Received January 4 2013, Revised April 22 2013, Accepted June 4 2013
Copyright ⵑ 2013 by the Korean Society of Civil Engineers
ǣǣȀȀǤǤȀͳͲǤͳʹͷʹȀ ǤʹͲͳ͵Ǥ͵͵ǤͶǤͳͶͺͻ Ǥ ǤǤ
⡳Ⳣ♿㚚Ἲ#ኞᷢ㬚#⍆㦪㰒#㝞♪♪′ⴖ#⍆⬆ⷓ⛯#㬲⛛
ࢢ ȵ୍ก
Kim, Yong Min*, Kim, Jaehong**
Instability Analysis of Unsaturated Soil Slope Considering Wet Condition
ABSTRACT
The monolithically coupled finite element analysis for a deformable unsaturated soil slope is performed to investigate the effect of antecedent rainfall which is assumed by initial conditions varying degree of saturation (36, 51, 77%) in finite element analysis. The distributions of matric suction and deformation on slope surface obtained from numerical simulation show the instability of antecedent rainfall-induced unsaturated soil slope. Moreover, the numerical analysis using Drucker-Prager model can be checked if a soil slope has reached failure (trial failure criterion f
tr>0, plastic behavior) or not (trial failure criterion f
tr< 0, elastic behavior). It is found that displacement of slope surface layer increases and the matric suction on soil slope decreases with an increase of initial degree of saturation by antecedent rainfall. Especially, the matric suction of the soil slope in dry condition (S=36%) rapidly decreases rather than that in wet condition (S=51%) at the same rainfall duration. The results of the trial failure criterion ( f
tr> 0) show slope instability in the toe region and surface of the slopes.
Key words : Antecedent rainfall, Hydro-mechanical analysis, Finite element analysis, Drucker-Prager model, Slope stability
Ⅹಾ
ᅙᩑǍᨱᕽ۵ᇩ⡍⪵ᔍ໕ᨱᕽᖁ⧪vᬑ᮹ᩢ⨆ᮥŁಅ⦹Łᯱḡၹ᮹Ⅹʑ⡍⪵ࠥෝ3aḡ(36, 51, 77%)ಽᖅᱶ⦹ᩍᙹญ⦺ᱢ-ᩎ⦺ᱢ࠺ᮁ
⦽ᗭ⧕ᕾ(monolithically coupled finite element analysis)ᮥᙹ⧪⦹ᩡ݅. ᖁ⧪vᬑᨱ᮹⦽ᇩ⡍⪵ᔍ໕᮹ᇩᦩᱶᖒᮡᔍ໕ԕ༉š⯂ᙹಆ
ᇥ⡍᪡ᔍ໕⢽⊖᮹ᄡ᭥ෝ☖⦹ᩍ⪶ᯙ⧁ᙹᯩ݅. ੱ⦽Drucker-Prager model᮹⧎ᅖĞĥʑᵡ(trial failure criterion)ᮥᱢᬊ⦹ᩍvᬑ⋉⚍
ᨱ᮹⦽ᇩ⡍⪵ᔍ໕᮹┥ᖒၰᗭᖒÑ࠺ᮥ❭ᦦ⦹ᩡ݅. əđŝ, ᖁ⧪vᬑᨱ᮹⦽ḡၹ᮹Ⅹʑ⡍⪵ࠥaⓕᙹಾvᬑᨱ᮹⦽ᄡ᭥aⓍíၽᔾ⦹
໑༉š⯂ᙹಆੱ⦽qᗭ⦽݅. ✚⯩, ༉š⯂ᙹಆᮡḡၹᯕÕ᳑⧁ᙹಾíqᗭ⦹໑, Ⅹʑᨱ࠺ᯝ⦽༉š⯂ᙹಆᮥw۵ᔍ໕⢽⊖ᨱᕽ⡍⪵ࠥ
a᯲ᮡḡၹᯝᙹಾෙ༉š⯂ᙹಆqᗭෝᅕᩡ݅. ⣮⪵☁᮹┥ᖒŝᗭᖒÑ࠺ᮥǍᇥ⦹ʑ᭥⧕ᔍᬊࡽDrucker-Prager modelᮥ☖⧕ᔍ ໕❭ƕa᯲ࡹ۵⧎ᅖĞĥḡᱱᮥ⪶ᯙ⧁ᙹᯩᨩ݅.
áᔪᨕ ᖁ⧪vᬑ, ⋉⚍Ñ࠺⧕ᕾ, ᮁ⦽ᗭ⧕ᕾ, Drucker-Prager model, ᔍ໕ᦩᱶᖒ
1. ᕽು
↽ɝᬑญӹ۵vᬑᨱ᮹⦽ᔍ໕❭ƕa᷾a⦹໕ᕽᔑᔍ┽ၰ☁ᕾඹᨱ᮹⦽ᯙၰᰍᔑ⦝⧕a᷾aࡹŁᯩᮝ໑, ʑ⬥ᄡ⪵ᨱ
᮹⦽Ḳᵲvᬑ᮹ၽᔾኩࠥa᷾aࡹ໕ᕽə᭥⨹ᖒᮡɪĊ⯩᷾a⦹Łᯩ۵⇵ᖙᯕ݅. ✚⯩ᬑญӹ᮹Ğᬑᨱ۵ǎ☁໕ᱢ᮹᧞70%a
ݓъėॡ
ᔑḡಽࢹ్ᩍᯩʑভྙᨱvᬑ⋉⚍ᨱ᮹⦽᧶ᮡᔍ໕❭ƕa
ݡᇡᇥᮥ ᯕŁ ᯩ݅(Kim et al., 2002; Jeong et. al., 2008;
Kim et al., 2012).
ᯝၹᱢᮝಽvᬑaᯝᱶʑeḡᗮࡹ໕ḡၹ᮹⧉ᙹእa᷾a⧉
ŝ ࠺ᨱ ḡၹ᮹ ᱥ݉vࠥෝ ᷾a┅۵ ༉š⯂ᙹಆ(matric suction)ᯕqᗭ⦹ʑভྙᨱᔍ໕᮹ᇩᦩᱶᖒᯕ᷾a⦹íࡽ݅(Ng and Shi, 1998; Kim et. al., 2004; Jeong et al., 2009). ੱ⦽
vᬑ⋉⚍ᨱ᮹⦽ᔍ໕᮹ᇩᦩᱶᖒᮡᖁ⧪vᬑᨱⓑᩢ⨆ᮥၼí
ࡽ݅. ᔍ໕ᯕÕ᳑⦽ᔢ┽ಽḡᗮࡹ໕⢽⊖ŝᔢᇡᨱᕽᯙᰆɁᩕᯕ
ၽᔾࡹ໑ᇩ⡍⪵ᔢ┽᮹⚍ᙹĥᙹ۵᯲ᦥḥ݅. əญŁḡ⦹ᙹ᭥۵
ᱱᱱԏᦥḡ໑ᔍ໕ᱥℕ᮹↽ݡ༉š⯂ᙹಆᮡ᷾a⦹ᩍᇡ⦹ᵲ ᯕa⧕ḡḡᦫ۵⦽ᇩ⡍⪵ᔍ໕ᮡᦩᱶ⦽ᔢ┽ಽ᳕ᰍ⦽݅. ᯕ్⦽
Õ᳑⦽ᔢ┽ᨱᕽᖁ⧪vᬑaၽᔾ⦹໕ᯙᰆɁᩕᯕၽᔾ⦽ḡᱱᮝ ಽྜྷᯕʫᙺᯕ⋉⚍⦹ᩍ༉š⯂ᙹಆᮡɪĊ⯩qᗭࡹŁᇩ⡍⪵
⚍ᙹĥᙹ۵ᱱ₉᷾aࡽ݅. ᖁ⧪vᬑᯕ⬥ᨱḲᵲvᬑaၽᔾ⧁
Ğᬑᨱ۵ᔍ໕᮹ᱥ݉vࠥ qᗭ᪡ᇩ⡍⪵⚍ᙹĥᙹ᷾aಽᯙ⦽
vᬑ⋉⚍a ᷾a⦹í ࡹᨕ ᔍ໕ᯕ ๅᬑ ᇩᦩᱶ⦽ ᔢ┽a ࡽ݅
(Rahardjo et al., 2009).
ᬑʑၽᔾࡹ۵ᔍ໕❭ƕ۵ᖁ⧪vᬑᨱⓑᩢ⨆ᮥၼḡᦫ۵݅
۵ᩑǍđŝaᯝᇡၽ⢽ࡽᱢᮡᯩᮝӹ(Brand, 1984; Pitts, 1985), Tan et al. (1987)ŝChatteriea (1989)۵ʑ᳕ᩑǍđŝ᪡ᔢၹࡹí
ᖁ⧪vᬑaᔍ໕❭ƕᨱⓑᩢ⨆ᮥၙ⊽݅Łᇥᕾ⦹ᩡ݅. Wei et al. (1991)ᮡᖁ⧪vᬑ᮹ḡᗮeŝvᬑaၽᔾ⦹ḡᦫᮡeᯕ
ᔑᔍ┽ၽᔾŝᩑšᯕᯩ݅Łᇥᕾ⦹ᩡ݅. Morgenstern (1992)ᮡ
ḡၹ᳑Õᨱᕽᖁ⧪vᬑ᮹ᩢ⨆ᯕᕽಽ݅ᮥᅕŁ⦽ၵᯩ݅.
ੱ⦽Rahardjo et al. (2001)ᮡᖁ⧪vᬑᨱݡ⦽ᩢ⨆ᮥ⇵aᱢᮝಽ
ᇥᕾ⦹ʑ᭥⦹ᩍvᬑᨱෙeɚᙹᦶᄡ⪵ෝᰆʑᱢᮝಽšₑ⦹
ᩍ, ᖁ⧪vᬑaᔍ໕ᇩᦩᱶᖒᨱᵲᬊ⦽ᩎ⧁ᮡ⦽݅Łᇥᕾ⦹ᩡ݅.
ᯕ᪡ zᯕ ᇩ⡍⪵ ᔍ໕ᨱᕽ۵ ᖁ⧪vᬑᨱ ᮹⧕ ḡၹ᮹ Ⅹʑ
vࠥa᳭ᬑࡹŁᦩᱶᖒᨱⓑᩢ⨆ᮥၙ⊹ʑভྙᨱᔍ໕❭ƕෝ
ᩩᔢ⦹Łᯕᨱᱢᱩ⦹íݡ᮲⦹ʑ᭥⧕ᕽ۵ᇩ⡍⪵ḡၹ᮹༉š⯂
ᙹಆŝḡၹ᮹Ⅹʑ⡍⪵ࠥᔢ┽ෝ⪶ᯙ⦹ᩍ༉š⯂ᙹಆᯕᔍ໕᮹
ᦩᱶᖒᨱၙ⊹۵ᩢ⨆ᮥȽ⧁⦥aᯩ݅. ᕽᅙᩑǍᨱᕽ۵
ᖁ⧪vᬑᨱෙᇩ⡍⪵ᔍ໕᮹ᇩᦩᱶᖒᮥᇥᕾ⦹ʑ᭥⧕⋉⚍᪡
⮺᮹ Ñ࠺ ⧕ᕾᮥ ࠺ᨱ ᙹ⧪⦹Ł eɚᙹᦶŝ ᄡ⩶ᯕ ᩑĥࡽ
ᮁ⦽ᗭ⧕ᕾ(Kim, 2010)ᮥᙹ⧪⦹ᩡᮝ໑, ᇩ⡍⪵☁᮹ᙹญ⦺ᱢ
✚ᖒᮥݡ⢽⦹۵⧉ᙹ✚ᖒłᖁᝅ⨹ᮥ☖⧕ᕽᬙḡᩎᨱᇥ⡍ࡹᨕ
ᯩ۵ᇩ⡍⪵⣮⪵☁᮹✚ᖒჵ᭥ෝđᱶ⦹ᩡ݅. ᮁ⦽ᗭ⧕ᕾᮥ
☖⧕ḡၹԕ༉š⯂ᙹಆᄡ⪵᪡ᔍ໕⢽⊖ḡၹ᮹ᄡ᭥ෝᇥᕾ⦹ᩍ
ᔍ໕᮹ᇩᦩᱶᖒᮥ❭ᦦ⦹ᩡᮝ໑, Drucker-Prager model᮹⧎ᅖ Ğĥʑᵡ(trial failure criterion)ᮥᱢᬊ⦹ᩍᔍ໕❭ƕa᯲ࡹ۵
⧎ᅖĞĥḡᱱᮥ ⪶ᯙ⦹ᩡ݅.
2. ᇩ⡍⪵⣮⪵☁ḡၹ᮹⧉ᙹ✚ᖒłᖁ
⧉ᙹ✚ᖒłᖁ(soil-water characteristic curve)ᮡᇩ⡍⪵ḡၹ ᮹ ᙹญ⦺ᱢ ✚ᖒŝ ᩎ⦺ᱢ ✚ᖒᮥ đᱶ⦹۵ Łᮁ⦽ ྜྷᖒᯕ
⧁ᙹᯩᮝ໑, ݡᇡᇥᝅԕᝅ⨹ᮥ☖⧕ℕᱢ⧉ᙹእᨱෙ༉š⯂ᙹ ಆ᮹šĥಽӹ┡ԝᙹᯩ݅. ᅙᱩᨱᕽ۵ᬑญӹ᮹ᔑḡ᮹ݡᇡᇥ
ᮥ₉ḡ⦹Łᯩ۵⪵v⣮⪵☁ḡၹ᮹ ⧉ᙹ✚ᖒłᖁᮥᔑᱶ⦹ʑ
᭥⧕ᕽᬙԕᩍ్ḡᩎ᮹⣮⪵☁ෝݡᔢᮝಽᝅ⨹ᮥᙹ⧪⦹ᩡᮝ໑, ᝅ⨹ᰆ⊹᪡ ᝅ⨹ႊჶᮡ ݅ᮭŝ z݅.
2.1 ਓୋ౿ࢫਏ߹
Figure 1ᨱӹ┡ԙၵ᪡zᯕᦶಆ᳑ᱩᰆ⊹(pressure panel), ᇡ⦝⊂ᱶᰆ⊹(volume tube), ᖙၚॵᜅⓍ(HAVE disks)॒ᮝಽ
Ǎᖒࡹᨕᯩ۵GCTS SWC-150 ᰆእ(Pham and Fredlund, 2004)
ෝᯕᬊ⦹ᩍᝅԕᝅ⨹ᮥᙹ⧪⦹ᩡ݅. Pressure chamber᮹׳ᮡ
ᦶಆᮥ ᯕᬊ⦹ᩍ ᰆe ᝅ⨹ᮥ ☖⧕ ⧉ᙹ✚ᖒłᖁᮥ ⊂ᱶ ⧁
ᙹᯩᮝ໑, ⪵v⣮⪵☁ᝅ⨹ᨱᱢ⧊⦽ᝅ✙ḩ༉௹ᬊ(300~500kPa) ᖙၚॵᜅⓍෝᔍᬊ⦹ᩡ݅. ⡍⪵ࡽഭᨱŖʑᦶᮥa⦹ᩍᇩ⡍
⪵ᔢ┽ಽอॅᨕ႑⇽ࡽeɚᙹ᮹ᇡ⦝۵pressure cell ၲᇡᇥᨱ
ᩑđࡽ 2}᮹ ቑ౼ᮥ ☖⧕ ᔑᱶ⧁ ᙹ ᯩ݅.
ℕᱢ⧉ᙹእෝᔑᱶ⦹۵ŝᱶᮡÕ᳑ŝᱶ(drying process)ŝ
႑⇽ࡽ eɚᙹa ⮺ ԕᇡಽ ݅ ⯂ᙹࡹ۵ ᮅŝᱶ(wetting process)ᮥ☖⧕ᔑᱶ⧁ᙹᯩ݅. ᝅᱽvᬑ⋉⚍⩥ᔢŝᮁᔍ⦽
ႊჶᯕᮅŝᱶᯕḡอ, Õ᳑ŝᱶᯕ⬥ᨱᮅŝᱶᮥ༉ᔍ⧕
⦹ʑ ভྙᨱ ᔢݚ⦽ eᯕ ᗭࡹŁ ᯕಆ⩥ᔢᨱ ᕽ ᧞
2kPaᱶࠥ᮹ ༉š⯂ᙹಆ ₉ᯕ(Huang et al., 2011)ෝ ᅕᯕʑ
ভྙᨱᅙᩑǍᨱᕽ۵Õ᳑ŝᱶᮥ☖⧕⧉ᙹ✚ᖒłᖁᮥᔑᱶ⦹
ᩡ݅. ᯕಆ⩥ᔢᮥw۵⧉ᙹ✚ᖒłᖁᮡ⡍⪵ℕᱢ⧉ᙹእa݅
íӹ┡ӹŖʑ⧉᯦⊹᪡ᇩ⡍⪵⚍ᙹĥᙹᨱᩢ⨆ᮥၙ⊹ʑࠥ
⦽݅(Kim et al., 2012).
ᅙ ᝅ⨹ᨱᕽ۵ ഭ᮹ ↽ᗭ݉᭥ᵲప(ⱗmin)ŝ ↽ݡ݉᭥ᵲప(ⱗ
max)ᮥᔑᱶ⦹Ł50% ᔢݡၡࠥಽᙹ⧪⦹ᩡ݅. ੱ⦽Ɂ॒⦽݅ḱᮥ
᭥⧕ 3݉ ݅ḱჶᮥ ᯕᬊ⦹ᩡᮝ໑, ᔢ⨆⋉⚍ෝ ᯕᬊ⦹ᩍ ഭෝ
⡍⪵⎑݅(Lee et al., 2007).
2.2 ਏ߹ଭࠤ-ࢄࠤୡন
ᅙᩑǍᨱᕽ۵ᕽᬙḡᩎᨱᇥ⡍⦹۵⪵v⣮⪵☁ෝ₥≉⧕3aḡ
⮺ ഭॅᨱ ݡ⧕ ⧉ᙹ✚ᖒłᖁ ᝅ⨹ᮥ ᙹ⧪⦹ᩡ݅. b ഭ᮹
ʑᅙᱢᯙྜྷญᱢ✚ᖒॅᮡTable 1ᨱᱶญ⦹ᩡᮝ໑, Figure 2ᨱ
ഭᨱ ݡ⦽ ᯦ࠥᇥ⡍łᖁᮥ ࠥ⦹ᩡ݅.
⧉ᙹ✚ᖒłᖁᮡ☁พᯱ᮹᯦Ğŝ᯦ࠥᇥ⡍, ǍᖒᖒᇥəญŁ
eɚእ॒ᨱŖʑ⧉᯦⊹(air-entry value)᪡ඹℕᱢ⧉ᙹእ
valve valve Low Pressure
Low High
PressureHigh Fredlund SWCC Device
Cell
Top Plate
Bottom Plate Air
Pres sure Volume Tube
350mm 300mm
585m m
(a) Schematic diagram of SWCC apparatus (b) SWCC apparatus
Fig. 1. GCTS pressure plate apparatus
Table 1. Physical properties of weathered soils
Property Weathered granite soil (Seoul area, 3 sites)
Specific gravity ( G
s) 2.64
Max. dry unit weight ( 쩂
d max, kN/m
3) 17.5
Min. dry unit weight ( 쩂
d min, kN/m
3) 13.5
Uniformity coefficient (C
u) 9.3
Coefficient of gradation( C
c) 0.77
USCS SP
(residual volumetric water content)a bb ݅í ӹ┡ӽ݅.
ᔍḩ☁ᨱ እ⧕᯦Ğᯕ᯲ᮡᝅ✙ḩ⮺ᯕӹᱱ☁ḩ⮺ᮥ⡍⧉⦹۵
ഭ۵ ࠺ᯝ⦽ ༉š⯂ᙹಆ᮹ ᄡ⪵ᨱ ෙ ℕᱢ⧉ᙹእ᮹ ᄡ⪵a
᯲íӹ┡ӽ݅. ᅙᩑǍᨱᕽ۵݅᧲⦽᯦ࠥᇥ⡍, ݅ḱ⧉ᙹእ, eɚእ
ෝaḥഭෝᕽᬙᩍ్ḡᩎᨱᕽ₥≉⦹ᩍᩍ్⮺ᮝಽǍᇥ⦹ḡ
ᦫŁ3aḡ᮹⧉ᙹ✚ᖒłᖁᮥᔑᱶ⦹ᩡ݅. əญŁvan Genuchten (1980)᮹ᱽᦩᮥᯕᬊ⦹ᩍ⠪Ɂᱢᯙ⧉ᙹ✚ᖒłᖁᮥӹ┡ԕᨩ݅
(Figure 3). ↽ᗭᯱჶᮥ☖⦹ᩍݡ⢽ᱢᯙ⧉ᙹ✚ᖒłᖁᮥᔑᱶ⦹
ᩡᮝ໑, ᔢšĥᙹ(R2)a 97.1%ᨱ ɝᱲ⦽ ᪅₉ෝ w۵݅.
ඹℕᱢ⧉ᙹእ۵༉š⯂ᙹಆᯕⓑ᳑Õᨱᕽ⊂ᱶ⧁ᙹᯩʑ
ভྙᨱ ᝅ⨹ᰆእ ၰ ᝅ⨹ᩍÕ᮹ ᱽ⦽ᮝಽ ඹℕᱢ⧉ᙹእෝ
⪶ᯙ⦹ḡ༜⦹ᩍ‘0’ᮝಽaᱶ⦹ᩡ݅. ᝅ⨹ᮥ☖⧕ᔑᱶࡽ⧉ᙹ
✚ᖒłᖁᮡTable 2ᨱӹ┡ԙၵ᪡zᯕ⧉ᙹ✚ᖒłᖁ᮹Ŗʑ⧉
Fig. 3. Soil-water characteristic curve of weathered soils
Table 2. Curve fitting parameters of the SWCC and permeability (RETC, 2009)
Curve Fitting Parameters van Genuchten (1980) 쩀 = 0.0412 /kPa n = 2.07 m = 0.517 Saturated volumetric water
content (쩇
s) 0.39
Residual volumetric water
content ( 쩇
r) G0.0
Correlation factor ( R
2) 0.9719 Saturated permeability (k
s) 4.67 × 10
-6m/sec
᯦⊹۵ⱕ, łᖁ᮹Ğᔍ۵n, ඹ⧉ᙹእᨱݡ⦽ĥᙹ۵mᮝಽ
⢽⦹ᩡᮝ໑, ᇩ⡍⪵⚍ᙹĥᙹᔑᱶၰ⋉⚍⧕ᕾ᮹᯦ಆߑᯕ░
ಽ ᔍᬊ⦹ᩡ݅.
3. ᮁ⦽ᗭ⧕ᕾ
3.1 ࢼࢺ୨ਐ
ᯱᩑᔢ┽᮹ᇩ⡍⪵☁۵☁พᯱ(ns), ྜྷ(nw),Ŗʑ(na) 3ᔢᮝಽ
Ǎᖒࡹᨕᯩᮝ໑, ℕᱢእ(1 = ns + nw + na)ᨱbᗭॅᮥ
Ǎᇥ⦹۵Mixture Theory (Coussy, 2010)aḡ႑ႊᱶ᮹ɝeᮥ
ᯕŁᯩ݅. ᇩ⡍⪵☁᮹ḡ႑ႊᱶᮡ☁พᯱ᮹ᄡ᭥(u)᪡☁พᯱ
ԕᇡᨱᕽၽᔾࡹ۵eɚᙹᦶ(Pw)ᯕၙḡᙹᯕ໑, ྜྷℕᨱ᯲ᬊ⦹۵
༉ु ⯹ॅᮡ ⠪⩶ᮥ ᯕ്݅۵ ⠪⩶ႊᱶ(balance equation)ŝ
ྜྷℕ᮹ ḩపᯕ eᨱ ᄡ⦹ḡ ᦫ۵݅۵ ḩపᅕ᳕᮹ ჶ⊺
(balance of mass)ᮥ ʑᅙaᱶᮝಽ ⦹Ł ᯩ݅.
ᇩ⡍⪵☁ԕྜྷ᮹ᯕ࠺ᮡ⮺᯦ᯱ᮹ᬡḢᯥᨱ᳦ᗮࡹʑভྙᨱ
⮺᯦ᯱᨱݡ⦽⠪⩶ႊᱶ(balance of linear momentum)ᮝಽᇡ
░ᄡ᭥ෝĥᔑ⧁ᙹᯩᮝ໑, ᇩ⡍⪵☁ԕྜྷ᮹⮱ᮥ❭ᦦ⦹۵
⠪⩶ႊᱶ(balance of mass of the water)ᮝಽᇡ░eɚᙹᦶᮥ
ĥᔑ⧁ᙹᯩ݅. ᕽᅙᩑǍᨱᕽ۵⋉⚍ᨱ᮹⦽ḡၹ᮹Ñ࠺ŝ
ᙹญ⦺ᱢ-ᩎ⦺ᱢ(hydro-mechanical)✚ᖒᮥ࠺ᨱŁಅ⧁ᙹᯩ
۵ᮁ⦽ᗭ⧕ᕾ⎵ऽෝǍᖒ⦹ᩡᮝ໑ḡ႑ႊᱶᮡ݅ᮭŝz݅
(Kim, 2010).
0 g
DIV
σ
+ρ
= (1)w
S v v
ws p
n S + DIV = − DIV ~
∂
− ∂ &
(2)ᩍʑᕽⱦ۵ᱥ᮲ಆ, p۵ၡࠥ, g۵ᵲಆaᗮࠥ, n
1)
ᮡeɚශ, S۵ ⡍⪵ࠥ, s۵ ༉š⯂ᙹಆ, ṗw۵ eᨱ ෙ eɚᙹᦶ, v۵⮺᯦ᯱ᮹⮱ᗮࠥ, ṽw۵ᝅᱽྜྷ᮹ᗮࠥ(vw)᪡⮺᯦ᯱ᮹ᗮࠥ(v) ᮹ ₉ᯕෝ ӹ┡ԙ݅.
ੱ⦽᮲ಆ⧕ᕾᮥ᭥⧕⦥⦽ᮁ⬉᮲ಆǍᖒႊᱶŝᇩ⡍⪵☁
ԕ ྜྷ᮹ ⮱ᮥ ❭ᦦ⧁ ᙹ ᯩ۵ Darcy’s law۵ Eq. (3)ŝ Eq.
(4)᪡zᯕӹ┡ԝᙹᯩ݅(van Genuchten, 1980; Coussy, 2010).
(3)
) g )(
,
~ (
wRw
w
k
wS n p
v = −∇ + ρ
(4)ᩍʑᕽce۵┥ᖒĥᙹ, ⱙ ᮡᄡ⩶ᮉ, kw۵ᇩ⡍⪵☁⚍ᙹĥᙹ, x۵ᮁ⬉᮲ಆĥᙹ, ᯙᰆಆ(tension)ᮥ‘+’, ᦶ⇶ಆ(compression)
ᮥ ‘-’ sᮥ ӹ┡ԕ໑, ᅙ ᙹ⊹⧕ᕾᨱᕽ۵ ᮁ⬉⡍⪵ࠥ(effective degree of saturation, Se)ෝᱢᬊ⍽⧕ᕾᮥᙹ⧪⦹ᩡ݅. pwr (=
pw / nw)۵ྜྷ᮹ᝅᱽၡࠥෝӹ┡ԕ໑, eᨱෙ⋉⚍᪡ḡၹ᮲ಆ
ᔑᱶᨱ⦥⦽eɚශ, əญŁᱥℕၡࠥ۵Eq. (5)᪡Eq. (6)ᮝಽᇡ
░ ĥᔑ⧁ ᙹ ᯩ݅.
) ( tr 1 and
1 v v
v v
n nn
n
ε ε
ε ε
Δ = ΔΔ +
Δ
= +
+ (5)
wR
sR
n d S
d
n ρ θ ρ
ρ = [ 1 − ( )] + ( ) ( )
(6)ᩍʑᕽ⋉⚍ੱ۵ᇡ⦹ᵲᨱ᮹⧕ᄡ⪵⦹۵eɚශᮡℕᱢᄡ⩶
ශ(ⶸⱙv)᮹⧉ᙹᯕ໑, ᱥℕၡࠥᩎeᨱෙeɚශ(n(d))ŝ
⡍⪵ࠥ(S(ⱜ))ᨱ đᱶࡽ݅.
1) ‘n’ᮡ⧉ᙹ✚ᖒłᖁ᮹ᝅ⨹ᱶᙹಽ‘n’ᮡeɚශಽ⢽ෝǍᇥ⦹ᩡᮭ.
3.2 କ૬ীැজଡੵճࠤஶ
ᮁ⦽ᗭ⧕ᕾᮥ᭥⦹ᩍ⮺᮹ᄡ᭥᪡ԕᇡeɚᙹᦶᮥĥᔑ⧁
ᙹᯩ۵እᖁ⩶᧞⩶(coupled nonlinear weak form)ᮥᮁࠥ⦹ʑ
᭥⧕aᵲඹ⧎ჶ(method of weighted residual)ᮥᱢᬊ⦹ᩡᮝ ໑, 2aḡၙḡᙹᨱݡ⧕⧪⧉ᙹ(matrix form) ⩶┽ᯙᮁ⦽ᗭ
ႊᱶᮝಽ⢽⩥⦹ʑ᭥⧕ᕽ⩶ᔢ⧉ᙹ(shape function)ෝᱢᬊ⦹
ᩡ݅. bᗭ(element)ॅᯕaḡŁᯩ۵ԕಆ(internal force)Ł
ಆ(external force) ᄂ░ॅ᮹ ⠪⩶᳑Õᨱ ᮹⦹ᩍ ĥᔑ sॅᮥ
٥ᱢ┅໕ᩑĥࡽእᖁ⩶⡍ྜྷᖁၙᇥႊᱶ(coupled nonlinear parabolic partially differential equation)ᮥ ᱥℕ ᗭᨱ ݡ⧕
Eq. (7)ŝ zᮡ ⧪ ⩶┽ಽ ⢽⩥⧁ ᙹ ᯩ݅.
) ( ) ( )
(D D F D F D
C &+ INT = EXT (7)
ᩍʑᕽ
⎥ ⎦
⎢ ⎤
⎣
= ⎡ θ D d
,
⎥ ⎦
⎢ ⎤
⎣
= ⎡ θ&
&
& d
D
,⎥ ⎦
⎢ ⎤
⎣
⎡
= −
) , ( )
(
0 ) 0
(
, 1,θ
θ
θθ
K d
D K
C
dINT INT,
⎥ ⎦
⎢ ⎤
⎣
⎡
−
= −
) , (
) ( ) , ) (
(
, 2, ,θ θ θ
θ θ
d F
F d D F
F
INTINT d INT
d INT
,
⎥ ⎦
⎢ ⎤
⎣
= ⎡
dEXTEXTEXT
F d D F
F
,,
( , )
)
(
θθ
ᮥ ᮹ၙ⦽݅.
Backward Euler ႊჶᮥᯕᬊ⦽ԕᱽᱢႊჶ(fully-implicit time integration)ᮥ ☖⧕ ᭥ Eq. (7)ŝ zᮡ ⧪ႊᱶᮥ Newton- Raphson ႊჶᮥᔍᬊ⦹ᩍၙḡᙹᯙḡၹ᮹ᄡ᭥᪡ԕᇡeɚᙹᦶ
ᮥ ĥᔑ⧁ ᙹ ᯩᮝ໑, Eq. (8)ŝ Eq. (9)ಽ ݅ ⢽⩥ࡽ݅.
) ( )
( )
( +1 +1+ +1 = EXT n+1 INT n
n
n D F D F D
D
C & (8)
) ( )
( )
(Dn+1 Dn+1+K Dn+1 Dn+1= FEXT Dn+1
C & (9)
⧪ႊᱶᮡNewton-Raphson ႊჶᨱ᮹⧕ᖁ⩶⪵ࡽ⩶┽ಽ
⩥ᰍ e݉ĥ(tn+1)ᨱᕽ ⧕ᕾᯕ ᙹ⧪ࡹ໑ ԕಆᨱ ݡ⧕ᕽ FINT (Dn+1)GK (Dn+1)Dn+1ᯥᮥaᱶ⧁ᙹᯩ݅. đŝᱢᮝಽᙹญ⦺ᱢ- ᩎ⦺ᱢ ⮺᮹ Ñ࠺ᮥ እᖁ⩶ ᱲɝ ႊჶᯙ ၹ ԕᱽᱢ ႊჶ (semi-implicit time integration)ᮝಽ đŝෝ ࠥ⇽⦹ᩡ݅(Kim, 2010).
4. vᬑ✚ᖒᨱෙᇩ⡍⪵⣮⪵☁ᔍ໕᮹ᙹ⊹⧕ᕾđŝ
4.1 টෘԳࠜճߙࡦւใߚଭ࣡ฃ
vᬑᨱ᮹⦽ᔍ໕᮹ᇩᦩᱶᖒᮡᬑʑᖁ⧪vᬑᨱ᮹⧕Ⓧí
᳭ᬑࡽ݅. ☁ᕾඹၽᔾŝzᮡฯᮡእ┩໕❭ƕ᮹ᔍಡෝᔕ⠕ᅕ໕,
Õ᳑⦽ᔢ┽ᨱᕽԕญ۵⡎ᬑᅕ݅۵ᖁ⧪vᬑಽ ⡍⪵ࠥa׳ᮡ
ḡၹ᳑Õᨱᕽԕญ۵vᬑᨱᕽᔍ໕❭ƕ᮹᭥⨹ᖒᯕ׳݅۵äᮥ
ᙹ ᯩ݅. ᖁ⧪vᬑa ၽᔾ⦽ ᔍ໕ᨱᕽ۵ ᇩ⡍⪵ ⚍ᙹĥᙹa
ዉḡŁ༉š⯂ᙹಆᮡɪĊ⯩ᱡ⦹ࡽᔢ┽ᨱᕽ ᔍ໕᮹⢽⊖ᯕ
vᬑಽᯙ⦹ᩍዉญ⡍⪵ࡹʑᛞíࡽ݅. ḡၹ᮹⡍⪵⚍ᙹĥᙹᅕ݅
ⓑvᬑvࠥaᯝᱶ⦽vᬑḡᗮeᮥw۵݅໕ ᔍ໕᮹⢽⊖᮹
⡍⪵۵ʫᨕḡŁ, ᔍ໕ᇶƕᨱݡ⦽᭥⨹ࠥ۵ᬒ⍅ḡíࡽ݅.
ᅙᩑǍᨱᕽᨙɪ⦽ᮁ⦽ᗭ⧕ᕾ⥥ಽəఉᮥᔍᬊ⦹ᩍ, ᖁ⧪v ᬑ᳑ÕᮡⅩʑᔍ໕᮹⡍⪵ࠥ᳑Õᮥ36, 51, 77%᮹3aḡ᳑Õᮝ ಽ ӹ٥ᨕᕽ vᬑ ᔍ໕ԕ᮹ ༉š⯂ᙹಆ᮹ ᄡ⪵᪡ ❭ƕၽᔾ
a܆᭥⊹᮹ᄡ᭥ෝ⪶ᯙ⦹ᩡ݅. bbEq. (10)ŝEq. (11)ᨱᕽౝ, van Genuchten (1980)᮹⧉ᙹ✚ᖒłᖁŝᇩ⡍⪵⚍ᙹĥᙹႊᱶ
ᮥ bb ᔍᬊ⦹ᩍ ᇩ⡍⪵☁᮹ ⋉⚍✚ᖒᮥ ᩩ⊂⦹ᩡ݅.
1) 1
} (
) ( 1
{ n n
s + s − −
=
θ α
θ
(10)[
1 (1 1/S1/m)m]
2S k
kw = s⋅ − − (11)
ᩍʑᕽⱜ۵ℕᱢ⧉ᙹእ(volumetric water content), ⱜs۵⡍⪵
ℕᱢ⧉ᙹእ, ⱕ۵ Ŗʑ⧉᯦⊹(air-entry value)ᨱ ݡ⦽ ᝅ⨹ᱶᙹ, nᮡłᖁ᮹Ğᔍᨱݡ⦽ᝅ⨹ᱶᙹ, əญŁmᮡඹℕᱢ⧉ᙹእᨱ
ݡ⦽ᝅ⨹ᱶᙹᯕ݅. ⣮⪵☁᮹⡍⪵⚍ᙹĥᙹ(4.67 × 10-6m/sec)ᅕ
݅ⓑvᬑvࠥ(5.56 × 10-6m/sec)ෝᔍᬊ⦹ᩍᇩ⡍⪵ᔍ໕᮹⢽໕ ᯕ᪥ᱥ⡍⪵aa܆⦹ࠥಾⅩʑ᳑Õᮥᱢᬊ⦹ᩡ݅. ᮁ⦽ᗭ⧕ᕾ ᨱᕽ۵ᖁ⧪vᬑෝŁಅ⦹ʑ᭥⧕Ⅹʑ⡍⪵ࠥ᳑ÕᮥÕ᳑ᔢ┽ᨱ ᕽ ᮅᔢ┽᮹ 3aḡ ᳑Õ(36, 51, 77%)ᮝಽ ӹ٥ᨕᕽ ᇩ⡍⪵
⋉⚍⧕ᕾŝ༉š⯂ᙹಆ᮹ᄡ⪵ෝ⪶ᯙ⦹ᩡ݅. vᬑ✚ᖒᮡvᬑv
ࠥ᪡ ḡᗮeᮝಽ ӹ٭ ᙹ ᯩ۵ߑ, vᬑvࠥ۵ ᯝᔢᨱᕽ ᯱᵝ
ၽᔾࡹ۵20mm/hrಽḡၹᮥ⡍⪵┍ᙹᯩ۵Ⓧʑಽᖅᱶ⦹ᩍ
᧶ᮡᔍ໕❭ƕෝᮁၽ⧁a܆ᖒᯕᯩ۵᳑Õᯕ݅. vᬑḡᗮeᮡ
vᬑvࠥᨱ᮹⧕ ᔍ໕ ⢽⊖ᇡaᯥ᮹ ʫᯕʭḡ ᪥ᱥ⡍⪵ࢁ ᙹ
ᯩ۵ 24eᮝಽ aᱶ⦹ᩡ݅
Figure 4᪡zᯕᔍ໕׳ᯕa20mಽᯝᱶ⦹໑, ᔍ໕᮹Ğᔍ۵
40°ಽđᱶ⦹Łᔍ໕ᱥℕᩢᩎᮡ॒ႊ(isotropic)ᯙḡၹᮝಽaᱶ
⦹ᩡᮝ໑, ᳭ᬑ⊂໕ŝ⦹ᇡĞĥ໕ᮡᇩ⚍ᙹĞĥ᳑Õᮝಽᯕᨱ
ᙹḢ⦽ႊ⨆ᮝಽ۵Ŗʑ᪡ྜྷ᮹⮱ᯕၽᔾ⦹ḡᦫ۵݅Łaᱶ⦹
ᩡ݅. ੱ⦽᳭ᬑ⊂໕ᮡᙹ⠪ႊ⨆ᄡ᭥ෝ, ⦹ᇡĞĥ໕ᮡᙹ⠪ŝ
ᩑḢႊ⨆᮹ᄡ᭥ෝǍᗮ⦹ᩡ݅. Ⅹʑḡ⦹ᙹ᭥۵Õʑෝaᱶ⦹
ᩍᔍ໕ၵ݆ᨱᕽ10m ׳ᯕᨱᕽᙹ⠪⦹í᭥⊹⦹ᩡŁ, ḡ⦹ᙹ᭥
ᔢᇡᯙᇩ⡍⪵☁ᩢᩎᨱᕽ׳ᯕᨱ↽ݡ༉š⯂ᙹಆᮥ60kPa,
↽ᗭ20kPaಽaᱶ⦹ᩡ݅(Kim et al., 2012). Table 3ᮡ⧕ᕾᨱ
Fig. 4. Soil slope mesh and initial condition
Table 3. Mechanical properties of weathered soils and input parameters
Lame parameter ( 쩊, 쩋) 29×10
6, 7×10
6Pa Effective friction, dilatancy angle and cohesion
( 쩤', 쩗', c') 32°, 10°, 5kPa
Plasticity parameters ( h
c, 쩁) 10MPa, -1(TC) Solid real density ( p
sR) 2700 kg/m
3Water real density (p
wR) 1000 kg/m
3Air real density ( p
aR) 1.2 kg/m
3Viscosity of water ( 줂
w) 10
-3Pa·s Initial porosity ( n = n
w+ n
a) 0.4 (=0.2+0.2) Permeability at saturation (k
s) 4.67×10
-6m/sec Rainfall intensity and duration 20mm/hr, 24hr
-50 -50
-50
-50 -50
0
0
0 0
0 0
50
50
50 5
50 50 50
00
100
100
100 100 100
150 50
200 200
250 250
300 300
350 50
Fig. 5. Distribution of matric suction in soil slope with S=36%
0 0
0
0 0
0
50 50
50 50
50 50 50
100
100 100 100
150 150 00 200
Fig. 6. Distribution of matric suction in soil slope with S=51%
-20 -20
-20 -20
-20 -20 -20
-20 -20 -20
0
0
0
0
0 0 0
20
20
20 20
20 20 20
40 40
40 40 40
60
60 60 60
80 80 80
100 100 100
Fig. 7. Distribution of matric suction in soil slope with S=77%
ᔍᬊࡽ vࠥᱶᙹ᪡ ⮺᮹ ྜྷญᱢ ᖒḩᮥ ӹ┡ԕŁ ᯩ݅.
Figure 5۵ vᬑಽ ᯙ⦽ ᔍ໕ ԕ᮹ ༉š⯂ᙹಆ ᇥ⡍ᨱ ݡ⧕
ษḡส ݉ĥෝ ᅕᩍᵝŁ ᯩ݅. ᮁ⦽ᗭ⧕ᕾᨱ ᮹⦽ meshᨱᕽ
⦹ӹ᮹ᗭ۵ᄡ᭥ෝ᭥⦽9}ᱩᱱŝeɚᙹᦶᮥ᭥⦽4}ᱩᱱᮝ ಽǍᖒࡽ݅. ᖁ⧪vᬑ᮹Ⓧʑᨱݍḡ۵Ⅹʑᔍ໕᮹⡍⪵ࠥ
a ⍅ḱᨱ ༉š⯂ᙹಆ᮹ Ⓧʑࠥ ᯲ᦥḡŁ ᯩᮭᮥ ⪶ᯙ⧁
ᙹᯩ݅(Figures 5-7). ⡍⪵⚍ᙹĥᙹᅕ݅ⓑvᬑvࠥ᮹᳑Õᯕအ ಽ, ᔍ໕᮹⢽⊖᮹༉š⯂ᙹಆᮡɪĊ⯩qᗭ⦹ᩍ(ᷪ, 0ᨱᱲɝ) ᔍ໕⢽⊖ᯕᱱ₉ᱢᮝಽ⡍⪵ࡹŁᯩᮭᮥ⪶ᯙ⧁ᙹᯩ݅. Contour ಽ⪶ᯙ⦹۵༉š⯂ᙹಆ᮹ᄡ⪵ෝḢᱲ⪶ᯙ⦹ʑ᭥⦹ᩍFigure 4ᨱᕽӹ┡ԙäŝzᯕAᗭ᪡Bᗭᨱᕽ༉š⯂ᙹಆ᮹ᄡ⪵ෝ
⡍⪵ࠥᨱ Figures 8-9ᨱ ӹ┡ԕᨩ݅.
ᯝၹᱢᮝಽvᬑಽᯙ⦽ᔍ໕᮹ᇩᦩᱶᖒᮡ⋉⚍ᙹಽᯙ⦽⢽⊖
⡍⪵ಽᇡ░᯲ࡽ݅. ə௹ᕽᔍ໕᮹⢽⊖᮹ᔢ݉(A element)ŝ
⦹݉(B element)᮹༉š⯂ᙹಆ᮹ᄡ⪵ෝእƱ⦹Łᯱ⦹ᩡ݅. b
ᗭᨱᕽ⊂ᱶ⦽༉š⯂ᙹಆᮡ4}᮹ᱩᱱ(node)ᨱᕽᮡ⠪Ɂs
ᮥᔍᬊ⦹ᩍࠥ⦹ᩡ݅. vᬑᔢ݉ᨱ᭥⊹⦽Aᗭᨱᕽ⡍⪵ࠥ
ᨱෙ༉š⯂ᙹಆ᮹qᗭ۵aᰆÕ᳑⦽ᔍ໕(⡍⪵ࠥ36%)᮹
Ğᬑqᗭ᮹⡎ᯕaᰆⓍíӹ┡ԍᮝӹ, ᖁ⧪vᬑಽᯙ⦽ᮅ⦽
ᔍ໕᳑Õ(⡍⪵ࠥ 51%, 77%)ᮡ ༉š⯂ᙹಆ᮹ qᗭ⡎ᮡ ᯲ᮝӹ
Ⅹʑ᳑Õᇡ░wŁᯩ۵⧉ᙹపᨱ᮹⧕⡍⪵ࡹʑ᭥⦽a܆ᖒᮡ
aᰆ ׳ᦹ݅.
Fig. 8. Variation of matric suction at A element for a day
Fig. 9. Variation of matric suction at B element for a day
Fig. 10. Vertical displacement at A element and horizontal displacement at B element
ᔍ໕᮹⦹݉(B element)ᨱᕽ۵ḡ⦹ᙹ᭥ᇡ░ᔢᇡಽᱱ₉ᱢᮝ ಽ᷾a⦹۵༉š⯂ᙹಆ᳑Õভྙᨱ⡍⪵ࠥ36%(↽ݡ༉š⯂ᙹಆ
60kPa)᪡⡍⪵ࠥ51%(↽ݡ༉š⯂ᙹಆ40kPa)ᨱ⧕ݚ⦹۵Bᗭ ᮹4}ᱩᱱ᮹Ⅹʑ⠪Ɂ༉š⯂ᙹಆᮡݡఖ25kPaಽzᮡⓍʑᨱᕽ
᯲⦽݅. vᬑaᰆÕ᳑⦽ᔍ໕᮹ᔢ┽ᨱᕽ⢽⊖᮹༉š⯂ᙹಆ
ᮡɪĊ⦹íqᗭ⦹ʑভྙᨱ⡍⪵ࠥ36%ᯙᔍ໕᳑Õᨱᕽ᳡
ෙ༉š⯂ᙹಆ᮹qᗭෝᅕᯕŁᯩ݅(Figure 9). ᔍ໕⢽⊖ᨱᕽ
Ⅹʑ⡍⪵ࠥaԏᮥᙹಾvᬑಽᯙ⦽༉š⯂ᙹಆ᮹qᗭ۵᯲ᦥḡ ӹ, ⢽⊖ᨱᕽzᮡ༉š⯂ᙹಆᮥwŁᯩ݅໕ḡၹᯕÕ᳑⦽ᔢ┽ᯝ ᙹಾqᗭ⦹۵እᮉᮡᬒዉḥ݅۵äᮥ⪶ᯙ⧁ᙹᯩ݅.
4.2 Գଭॷଭ࣡෴ր൞֏
ᖁ⧪vᬑಽᯙ⦽ḡၹ᮹Ⅹʑ⡍⪵ࠥ᳑Õᮥ36, 51, 77% 3aḡ ಽǍᇥ⦹ᩍᔍ໕⢽⊖᮹ᔢ݉ŝ⦹݉᮹᭥⊹ᨱᕽᙹḢၰᙹ⠪ᄡ᭥
ෝᮁ⦽ᗭ⧕ᕾ⥥ಽəఉᮥᯕᬊ⦹ᩍĥᔑ⦹ᩡ݅. ᔍ໕⢽⊖ᔢ݉
(A element)ŝ⦹݉(B element)ᮡvᬑಽᯙ⧕ḡၹᯕÑ࠺⦹۵
᭥⊹ෝ❭ᦦ⧁ᙹᯩᮝ໑⢽⊖❭ƕෝᩩᔢ⧁ᙹᯩ۵Ŕᯕ݅. b
ᗭ᮹ 9}᮹ gauss points ᵲᨱᕽ ᵲᦺᨱ ᭥⊹⦽ gauss point đŝsॅᮥᯕᬊ⦹ᩍ⠪Ɂᱢᯙᄡ᭥ෝӹ┡ԕᨩ݅. Aᗭ᮹ᙹḢ ᄡ᭥(y-direction)᪡Bᗭ᮹ᙹḢၰᙹ⠪ᄡ᭥(x-direction)ෝ⊂
ᱶ⦹ᩍ Ⅹʑ ⡍⪵ࠥ ᳑Õᨱ ෙ Ñ࠺᮹ ₉ᯕᱱᮥ ⪶ᯙ⦹ᩡ݅.
Figure 10ᮡAᗭ᮹ᙹḢᄡ᭥᪡Bᗭ᮹ᙹ⠪ᄡ᭥ෝᅕᩍᵡ݅.
ᔍ໕ᔢ݉᮹ᙹḢᄡ᭥᪡⦹݉᮹ᙹ⠪ᄡ᭥۵bb᳭⢽⇶᮹‘-’ႊ⨆
ᯙᦥ௹᪡ႊ⨆ᮝಽÑ࠺⦽݅. ḡၹ᮹Ⅹʑ⧉ᙹపᯕฯᮥᙹಾ
b᭥⊹ᨱᕽၽᔾ⦹۵ᄡ᭥۵⍅ḡŁᯩᮭᮥᙹᯩ݅. ၹ໕ᨱ
BᗭᨱᕽᙹḢᄡ᭥۵vᬑ᯲ŝ࠺ᨱ݉᭥ᵲప᮹ᔢᮝಽ
ᔍ໕᮹᭸ႊ⨆ᯙ‘+’ႊ⨆ᮝಽÑ࠺⦽⬥ᨱᱱ₉ᱢᮝಽ⋉⚍ᙹ᮹
Fig. 12. Vertical displacement at the top elements of soil slope
-150 -100
-100 -50 -100
-50
-50 -50
0
0
0 0
50
50
50
100
100
150
150
200
00 250 00
Fig. 13. Distribution of trial failure criterion using DP model in S=36%
-150 -100
-100 -50 -100
-50
-50 -50
0
0
0 0
50
50
50
1
100
0
150
150
200
00 250 300
Fig. 14. Distribution of trial failure criterion using DP model in S=51%
-200 -100
-100
-100 0
0
0 0
10
100
0
200
200 300
Fig. 15. Distribution of trial failure criterion using DP model in S=77%
ḥ⧪ᮝಽᬱ᭥⊹ಽࡹ࠭ᦥ᪉݅. ə్ӹ, ⡍⪵ࠥaᱽᯝⓑ77%᮹
Ⅹʑḡၹᔢ┽ᨱᕽ۵Bᗭᨱᕽၽᔾ⦹۵ᙹḢᄡ᭥aᬱᔢ┽ಽ
⫭ᅖࡹḡ ᦫŁ ᧞e ᭸ႊ⨆ᮝಽ Ñ࠺ࡹᨕ ᯩ۵ äᯕ ⪶ᯙࡽ݅.
Ⅹʑ⧉ᙹእa᯲ᮡ᳑Õᅕ݅ᙹḢᄡ᭥ᨱݡ⦹ᩍ⪶ᩑ⦽₉ᯕᱱᮥ
ᅕᯕŁ ᯩᨩ݅.
Figure 12۵ᔍ໕ᔢ݉ᨱ᭥⊹⦽16}᮹ᗭᨱᕽၽᔾ⦽ᙹḢᄡ
᭥ෝ ӹ┡ԕᨩ݅. ᔍ໕᮹ ŝ ᪅ෙ᮹ Ğĥ᳑Õᮝಽ x⇶᮹
ᄡ᭥aၽᔾ⦹ḡᦫࠥಾŁᱶ⎑ᮝӹy⇶᮹ᙹḢᄡ᭥ෝšₑ⦽
đŝ, ᔍ໕ᔢ݉ᦩᮝಽᯕ࠺⧁ᙹಾྜྷ᮹⦹ᵲŝ⋉⚍aฯᦥḡအ ಽ ᙹḢᄡ᭥a ᷾a⦽݅. ᯕ۵ ┥ᖒᨱᕽ ᗭᖒ Ñ࠺ᮝಽ í
ḥ⧪⦹۵ ᬱᯙᯕʑࠥ ⦹݅(Figures 13-15).
4.3 Drucker-Prager (1952) ࡦ܄ଡଲॷ൞֏ंজ
ᔍ໕᮹❭ƕᔢ┽ෝ⪶ᯙ⦹ʑ᭥⦹ᩍDrucker-Prager (DP) ᗭᖒ༉ߙᮥ ᱢᬊ⦹ᩍ ❭ƕᨱ ᱲɝ⦹ᩡ۵ḡ ⠪a⧁ ᙹ ᯩ݅. ḡၹᮡ
࠺ḩ⦽(isotropic) ᰍഭಽaᱶ⦹ᩍᖁ⩶┥ᖒ⧕ᕾ(linear elasticity) ᮝಽ ᯕᨕᲭḡอ, ᮲ಆĞಽෝ⇵ᱢ⦹ʑ᭥⧕ᕽEq. (12)ŝzᮡ
DP ༉ߙᮥ ᔍᬊ⦹ᩍ ⧎ᅖ ᮁྕෝ ❭ᦦ⦹ᩡ݅.
(12)
ᩍʑᨱᕽs۵⇶₉᮲ಆ(deviatoric stress), p'۵⠪Ɂᮁ⬉᮲ಆ (mean effective sterss), c'۵ᮁ⬉ᱱ₊ಆ, 쩤'۵ᮁ⬉ԕᇡษₑb, ⱖ= -1ᮡDP༉ߙᮥMohr-Coulomb ᔝ⇶ᦶ⇶vࠥ(TC)ෝӹ┡ԕ ۵⧎ᅖĞĥ໕ᮥ᮹ၙ⦹໑, ⱖ= 1ᮡMohr-Coulomb ᔝ⇶ᯙᰆvࠥ
(TE)ෝӹ┡ԕ۵⧎ᅖĞĥ໕ᨱ⧕ݚ⦽݅. ⧎ᅖ( f, yield function) ĥᔑᨱf < 0۵┥ᖒÑ࠺ᮥ᮹ၙ⦹໑, f G0۵ᗭᖒÑ࠺ᮥ
᮹ၙ⦽݅. Eq. (3)ŝzᯕᮁ⬉᮲ಆᮡ༉š⯂ᙹಆ⧉ᙹಽǍᖒࡹᨕ
ᯩʑভྙᨱᇩ⡍⪵☁᮹┥ᖒÑ࠺ᮥ⧕ᕾ⧁ᙹᯩ݅. DP༉ߙᮡ
ᙹ⊹⧕ᕾᔢᨱᕽᗭᖒᄡ⩶ᮥḥ⧪┅ḡ༜⦹Ł, ݉ḡ❭ƕ⠪aᨱ
ݡ⦽ʑᵡᮝಽᔍᬊࡹᨩ݅. ə௹ᕽ, ☁พᯱෝᩑᗮℕಽaᱶ⦹۵
ᮁ⦽ᗭ⧕ᕾᨱᕽftrⴍ0 (trial failure function) ᮹ၙ۵ᗭᖒÑ࠺
ᯕ᯲⦹۵ᙽeᮝಽaᱶ⧁ᙹᯩ݅. ᙹ⊹⧕ᕾđŝಽ៉, Figures 13~15۵ 3aḡ ⡍⪵ࠥᨱ ၽᔾ⧁ ᙹ ᯩ۵ ᔍ໕᮹ ❭ƕa
ḥ⧪⦹۵ ŝᱶᮥ ⪶ᯙ⧁ ᙹ ᯩ݅.
ᙹ⊹⧕ᕾđŝᨱᕽᙹᯩॐᯕ, ᇩ⡍⪵☁ᔍᔍ໕᮹༉š⯂ᙹಆ ᯕ᯲ᦥḩᙹಾ, ੱ۵⡍⪵ࠥaⓕᙹಾᔍ໕᮹ᇩᦩᱶᖒᮡ⍅ḥ݅
(contour᮹ ᔪʵಽ ᱶࠥ᮹ ₉ᯕa Ǎᇥ). ⡍⪵ࠥa ᷾a⦹໕ᕽ
⧎ᅖĞĥᇥ⡍ࠥ۵ᔍ໕᮹ᔢ݉ŝ⢽⊖ᇡ᭥ᨱᗭᖒᩢᩎᮝಽᄡ⧕
aŁ ᯩᮭᮥ ⪶ᯙ⧁ ᙹ ᯩ݅. ⡍⪵ࠥ 77%ᨱ ⧕ݚ⦹۵ Figure 15ᨱᕽ۵ᔍ໕⢽⊖ᵲeᇡɝᨱᕽᔍ໕❭ƕa܆ᖒᯕ⪶ᯙࡹ۵
ᗭᖒᩢᩎᮝಽftr = 0ᯥᮥ⪶ᯙ⧁ᙹᯩᨩ݅. Figure 10ŝFigure 11ᨱᕽౝ ᙹḢŝ ᙹ⠪ᄡ᭥ᨱᕽ ᙹ ᯩॐᯕ ḡၹ᮹ 36%᪡
51%᮹Ⅹʑ⡍⪵ࠥ᳑Õᅕ݅۵77%᮹⡍⪵ࠥ᳑Õᨱᕽᔍ໕᮹
ᇩᦩᱶᖒᮥᇥ⦹í⪶ᯙ⦹ᩡŁ, ⧎ᅖĞĥ໕ᨱᱲɝ⦹۵ᔍ໕ԕ᮹
contourᨱᕽࠥ⪶ᯙ⧁ᙹᯩᨩ݅. ᔍ໕ᔢ݉(Figure 15)ᨱᕽࠥFigure 12᪡zᯕᄡ᭥aaᰆⓍíᯝᨕӹ۵ḡᱱᨱᕽ❭ƕᩢᩎ( f trⴍ
0)ᮥ ⪶ᯙ⧁ ᙹ ᯩᨩ݅.
5. đು
ᅙᩑǍ۵ᕽᬙḡᩎᨱᕽ₥≉⦽3᳦ඹ᮹⪵v⣮⪵☁ෝݡᔢᮝಽ
ᇩ⡍⪵☁✚ᖒᮥđᱶ⦹Ł, ᖁ⧪vᬑᨱݡ⦽ᇩ⡍⪵⣮⪵☁ᔍ໕ᨱ
⋉⚍⦹۵vᬑᨱݡ⦹ᩍḡၹԕ᮹༉š⯂ᙹಆᄡ⪵᪡ᔍ໕᮹᭥⊹
ᨱෙᄡ⩶✚ᖒ, Drucker-Prager ᗭᖒ༉ߙᮥᯕᬊ⦹ᩍ⧎ᅖĞĥ ᨱࠥݍ⦹۵ḡၹ᮹❭ƕ᭥⊹ෝ❭ᦦ⦹ᩡ݅. ᮁ⦽ᗭ⧕ᕾ⥥ಽə ఉᮥᯕᬊ⦹ᩍvᬑᔍ໕᮹ᇩᦩᱶᖒᨱݡ⦽ᩑǍđŝෝ݅ᮭŝ
zᯕ ᮥ ᙹ ᯩᨩ݅.
(1) ᖁ⧪vᬑಽ ᯙ⦽ ᔍ໕ Ⅹʑ᳑Õᮥ ⡍⪵ࠥ Ⓧʑಽ aᱶ⦹ᩍ
}ၽ⦽ᮁ⦽ᗭ⧕ᕾ⥥ಽəఉᮥᔍᬊ⦹ᩍ⧕ᕾ⦽đŝ, vᬑಽ
ᯙ⦽ᔍ໕⢽⊖᮹༉š⯂ᙹಆᮡᮅᔢ┽(׳ᮡ⡍⪵ࠥ)ᯝভ
۱ญíqᗭ⦹ḡอ, Õ᳑⦽᳑Õ(ԏᮡ⡍⪵ࠥ) ᯝᙹಾ༉š⯂ᙹ ಆᮡ vᬑ ⋉⚍ᨱ ᮹⧕ ɪĊ⦹í qᗭ⧉ᮥ ᙹ ᯩᨩ݅.
(2) ᔍ໕ᔢᇡ᪡⦹ᇡ᭥⊹ᨱᕽᙹḢၰᙹ⠪ᄡ᭥ෝ⊂ᱶ⦽đŝ, ḡၹ᮹Ⅹʑ⡍⪵ࠥaⓕᙹಾᄡ᭥aᱱ₉ᱢᮝಽⓍí᷾a⧉ᮥ
ᙹᯩᨩŁ, ✚⯩ᔍ໕⢽⊖⦹ᇡ(B element)᮹ᙹḢᄡ᭥ᨱᕽ ۵ ⡍⪵ࠥa ⓕ ভ(ᖁ⧪vᬑಽ ฯᯕ ᮅ⦽ ᔢ┽, S=77%), vᬑ݉᭥ᵲప᮹ᔢᮝಽⓑᙹḢÑ࠺ᯕၽᔾ⦽⬥, e
ĞŝᨱࠥəÑ࠺ᮡᬱᔢ┽ಽ⫭ᅖࡹḡᦫŁᇩᦩᱶᖒᮥᅕᩡ݅.
(3) vᬑ ⋉⚍ಽ ᯙ⦽ ᔍ໕᮹ ᇩᦩᱶᖒᮥ ❱݉⦹ʑ ᭥⦹ᩍ
Drucker-Prager ᗭᖒ༉ߙᮥᯕᬊ⦽⧎ᅖĞĥsॅᮥ⧕ᕾ⦽
đŝ, ᔍ໕ᔢ݉ŝ⢽⊖ᨱᕽᗭᖒÑ࠺ᮝಽᄡ⧕a۵༉ᮥ
⪶ᯙ⧁ᙹᯩᨩᮝ໑, Ⅹʑ⡍⪵ࠥaaᰆⓑᔍ໕(S=77%)ᨱᕽ
⢽⊖᮹⡍⪵aᱽᯝዉญ᯲ࡹŁᔍ໕❭ƕ᮹a܆ᖒ( ftr ⴍ 0) ੱ⦽ ⪶ᯙ⧁ ᙹ ᯩᨩ݅.
qᔍ᮹ɡ
ᅙᩑǍ۵2011֥ࠥᱶᇡ(ၙ௹᳑ŝ⦺ᇡ)᮹ᰍᬱᮝಽ⦽ǎᩑ Ǎᰍ݉(No. 2011-0030842)᮹ḡᬱᮥၼᦥᙹ⧪ࡹᨩᮝ໑, ᯕᨱ
ʫᮡ qᔍෝ ऽพܩ݅.
