Limit Analysis of Stability of Circular Foundation Pit

Transcription

1 Limi Anlyi of Sbiliy of Ciul Foundion Pi, Wu Simei Sool of Civil Engineeing Sndong Univeiy Sool of Civil Engineeing, Sou Cmpu of Sndong Univeiy, Jingi Rod 73#, Jinn, 561 Cin p:// Ab: - Ciul foundion pi ofen ppe in ivil engineeing. In ode o obin e iil dep of e non-uppoed iul foundion pi, e uppe-bound meod in pliiy meni w employed. Te umed lip ufe in nlyi w e oionl logpil ufe. Te kinemilly dmiible veloiy field w obined oding o e oied flow ule fo Coulomb meil, nd e opimizion model of e iil dep w eblied nd olved wi SQP opimizion lgoim. Te viion of e iil dep wi e lope ngle, e io of dep o diu of pi nd e inenl fiion ngle of oil wee udied. Te effe of e iul foundion pi mke e iil dep lge n e iil eig of e plne lope; oweve, wen e io of dep o diu of pi ppoe zeo, e uppe-bound oluion of e fome ppoe of e le. If e io of dep o dimee of pi i le n 1, e effe my be ignoed nd e foundion pi n be nlyzed e plne lope wi e meod of lie. Compion beween uppe-bound oluion(ubs), e oluion fom ppoxime lip line eoy(sls) nd finie diffeene oluion(fds) owed UBS i le n SLS nd lge ligly n FDS. Key-Wod: - iul foundion pi; lope; iil dep; limi nlyi; uppe-bound meod; effe 1 Inoduion Limi nlyi eoy i n impon bn of pli meni. I w developed fom e mel pli eoy nd ledy been exended o ok nd oil meni now. Limi nlyi w ued o olve ome engineeing poblem, u lope biliy nd limi lod[1-8]. I onin wo kind of bi meod, i.e. e uppe-bound meod nd e lowe-bound meod. Bed on uppe-bound eoem, e uppe-bound meod need o ebli e kinemilly dmiible filue menim nd veloiy field in dvne. Te veloiy field mu mee moion boundy ondiion nd oied pli flow ule. Bed on lowe-bound eoem, e lowe-bound meod need o e up illy dmiible e field wi mu ify equilibium equion, e boundy ondiion nd no diobey e filue ieion wi i Mo- Coulomb filue ieion fo ok nd oil. Limi nlyi n give e definie bound of ome poblem u e lope iil eig nd e pile being piy[1-8]. Howeve, e oluion fom igid limi equilibium meod, wi i noe nlyi meod ued exenively in geomeni, i diffiul o ell i i n uppebound oluion o lowe-bound oluion. Fo Some filue menim ued in limi equilibium meod, e oeponding kinemilly dmiible veloiy field fo limi nlyi n be obined oding o viul wok piniple. So e limi equilibium oluion fom ee filue menim e uppe-bound oluion, ju e Sm meod wi i n impon meod olving e fey fo of lope. Bu fo Some filue menim ued fequenly in limi equilibium meod, e oeponding kinemilly dmiible veloiy field n be e up, o e limi equilibium oluion en uppebound oluion, ju e veil lie meod nd e iul lide meod ued fo e biliy nlyi of lope. Fo veil lie meod, inelie foe n ify Mo-Coulomb filue ieion. Fo Coulomb meil obeying oied flow ule, e ngle beween veloiy jump veo nd e ngen of e lip ufe ould be equl o e inenl fiion ngle of meil, bu fo e iul igid lide menim, we n e up ny veloiy field ifying i. ISS: Iue 1, olume, Deembe 7

2 In ddiion, e limi equilibium oluion n be ued e lowe-bound oluion beue e e field in igid body i no known. Te uppe-bound meod i pplied moe exenively n e lowe-bound meod beue e eblimen of illy dmiible e field i e diffiul. Wile olving e poblem wi e uppe-bound meod, vlid filue menim i umed fily, nd en e inenl enegy diipion e nd e wok done by exenl lod e luled epeively nd equed wi e oe. Tu eil of uppe-bound oluion oeponding o e peifi menim e obined, nd finlly, e opimum uppe-bound oluion n be goen by employing opimizion enology[8]. Donld I nd Cen Z Y udied e biliie of pln in nd ee-dimenion lope uing igid blok nlionl filue menim[1-]. Muff obined e lel being piy of pile wi ee-dimenionl defoming menim[3]. Howeve, limi nlyi n be ued o olve e xiymeil poblem ye. Axiymeil poblem ppe fequenly in geonil engineeing u e biliie of iul foundion pi nd mo in-iu e u CPT nd SPT. Te im of i ppe i o olve e non-uppoed iil dep of e iul foundion pi o demone e ppliion of uppe-bound meod o xiymeil poblem. Te iul foundion pi i ofen nlyzed e plne in poblem wen e io of dep o diu of pi i lge enoug. Wen e io of dep o diu of pi i mll, oweve, e plne in oluion i no ue beue e effe of oil m lgely enne e biliy of e pi. In i ppe, limi nlyi meod will be employed o olve e iil dep of e iul foundion pi. Bed on e wok of Верезанцев В Г[9], limi equilibium oluion w lo obined nd omped wi uppe-bound oluion. In nlyi, oil i umed omogeneou nd ioopi, nd e filue i xiymeil. liding m nd long e lip ufe i le n e wok e done by e exenl lod. Ω σ * ij ε ij dω + σ Γε ΓdΓ W + T (1) Γ wee σ ij nd σ Γ denoe e of ee wiin e liding m egion Ω nd on poenil lip ufe Γ in pli limi e. ε ij, ε Γ nd denoe e of kinemilly dmiible in e wiin e liding m, on e poenil lip ufe Γ nd e veloiy field of e m, epeively. W nd T e e weig of liding m nd e exenl foe ing on e liding m, epeively. Fig.1 Sliding m of lope. Filue Menim A kind of lope, e ommon filue ufe of iul foundion pi i logpil ufe, ju own in fig.. Te oing ene of filue fe i on e xi of foundion pi. Te funion of logpil ufe i: e ( ) n φ Wee φ i inenl fiion ngle of oil, pol diu of poin b wee. () i Poblem Fomulion.1 Uppe-bound Teoem Wen lope liding m bove poenil lip ufe i in pli limi e (Fig.1), nd wen ny kinemilly dmiible veloiy field i inodued o e liding m, bed on uppebound eoem, e enegy diipion wiin e Fig. Filue menim of iul foundion pi ISS: Iue 1, olume, Deembe 7

3 Te pe of foundion pi n be deibed wi e io of dep o boom diu D / R nd lope ngle α of foundion pi, own in fig.. One ey e known, e dimenion of e wole filue zone n be deemined by wo independen vible: e eii ngle nd e dep of foundion pi D. In Fig., L i e leng of b ; β i defined pli ngle i e ngle beween lip ufe nd pi wll lope oe. Le A D / nd B L /. Ten equion (3)-(5) n be deived fom Fig.1: (3) (4) (5) (6) ( ) n φ A in e in B o o e β ( ) n φ A φ + α π / Te funion of e line in in Te funion of e line b i: i: oα in + nα (o B) in + nα o (7) (8) Te pol ngle of e poin o B o in Aoding o geomeil elionip, n be obined fom e following equion: (9) D R in e ( ) nφ ( ) nφ in e i: o in, nd ψ e dil, ngenil nd iumfeenil veloiy omponen in peil oodine yem (,, ψ ), epeively. Aoding o e oied pli flow ule, ψ on e logpil filue ufe. Hee we ume fue ψ in e wole defoming egion. n be deemined oding o e oied pli flow ule. In peil oodine, eno- in-e e wien : (1) ε 1 ε o εψ 1 γ ( γ ψ γ ψ ) Beue γ ψ γ ψ, ε ψ i pinipl in e. Aoding o e nomliy ofε, ε nd ε ψ, we know ε nd ε e lo pinipl in e. Subequenly we n know: (11) γ 1 ( ) Fo Coulomb meil ming oied flow ule, Cen W F gve e following equion[8]: (1) T ε + ε φ wee T n ( π 4 ), ε nd ε e e ummion of pinipl enionl in e nd e ummion of pinipl ompeive in e, epeively. oe enionl in e e poiive in i ppe. If φ, e following equion n be obined fom Eq.(1): ε + ε (13) `.3 eloiy Field ISS: Iue 1, olume, Deembe 7

4 Eq.(13) efle e inompeibiliy of Te meil nd i uied fo undined e of ued oil m. Beue ε, we know fom equion (1) ee i mximum in e ε mx nd minimum in eε min in ε ndε ψ. In ode o obin veloiy field, we udied e following wo e., ε ε min (1) ε ψ ε mx (1) (13) Fom Eq.(1) nd Eq.(1), we obin: o + T Fom Eq.(11) nd Eq.(1), we obin: T o + Te pil diffeenil equion bove n be olved uing e vible epion meod. Le f ( ) g( ), nd ubiue i ino Eq.(13) nd obin: (14) C 1 f ( ) g ( ) n C f ( ) g( ) T wee i onn. Solving e wo odiny diffeenil equion in Eq.(14), f () nd g ( ) n be obined. Te finl expeion of i: (15) ( ) ( 1 1 ) C in TC f g C wee C i onn. Subiuing Eq.(15) ino Eq.(11) o Eq.(1) led o C 1 1, u Eq.(15) n be wien (16) C in T Beue ε ψ ε mx >, we n know C > fe ubiuing Eq.(16) ino Eq.(1). A e me 1 π ime, < <. So we n know >. Ti indie oil m lide down nd doen diobey e ue pyil e., ε ψ ε min () ε ε mx Uing e me meod, we obin: T C in 1 (17) Beue ε ψ ε min <, C < n be known fe ubiuing Eq.(17) ino Eq.(1). A e me π ime, < <. So we n know <. Ti implie oil m lide up nd diobey e ue pyil ondiion. ow we know e mximum nd minimum pinipl in e e ε ψ ndε, epeively, nd i deemined uniquely by Eq. (16)..4 Inenl Enegy Diipion Re nd Exenl Foe Wok Re Enegy diipion ou on e lip ufe nd in e defoming zone. Hee wok i done by gviy of oil only beue ee e no oe exenl foe..4.1 Enegy Diipion Re Te inenl enegy diipion e inlude oe on filue ufe nd in pliiy defoming zone. Aoding o e lieue [8], e enegy diipion e of uni e on filue ufe i luled wi e following equion: (18) E 1 wee i oeion of oil; i ngenil veloiy jump o e filue ufe. Te ol enegy diipion e D 1 on filue ufe i obined by ineging Eq.(18) long e ufe i goen by oing e logpil line b bou e xe of foundion pi, own in Fig.: wee b D π o / oφe d (19) 1 b 1 i obined by Eq.(). ISS: Iue 1, olume, Deembe 7

5 Lieue [8] lo gve e expeion of e inenl enegy diipion e of uni volume in pliiy defoming zone: () E T One veloiy field i known, in e field n be obined oding o geomei equion. Bu i diffiul o obin e nlyi expeion of pinipl in e nd i limi e ppliion of Eq.(). In oogonl oodine yem { x, y, z}, if { x, y, z } of defoming egion i known, e pli noml in e field { ε x, ε y, ε z } n be obined oding o geomey equion. Fo oogonl oodine yem, ee i: ε ε + ε + ε ε + ε + ε x y z 1 3 ε + ε ε (1) wee ε i ( i 1,,3 ) e pinipl pli in e, ε i e bulk in e, ε nd ε e own in Eq.(1). Fom Eq.(1) nd Eq.(1), ε nd ε n be obined: ε ε () 1 T ε ε (3) 1 1/ T Fom Eq.() nd Eq.(), new expeion of e enegy diipion e pe uni volume n be obined: E oφ ε (4) Eq.(4) implie e enegy diipion e pe uni volume n be expeed wi e funion of pli bulk in e. In Eq.(4), oφ i e oodine of veex of e ig exgonl pymid woe ufe epeen Mo-Coulomb yield ufe in pinipl e pe. Wen φ, Coulomb meil will end o Te meil, o φ nd ε. Fo Te meil, e inenl enegy diipion e pe uni volume i: E ε (5) mx Comping Eq.(4) wi Eq.(5), e following equion n be obined: limoφε ε (6) φ mx Beue e pli bulk in e n be obined diely fom veloiy field nd geomey equion, Eq.(4) povide genel olving meod of e enegy diipion e pe uni volume. Te meod implifie e olving poe of e enegy diipion e. Te enegy diipion e of uni volume in defoming egion of iul pi n be obined fom Eq.(1) nd Eq.(4): E (7) C(1 T )oφ o in Te ol inenl enegy diipion e in defoming egion D n be luled by ineging Eq.(7) in e nnul domin i obined by oing e e b bou e xe of foundion pi, own in Fig.: D + d bd b π π T oe dd oe dd (9) wee b nd e obined by Eq.(6) nd Eq.(7), epeively, bd nd d e obined by Eq.()..4. Exenl Foe Wok Re Fo e foundion pi udied in i ppe, e gviy of oil i e only exenl foe. Te gviy wok e of uni volume oil i expeed: (3) w γ o wee γ i oil bulk deniy. Te ol gviy wok e of oil m W n be obined by ineging Eq.(3) in e nnul ISS: Iue 1, olume, Deembe 7

6 domin i obined by oing e e b bou e xe of foundion pi, own in Fig.: W + bd b d π π ow dd ow dd (31) wee nd e obined by Eq.(6) nd b Eq.(7), epeively, nd e obined by bd Eq.(). Te inegion bove ve no nlyi oluion, o we need o employ numeil meod o olve em..5 Memi Model Aoding o uppe-bound eoem, ee i (3) D D W 1 + d 3 Reul nd Anlyi 3.1 Ciil Heig Wen D / R i equl o.1 nd 1., epeively, e viion of wi φ nd α e owed in Fig.3. inee wi e inemen of φ. Moeove, e bigge φ i, e moe pidly vie. deee wi e inemen of α, fuemoe, e mlle α i, e moe pidly vie. Fig.4 ow viion of wi D / R nd φ wen α 9. inee wi e inemen of D / R. Moeove, wen D / R i vey mll, e viion of i lo mll. Ti ow e effe of oil m lgely enne e biliy of e foundion pi. Te iil dep of foundion pi n be expeed : D/R 1 (33) D γ f ( ) φ / ( ) In ode o obin e minimum uppe-bound oluion of iil dep, we need o olve e minimum of funion f ( ). Le min f ( ), wee i dimenionle vible, independen of nd γ nd only eled o φ, α nd D / R. Te memi model ued o olve e minimum uppe-bound oluion of iil dep of foundion pi i follow: () D/R 1 D/R.1 φ / ( ) (34) min f ( ) B (b) D/R.1 Fig.3 ying uve of wi φ nd α Te onin ondiion in Eq.(34) mke ue poin b i on e ig of poin, own in Fig.. Be on SQP opimizion lgoim, we ued mlb ofwe o olve e minimum uppe-bound of iil dep of iul foundion pi. ISS: Iue 1, olume, Deembe 7

7 α 9 Rigid zone Filue ufe β φ Rigid zone Fig.4 ying uve of wi D/R nd φ Fig.5 ow viion of eii ngle wi φ nd α. inee wi e inemen of φ nd α. One nd e known, oe eii ngle nd nd eii dimenion u, L nd o on n ll be obined. Tu e loion nd pe of e lip ufe n be deided. Wen D / R i vey mll, e iul foundion pi n be nlyzed e plne lope. In limi nlyi of plne lope, wo kind of filue menim e uully ued, i.e. igid blok nlionl nd oionl menim, own in fig.6 nd fig.7, epeively. In ble 1, of veil iul foundion pi wi D / R.1 nd veil plne lope fom diffeen filue menim e given, epeively. Wen D / R i vey mll, i n be found of iul foundion pi ppoxime of pln lope fom igid blok nlionl menim, nd bo of em e ligly gee n fom igid blok oionl menim. /( ) D/R D/R 1 φ /( ) Fig.5 ying uve of wi φ nd α Fig.6 Rigid blok nlionl filue menim Aoding o e eul ompued wi e meod popoed in i ppe, we lo know wen D / R. Ti implie e oionl logpil ufe degenee o e iul uned ufe. A e me ime, e iil lip ngle β oeponding o D i e me e iil vlue of β own in Fig.5 nd bo e equl o ( π / 4 φ / ). Bu i doen imply e pliiy defoming egion degenee o igid egion beue inenl enegy diipion e expeed by Eq.(4) in equl o zeo. Tble 1 obined fom ome filue menim φ α Plne lope wi igid blok nlionl filue menim Rigid zone Plne lope wi igid blok oionl filue menim Ciul foundion pi of D/R.1 wi xiymeil filue menim φ Rigid zone Logpil ufe Fig.7 Rigid blok oionl filue menim ISS: Iue 1, olume, Deembe 7

8 Te biliy of plne lope ofen i nlyzed wi veil lie meod, e.g. Jnbu meod. Fo pln lope, wen φ, fom Jnbu meod i equl o 6.1. I i vey imil o e uppe limi oluion of iul foundion pi woe D / R i.. So e effe of e foundion pi my be ignoed nd i n be egded pln lope nd nlyzed wi e lie meod wen e io of dep o dimee of pi i le n Slip Sufe Knowing nd, e pe nd loion of lip ufe n be obined. Hee, ume d nd e e oizonl nd veil dine of ny poin on lip ufe fom foundion pi wll oe, ju poin own in Fig.. In ode o plo lip ufe fom diffeen e on digm, nomlize d nd wi D nd obin dimenionle vible d / D nd / D, epeively. Fig.8 ow e effe of D / R on lip ufe of ylindil foundion pi wen φ. Wi e inee of D / R, lip ufe ink inwd. 4 Diuion Fig.9 Effe of φ on lip ufe Te oluion fom lip-line eoy (SLS) nd e oluion fom finie diffeene meod (FDS) wee ompued o omped wi e uppe-bound oluion (UBS). 4.1 Compion of UBS wi SLS Uing xiymeil lip-line eoy, Верезанцев В Г obined e ppoxime ive e peue on eining wll of ylindil foundion pi [9], own in e following equion: p T R λ R λ γ R [1 ( ) 1 ] + o φ( ) T ( 35) λ 1 Z Z wee γ i oil bulk deniy, φ nd e inenl fiion ngle nd oeion of oil, epeively, R π φ i diu of ylindil pi, T n ( ), 4 λ nφ / T, Z R + z T nd z i e dep unde gound, own in Fig.1. Fig.8 Effe of D / R on lip ufe Fig.9 ow e effe of φ on lip ufe of ylindil foundion pi wen D / R.8. Wi e inee of φ, lip ufe ink inwd. O x z Fig.1 Cylindil foundion pi ISS: Iue 1, olume, Deembe 7

9 Le R, we n obin: p γ zt T (36) Fom Eq.(36), e expeion of ive e peue i een o be e me Rnkin fomul fo plne poblem wen R i vey lge. In ode o olve e iil dep of ylindil foundion pi, following nion meod of Tezgi nd pek[1], e ive e peue i ineged long e dep nd equing e inegl zeo: D p dz (37) Fom Eq.(35) nd Eq.(37), we n obin: ( A B)( λ )( λ 1) T D [ BT ( λ) + A( λ 1)] R + [ BT ( λ) R + A( λ 1)( R λ ( ) 1 T D + R T D + R)] (38) T wee A γr, B oφ. λ 1 SLS of iil dep of ylindil foundion pi n be obined by olving Eq.(38) wi numeil meod. Compion w mde beween UBS nd SLS, own in Fig.7. Fo ylindil foundion pi, i n be een UBS i ligly gee n SLS wen φ, oweve, in oe e, e fome i lwy le n e le. UBS SLS Auming 1 nd e UBS nd SLS of, epeively, e elion of 1 nd n be eblied. Fo ylindil foundion pi, (39) D f (, γ, R, φ) omlize wo ide of Eq.(39) nd we n obin: (4) (4) Dγ Rγ g(, φ) Fom Eq.(33) nd Eq.(34), we know Rγ 1 D / R Fom Eq.(4) nd Eq.(41), we n obin 1 g(, φ) D / R 1 (41) Eq.(4) give e elion beween nd. Fo D / R.1 ~ 1., e elion beween nd i e D / R φ 1 (43) Fig.1 ow e fiing effeive of Eq.(43). Te elion beween nd n be expeed well wi Eq.(43). 1 φ /( ) Fig.11 Compion beween UBS nd SLS of iil dep Fig.1 Fiing effeive digm ISS: Iue 1, olume, Deembe 7

10 4. Compion of UBS wi FDS FLAC/SLOPE pogm w ued o ompue e numeil oluion of. Ti pogm i miniveion of FLAC(F Lgngin Anlyi of Coninu) i povided by I Conuling Goup, In. FLAC/Slope i deigned peifilly o pefom fo-of-fey lulion fo lope biliy nlyi. Beide wo-dimenionl plnein nlyi, xiymmei nlyi n lo be pefomed wi FLAC/Slope. FLAC/Slope povide n lenive o diionl limi equilibium pogm o deemine fo of fey. Limi equilibium ode ue n ppoxime eme - ypilly bed on e meod of lie -in wi numbe of umpion e mde (e.g., e loion nd ngle of inelie foe). Sevel umed filue ufe e eed, nd e one giving e lowe fo of fey i oen. Equilibium i only ified on n idelized e of ufe. In on, FLAC/Slope povide full oluion of e oupled e/diplemen, equilibium nd oniuive equion, ju FEM[11-13]. Given e of popeie, e yem i deemined o be ble o unble. By uomilly pefoming eie of imulion wile nging e eng popeie (e eng eduion enique), e fo of fey n be found o oepond o e poin of biliy, nd e iil filue ufe n be loed. Aoding o eng eduion enique, eng pmee ued in biliy nlyi n be wien :, F nφ n φ (44) wee F i fey fo, nd φ e mobile oeion nd mobile inenl fiion ngle of oil, epeively. Wen iil filue ppen unde ndφ, F n be obined fom Eq.(44). In ode o obin e iil dep of ylindil pi, nge e dep gdully nd obin e oeponding fey fo. Wen e fey fo i ppoximely 1., e dep i e iil one. Fig.13 illue e ompuing poe in wi fey fo vie wi dep. In Fig.13, γ k/m 3, 1kP, φ 3 nd R.5m. F Fig.13 D-F uve Fom Eq.(33) nd Eq.(34), we obin Dγ (45) Some numeil eul of iil dep of ylindil pi wee lied in ble nd omped wi UBS. I n be een UBS of i lge ligly n FDS. Tble Compion of UBS wi FDS φ 3 1 D/R FDS UBS Filue ufe n be obined long wi D. Fo e e of γ k/m 3, 1kP, φ nd R.5m, e filue of pi i illued in Fig.14. Te filue ufe i een o ppo e oionl logpil ufe umed in limi nlyi in i ppe. ISS: Iue 1, olume, Deembe 7

11 JOB TITLE :. FLAC/SLOPE (eion 5.) 8-Ap-8 14:19 LEGED Fo of Sfey 1.1 Mx. e in-e 5.E-7 1.E-6 1.5E-6.E-6.5E-6 3.E-6 3.5E-6 4.E-6 4.5E-6 Conou inevl 5.E-7 (zeo onou omied) Boundy plo 5E eloiy veo mx veo 1.167E-5 E (*1^1) Fig.14 Filue ufe 5 Conluion Auming oil m i omogeneou nd ioopi nd e filue model of iul foundion pi i xiymeil, kinemilly dmiible filue menim nd veloiy field wee eblied oding o oied pli flow ule. Uing em, e biliy of e iul foundion pi w nlyzed nd e limi uppebound oluion of iil eig w obined. Te following e ome impon onluion: (1) Ciil dep inee wi e deemen of lope ngle of foundion pi nd e inemen of e io of dep o diu nd inenl fiion ngle of oil. () Te effe of iul foundion pi mke i iil dep gee n e iil eig of plne lope. Howeve, e effe i unonpiuou wen e io of dep o dimee of pi i le n 1. (3) Wen e io of dep o diu i vey mll, e uppe-bound oluion of iul foundion pi i vey loe o e one of plne lope obined fom igid blok nlionl filue menim. Moeove, on e xiymeil ufe, e oionl logpil filue line degenee o ig line, nd e iil lip ngle i e me plne lope fom igid blok nlionl menim, oweve, e pliiy defoming zone in igid ye. (4) UPS of e iil dep i le n SLS nd lge ligly n FDS. In i ppe, e iil dep of e nonuppoed iul foundion pi w udied; oweve, fue eee e ugen on e biliy of iul pi einfoed, fo exmple, by oil nil. (*1^1) Aknowledgemen Ti wok w uppoed by Cine ul Siene Foundion(o.57856) nd ul Siene Foundion of Sndong Povine, Cin(o. Q6F). Refeene: [1] Donld I, Cen Z Y. Slope biliy nlyi by e uppe-bound ppo: fundmenl nd meod. Cndin Geoenil Jounl, ol.34, o.6, 1997, pp [] Cen Z Y, Wng X G, Hbefield C, e l. A ee-dimenionl lope biliy nlyi meod uing e uppe-bound eoem P 1: eoy nd meod. Inenionl Jounl of Rok Meni nd Mining Siene, ol.38, o.6, 1, pp [3] Muff J D, Hmilon J M. P-ulime fo undined nlyi of lelly loded pile. Jounl of Geoenil Engineeing, ol.119, o.1, 1993, pp [4] Dee A, Deouny E. Limi lod in nlionl filue menim fo oiive non-oiive meil. Geoenique, ol.43, o.3, 1993, pp [5] Oni J, Oii H nd Ymmoo K. Being piy nlyi of einfoed foundion on oeive oil. Geoexile nd Geomembne, ol.16, 1998, pp [6] Zeng X, Booke J R, Ce J P. Limi nlyi of e being piy of fiued meil. Inenionl Jounl of Solid nd Suue, ol.37,, pp [7] Pob A, Zo A, Kobyi M nd Kiid T. Uppe bound eime of led einfoed oil eining wll. Geoexile nd Geomembne, ol.18,, pp [8] Cen W F. Limi nlyi nd oil pliiy. Amedm, e eelnd: Elevie Publiing Co., 1975 [9] Верезанцев В Г. Axil-ymmeil limi equilibium poblem of looe medi. Beijing: Cin Aieue nd Building Pe, 1981 [1] Tezgi K, Pek R B. Soil Meni in Engineeing Pie[M]. ew Yok: Wiley, 1967 [11] Lin, Suliu, Ann S. Adpive Finie Elemen Anlyi fo Soluion of Complex Engineeing Poblem. We Tnion on Applied nd Teoeil Meni, ol.1, 6, pp [1] Bbiei A, Cei A. Anlyi of mony olumn by 3D FEM omogeniion poedue. We Tnion on Applied nd Teoeil Meni, ol., 6, pp.4-51 ISS: Iue 1, olume, Deembe 7

12 [13] Dubvk M. On dimenionl eduion in mulile, finie elemen nd omii nlyi in olid meni. We Tnion on Applied nd Teoeil Meni, ol.1, 6, pp.16-4 ISS: Iue 1, olume, Deembe 7