Development of Numerical Method for Large Deformation of Soil Using Particle Method

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㢹㣄ⷉ㡸G 㢨㟝䚐G 䋔㇠㢌G ␴ⷴ䝉G 䚨㉑ⷉG ᵐⵐ 68 ISSN 1229-2427 (Print) ISSN 2288-646X (Online) http://dx.doi.org/10.7843/kgs.2013.29.12.35 㻤ᛵ㍈■ᙽ㻡䀔ᬄⓀ㍙# # ㆤ29᜔# 12㿀# 2013ᫌ# 12ぜ# pp. 35 G 44

JOURNAL OF THE KOREAN GEOTECHNICAL SOCIETY Vol.29, No.12, December 2013 pp. 35 G 44

⫐⫛ῠ⪏# ⪿⧴㢧# 㓫⏷⪣# ᙋ㥠# 㢿⑨ῠ# ཧὧ

Development of Numerical Method for Large Deformation of Soil Using Particle Method

⚌ ⮨ ⵔ¹ Park, Sung-Sik

㇫ ḻ 䁻² Lee, Do-Hyun

᠃ ♳ 䂯³ Kwon, Min-Ho

Abstract

In this study, a particle method without using grid was applied for analysing large deformation problems in soil flows instead of using ordinary finite element or finite difference methods. In the particle method, a continuum equation was discretized by various particle interaction models corresponding to differential operators such as gradient, divergence, and Laplacian. Soil behavior changes from solid to liquid state with increasing water content or external load. The Mohr-Coulomb failure criterion was incorporated into the particle method to analyze such three-dimensional soil behavior.

The yielding and hardening behavior of soil before failure was analyzed by treating soil as a viscous liquid. First of all, a sand column test without confining pressure and strength was carried out and then a self-standing clay column test with cohesion was carried out. Large deformation from such column tests due to soil yielding or failure was used for verifying the developed particle method. The developed particle method was able to simulate the three-dimensional plastic deformation of soils due to yielding before failure and calculate the variation of normal and shear stresses both in sand and clay columns.

⧟# # # ⴋ

ٍ҆ĵقԴəࢹԐڮʴęÏڹʂѺ঍३Եںڦ३şܕڮॢڅՙѪۋǣڮॢ޲қѪęɵνüۙεԐڌॠݓ

؍əۓۙѪںԐڌॠٕڷ϶, ۓۙѪڹĵѕ, ьԓ, ͆॔͆֨؋ęÏڹйқٍԓۙقʂڿॠəۓۙÂԜ঒ۚڌϿʝں

ۋڌॠيٍ՚ߕۆݓѕѓ܁֩ںۋԓজॠٕɰ. ٽҙॠܼۋǣ॥սҼݒÀق˰͆ČߕقԴڮߕԜࢗͿѺॠə

৚ۆ3޲ڙʂѺ঍äʴں३Եॠşڦ३şܕۓۙѪق৚ۆࣷĨԜࢗεČͲॣսەəMohr-Coulomb ࣷĨşܵں

ʪۓॠٕڷ϶, ৚ۆࣷĨۋۻۆ२҄ۋǣąজইԜڷͿۍॢʂѺ঍ڹ৚ں۾ՁڮߕͿÀ܁ॠي३Եॠٕɰ. Òь ʽۓۙѪڹϤ۹ĵ՚ؓۋۚڌॠݓ؍ČÌʪÀ0ۍϿ͒şˊңĨ֬ॹĀęεۋڌॠيêݒॢɰڼ, ۾޳ͳں

ÀݓČۙςۋÀɠॢ۾ࢹşˊңĨ֬ॹں֬֨ॠي৚ۆ२҄ۋǣࣷĨͿۍॢʂѺ঍ں֨бͪۋՎॠٕɰ. Òьʽ

ۓۙѪڹϿ͒ٮ۾ࢹۆ3޲ڙʂѺ঍äʴںڮԐॠó۞ٚࠑॠٕڷ϶, ࣷĨۋۻقьԦॠə৚ۆ२҄ڷͿۍॢ

ՙՁѺ঍ق˰δ৚şˊǴۆսݔфۻɳڿͳѺজʪćԓॣսەؽɰ.

Keywords : Particle method, Soil flow, Large deformation, Failure criterion, Stress

1ⲯ㬦⭪, ᅗ∛រ㧳ᇪ# ᄎ㈯㘺ὃᆏ㧳√# 㘺ὃᆏ㧳Ⲟᆏ#ⴊᇪ⚲ (Member, Assistant Prof., Dept. of Civil Engrg., Kyungpook National Univ., Tel: +82-53-950-7544, Fax:

+82-53-950-6564, [email protected], Corresponding author, ᇪ❺ⲚⰪ)

2⋞㬦⭪, ᅗ∛រ㧳ᇪ# ᄎ㈯㘺ὃᆏ㧳√# 㘺ὃᆏ㧳Ⲟᆏ# 㧳╆ᆖⲯ# (Undergraduate Student, Dept. of Civil Engrg., Kyungpook National Univ.) 3⋞㬦⭪, ᅗ╛រ㧳ᇪ# 㘺ὃᆏ㧳ᆖ# ᇪ⚲, ᆏ㧳⪊ሆ⭪# (Prof., Dept. of Civil Engrg., Gyeongsang National Univ.)

*→# ᘖῒ⩪# រ㧶# 㘺⯲Ḗ# ⭪㧲ᜮ# 㬦⭪⯚# 2014ᗞ# 6⭮# 30Ⱆዦ⹚# ኒ# ᕎ⭃⯞# 㧳㬦ᳶ# ↎ᕎⶖ❶ዊ# ₮ᰧᝢ᝾. ⲚⰪ⯲# ᄚ㘺# ᕎ⭃ᆖ# 㨂፲# ᘖῒ⹫⩪# ᄦⱆ㧲⪆# ᥶ṗᝢ᝾.

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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69 䚐ạ㫴ⵌḩ䚍䟀⊰ⱬ㬅G G 㥐Y`ỀG 㥐XY䝬

j

r

e

i

Ilj1#41#Sduwlfoh#dssur{lpdwlrq#xvlqj#nhuqho#ixqfwlrq#lq#PSV#phwkrg

1. ⑧# ᮫

ٽҙǣǴҙۆڿͳѺজق˰δ৚ۆäʴںٚࠑॠ şڦॠيԐڌʼəս࠘३ԵѓѪڹ৚ںٍ՚ߕͿÂ

ܳॠəڮॢڅՙѪ(Finite element method, FEM)ęڮॢ

޲қѪ(Finite difference method, FDM), ŔνČ৚ںۋ ԓߕͿÂܳॠəÒѻڅՙѪ(Discrete element method, DEM)ۋʂशۺۋɰ. ڮॢڅՙѪęڮॢ޲қѪڹݓъ ںڅՙ(element)ǣüۙ(grid)Ϳǣɂˏ࢏ՙՁϿʝں

Ҽ΅ॢ৚ۆĵՁ֩ںۋڌॠيɰتॢٍĵқآӼχ

؉ɦ͆֬ИقԴʪȇνԐڌʼČەɰ. ॠݓχԐϸң Ĩ, ԓԐࢗ, ٍأݓъڵş, ؚԜজͿۍॢࠑѓڮʴˣ ęÏڹݓъۆʂѺ঍ęࣷĨ঳৚ۆ৔ζęÏڹ३ ԵقԴüۙεԐڌॣąڍüۙÀƚۋäǣˏࣥνş

˺ЛقćԓʼəѺڦقॢćÀەںӼχ؉ɦ͆Ѻ঍

ڷͿۍॢćԓ܁ঝʪÀই۹০̆رݓóʽɰ. ॢठ

üۙεԐڌॠݓ؍əÒѻڅՙѪڹݓъĵܓНۆʂ Ѻ঍३ԵۋÀɠॠݓχ, ڮॢڅՙѪۋǣڮॢ޲қѪę

Ïۋ৚ۆĵՁěćεۺڌॠşÀرͷɰ. ࣢০ێҙۓ ͳѺսə৚ۆНՁ࠘ٮəěćػş˺ЛقѺսĀ܁

ۋؙϔϿ঒ॠɰəɳ۾ۋەɰ. ˰͆ԴڮॢڅՙѪۋ ǣڮॢ޲қѪŔνČÒѻڅՙѪęÏڹۻࣀۺۍս

࠘३ԵѓѪںۋڌॠيݓъėॡقԴێرǣəʂѺ

঍Л܃ε३ԵॠəʚəॢćÀەɰ.

ۓۙѪ(Particle method)ڹٍ՚ߕًॡۆս࠘֨бͪۋ ՎѪڷͿٍ՚ߕۆäʴںڮॢÒۆۓۙڏʴڷͿćԓ ॠəѓѪۋɰ. ʂशۺۍۓۙѪقəLucy(1977)ÀÒь

ॢSPH(Smoothed Particle Hydrodynamics)ѪęKoshizuka et al.(1995)ۋÒьॢMPS(Moving Particle Semi-implicit) Ѫۋەɰ. ۓۙͿ३ԵًٖںĵՁॠдͿėÂۋԞ΄

ó঍ՁʼəʂѺ঍Л܃३Եقۺ०ॠيڮߕۆқَ

̚ə०ߕٮÏڹҼԸ঍ۙڮशϸں֨бͪۋՎॠə

ڮߕًॡقԴ١͒ۻҙࢢٍĵʼؽɰ(Daly et al., 1965).

̚ॢۓۙѪڹࣷʪǣݓݕ३ێۆäʴ३Ե, ڮߕٮԸ чęÏڹĵܓНęۆԜ঒ۚڌˣڮߕۆ৔ζں३Ե ॠəڮߕًॡęČߕۆࣷĨǣŒَęÏڹٍ՚ߕۆ

ʂѺ঍ˣČߕًॡқآقȇνڿڌʼČەɰ. ॠݓχ

ۋٮÏڹۤ۾ںÀݕۓۙѪںڮߕٮČߕۆՁݗں

ʴ֨قÀݓČەə৚ۆʂѺ঍Л܃قۺڌॠşڦॢ

ٍĵə߯Ŗێ҆ۆϽϽٍĵۙقۆ३֨ۚʼؽɰ(Bui et al., 2008; Maeda and Sakai, 2004; Naili et al., 2005).

ۋ˞ʂҙқۆٍĵۙ˞ڹLiu and Liu(2003)Àڮߕф

ČߕًॡقۺڌॠşڦॠيÒьॢSPHѪںц࢖ڷͿ

ݓъėॡқآقڿڌॠşڦॠي৚ۆĵՁ֩ں߸À ॠəٍĵεݕॱॠČەɰ. Maeda et al.(2006)ڹ৚ę

НۆԜ঒ۚڌںČͲॢϿ͒ݓъۆࠞ࣊ࣷĨ३Եٍ

ĵε֬֨ॠٕɰ. ߯ŖBui et al.(2008)ڹڮߕًॡںڦ

ॢSPHѪںşߣͿ࢏ٰۻՙՁϿʝۍDrucker-PragerϿ ʝںҼ΅ॢ৚ۆʂशۺۍ࢏ՙՁϿʝۍս܁Cam-Clay Ͽʝūݓۺڌॠٕڷ϶, ৚ęНۆԜ঒ۚڌěćūݓ

ČͲॠٕɰ. ۋٮÏۋBui et al.(2008)ۋÒьॢݓъė ॡڌSPHѪڹԐϸңĨ, ࢹԵΪ३ԵۆÀɠՁں҃ي

ܳؽɰ. ॢठMPSѪڹԸчėॡ, ࢹЀėॡ, ڙۙͳėॡ

ˣۆėॡЛ܃قԴҼؓ߹Ձڮߕε३Եॠşڦॠي

SPHѪںѺ঍ॢѓѪڷͿۓۙÂۆԜ঒ۚڌںϿʝτ ॠəѓ֩ق޲ۋÀەɰ.

ٍ҆ĵقԴəࢹԐۆʂѺ঍३Եںڦ३MPSѪق

şߣॠيMohr-Coulomb ࣷĨşܵںप॥ॢ3޲ڙڮ ʴ३ԵѪںÒьॠٕڷ϶, Ͽ͒şˊңĨ֬ॹę۾ࢹۆ

ێ߹ؓ߹֨ॹں֬֨ॠي҆३ԵѪۆۺڌՁںêࢹ ॠٕɰ. ۋٮÏڹࢹԐʂѺ঍३ԵѪڹşܕۆ৚ۆ

ࣷĨͿۍॢࢹԐۆѺ঍܁ʪǣڮʴѩڦεս࠘३Ե

ѓѪڷͿ܁ঝॠóٚࠑॠيݓъĵܓНԺćقъٖ

ॠäǣşܕĵܓНں҃ÌॠيݓъۆʂѺ঍ڷͿۍ

ॢक़३εѓݓॣսەɰ.

2. ⫐⫛ῠ(Particle method): MPSῠ

ۓۙѪۆş҆ۺۍę܃əüۙεԐڌॠݓ؍Čر̎

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㢹㣄ⷉ㡸G 㢨㟝䚐G 䋔㇠㢌G ␴ⷴ䝉G 䚨㉑ⷉG ᵐⵐ 6:

Ilj1# 51# Judglhqw# prgho# ehwzhhq# sduwlfohv# lq# PSV# phwkrg

ó йқѓ܁֩ں ۋԓজॣ ìۍÀۋɰ. ҆ ٍĵقԴə

Ҽؓ߹Ձڮߕۆ৔ζٚࠑںڦॠيڮߕėॡқآقԴ

Òьʽս࠘३ԵѪۍMPSѪںԐڌॠٕɰ. MPSѪڹĵ ѕ(Gradient), ьԓ(Divergence), ͆॔͆֨؋(Laplacian)ę

ÏڹйқٍԓۙقʂڿॠəۓۙÂԜ঒ۚڌϿʝں

ۋڌ३Դٍ՚ߕۆݓѕѓ܁֩ںۋԓজॠٕɰ. Áۓ

ۙÀČڮۆНν͟ںÀݓČےćäνǴقԴەəɰ δۓۙٮНν͟ںܳČыəɰ. ۋ˺ܳČыəНν͟

ۆࡾşəFig. 1قԴǣࢍǣˢۋۓۙÂäνق˰͆

Àܼ࠘εÍəɰ. ۓۙѪقԐڌʼəݓѕѓ܁֩, ܳڅ

ٍԓۙфڿͳćԓѓѪڹɰڼęÏɰ.

2.1 ⴋύὴ⭠☨

Ҽؓ߹Ձ۾Ձڮʴقʂॢݓѕѓ܁֩ڹٍ՚ѓ܁֩

(֩(1))ęNavier-Stokes ѓ܁֩(֩(2))ۋɰ. Navier-Stokes ѓ܁֩ڹ՚ʪѺজεĵॠşڦॢ֩ڷͿؓͳ२, ۾Ձ २, ŔνČٽͳ२ۆ०ڷͿĵՁʽɰ.

ćƒ

Ň

á × (1)

ćƒ

^ćĈƓ áà ćŇ

Îx©â Ńx^ÏćĈƓâ ćĈ (2)

يşԴ, 쩐əнʪ, tə֨Â, ^ćĈƓə՚ʪѯࢢ, xəĵѕٍ

ԓۙ, Pəؓͳ, Ńəʴ۾Ձćս, ^ćĈəٽͳںǣࢍǶɰ.

2.2 ⫐⫛ཏ# ␌㦃⫝̸⧴# ᷳᛃ

2.2.1 Kernel function

Kernel functionڹ֩(3)ęÏۋۓۙÂۆäνƐق

˰͆ۓۙÂԜ঒ۚڌقÀܼ࠘°ÞƐßں˃şڦॢ॥

սۋɰ. ݌, ۓۙÀÀūڐս΀Àܼ࠘À࠶ݓČъʂͿ

˃ۓۙۆäνÀ߿қ০ϥرےćäνƐƃ ۋԜۋʼϸ

Àܼ࠘εИ֨ॣսەڷдͿ0ڷͿ҆ɰ.

°ÞƐß á Ē ē Ĕ

ĕ ĕ

ćƐ Ɛƃ

à Î Þ× = Ɛ ï Ɛ_ƃß

× ÞƐ_ƃï Ɛß (3)

֩(3)ڹKoshizuka and Oka(1996)À܃؋ॢKernel functionڷͿ˃ۓۙԐۋۆäνÀϔڍÀūڐ˺Ԝ

঒ۚڌÀܼ࠘ÉۋИॢʂͿݒÀॠə࣢ݜۋەɰ. ҆

ٍĵقԴəڮʴۋࡾóü͵ॠݓ؍ڹ֨бͪۋՎق Դ҃ɰ؋܁ۺۍؓͳںصşڦ३֩(3)ںʂ֪ॠي

Jeong(2008)ۋ܃؋ॢ֩(4)ٮÏڹKernel functionں

Ԑڌॠٕɰ. ۋ֩ںۋڌॠϸۓۙÂäνÀÀūڗݗ

˺Àܼ࠘À࣢܁ÉڷͿսͶॢɰ.

°ÞƐß á Ē ē Ĕ

ĕ ĕ

ÞÎ à ćƐƃ

Ɛß^ÐÞÎ â ćƐƃ

Ɛß^Ð Þ× = Ɛ ï Ɛ_ƃß

× ÞƐ_ƃï Ɛß (4)

يşԴ, ےćäνƐƃۆÉڹۓۙÀथŒäνقےۆۆ

É2.1, 3.1 ˣۆÉںĕ३ćԓॢɰ.

2.2.2 ៣⚧┟ᷯ(Gradient model)

ۓۙƇٮƈÀÁÁНν͟ŋƇٮŋƈεÀݓČەɰČ

ॣ˺Fig. 2ٮÏۋۓۙƇٮܳѺۓۙƈ Ԑۋۆĵѕѯ ࢢεÞŋ_ƈà ŋ_ƇßÞćĈƐ_ƈà ćĈƐ_Ƈßîç^ćĈ^Ɛƈà ćĈƐ_Ƈç^ÏͿćԓॢɰ. يşقKernel functionڷͿÀܼ࠘εĕ३ܳČۓۙÀيͦێąڍۻ ߕۆ०ںߣşѕ࠘قԴۓۙսнʪۍƌ^×Ϳǣɀر

֩(5)ٮÏۋĵѕεćԓॢɰ.

ï xŋ ð_Ƈá ćƌ^× Ƃ

ā

_{ƈd Ƈ}^ƙ

Ɯƚć

ç^ćĈ^Ɛƈà ćĈƐ_Ƈç^Ï

ŋ_ƈà ŋ_Ƈ

ÞćĈƐ_ƈà ćĈƐ_ƇßƕÞç^ćĈ^Ɛƈà ćĈƐ_Ƈç^ß^ƛ

Ɲƞ⁽⁵⁾

يşԴ, dəćԓėÂۆ޲ڙۋɰ. Ƈۆۓۙսнʪε

ǣࢍǴə ƌ_ƇəɰڼęÏۋ܁ۆॢɰ.

ƌ_Ƈá

ā

_{ƈd Ƈ}^{ƕ Þ}^ç^ćĈ^Ɛ^ƈ^{à ćĈ}^Ɛ^Ƈ^ç^ß⁽⁶⁾

Ҽؓ߹ՁܓæقԴəнʪÀێ܁ॠдͿۓۙսн

(4)

6; 䚐ạ㫴ⵌḩ䚍䟀⊰ⱬ㬅G G 㥐Y`ỀG 㥐XY䝬

Ilj1# 61# Odsodfldq# prgho# lq# PSV# phwkrg Ilj1# 71# Iorz# fkduw# iru# fdofxodwlrq

ʪʪێ܁३آॢɰ. ƌ^×ÀŔşܵۋʼəۓۙսнʪ Ϳۓۙۆߣşѕ࠘ԜࢗقԴĀ܁ʼ϶ćԓę܁قԴ

Ѻॠݓ؍əČ܁Éۋɰ. ƌ_ƇəۓۙƇۆےćäν؋ق Դۓۙۆѕ࠘ԜࢗεǣࢍǴəÉڷͿےćäνǴق

ۓۙÀψںս΀, ŔνČےćäνǴقԴʪܼ֮ۓۙ

ÀūۋقۓۙÀψۋқप३ەںս΀Éۋ࠶ݕɰ.

ƌ^×Éڹۻߕۓۙѕ࠘ۆʂऺÉڷͿۋٮҼİॠيƌƇ

ÉۋࡾɰϸۓۙÀӓӓॢìڷͿۚɰϸɗ֡ॢԜࢗ

ۍìڷͿࣺɳॣսەɰ. ˰͆Դƌ^×əۻߕۓۙѕ࠘

ε۞ʂशॣսەəÉڷͿ܁३آॠ϶, ٍ҆ĵقԴ ə֨Βۆܼ֮قەəۓۙεƌ^×ۆşܵۓۙͿԐڌॠ

ٕɰ.

2.2.3 ⇳㾃⇳ⵓ⾿# ┟ᷯ(Laplacian model)

͆॔͆֨؋(x^Ï)ڹ֩(7)ęÏۋНνۺڷͿঝԓں

ۆйॠдͿFig. 3قԴǣࢍǣˢۋۓۙѪقԴ͆॔͆

֨؋ϿʝڹۓۙiÀÍəНν͟ۆێҙεܳڦۓۙ

jͿқѕ३ܳəًॣںॢɰ.

ï x^Ïŋ ð_Ƈá ć Łƌ^×

ÏƂ

ā

_{ƈd Ƈ}^ã^Þŋ^ƈ^{à ŋ}^Ƈ^ßƕÞ^ç^ćĈ^Ɛ^ƈ^{à ćĈ}^Ɛ^Ƈ^ç^ß^ä⁽⁷⁾

يşԴ, Łə֩(8)ęÏۋ३Ե३ٮࣀćۺۍঝԓۆ

܁ʪεێ࠘ॠóॠşڦॢćսۋɰ.

Ł á ć

ā

ƈd Ƈ

ƕç^ÞćĈ^Ɛƈà ćĈƐ_Ƈßç

ƈd Ƈ

ā

ç^ćĈ^Ɛƈà ćĈƐ_Ƈç^Ï^ƕç^ÞćĈ^Ɛƈà ćĈƐ_Ƈßç

(8)

Łə֨Âęۓۙۆڦ࠘ق˰͆ÉۆѺজÀϔڍۚڷ дͿČ܁ʽÉںԐڌॣսەɰ.

2.2.4 ⢻⿌㠌⮨# ┟ᷯ

MPSѪۆҼؓ߹Ձڮߕۆړݔےڹ˃ɳćͿǣɂ ɰ. 1ɳćقԴə۾Ձ२ęٽͳ२(ܼͳˣ)ۆćԓںࣀ ३ۓۙۆܼÂ՚ʪ(^ćĈƓ_Ƈ^Ý)ٮܼÂڦ࠘(^ćĈƐ_Ƈ^Ý)ε֩(9)ٮÏ ۋćԓॢɰ.

ćĈƐƇÝáćĈ ƐƇƌâ ĢƒćĈ

ƓƇÝ

(9)

2ɳćقԴəϤ۹ؓͳۆप؉բѓ܁֩ں֩(10)ęÏ ۋćԓॢɰ.

ï x^Ï©ð_Ƈ^{ƌ âÎ}áà ćĢƒ^Ï Ň ćƌ^×

ƌ_Ƈ^Ýà ƌ^×

(10)

ƌ_Ƈ^Ýə1ɳćقԴćԓॢܼÂڦ࠘εۋڌॠيĵॢۓۙ

սнʪۋɰ. ֩(10)ۆڍѺڹۓۙսнʪۆ޲ۋͿ

ؓͳۆঝԓ(x^Ï)ںǣࢍǴČەɰ. ܟѺڹ͆॔͆֨؋

Ͽʝۍ֩(7)ͿǣࢍǷսەرٍς1޲ѓ܁֩ڷͿؓ

ͳںĵॣսەɰ. ĵॢؓͳں֩(10)قʂۓॠي՚

(5)

㢹㣄ⷉ㡸G 㢨㟝䚐G 䋔㇠㢌G ␴ⷴ䝉G 䚨㉑ⷉG ᵐⵐ 6<

Ilj1# 81# Prku*v# flufoh

Ģ ƒï x©^{ƌ âÎ}ð_Ƈ (11)

ï x©^{ƌ âÎ}ð_Ƈə֩(5)εԐڌॠيĵॣսەɰ.

߯ܛۺڷͿ֨Án+1قԴۆۓۙۆ՚ʪٮڦ࠘əɰڼ ęÏۋĵ३ݕɰ.

ćĈƓ_Ƈ^{ƌ âÎ}áćĈ Ɠ_Ƈ^ÝâćĈ

ćĈƐ_Ƈ^{ƌ âÎ}áćĈ Ɛ_Ƈ^Ýâ ĢƒćĈ

ۓۙѪۆۻߕۺۍڿͳćԓ৔ζʪəFig. 4ٮÏɰ.

2.3 ⪜᭰# ဏ⏻

2.3.1 ⲏ㐸# ⚆# ⲏ㹀# ㇈⊜

ۓۙi+1ۆսݔڿͳň_{Ƈ âÎ}ںćԓॠşڦ३ۓۙi+1 ڹ֩(14a)ٮÏۋ֪ۙۆцͿڦۓۙiۆڿͳň_Ƈقχ

ٖॳںыəɰČÀ܁ॢɰ. Ͽ˜ۓۙə֪ۙۆИóχ

ࢂцͿ؉͒ۓۙقڿͳںۻɵॢɰ. ˰͆Դशϸق

ەəۓۙۆڿͳڹ0ۋɰ. ŔνČϿ˜ۓۙə֪ۙق óٖॳںܳəۓۙۆڿͳںۻɵыəɰ.

ňƇ âÎá ň_Ƈâ Þƕ_ƇîƇß (14a)

ň_{Ƈ âÎ}á ňƇâ ÞƕƇîƇß â ÞƇîƇß (14b)

يşԴ, ňəڿͳ, ƕəۓۙۆИó, Aəɳϸۺںǣ

ࢍǶɰ. ۓۙۆИóəۓۙۆϿتںۓۙÂथŒäν (dis)εݓζওڹॢѺڷͿÀݓəĵ঍̚ə܁ԐÁ঍

ڷͿÀ܁ॠيĵॣսەɰ. ֩(14b)ə৚ۆܼۙۋٽ قٽͳۋۚڌॣ˺ڿͳںĵॠə֩ڷͿFəѺ঍՚

ʪق˰δ৪ںǣࢍǴ϶, Aə৪ںыəϸۺںǣࢍǶ ɰ. ۓۙۆսथڿͳڹۓۙۆսݔڿͳقভࢹؓćս

kεĕॠيĵॣսەɰ.

2.3.2 Mohr-Coulomb 㶃ឫᢧ㍷

৚ۓۙۆࣷĨيҙəFig. 5ٮÏڹMohr’s Circleں

ÏۋĵॣսەČ

ŉ_À´Ë á ćÏ

(16)

ۋ˺ࣷĨϸقԴۻɳڿͳŉəɰڼęÏۋĵॣս

ەɰ.

ŉ_ƄəɰڼęÏɰ.

݌, ࣷĨϸقԴۆۻɳڿͳŉٮࣷĨÀێرǨ˺ۆۻ ɳڿͳŉ_ƄÀ ێ࠘ॣ ˺ ۻɳࣷĨÀ ێرǣə ìڷͿ

҆ɰ.

2.3.3 㿤❬(Yielding) ⚆# ᜴䃋✌㣐(Hardening rule) ێъۺڷͿ৚ڹFig. 6ęÏۋۻɳڿͳۋMohr-

(6)

Ilj1# 91# Jhqhudo# vwuhvv0vwudlq# ehkdylru# ri# vrlov

313# vhf 319# vhf 415# vhf

Wrs# ylhz

Vlgh# ylhz

Ilj1# :1# Uhvxow# ri# vdqg# froxpq# idloxuh# whvw

CoulombęÏڹࣷĨşܵقʪɵॠşۻقʪ Ѻ঍ę

ʴ֨ق२҄ڷͿۍॢՙՁѺ঍ۋьԦॢɰ(Park et al., 2005). ॢठٍ҆ĵقԴəşܕۆٍ՚ߕًॡقԴԐ ڌʼəɰՙ҄ۡॢ२҄фąজѪ࠙(Park et al., 2005) ںۓۙѪقۺڌॠş҃ɰə৚ۆäʴں۾ՁڮߕͿ

À܁ॢɰڼۋٮÏڹ२҄ڷͿۍॢąজইԜں֨б

ͪۋՎॠČۙॠٕɰ. ٍ҆ĵقԴ৚ۆࣷĨۋۻقь ԦॠəՙՁѺ঍ں֨бͪۋՎॠşڦ३Ԑڌॢʴ۾

Ձćսə0.13Ƌ^ÏîƑٮÏڷ϶, ৚ۋࣷĨşܵںχܔॣ

ąڍق৚ۆ۾Ձۋՙ֬ʽɰ.

3. ᷳ᫣ᅻឰ#⃠ၿ☯㤣#὚#⫐⫛ῠ#☧ỷ᭓⪿⒣#↏ႛ

æܓॢϿ͒قĵ՚ؓۋۚڌॠݓ؍ںąڍܼۙقۆ ३৙͠ǴνəʂѺ঍Ԑͻεۦইॠٕڷ϶, ۋε3޲ڙ

ۓۙѪںۋڌॠي֨бͪۋՎॠٕɰ. 3޲ڙڷͿ৔βə

৚ۆڮʴں֨Âق˰͆Ҽİॠşڦ३Mohr-Coulomb

ࣷĨşܵںĀ܁ॠəÌʪ܁սۍ۾޳ͳęυ޶Áڹ

Ͽ˃0ڷͿԺ܁ॠيϿ͒şˊۋÌʪÀػəڮߕߌͤ

ڮʴॠəę܁ں3޲ڙۺڷͿ֨бͪۋՎॠٕɰ.

3.1 ᷳ᫣ᅻឰ# ⃠ၿ☯㤣

Ͽ͒şˊңĨ֬ॹقԴəݔą28cm, ȭۋ50cmۍ

؉ࡾπڙşˊǴقǤʴÌϿ͒εȏČFig. 7ęÏۋ

Ӈδ՚ʪͿ٤͹ں˺ڙşˊǴϿ͒À৙͠Ǵνʪ΀

֬ॹॠٕɰ. Fig. 7ڹ0.0ߣ, 0.6ߣ, 1.2ߣقԜҙ(top view) ٮ܁ϸ(side view)قԴ҆Ͽ͒şˊۆ৔βəϿ֥ں

ٖ߭ॢԐݕۋɰ. ۾ՁۋػڷϸԴĵ՚ؓۋۚڌॠݓ

؍ںąڍر̃ॢÌʪʪÀݓݓ؍əϿ͒şˊۋӇβ ó৙͠ǴνϸԴ1.2ߣقफ120cm, ȭۋ(ܼؖҙқ) 15cm ͿѺॠٕɰ.

3.2 ᷳ᫣ᅻឰ# ⃠ၿ☯㤣# ☧ỷ᭓⪿⒣

֨Âق˰δϿ͒şˊңĨইԜںۓۙѪڷͿ֨б

ͪۋՎॠٕɰ. Fig. 8ڹϿ͒şˊۋңĨʼəę܁ںڦ قԴǴͲɰ҇ąڍ(top view)ٮؘقԴ҇ąڍ(side view)قϿ͒şˊۆ3޲ڙäʴںۓۙѪۋ֨бͪۋՎ ॠČەɰ. ֨Âۋݓǫق˰͆Ͽ͒À۾۾৙͠Ǵνϸ ԴܳѺڷͿঝʂʼرÀəìں҇սەɰ. şܕڮߕ ३ԵقԐڌʼəۓۙѪ॒ͿŔ͖ڹʂҙқ2޲ڙ३Ե χÀɠॠݓχ, ٍ҆ĵقԴÒьܼۍ॒ͿŔ͖ڹࢹԐ ۆʂѺ঍ۋǣڮʴں3޲ڙۺڷͿ֨бͪۋՎॠдͿ

ܘʌ܁ঝॢąćܓæںս࠘३Եقъٖॣսەɰ.

(7)

㢹㣄ⷉ㡸G 㢨㟝䚐G 䋔㇠㢌G ␴ⷴ䝉G 䚨㉑ⷉG ᵐⵐ 74

Wlph# @# 3# vhf Wlph# @# 319# vhf Wlph# @# 415# vhf

Wrs#

ylhz

Vlgh#

ylhz

Ilj1# ;1# Vlpxodwlrq# ri# vdqg# froxpq# idloxuh# whvw

Ilj1# <1# H{shulphqwdo# vlpxodwlrq# ri# odujh# ghirupdwlrq# ri# fod|# froxpq

Fig. 8ڹFig. 7ۆ֬ॹĀęٮҼİۺ۞ێ࠘ॠəìڷ ͿǣࢍǮɰ. ҆३ԵقԴəϿ͒şˊۆÌʪε0ڷͿ

҃ؕş˺Лقڿͳćԓ҃ɰə3޲ڙѺ঍঍ࢗχں

ҼİॠČۙॠٕɰ.

4. ⭛㓫ᅻឰ#⃠ၿ☯㤣#὚#⫐⫛ῠ#☧ỷ᭓⪿⒣#↏ႛ

۾޳ͳڷͿۙςۋÀɠॢ۾ࢹεۋڌॠي֬ǴңĨ

֬ॹ(ێ߹ؓ߹֨ॹ)ں֬֨ॠٕɰ. ڙşˊϿتۆ۾ࢹε

Ձ঍ॢɰڼॠҙقԴॠܼںÀॠيңĨ֨ࢅə֬ॹں

֬֨ॠٕڷ϶, ۋεٍ҆ĵقԴÒьॢۓۙѪںۋڌ ॠيࣷĨۋۻۆՙՁѺ঍ں֨бͪۋՎॠČۙॠٕɰ.

̚ॢ۾ࢹǴҙقۚڌॠəڿͳԜࢗÀ Mohr-Coulomb

ࣷĨşܵںχܔॣąڍقəҼ۾ՁڮߕٮÏڹʂѺ

঍ۋьԦॠóʽɰ. ս࠘३Եقज़څॢۓͳѺսə۾

ࢹҼܼ2.65, ভѓॳࢹؓćս0.5, ؓ߹՚ʪŔνČؘ

ԴسśॢMohr-Coulomb ࣷĨşܵۆÌʪ܁սۋɰ. ҆

ٍĵقԐڌॢ۾ࢹεstiff clayͿࣺɳॠيҼѕսۻ ɳÌʪəDas(2010)ε޷Čॠي۾޳ͳ100kPaęυ޶

Á0ʪεԐڌॠٕɰ. Ŕшقʪćԓę܁قज़څॢۓ ͳѺսͿəƌ^×ۆÉওڹƌ^×ÉćԓقԐڌॣۓۙۆ

ѥ঒Àज़څॢʚƌ^×Éۋۻъۺۍս࠘३ԵĀęقࢀ

ٖॳںй࠘дͿܳۆ३ԴԸ࢘३آॠ϶, ٍ҆ĵقԴ əƌ^×ÉڷͿ4.879εԐڌॠٕɰ.

4.1 ⭛㓫ᅻឰ# ⃠ၿ☯㤣

ٍ҆ĵقԐڌॢ۾ࢹəݕ३ۍŖ३ًقԴ޽ࠄॢ

३Ձ۾ࢹͿݔą5cm, ȭۋ10cmۆڙşˊϿتۆ۾

ࢹεՁ঍ॢɰڼॠҙقԴɗο՚ʪ(10mm/min)Ϳॠ

ܼںÀॠيңĨ֬ॹں֬֨ॠٕɰ. Fig. 9ə֬ॹ֨

ۚ঳30, 90, 180, 360ߣێ˺۾ࢹşˊۆѺ঍Ͽ֥ں

(8)

Ilj1# 431# Qxphulfdo# vlpxodwlrq# ri# odujh# ghirupdwlrq# ri# fod|# froxpq

Ilj1# 441# Yduldwlrq# ri# vkhdu# vwuhvv# rq# fod|# froxpq# fdofxodwhg# e|# sduwlfoh# phwkrg# +lq# nj2p⁵,

҃يܳČەڷ϶, 360ߣąęॠٕں˺əȭۋÀ10cm قԴ4cmūݓؓ߹ʼؽɰ.

4.2 ⭛㓫ᅻឰ# ⃠ၿ☯㤣# ☧ỷ᭓⪿⒣

Fig. 10ڹ֨Âق˰δ۾ࢹşˊۆңĨę܁ں֨б

ͪۋՎॠČەڷ϶, Fig. 9ۆ֬ॹĀęٮڮԐॠóѺ঍

ʼəìں؎սەɰ. սݔڿͳڹر̅ݓ۾قۚڌॠ ə৚ۆИóٮ߹Ѻ঍ق˰͆ۓۙقۚڌॠə ৪ (F=ma)ں०ॠي֩14(b)ٮÏۋćԓॠٕڷ϶, ۻɳڿ ͳڹ֩(17)ق˰͆ࣷĨϸقԴۆۻɳڿͳڷͿćԓ ॠٕɰ. Fig. 11ڹ߹Ѻ঍ق˰δ۾ࢹşˊशϸقۚڌ ॠəۻɳڿͳѺজεǣࢍǴČەɰ.

Fig. 12ə۾ࢹşˊۋѺ঍ʿق˰ܼ͆Â(Middle) ݓ

۾ۆǴҙ(center)ٮٽҙ(surface)εҼ΅ॠيԜҙ(Top) ٮॠҙ(Bottom)ۆǴҙقۚڌॠəսݔڿͳфۻɳ ڿͳۆѺজεҼİॠČەɰ. يşԴǴҙəɳϸۆܼ

ؖݓ۾قەəۓۙۆڿͳں, ٽҙəÀۤцŶޅقە əۓۙۆڿͳںǣࢍǶɰ. Fig. 12(a)əܼÂݓ۾ۆǴ ҙٮٽҙսݔڿͳ(Axial stress)ęۻɳڿͳ(Shear stress) ںҼİॠČەڷ϶, Ǵҙقۚڌॠəڿͳڹ60mm Ѻ

঍ūݓݓ՚ۺڷͿݒÀॠəąॳں҃ۋݓχ, ٽҙق

ۚڌॠəڿͳڹ50mm Ѻ঍قԴҙࢢԴԴ০Çՙॠə

ąॳںٕ҃ɰ. ۋə৚ۓۙÀшڷͿнͲǣϸԴٽ ͳقۆॢٖॳۋܶر˞ؽş˺ЛڷͿ҃ۍɰ. ŔνČ

ॠҙقۚڌॠəڿͳۋԜҙقۚڌॠəڿͳ҃ɰɰ ՙࢀìں؎սەɰ. ۾ࢹşˊۆ؉͒ҙқۆڿͳۋ

۾޲ݒÀॠϸԴMohr-Coulomb ࣷĨşܵںχܔॠݓ ə؍ݓχ२҄ڷͿۍॢݓ՚ۺۍՙՁѺ঍ۋьԦॠ

ٕɰ.

ٍ҆ĵقԴÒьॢۓۙѪقMohr-Coulomb ࣷĨş

ܵںۺڌॠٕڷ϶, ̚ॢ৚ں۾ՁڮߕͿÀ܁॥ڷͿ

׆۾ࢹşˊǴڿͳݒÀق˰δ२҄ۋǣąজইԜق

˰δʂѺ঍ں֨бͪۋՎॣսەؽɰ. ॢठPFCٮÏ ڹÒѻڅՙѪʪۋٮڮԐॢʂѺ঍३ԵۋÀɠॠݓ χMohr-Coulomb ࣷĨşܵۋǣ२҄фąজѪ࠙ęÏ ڹՙՁۋ΁ںۺڌॠşقəرͲڏ۾ۋەɰ.

5. ࿻# ᮫

ٍ҆ĵقԴəڮߕۆɰتॢąć࠘Л܃قۺڌʼə

ۓۙѪں॥սҼݒÀق˰͆ČߕقԴڮߕͿäʴॠə

ࢹԐۆʂѺ঍३ԵقۺڌॠČۙॠٕɰ. ۓۙѪڹʂ Ѻ঍३ԵۋÀɠॠϸԴʪ৚ۆĵՁ֩ʪۓۋÀɠॠɰ əۤ۾ۋەɰ. ˰͆Դٍ҆ĵقԴəMohr-Coulombę

ÏڹࣷĨşܵںۓۙѪقս֩জॠٕڷ϶, ۋεۋڌ ॠيϿ͒ٮ۾ࢹۆңĨ֬ॹĀęε֨бͪۋՎॠي

ÒьʽۓۙѪۆʂѺ঍३ԵقʂॢۺڌՁںêࢹॠ

ٕɰ. ŔĀęəɰڼęÏɰ.

(9)

㢹㣄ⷉ㡸G 㢨㟝䚐G 䋔㇠㢌G ␴ⷴ䝉G 䚨㉑ⷉG ᵐⵐ 76 Ilj1# 451# Yduldwlrq# ri# qrupdo# dqg# vkhdu# vwuhvv# rq# fod|# froxpq# fdofxodwhg# e|# sduwlfoh# phwkrg=# +d,# plggoh/# +e,# wrs# dqg# erwwrp

(1) ৚قۚڌॠəսݔڿͳڹۓۙÂ՚ʪѺজٮۙ

ܼںۋڌॠيćԓॠٕڷ϶, սथѓॳڷͿۚڌॠ əڿͳڹভѓॳࢹؓćսεۋڌॠيćԓॠٕɰ.

(2) Mohr-Coulombۆ ࣷĨşܵں ۓۙѪق ʪۓॢ ɰ ڼćԓʽ৚ۆڿͳԜࢗÀMohr-CoulombۆࣷĨ şܵںχܔॣąڍࣷĨͿۍॢʂѺ঍ۋьԦॠ ʪ΀ॠٕɰ. ॢठࣷĨşܵںχܔॠşۋۻقə

ۓۙѪۆ ۾Ձ२ں ۋڌॠي २҄ڷͿ ۍॢ ৚ۆ

ՙՁäʴں֨бͪۋՎॠٕɰ.

(3) ÒьʽۓۙѪڹĵ՚ؓۋۚڌॠݓ؍ČÌʪÀ0 ۍϿ͒şˊۆ3޲ڙ৔ζۋǣ۾޳ͳںÀݕ۾ࢹ şˊۆێ߹ؓ߹ڷͿۍॢ3޲ڙʂѺ঍ęڿͳѺ জεćԓॣսەؽɰ. ۓۙѪ३ԵĀęə֬ॹĀ ęٮҼİۺێ࠘ॠٕɰ.

(4) ٍ҆ĵقԴÒьॢۓۙѪڹॠܼݒÀق˰δڿ ͳćԓۋÀɠॣӼ؉ɦ͆ۋͿۍॢʂѺ঍֨б

ͪۋՎۋÀɠॠٕɰ. ˰͆ԴÒьʽۓۙѪڹԐ ϸңĨٮ Ïڹ ɰتॢ ݓъėॡ ʂѺ঍ ٚࠑقʪ

টڌÀɠॣìڷͿࣺɳʽɰ.

ཛ⏷⪣# ᅋ

ۋȦЛڹ2013țʪ܁ҙ(İگҙ)ۆۦڙڷͿॢĶٍ

ĵۦɳۆ şߣٍĵԐغ ݓڙں ы؉ սॱʼؽڷ϶(No.

2010-0023540), ێҙə 2012țʪ ܁ҙ(й͒޻ܓęॡҙ) ۆۦڙڷͿॢĶٍĵۦɳ-ėė҄ݓ؋ۻٍĵԐغۆݓڙ ںы؉սॱʼؽڷ϶(No. 2012M3A2A1050982) ۋقÇ Ԑ˚ςɦɰ.

(10)

⾃ါẃ㤗# (References)

1. Bui, H.H., Fukagawa, R., Sako, K., and Ohno, S. (2008), “Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic-plastic soil constitutive model”, International Journal for Numerical and Analytical Methods in Geomechanics, Vol 32, pp 1537-1570.

2. Daly, B.J., Harlow, F.H., Welch, J.E., Wilson, E.N., and Sanmann, E.E. (1965), Numerical fluid dynamics using the Particle-and-Force Method, LA-3144.

3. Das, B. M. (2010), Principles of Geotechnical Engineering, Cengage Learning, USA.

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Received : August 1^st, 2013 Revised : October 15^th, 2013 Accepted : October 25^th, 2013