly by er nd prly by ir. Sred soil is o phse eril consising of solid seleon nd er filled oids []. When n increen of sress is sddenly pplied o n eleen of sred soil, here is n insnneos increse in pore pressre nd excess pore pressre deelops. The process of consolidion nder one-diensionl s firs inesiged by Terghis in[] nd s exended o hree diensionl by Bios in [3]. Boh hors greed h he flo of pore er s goerned by Drcy s l nd he response of soil seleon s elsic. Finie eleen forlion bsed on one-diensionl ideliion hs been esblished nd he nericl chrcerisics he been exined. Desi [4] copred o lernies for solion of one-diensionl consolidion hich sed finer esh ih loer order pproxiion fncion nd sed corser esh ih higher order pproxiing fncion. They fond h, in ers of ccrcy, he liner odel shos beer greeen ih he closed for solion copred o he cbic odel. An exinion of copionl ie, he cbic odel s fond o be bo for ies longer hn for liner odel de o he greer degree of conneciiy hs incresing he bnd idh of he eqion se solion. The pplicion of he finie eleen ehod o he solion of consolidion eqion s considered by Sndh nd Wilson s in [5]. Since hen, nber offinie eleen forlionsfor he consolidion of elsic eril he ppered. Wn NrFirds: ser s reserch sden, Fcly of Ciil Engineering, UniersiiTenologi lysi (UT), lysi. E-il: firsh87@gil.co ishohd:ecrer,fcly of Ciil Engineering, UniersiiTenologi lysi (UT), lysi. E-il: hish@.y IJSER hp://.ijser.org These inclde he ors ofcrer s in [6], nd enende s in [7]. Solion echniqes for finie eleen nlysis of consolidion re slly bsed on firs order, iplici inegrion ehods. The bcrd Eler schee is idely sed in boh liner nd nonliner sdies. The sbiliy of consolidion nlyses hs receied considerble enion by inesigors nd ny inerpolion schees he been proposed for he ie doin. The sbiliy nd ccrcy of firs order inegrion schees hs been inesiged by Booer nd Sll s in []. They proed h he inegrion preer, θ s be less hn.5 in order for he solion schee o be ncondiionlly sble. A flly explici ehod is obined hen sing θ eql o ero. Vereer nd Verrji [8] inesiged he ie seps shold no be de oo sll o oid oscillion in he pore er pressre nd hey conclded h loer lii on he ie sep, bsed on he coefficien of consolidion nd he esh sie, cold redce he occrrence of oscillory resls. In ddiion, Abid nd Pyrh [9] fond h he explici schee is no recoended nd iplici schee fond o be sisfcory nd is recoended. ng nd Griffihs [] re-exined copled, ncopled nd he Terghi one-diensionl consolidion heories sing he finie eleen ehod. They conclded, for lyered soil syse, cobining he coefficien of perebiliy nd coefficien of copressibiliy ino single coefficien of consolidion ill gie rong excess pore er pressre disribion. Recenly, Osn [] fond o he ncopled consolidion nlysis in excellen greeen ih he copled nlysis nd he ncopled nlysis led o reliely sipler clclion procedre copred o copled nlysis. Oher resercher lso conclded h for he cse here he flo occrs in one direcion hen hydrlic condciiy is consn ih ie, he riion of pore pressre ih ie clcled fro ncopled nlysis becoes idenicl o h clcled sing Terghi s ncopled nlysis []. In his pper, forlion of one diensionl consolidion is considered. The finie eleen is eployed o chiee nericl solion, in conjncion ih finie difference

2 Inernionl Jornl of Scienific & Engineering Reserch,Vole 3, Isse 5, y- ISSN ie-sepping lgorih o define he rnsien nre of he proble. TEORY OF CONSOIDATION The heory, hich is expressed in heicl for, becoes he bsis for ll he sdies of consolidion s ell s oher sbjecs reled o he deforion of he flo of flids hrogh he poros edi. The folloing sspions re essenil for he generl deelopen of he consolidion heory s firs gien by Terghi:[3]. Soil in he consoliding lyer is hoogenos.. Soil is copleely sred (S=%). 3. Copressibiliy of eiher er or soil grins is negligible. 4. Srins re infiniesil. An eleen of diension dx, dy, nd hs he se response s one ih diensions x, y, nd. 5. Flo is one-diensionl. 6. Copression is one-diensionl. 7. Drcy s l is lid (=i). 8. Soil properies re consns. 9. The oid rio, e s. pressre, p response is liner. For one-diensionl flo (in he ericl direcion or ), he oleric flo is defined s h dv dxdy () d The eleen ole is dx dy nd he pore ole is (dx dy) [e / (+e)]. All ole chnges V re he pore ole chnges fro sspion 3, so he ie re of ole chnge s e dv dxdy () e d Since he (dx dy) / (+e) is he consn ole of solids, Eqion () cn be rerien s [(dx dy) / (+e)] ( e / ). Eqing his ino Eqion (): h e e (3) e Only pressre hed in excess of he hydrosic hed ill cse flo (nd ole chnge) nd since h = Δ/ Eqion (3) cn be rerien s e (4) e Fro he slope of he liner pr of n riheic plo of oid rio, e or srin, ε erss pressre, p, he coefficien of copressibiliy, nd he copressibiliy rio, cn be defined s e de p dp (5) d ' (6) p dp Wih negie sign ignored. Before ny pore pressre dissipes, dp = d, so de = eqion (4) o gie e d, hich cn hen be sbsied ino The brceed ers cn be defined s he coefficien of consolidion, c e or And he coefficien of ole copressibiliy, (nd inrodcing he iniil in si oid rio, for e) s e ' (8) (7) hs he nis of sress-srin odls (P or p). I is ofen referred o s he consrined odls nd is he odls of deforion, Es. Rerie eleen nlysis. c ( e) Rerie Eqion (7) s (9) in for sible for finie () () The solion of Eqion () is no riil nd ses of Tylor series expnsion. I is s follos: 4 i N sin N e N c () This eqion is generl nd pplies for ny cse of iniil hydrosic pressre, in hicness of soil, nd refers o dringe ph. Since he coefficien of consolidion, consn nd ie, is liple of, diensionless ie fcor, T cn be defined s ci T (3) The erge degree of consolidion s in he folloing eqion: U N c 8 e (4) N 3 FINITE EEENT FORUATION To derie he bsic rix eqions for consolidion, he sring poin is he differenil eqion. The goerning differenil eqion for consolidion process cn be rien s (5) is IJSER hp://.ijser.org

3 Inernionl Jornl of Scienific & Engineering Reserch,Vole 3, Isse 5, y- 3 ISSN The spil discreiion of he proble is chieed by sing liner pproxiion fncion s shon in Figre. The cly lyer is diided ino nifor eleens. The eleen hs o nodes ih degree of freedo. Since in his pper, he ncopled pproch is considered, he secondry rible (seleen) ill be clcled seprely once he priry rible (pore pressre) hs been obined s in [9]. The pore pressre is liner riion long he eleen nd cn be expressed s: N N (6) here nd denoe he nodl pore pressre ssocied ih he shpe fncion of he eleen; N, N (7) i i Eqion (9) becoes: Ni Ni N i () i Rerrnge he eqion: i K i K i i here: is he lengh of he eleen, Siffness rix, K ss rix, i, () () rix eqions of eleens he been obined for he cse of one-diensionl consolidion. Bsed on h, he progr is coded sing ATAB in his pper. Fig..Finie eleen discreiion of one-diensionl consolidion. When Eqion (6) is sbsied ino he differenil eqion of consolidion, he folloing eqion cn be obined. T NN (8) Folloing he sndrd Glerin s eighed residl pproch, solion of pril differenil eqion of consolidion cn be chieed by liply he hole Eqion () ih shpe fncion of pore pressre nd, hen inegre i oer he eleen. The need o pply inegrion by pr is iporn in order o decrese he order of he goerning differenil eqions. The residls re se ero, he folloing eqions cn be obined [4]. N N i (9) Generlly, finie difference solion o he ie inegrion hs been sed. Booer nd Sll [] sggesed h θ ½ cn eep he sbiliy of inegrion schee. The iplici ie inegrion (Eler bcrd) is sed nd θ is doped here. Ter deriies of ie y be rien in finie difference for s: 4 NUERICA EXAPE TABE ATERIA PROPERTIES The cse is cly lyer of hicness 7 sbjeced o ericl pressre of P nd inined consn ih ie. Dringe is lloed fro he op nd boo srfce. The cly lyer is diided ino nifor eleens. And he ie sep of =.5 dy s sed. The folloing eril properies re ssed: Figre (), (b), (c), nd (d), shon plo of he excess pore pressre ih deph fro he nericl solion nd fro Terghi s solion, for differen ies fer pplied of loding, i.e. dy, dys, 365 dys nd 6 dys. The resls sho ery good greeen beeen he nericl nlysis nd he exc solion. IJSER hp://.ijser.org

4 Inernionl Jornl of Scienific & Engineering Reserch,Vole 3, Isse 5, y- 4 ISSN Fig. ().Coprisons of excess pore pressre long deph ie, = dy. Fig. (d).coprisons of excess pore pressre long deph ie, =6 dys. Figre 3 is grph of he erge degree of consolidion, U erss he diensionless ie fcor, T. Siilr resls of degree of consolidion ere obined fro he nericl solion nd he exc solion of Terghi s heory. Fig. (b).coprisons of excess pore pressre long deph ie, = dys. Fig. 3.Aerge degree of consolidion, U erss ie fcor, T: coprison beeen Terghi s exc solion nd FE. Fig. (c).coprisons of excess pore pressre long deph ie, =365 dys. 5 CONCUSION In his sdy, he proble of he consolidion of n elsic sred soil s soled ih sccess sing finie eleen ehod. A forlion consisen ih he Terghi s onediensionl heory hs been deeloped nd i hs been resriced o he cse of soils ih liner elsic. The nericl lgorihs ipleened in his pper re efficien de o he resl hich shos good greeen ih he Terghi s exc solion. Frher or is crrenly ndergoing in order o exend his heory o he nlysis of soils ih non-elsic behior nd copled consolidion pproch. IJSER hp://.ijser.org

5 Inernionl Jornl of Scienific & Engineering Reserch,Vole 3, Isse 5, y- 5 ISSN REFERENCES [] J.R. Booer nd J.C. Sll, Finie Eleen Anlysis of Priry nd Secondry Consolidion, Inernionl Jornl Solids Srcres, ol.3, pp.37-49, 977. [] K. Terghi, Die Berechnng der Drchlssigeisiffer des Tones sdeverlf der hydrodynischenspnnngsersceinngen, Originlly pblished in 93 nd reprined in Fro Theory o Prcice in Soilechnics, John Wiley nd Sons, Ne Yor, 33-46, 96. [3].A. Bio, Generl Theory of Three-Diensionl Consolidion, Jornl of Applied Physics, ol., pp.55-64, Feb 94. [4] C.S. Desi nd.d. Johnson, Elion of To Finie Eleen Forlions for One-Diensionl Consolidion, Copers nd Srcres, ol., pp , 97. [5] R.S. Sndh nd E.. Wilson, Finie Eleen Anlysis of Flo in Sred Poros Elsic edi, Jornl of Engineering echnics Diision, ASCE, ol.95, isse 3, pp.64-65, 969. [6] J.P. Crer, J.C. Sll, nd J.R Booer, A Theory of Finie Elsic Consolidion, Inernionl Jornl Solids Srcres, ol.3,pp , 977. [7] C. enende, P.J.G. Nieo, F.A. Oreg, nd A. Bello, heicl odelling nd Sdy of he Consolidion of n Elsic Sred Soil ih n Incopressible Flid by FE, heicl nd Coper odelling, ol.49, pp.-8, 9. [8] P.A. Vereer nd A. Verriji, An Accrcy Condiion for Consolidion by Finie Eleens, Inernionl Jornl for Nericl nd Anlyicl ehods in Geoechnics, ol.5, pp.-4, 98. [9] Abid,.. nd I.C. Pyrh, Gidelines for Using he Finie Eleen ehod o Predic One-Diensionl [] J. ng nd D.V. Griffihs, One-Diensionl Consolidion Theories for yered Soil nd Copled nd Uncopled Solions by he Finie Eleen ehod, Geoechniqe, 9, doi:.68/geo.8.p.38. [] A.S. Osn, Coprison beeen Copled nd Uncopled Consolidion Anlysis of Rigid Sphere in Poros Elsic Infinie Spce, Jornl of Engineering echnics, ol. 36, No. 8, pp.59-64, Ags, doi:.6/_asce_e [] G.C. Sills, Soe Condiions nder Bio s Eqion of Consolidion Redced o Terghi s Eqion, Geoechniqe, ol.5, isse, pp.9-3, 975. [3] S. elny, Applied Soil echnics, John Wiley & Sons, Inc., oboen, Ne Jersey, pp.3-35, 7. [4] J.C. i, W.B. Zho, nd J.. Zi, Reserch on Seepge Force Inflence on One-Diensionl Consolidion, Unsred soil, Seepge, nd Enironenl Geoechnics (GSP 48), pp.3-9, ASCE 6. IJSER hp://.ijser.org