** ᩑᖙݡ⦺Ʊ ☁༊⪹ĞŖ⦺ŝ ၶᔍŝᱶ ([email protected])
Received June 6 2012, Revised July 11 2012, Accepted April 1 2013
Copyright ⵑ 2013 by the Korean Society of Civil Engineers
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)
ǣǣȀȀǤǤȀͳͲǤͳʹͷʹȀ ǤʹͲͳ͵Ǥ͵͵Ǥ͵ǤͳͲͳ Ǥ ǤǤ
⍆㦪㰒#㩋㰒㝞#♪′ⴖ#ኾ㴟⟖ᷣ#⍂㦪⮎#ᢾ㬚#⽾⇖⸮ሲኺ#ᇓⱮᇓᦂⴖ#⮿㭣
ࢢ ȵଲֈ ȵ୨ฅ
Kim, Yong Min*, Lee, Kwang Woo**, Kim, Jung Hwan***
Influence of Soil Characteristic and Rainfall Intensity on Matric Suction of Unsaturated Weathered Soil Slope
ABSTRACT
The monolithically coupled finite element analysis for a deformable unsaturated soil slope is performed to investigate matric suction distribution on a soil slope subjected to rainfall infiltration, which can consider the hydraulic-mechanical characteristics for the analysis. The soil-water characteristic curves (SWCC) are experimentally determined to estimate three types of hydraulic properties of domestic areas. Based on the physical properties, the distribution of matric suction is investigated by considering the major factors, such as soil conditions, rainfall intensities, and slope angles. It is found from the results of this study that the matric suction rapidly decreases with an increase in rainfall intensity, regardless a slope angle. The slope surface is more easily saturated when its saturated hydraulic conductivity is smaller than rainfall intensity, and for the case of multi-layered soil slope, hydraulic characteristics of slope surface has a significant influence on matric suction distribution.
Key words : Unsaturated soil, Seepage analysis, Slope failure, Soil-water characteristic curve, Matric suction, Finite element analysis
Ⅹಾ
ᅙᩑǍᨱᕽ۵vᬑ⋉⚍ᨱ᮹⦽ᔍ໕᮹༉š⯂ᙹಆᇥ⡍✚ᖒᮥᇥᕾ⦹ʑ᭥⦹ᩍᙹญ⦺ᱢ-ᩎ⦺ᱢ✚ᖒᮥŁಅ⦽࠺ᩑĥ⧕ᕾᮥᙹ⧪⦹ᩡ݅.
ᯕෝ᭥⧕, ǎԕ3aḡḡᩎᨱᕽ₥≉⦽⣮⪵☁ෝݡᔢᮝಽ⧉ᙹ✚ᖒłᖁ(SWCC)ᮥᔑᱶ⦹ᩡᮝ໑, ᯕෝ☁ݡಽḡၹ᳑Õ, vᬑ✚ᖒ, ᔍ໕Ğᔍ ᨱෙ༉š⯂ᙹಆ᮹ᄡ⪵ෝš⊂⦹ᩡ݅. əđŝ, vᬑvࠥa᷾a⧉ᨱᔍ໕ԕ᮹༉š⯂ᙹಆᮡɪĊ⯩qᗭ⦹۵Ğ⨆ᯕӹ┡ԍᮝ໑, ᔍ ໕Ğᔍᨱ۵ⓑᩢ⨆ᮥၼḡᦫ۵äᮝಽӹ┡ԍ݅. ੱ⦽vᬑvࠥᅕ݅⡍⪵⚍ᙹĥᙹa᯲ᮡḡၹᮡ⢽⊖ᨱᕽ⡍⪵aᛞíᯝᨕӹ۵äᮥ⪶ᯙ⦹
ᩡᮝ໑, ݅⊖ᮝಽ᳕ᰍ⦹۵Ğᬑᨱࠥᔍ໕⢽⊖ḡၹ᮹ᙹญ⦺ᱢ✚ᖒᯕ༉š⯂ᙹಆᇥ⡍ᨱⓑᩢ⨆ᮥᵝ۵äᮝಽӹ┡ԍ݅.
áᔪᨕ ᇩ⡍⪵☁, ⋉⚍⧕ᕾ, ᔍ໕❭ƕ, ⧉ᙹ✚ᖒłᖁ, ༉š⯂ᙹಆ, ᮁ⦽ᗭ⧕ᕾ
1. ᕽು
↽ɝᱥᖙĥᱢᮝಽʑ⬥ᄡ⪵ᨱ᮹⦽⡎ᬑ, ⡎ᖅ, ᛩ⟝┽⣮॒ŝzᮡᩩᔢ⦹ḡ༜⦽ᯕᔢʑ⬥⩥ᔢᯕӹ┡ӹŁᯩᮝ໑, Ḳᵲvᬑ᮹
ၽᔾኩࠥa᷾aࡹ໕ᕽᔍ໕❭ƕaኩჩ⦹íၽᔾ⦹Łᯩ݅. ✚⯩ᬑญӹ᮹Ğᬑᨱ۵ǎ☁໕ᱢ᮹᧞70%aᔑḡಽǍᖒࡹᨕᯩŁ, ᬑʑᨱၽᔾ⦹۵ᔍ໕❭ƕ۵vᬑ⋉⚍ಽᯙ⦽᧶ᮡ❭ƕaݡᇡᇥᯕʑভྙᨱᔑᔍ┽᪡☁ᕾඹ᮹ᵝ⦽ᬱᯙᯕࡹŁᯩ݅(᳑ᖒᮡŝ
ᯕ௹, 2000; ʡᰍ⪮ ॒, 2002; Jeong et. al., 2008; Kim et al., 2012).
ݓъėॡ
Õ᳑⦽ ḡၹᔢ┽ᨱᕽ vᬑa ᯝᱶ ʑe ࠺ᦩ ĥᗮ ḡᗮࡹ໕
ḡၹ᮹ ⡍⪵ࠥၰ ⧉ᙹእa ᷾a⦹íࡹŁ ᔍ໕ ԕ༉š⯂ᙹಆ (matric suction)ᯕ⩥ᱡ⯩qᗭ⦹ᩍᔍ໕᮹ᇩᦩᱶᯕ᷾a⦹í
ࡽ݅(Lambe, 1996; Ng and Shi, 1998). ੱ⦽ᇩ⡍⪵ᔍ໕ᨱᕽ۵
ᖁ⧪vᬑᨱ᮹⧕ḡၹ᮹Ⅹʑvࠥa᳭ᬑࡹʑভྙᨱᦩᱶᖒᨱ
ⓑᩢ⨆ᮥၙ⊽݅(Rahardjo et al., 2009). ᕽᔍ໕❭ƕෝᩩᔢ
⦹Łᯕᨱᱢᱩ⦹íݡ᮲⦹ʑ᭥⧕ᕽ۵ᇩ⡍⪵ḡၹ᮹༉š⯂ᙹಆ ŝ vᬑ✚ᖒŝ᮹ šĥෝ ⪶⯩ Ƚ⧁ ᙹ ᯩᨕ ⦽݅.
ʑ᳕᮹vᬑᔍ໕ᦩᱶ⧕ᕾᮡḡ⦹ᙹ᭥ෝʑᵡᮝಽᱡ໕ᇡ۵
⡍⪵ᔢ┽ಽ, ḡ⦹ᙹ᭥ᔢᇡ۵Õ᳑ᔢ┽ಽaᱶ⦽⬥ᵝಽ⦽ĥ⠪
⩶ჶᮥᯕᬊ⦹ᩡᮝӹ, ᯕ్⦽⧕ᕾᮡᇩ⡍⪵ᔍ໕❭ƕ᮹ᵝ⦽
ᬱᯙᯕࡹ۵ᩢ⨆ᯙᯱॅᮥᱢᱩ⯩Łಅ⧁ᙹᨧʑভྙᨱ⋉⚍ಽ
ᯙ⦽ᇩ⡍⪵ᔍ໕᮹Ñ࠺ᄡ⪵ෝ⧊ญᱢᮝಽŁಅ⦹ʑaᨕಖ݅.
ᕽᇩ⡍⪵☁᮹⋉⚍Ñ࠺ᮥ⧕ᕾᱢᮝಽᱲɝ⦹ಅ۵ᩑǍaḥ
⧪ࡹᨕ᪵݅(Lam et al., 1987). ↽ɝᨱ۵vᬑ⋉⚍ᨱ᮹⦽ḡ⢽໕
⡍⪵ʫᯕෝᩩ⊂⦹Łᔍ໕᮹ᦩᱶᖒᮥ⠪a⧁ᙹᯩ۵ᔢᬊ⥥ಽ əఉ(GEO-SLOPE, 2012; FEFLOW, 2012)ᯕ ॒ᰆ⧉ᨱ
ᯕෝ⪽ᬊ⦽ᔍ໕ᦩᱶ⧕ᕾᩑǍa⪽ၽ⯩ḥ⧪ࡹŁᯩ݅(Kim et al., 2004; ᱶᔢᖍ ॒, 2009; Rahardjo et al., 2010).
⦽⠙, ʑ᳕᮹ᔢᬊ⥥ಽəఉॅŝᙹ⧪ࡹᨕ᪉ᩑǍॅᮡ⋉⚍⧕ᕾ ŝᔍ໕᮹ᩎ⦺ᱢÑ࠺ᮥᇥญ⦹ᩍ⧕ᕾ⦹۵݉ĥ⧕ᕾ(staggered analysis)ᮥᵝಽᔍᬊ⦹ᩡ݅(Duncan, 1996). ə్ӹᝅᱽ⩥ᔢᨱ ᕽ۵ḡ⢽໕ᨱvᬑa ⋉⚍⦹໕ᕽḡၹ᮹eɚእෝᄡ⪵┅Ł
əಽᯙ⦹ᩍ⮺᮹ᄡ⩶ᯕᔾʑ໕ᕽᇩ⡍⪵⚍ᙹĥᙹaݍḡí
ࡽ݅. ᯕ్⦽Ñ࠺ᯕၹᅖࡹ໕ḡၹ᮹⋉⚍܆(infiltration rate)ᯕ
ݍḡíࡹŁḡ⢽໕ᮝಽᇡ░ᯝᱶ⦽⡍⪵ݡ(wetting band)a
⩶ᖒࡹᨕᔍ໕᮹᧶ᮡ❭ƕෝᮁၽ⦹۵ᬱᯙᯕࢁᙹᯩ݅(ʡᰍ⪮
॒, 2002; Kim et al., 2004; Jeong et al., 2008). ੱ⦽ᮁ⬉᮲ಆᮡ
⋉⚍ᨱෙᄡ⩶ŝ༉š⯂ᙹಆ᮹Ⓧʑᨱᄡ⪵⦹အಽ⋉⚍
⧕ᕾŝ ᔍ໕᮹ᄡ⩶ ၰ Ñ࠺⧕ᕾᯕ ᇥญࡹᨕ ᙹ⧪ࡹ໕ᱶ⪶⦽
ᔍ໕ԕ᮹⮺᮹vࠥ⧕ᕾᮥᙹ⧪⧁ᙹᨧíࡽ݅. Alonso ॒(1988)
ᮡᇩ⡍⪵☁᮹⮱-ᄡ⩶ᯕᩑšࡽ⧕ᕾႊჶᮥᱽᦩ⦹ᩍᯕෝᔍ໕
vᬑ ⋉⚍⧕ᕾᨱ ᱢᬊ⦹ᩡᮝ໑, Cho and Lee(2001)۵ ᯕ్⦽
ᩑĥ⧕ᕾᮥ☖⧕vᬑᨱ᮹⦽ᔍ໕᮹ᇩᦩᱶີ⍅ܩ᷹ᮥᩑǍ⦹ᩡ
Ł, ᮲ಆᰆ(stress field)ᮝಽᇡ░ᔍ໕᮹❭ƕ໕ŝᦩᱥᮉᮥᔑᱶ⦹
ᩍ ᔍ໕ ᦩᱶᖒ ⠪a ᜅ▽ᮥ }ၽ⦹ᩡ݅.
ᅙᩑǍᨱᕽ۵⋉⚍᪡⮺᮹Ñ࠺⧕ᕾᮥ࠺ᨱᙹ⧪⦹Łeɚᙹ ᦶŝ ᄡ⩶ᯕ ᩑĥࡽ ᮁ⦽ᗭ⧕ᕾᮥ ᙹ⧪⦹ᩡᮝ໑, ᇩ⡍⪵☁᮹
ᙹญ⦺ᱢ✚ᖒᮥݡ⢽⦹۵⧉ᙹ✚ᖒłᖁᝅ⨹ᮥ☖⧕ǎԕᵝᔑ ḡᨱᇥ⡍ࡹᨕᯩ۵ᇩ⡍⪵⣮⪵☁᮹✚ᖒჵ᭥ෝđᱶ⦹ᩡ݅. ᯕ
ෝၵ┶ᮝಽvᬑ᳑Õ, ḡၹ᳑Õ, ᔍ໕Ğᔍᨱෙᇩ⡍⪵⣮⪵☁
ᔍ໕᮹⋉⚍⧕ᕾᮥᙹ⧪⦹ᩍḡၹԕ༉š⯂ᙹಆ᮹ᄡ⪵ෝš⊂⦹
Łᯱ ⦽݅.
2. ᇩ⡍⪵⣮⪵☁ḡၹ᮹⧉ᙹ✚ᖒłᖁ
⧉ᙹ✚ᖒłᖁ(soil-water characteristic curve)ᮡᇩ⡍⪵ḡၹ ᮹Łᮁ⦽✚ᖒᯕ໑, ḡၹ᮹ᙹญ⦺ᱢ✚ᖒŝᩎ⦺ᱢ✚ᖒᮥđᱶ
⦹۵ๅᬑᵲ⦽ʑᅙྜྷᖒᯕ⧁ᙹᯩ݅. ᯝၹᱢᮝಽ⧉ᙹ✚ᖒł ᖁᮡḡၹᯕwŁᯩ۵ྜྷ᮹᧲ŝ༉š⯂ᙹಆ(matric suction)ŝ᮹
šĥಽᱶ᮹ࡽ݅. ḡၹᗮྜྷ᮹᧲ᮡᵲప⧉ᙹእ(gravimetric water content), ℕᱢ⧉ᙹእ(volumetric water content) əญŁ⡍⪵ࠥ
(degree of saturation)ಽ⢽⩥⧁ᙹᯩᮝӹ⋉⚍⧕ᕾᮥ᭥⧕ᕽ۵
ᇩ⡍⪵ḡၹ᮹Ǎᖒᗭᯙ⮺᯦ᯱ, ྜྷ, Ŗʑ᮹ᖙaḡǍᖒᖒᇥᮝ ಽŁಅࡹᨕ⦽݅. ᯕ్⦽✚Ḷভྙᨱ⡍⪵ḡၹ⧕ᕾᨱᕽᔍᬊࡹ
۵ᵲప⧉ᙹእ(Ww/Ws)ᅕ݅۵eɚŖʑෝŁಅ⦹۵ℕᱢ⧉ᙹእ (Vw/V ) }ֱᮝಽ⢽⩥⦹۵äᯕၵ௭Ḣ⦹݅. ə్အಽᇩ⡍⪵
ḡၹ᮹⧉ᙹ✚ᖒłᖒᮡℕᱢ⧉ᙹእ᪡༉š⯂ᙹಆšĥ᮹łᖁᮝಽ
⢽⩥⦽݅. ᕽᅙᱩᨱᕽ۵ᬑญӹ᮹ᔑḡ᮹ݡᇡᇥᮥ₉ḡ⦹
Łᯩ۵⪵v⣮⪵☁ḡၹ᮹⧉ᙹ✚ᖒłᖁᮥᔑᱶ⦹ʑ᭥⧕ᕽᬙ, ʡ⧕, ᩍᵝḡᩎ᮹⣮⪵☁ෝݡᔢᮝಽᝅ⨹ᮥᙹ⧪⦹ᩡᮝ໑, ᝅ⨹ᰆ
⊹᪡ ᝅ⨹ႊჶᮡ ݅ᮭŝ z݅.
2.1 ਓୋ౿ࢫਏ߹
⧉ᙹ✚ᖒłᖁᔑᱶᮥ᭥⦹ᩍഭ᮹༉š⯂ᙹಆᮥ᳑ᱩ⧁ᙹ
ᯩ۵GCTS SWC-150 ᰆእ(Pham and Fredlund, 2004)ෝᯕᬊ⦹
ᩡᮝ໑, Fig. 1ŝzᯕᦶಆ᳑ᱩᰆ⊹(pressure panel), ⦹ᵲᰍ⦹ᰆ
⊹(Loading shaft), ᇡ⦝⊂ᱶᰆ⊹(Volume tube), ᦶಆᖡ(pressure chamber), ᖙၚॵᜅⓍ(HAVE disks)॒ᮝಽǍᖒࡹᨕᯩ݅. Pres- sure chamber۵ ᝅ⨹ᰆ⊹ ᦩᨱ ᖙၚॵᜅⓍෝ ᯕᬊ⦹ᩍ ׳ᮡ
ᦶಆŝ ᰆe ᝅ⨹ᮥ ☖⧕ ⧉ᙹ✚ᖒłᖁᮥ ⊂ᱶ⧁ ᙹ ᯩࠥಾ
⦽݅. ᖙၚॵᜅⓍ۵ ༉š⯂ᙹಆᨱ čॽ ᙹ ᯩ۵ ᱶࠥa
݅ʑভྙᨱഭ᮹᳦ඹᨱbʑ݅ෙᦶಆ᮹ᖙၚॵᜅⓍ
ෝᔍᬊ⧕⦽݅(༉௹: 100~300kPa, ᝅ✙ḩ༉௹: 300~500kPa, ᱱ☁: 1500kPa). ᝅ⨹ʑ᮹ᱢᬊa܆⦽༉š⯂ᙹಆ᮹ჵ᭥۵0.1~
1500kPaᯕ໑, pressure cell ၲᇡᇥᨱ2}᮹ቑ౼ᮝಽᩑđࡹᨕ
ᯩᨕഭ᮹⧉ᙹእᄡ⪵ෝ⊂ᱶ⧁ᙹᯩ݅. ⧉ᙹ✚ᖒłᖁᮡ⣮⪵
☁ഭ᮹ᔢݡၡࠥᨱๅᬑၝq⦹íᄡ⦹ʑভྙᨱ(ᯕȽ⩥॒, 2007)ᅙᝅ⨹ᨱᕽ۵ഭ᮹↽ᗭ݉᭥ᵲప(γmin)ŝ↽ݡ݉᭥ᵲప (γmax)ᮥᔑᱶ⦹Ł50% ᔢݡၡࠥᨱ⧕ݚ⦹۵ഭపᮥđᱶ⦹ᩡ
݅. ੱ⦽Ɂ॒⦽݅ḱᮥ᭥⧕3݉݅ḱჶᮥᯕᬊ⦹ᩡᮝ໑, ᔢ⨆⋉⚍
ෝ ᯕᬊ⦹ᩍ ഭෝ ⡍⪵⎑݅.
⧉ᙹ✚ᖒłᖁᮡ⡍⪵ࡽഭᨱŖʑᦶᮥa⦹ᩍᇩ⡍⪵ᔢ┽ಽ
อॅᨕ႑⇽ࡽeɚᙹ᮹ᇡ⦝ෝ☖⧕ℕᱢ⧉ᙹእෝᔑᱶ⦹۵Õ᳑
ŝᱶ(drying process)ŝ႑⇽ࡽeɚᙹaԕᇡಽ⯂ᙹࡹ۵ᮅŝ ᱶ(wetting process)ᮥ ☖⧕ ᔑᱶ⧁ ᙹ ᯩ݅. ᮅŝᱶᮥ ☖⧕
(a) Schematic diagram of SWCC apparatus (b) SWCC apparatus Fig. 1. GCTS pressure plate apparatus
Table 1. Physical properties of soils
Property Seoul Yeoju Gimhae
Specific gravity (Gs) 2.64 2.65 2.68
Max. dry unit weight (γdmax, kN/m3) 17.5 16.5 15.8
Min. dry unit weight (γdmin, kN/m3) 13.5 13.3 12.2
Uniformity coefficient(Cu) 9.3 3.6 2.3
Coefficient of gradation(Cc) 0.77 1.11 0.3
USCS SP SW SW
Fig. 2. Grain-size distribution curves
⧉ᙹ✚ᖒłᖁᮥᔑᱶ⦹۵äᯕᝅᱽḡၹŖ⦺ᱢᯙྙᱽ᪡ᮁᔍ⦽
ႊჶᯕḡอÕ᳑ŝᱶᯕ⬥ᨱᮅŝᱶᮥ༉ᔍ⧕ ⦹ʑভྙᨱ
ᔢݚ⦽eᯕᗭࡽ݅. ᕽᅙᩑǍᨱᕽ۵Õ᳑ŝᱶᮥ☖⧕
⧉ᙹ✚ᖒłᖁᮥᔑᱶ⦹ᩡ݅. ᯕಆ⩥ᔢᮥw۵⧉ᙹ✚ᖒłᖁᮡ᧞
2kPaᱶࠥ᮹༉š⯂ᙹಆ₉ᯕෝᅕᯕ໑(Huang et al., 2011), ⡍⪵
ℕᱢ⧉ᙹእa݅íӹ┡ӹŖʑ⧉᯦⊹ӹᇩ⡍⪵⚍ᙹĥᙹᨱᩢ
⨆ᮥ ၙ⊽݅(ʡᰍ⪮ ॒, 2012).
ᅙ ᩑǍᨱᕽ۵ ᬑญӹ ᔑᦦḡᩎᨱ ᇥ⡍⦹۵ ⪵v⣮⪵☁ෝ
ݡᔢᮝಽʡ⧕, ᕽᬙ, ᩍᵝḡᩎᨱᕽ₥≉⦽3aḡ⮺ഭॅᨱ
ݡ⧕⧉ᙹ✚ᖒłᖁᝅ⨹ᮥᙹ⧪⦹ᩡ݅. ᝅ⨹ഭ۵☖ᯝᇥඹჶᔢ
SP, SW ĥᩕಽ⪶ᯙࡹᨩᮝ໑༉ुഭᨱᕽ4ჩℕ☖ŝᮉᯕ
90%ᯕᔢ, 200ჩℕ☖ŝᮉᯕ1%ၙอᮝಽ ӹ┡ԍ݅. bഭ᮹
ʑᅙᱢᯙྜྷญᱢ✚ᖒॅᮡTable 1ᨱᱶญ⦹ᩡᮝ໑, Fig. 2ᨱ3aḡ
ഭᨱ ݡ⦽ ᯦ࠥᇥ⡍łᖁᮥ ࠥ⦹ᩡ݅.
Fig. 3. SWCC for the three soils: experimental data and fitted curves
Table 2. Curve-fitting parameters for the SWCC and saturated permeability of the soils
Type Type A Type B
α 0.0423 /kPa 0.27 /kPa
n 2.06 3.1
Saturated volumetric
water content (θs) 0.41 0.36 Residual volumetric
water content (θr) 0.054 0.037 Saturated permeability (ks) 4.67 × 10-6 m/sec 5.12 × 10-5 m/sec
2.2 ऄඑฃණฃഠࢱଭনմটॺ୨
⧉ᙹ✚ᖒłᖁᮡ☁พᯱ᮹᯦Ğŝ᯦ࠥᇥ⡍, ǍᖒᖒᇥəญŁ
eɚእ॒ŝzᯕ⮺᮹᳦ඹᨱ݅ෙ⩶┽ෝᅕᯙ݅. ᯕłᖁ᮹
⩶┽ෝđᱶḴ۵ᵲ⦽ᗭ۵Ŗʑ⧉᯦⊹(air-entry value)᪡
ඹℕᱢ⧉ᙹእ(residual volumetric water content)ಽӹ٭ᙹ
ᯩ݅. ᇩ⡍⪵⣮⪵☁᮹✚ᖒᮥᔑᱶ⦹ʑ᭥⦹ᩍ݅᧲⦽᯦ࠥᇥ⡍,
݅ḱ⧉ᙹእ, eɚእෝaḥഭෝᕽᬙ, ᩍᵝ, ʡ⧕ḡᩎᨱᕽ₥≉
⦹ᩍ ᝅ⨹⦹ᩡᮝ໑, ᯦ࠥᇥ⡍᪡ eɚእෝ ᯙ᭥ᱢᮝಽ ᳑ᖒ⦹ᩍ
⇵aᝅ⨹ᮥᙹ⧪⦹ᩍ⧉ᙹ✚ᖒłᖁᮥ⢽⩥⦹ᩡ݅. Fig. 3ᮡ⮺ᨱ
ݡ⦽༉š⯂ᙹಆᨱ႑⇽ࡹ۵eɚᙹ᮹᧲ᮥ⊂ᱶ⦽đŝෝ
༉š⯂ᙹಆŝℕᱢ⧉ᙹእ᮹šĥಽ⢽⩥⦽⧉ᙹ✚ᖒłᖁᮥӹ┡ԙ
äᯕ݅.
☁พᯱ᮹᯦ĞᯕእƱᱢⓍŁɁ॒⦽ᔍḩ☁ḩ᮹ᩍᵝ⣮⪵☁۵
᯲ᮡ༉š⯂ᙹಆ᮹ᄡ⪵ᨱࠥⓑℕᱢ⧉ᙹእ᮹ᄡ⪵ෝᅕᯕŁ, ᔍ ḩ☁ḩᨱእ⧕᯦Ğᯕ᯲ᮡᝅ✙ḩᯕӹᱱ☁ḩᮥ⡍⧉⦹۵ᕽᬙ
ഭ۵࠺ᯝ⦽༉š⯂ᙹಆ᮹ ᄡ⪵ᨱෙℕᱢ⧉ᙹእ᮹ᄡ⪵a
ᱢŁᔍḩ☁ḩᨱእ⦹ᩍⓑඹ⧉ᙹእෝw۵݅. ᝅ⨹ߑᯕ░۵
van Genuchten(1980)᮹⧉ᙹ✚ᖒłᖁᮥᯕᬊ⦹ᩍ☖ᯝᇥඹჶ ᔢ SPᨱ ⧕ݚ⦹۵ ᕽᬙḡᩎ(Type A)ŝ SWᨱ ⧕ݚ⦹۵ ᩍᵝ, ʡ⧕ḡᩎ(Type B)᮹⧉ᙹ✚ᖒłᖁᮝಽǍᇥ⦹ᩡ݅. Table 2ᨱ
ӹ┡ԙၵ᪡zᯕ⧉ᙹ✚ᖒłᖁ᮹Ŗʑ⧉᯦⊹۵Type Aḡၹᯕ
Type Bḡၹᅕ݅᧞6.4႑Ⓧíӹ┡ԍᮝ໑, ⧉ᙹ✚ᖒłᖁ᮹ʑᬙ ʑෝđᱶḴ۵nsŝඹℕᱢ⧉ᙹእ۵bb1.5႑₉ᯕෝӹ┡ԕ ᨩ݅.
3. ⋉⚍⧕ᕾ
3.1 ࢼࢺ୨ਐ
ᯱᩑᔢ┽᮹⮺ᮡ☁พᯱ(ns),ྜྷ(nw), Ŗʑ(na)ಽǍᖒࡹᨕᯩᮝ ໑, ℕᱢእ(1 = ns+nw+na)ᨱbᗭॅᮥǍᇥ⦹۵mixture theory(Coussy, 2010)aᇩ⡍⪵☁ḡ႑ႊᱶ᮹ɝeᮥᯕŁ
ᯩ݅. ᇩ⡍⪵☁᮹ၙḡᙹ۵☁พᯱ᮹ᄡ᭥(u)᪡☁พᯱԕᇡᨱᕽ
ၽᔾࡹ۵eɚᙹᦶ(Pw)ᯕ໑, ᯕෝĥᔑ⦹ʑ᭥⧕ᕽ۵ࢱ}᮹ḡ႑
ႊᱶᯕ⦥⦹݅. ḡ႑ႊᱶᮡྜྷℕᨱ᯲ᬊ⦹۵༉ु⯹ॅᮡ
⠪⩶ᮥᯕ്݅۵⠪⩶ႊᱶ(balance equation)ŝྜྷℕ᮹ḩపᯕ
eᨱᄡ⦹ḡᦫ۵݅۵ḩపᅕ᳕᮹ჶ⊺(balance of mass)ᮥ
ʑᅙaᱶᮝಽ⦹Łᯩ݅. ᇩ⡍⪵ḡၹ᮹ᮁ⦽ᗭ⧕ᕾᮥ᭥⦹ᩍ
⋉⚍ᨱ᮹⦽ḡၹ᮹Ñ࠺ŝᙹญ⦺ᱢ-ᩎ⦺ᱢ(hydro-mechanical)
✚ᖒᮥ࠺ᨱŁಅ⦹ᩍ⧕ᕾ⧁ᙹᯩ۵ᙹ⊹⧕ᕾ⎵ऽෝǍᖒ⦹ᩡ
ᮝ໑ ḡ႑ႊᱶᮡ ݅ᮭŝ z݅(Kim, 2010).
ℌṙಽᇩ⡍⪵☁ԕྜྷ᮹ᯕ࠺ᮡ⮺᯦ᯱ᮹ᬡḢᯥᨱ᳦ᗮࡹʑ
ভྙᨱ⮺᯦ᯱᨱݡ⦽⠪⩶ႊᱶ(balance of linear momentum) ᮝಽᇡ░ᄡ᭥ෝĥᔑ⧁ᙹᯩ݅. ࢹṙಽᇩ⡍⪵☁ԕྜྷ᮹⮱
ᮥ❭ᦦ⦹ᩍeɚᙹᦶᮥĥᔑ⧁ᙹᯩ۵ྜྷᨱݡ⦽⠪⩶ႊᱶ
(balance of mass of the water)ᮝಽᇡ░eɚᙹᦶᮥĥᔑ⧁ᙹ
ᯩ݅. bb᮹⠪⩶ႊᱶॅᮡ݅ᮭEqs. (1) and (2)᪡z݅(Borja, 2004).
0 g
DIVσ+ρ = (1)
w S v vw
s p
n S + DIV =−DIV~
∂
− ∂ & (2)
ᩍʑᕽDIV(divergence)۵∇ෝ᮹ၙ⦹໑,σ۵ᱥ᮲ಆ, ρ۵
ၡࠥ, g۵ᵲಆaᗮࠥ, nᮡeɚශ, S۵⡍⪵ࠥ, s۵༉š⯂ᙹಆ, p& ۵ eᨱ ෙ eɚᙹᦶ, v۵ ⮺ ᯦ᯱ᮹ ⮱ ᗮࠥ,w v~w۵
ᝅᱽ ྜྷ᮹ ᗮࠥ(vw)᪡ ⮺ ᯦ᯱ᮹ ᗮࠥ(v)᮹ ₉ᯕෝ ӹ┡ԙ݅.
3.2 କ૬ীැজଡੵճࠤஶ
ᮁ⦽ᗭ⧕ᕾᮥ᭥⦹ᩍ⮺᮹ᄡ᭥᪡ԕᇡeɚᙹᦶᮥĥᔑ⧁
ᙹᯩ۵እᖁ⩶᧞⩶(coupled nonlinear weak form)ᮥᮁࠥ⦹ʑ
᭥⧕ aᵲඹ⧎ჶ(method of weighted residual)ᮥ ᱢᬊ⦹ᩡ
ᮝ໑, 2aḡၙḡᙹᨱݡ⧕⧪⧉ᙹ(matrix form) ⩶┽ᯙᮁ⦽
ᗭႊᱶᮝಽ⢽⩥⦹ʑ᭥⧕ᕽ⩶ᔢ⧉ᙹ(shape function)ෝᱢᬊ
Fig. 4. Soil slope mesh used for seepage analysis and ground water level
Table 3. Analysis cases
Soil type Slope angle Slope height Rainfall intensity Rainfall duration Water table
Type A 30°
10m
10mm/h
24~72hr Dry condition
Type B 45° 30 mm/h
- 60° 50 mm/h
⦹ᩡ݅. bᗭ(element)ॅᯕaḡŁᯩ۵ԕಆ(internal force)ŝ
ಆ(external force) ᄂ░ॅ᮹⠪⩶᳑Õᨱ᮹⦹ᩍĥᔑsॅᮥ
٥ᱢ┅໕ᩑĥࡽእᖁ⩶⡍ྜྷᖁၙᇥႊᱶ(coupled nonlinear parabolic partially differential equation)ᮥ ᱥℕ ᗭᨱ ݡ⧕
Eq. (3)ŝ zᮡ ⧪ ⩶┽ಽ ⢽⩥⧁ ᙹ ᯩ݅.
) ( ) ( )
(D D F D F D
C &+ INT = EXT (3)
ᩍʑᕽ =⎢⎣⎡θ⎥⎦⎤ D d
, ⎥⎦
⎢ ⎤
⎣
=⎡ θ&
&
& d
D , ⎥⎦
⎢ ⎤
⎣
⎡
= −
) , ( )
(
0 ) 0
( , 1,
θ
θ θ
θ K d
D K
C dINT INT
,
⎥⎦
⎢ ⎤
⎣
⎡
−
= −
) , (
) ( ) , ) (
( , 2, ,
θ θ θ
θ θ
d F
F d D F
F INT
INT d INT
d INT
, ⎥⎦
⎢ ⎤
⎣
=⎡ dEXTEXT
EXT
F d D F
F ,
, ( , )
)
( θ θ
ᮥ ᮹ၙ⦽݅.
Backward Euler ႊჶᮥᯕᬊ⦽ԕᰍᱢႊჶ(fully-implicit time integration)ᮥ☖⧕᭥Eq. (3)ŝzᮡ⧪ႊᱶᮥNewton- Raphson ႊჶᮥᔍᬊ⦹ᩍၙḡᙹᯙḡၹ᮹ᄡ᭥᪡ԕᇡeɚᙹᦶ
ᮥ ĥᔑ⧁ ᙹ ᯩᮝ໑, Eqs. (4) and (5)ಽ ݅ ⢽⩥ࡽ݅.
) ( ) ( )
( +1 +1+ +1 = EXT n+1 INT n
n
n D F D F D
D
C & (4)
) ( )
( )
( +1 +1+ +1 +1= EXT n+1 n
n n
n D K D D F D
D
C & (5)
⧪ ႊᱶᮡ Newton-Raphson ႊჶᨱ ᮹⧕ ᖁ⩶⪵ࡽ ⩶
┽ಽ⩥ᰍe݉ĥ(tn+1)ᨱᕽ⧕ᕾᯕᙹ⧪ࡹ໑ԕಆᨱݡ⧕ᕽ
1 1
1) ( )
( n+ ≈ n+ n+
INT D K D D
F ᯥᮥaᱶ⧁ᙹᯩ݅. đŝᱢᮝಽᙹญ⦺ᱢ -ᩎ⦺ᱢ⮺᮹Ñ࠺ᮥእᖁ⩶ᱲɝႊჶᯙၹԕᰍᱢႊჶ(semi- implicit time integration)ᮝಽđŝෝࠥ⇽⦹ᩡ݅(Kim, 2010).
4. vᬑ✚ᖒᨱෙᇩ⡍⪵⣮⪵☁ᔍ໕᮹ᙹ⊹⧕ᕾđŝ
ḡၹ᳑Õ, vᬑ✚ᖒ, ᔍ໕Ğᔍᨱ ෙ ᇩ⡍⪵ ⣮⪵☁ ᔍ໕᮹
༉š⯂ᙹಆᇥ⡍ᄡ⪵ෝᇥᕾ⦹ʑ᭥⧕ᕽ3ᰆᨱᕽᱽ⦽࠺⧕ᕾ
⥥ಽəఉᮥᯕᬊ⦹ᩍ⋉⚍⧕ᕾᮥᙹ⧪⦹ᩡ݅. ḡၹ᳑Õᮡ2ᰆᨱᕽ
ᨙɪ⦽ၵ᪡zᯕǎԕ3aḡḡᩎᨱᕽ₥≉⦽⣮⪵☁ෝݡᔢᮝಽ
⧉ᙹ✚ᖒłᖁᝅ⨹ᮥ☖⧕Type A᪡Type BಽḡၹᮥǍᇥ⦹ᩡ݅.
vᬑ✚ᖒᮡvᬑvࠥ᪡ḡᗮeᮝಽӹ٭ᙹᯩ۵ߑ, vᬑvࠥ۵
ᯝᔢᨱᕽᯱᵝၽᔾࡹ۵10mm/h᪡᧶ᮡᔍ໕❭ƕෝᮁၽ⧁a܆
ᖒᯕⓑ30mm/h, 50mm/hෝᔍᬊ⦹ᩡ݅(ᱶᔢᖍ॒, 2009). vᬑ ḡᗮe᮹Ğᬑᨱ۵vᬑvࠥᨱ᮹⧕ᔍ໕⢽⊖ᇡa᪥ᱥ⡍⪵ࡹ
۵eอⓝ⧕ᕾᮥᙹ⧕⦹ᩡᮝ໑əჵ᭥۵12e~72eᯕ݅.
ੱ⦽ᅙ⧕ᕾᨱᕽ۵Fig. 4᪡zᯕᔍ໕׳ᯕa10mಽᯝᱶ⦹໑, ᔍ໕᮹Ğᔍ۵30°, 45°, 60°ಽᄡ⪵⍽a໕ᕽ༉š⯂ᙹಆᨱၙ⊹
۵ᩢ⨆ᮥᇥᕾ⦹ᩡ݅. ᔍ໕ᱥℕᩢᩎᮡ॒ႊ(isotropic)ᯙḡၹᮝ ಽaᱶ⦹ᩡᮝ໑, ᳭ᬑ⊂໕ŝ⦹ᇡĞĥ໕ᮡᇩ⚍ᙹĞĥ᳑Õᮝಽ
ᯕᨱᙹḢ⦽ႊ⨆ᮝಽ۵Ŗʑ᪡ྜྷ᮹⮱ᯕၽᔾ⦹ḡᦫ۵݅Ł
aᱶ⦹ᩡ݅. ੱ⦽ ᳭ᬑ⊂໕ᮡ ᙹ⠪ႊ⨆ ᄡ᭥ෝ, ⦹ᇡ Ğĥ໕ᮡ
ᩑḢႊ⨆᮹ᄡ᭥ෝǍᗮ⦹ᩡ݅. Ⅹʑḡ⦹ᙹ᭥۵Õʑෝaᱶ⦹
ᩍᔍ໕ၵ݆ᨱᕽ3m ׳ᯕᨱᕽᇡ░7°ĞᔍෝaḡŁᖁ⩶ᱢᮝಽ
᷾a⦹۵äᮝಽᖅᱶ⦹ᩡᮝ໑, ḡ⦹ᙹ᭥ᔢᇡᯙᇩ⡍⪵☁ᩢᩎᨱ ᕽ׳ᯕᨱ↽ݡ༉š⯂ᙹಆᮥ-75kPaಽaᱶ⦹ᩡ݅(Rahardjo et al., 2007; Rahardjo et al., 2010). Table 3ᮡ⧕ᕾ᮹᳦ඹෝ
ӹ┡ԕŁᯩᮝ໑, Table 4۵⧕ᕾᨱᔍᬊࡽḡၹ᮹vࠥᱶᙹෝ
ᱶญ⦹ᩡ݅.
Fig. 5۵Type A(ks=4.67 × 10-6 m/sec)ḡၹ᮹45°Ğᔍෝw۵
ᔍ໕ᨱᕽeᨱෙ༉š⯂ᙹಆᄡ⪵ᇥ⡍ࠥᯕ໑, bb᮹ᗭ ۵9}ᱩᱱᮝಽǍᖒࡹᨕᯩʑভྙᨱ4}᮹༉ᕽญᱩᱱᨱᕽ
ᨕḥ༉š⯂ᙹಆ᮹⠪Ɂsᮝಽӹ┡ԕᨩ݅. Fig. 5ᨱᕽ᪡zᯕ
vᬑvࠥa ⓕᙹಾ ᔍ໕ ⢽⊖ ၰ ԕᇡḡၹᯕ ⡍⪵ࡹ۵ eᯕ
Table 4. Properties of soils used in numerical analysis
Type Type A Type B
Lame parameter (λ,μ) 29 × 106, 7 × 106 Pa Solid real density (ρsR) 2700 kg/m3 Water real density (ρ )wR 1000 kg/m3 Air real density (ρ )aR 1.2 kg/m3 Viscosity of water (ηw) 10-3 Pa·s
Initial porosity (n = nw+ na) 0.41 (=0.2+0.21) 0.38 (=0.18+0.2) Saturated permeability (ks) 4.67 × 10-6 m/sec 5.12 × 10-5 m/sec Residual degree of saturation (Sr) 0.131 0.08
(a) Element A (b)Element B
Fig. 5. Variation of matric suction with slope angle 45°(Type A)
(a) 10mm/h (element A) (b) Saturation time with slope angle
Fig. 6. Effect of slope angle for matric suction (Type A)
ṈᦥḡŁ༉š⯂ᙹಆᯕɪĊ⯩qᗭ⧉ᮥᙹᯩ݅. ✚⯩, ᔍ໕
ԕᇡ(element A)ᅕ݅ᔍ໕⢽⊖ᨱᕽ(element B) ༉š⯂ᙹಆᯕ
ዉญqᗭ⦹ᩍ⡍⪵ᔢ┽ᨱࠥݍ⦽݅. ᯕ۵Fig. 5(a)ᨱᕽ᪡zᯕ
ᔍ໕⢽⊖ᨱᕽԕᇡಽྜྷᯕ⋉⚍ࡹ۵ŝᱶᨱᕽ༉š⯂ᙹಆᯕqᗭ
ࡹ۵ eᯕ ḡᩑࡹʑ ভྙᯕ݅.
Fig. 6ᮡ ᔍ໕Ğᔍa ༉š⯂ᙹಆᨱ ၙ⊹۵ ᩢ⨆ᮥ ᇥᕾ⦹ʑ
᭥⦹ᩍType A ḡၹᨱᕽ᮹đŝෝእƱࠥ⦹ᩡ݅. vᬑvࠥa
10mm/h᮹Ğᬑᨱᕽ۵ḡၹ᮹⡍⪵⚍ᙹĥᙹᅕ᯲݅ᮡ᧲᮹እa
ԕಙʑভྙᨱ⡍⪵aၽᔾ⦽Ŕᮡᨧᨩᮝӹ(Fig. 6a), Fig. 6(b)ᨱ ᕽ᪡zᯕvᬑvࠥa10mm/hᯕᔢ᮹Ğᬑᨱᕽ۵ݡᇡᇥ⢽⊖ḡၹ
(a) Element A (b)Element B Fig. 7. Variation of matric suction in two types of soil (30mm/h)
Fig. 8. Vertical displacement on the top surface of slope ᯕ⡍⪵ࡹᨩᮝ໑, ᔍ໕Ğᔍᨱ۵ⓑᩢ⨆ᯕᨧ۵äᮝಽӹ┡ԍ݅.
vᬑḡᗮeᯕ᷾a⧉ᨱ༉š⯂ᙹಆᯕqᗭ⦹۵ᱶࠥ۵݅
ᗭ ₉ᯕa ᯩᮝӹ, ↽᳦ᱢᮝಽ ⡍⪵ࡹ۵ ᱱᯕ Ñ᮹ ࠺ᯝ⦹ʑ
ভྙᨱ ᔍ໕Ğᔍ۵ ༉š⯂ᙹಆ qᗭᨱ ⓑ ᩢ⨆ᮥ ᵝḡ ᦫ۵݅.
ᷪ, Ğᔍᨱෙᔍ໕᮹ᦩᱥᮉ᷾qᮡᩎ⦺ᱢšĥᨱ᮹⧕ᔍ໕⪽
࠺ಆᄡ⪵॒ᨱⓍí᳭ᬑࡹŁ, ༉š⯂ᙹಆᮡĞᔍᨱ᮹⦽ᩢ⨆ᅕ
݅ ḡၹ᳑Õᯕӹ vᬑvࠥᨱ ⓑ ᩢ⨆ᮥ ၼᦥ ᔍ໕᮹ ᦩᱶᖒᮥ
qᗭ┅íࡽ݅. Chen and Young(2006)ᮡᔍ໕᮹Ğᔍᨱ
⋉⚍ʫᯕ(infiltration depth)᮹₉ᯕaⓍḡᦫᮭᮥᅕŁ⦽ၵᯩ݅.
Fig. 7ᮡᔍ໕Ğᔍ30°ᨱᕽ᮹ḡၹ᳑Õᨱෙ༉š⯂ᙹಆ᮹
ᇥ⡍ࠥෝࠥ⦹ᩡ݅. Fig. 7(a)ᨱᕽ᪡zᯕ ᔍ໕ԕᇡ(element A)ᨱᕽ᮹ ༉š⯂ᙹಆᮡ ⚍ᙹĥᙹa vᬑvࠥᅕ݅ ⓑ Type B (ks= 5.12x10-5 m/sec)ḡၹᨱᕽíqᗭ⦹ᩡᮝ໑, Type A (ks= 4.67x10-6 m/sec)ḡၹᨱᕽ۵vᬑa᯲ࡽ18eᯕ⬥ᨱ
༉š⯂ᙹಆᯕqᗭ⦹ʑ᯲⦹ᩡ݅. ↽᳦ᱢᮝಽ۵⚍ᙹĥᙹav ᬑvࠥᅕ᯲݅ᮡType Aḡၹᨱᕽຝᱡ⡍⪵aၽᔾ⦹ᩡ݅. əญŁ
Fig. 7(b)ᨱࠥ⦽ၵ᪡zᯕType BḡၹᨱᕽⅩʑ༉š⯂ᙹಆᮡ
Type Aḡၹᨱእ⧕íqᗭ⦹۵Ğ⨆ᮥᅕᯕḡอType Aḡၹ
⢽⊖ᨱᕽຝᱡ⡍⪵ࡹᨩᮝ໑, eḡᩑ⩶ᔢᮡӹ┡ӹḡᦫᦹ݅.
Cho and Lee(2001)᮹ᩑǍđŝᨱᕽ۵⡍⪵⚍ᙹĥᙹavᬑvࠥ
ᅕ݅ ᯲ᮥ ভ ᔍ໕ ⢽⊖ᨱᕽ᮹ ༉š⯂ᙹಆᯕ ɪĊ⯩ qᗭ⦹ᩍ
ຝᱡ⡍⪵aၽᔾ⦽݅ŁᅕŁ⦹Łᯩ݅. ə౨ḡอᅙᩑǍđŝᨱᕽ ۵Ⅹʑ༉š⯂ᙹಆ(0~24e)ᯕ⚍ᙹĥᙹaⓑType B ḡၹᨱᕽ
íqᗭ⦹۵äᮝಽӹ┡ԍ݅. ᯕ్⦽༉š⯂ᙹಆ᮹ᄡ⪵۵
ḡၹ᳑Õᨱ݅íӹ┡ӹ໑, ᇩ⡍⪵ḡၹ᮹⚍ᙹĥᙹ᪡⧉ᙹ
✚ᖒłᖁ᮹Ŗʑ⧉᯦⊹, ඹℕᱢ⧉ᙹእ, ⡍⪵ℕᱢ⧉ᙹእᨱⓍí
᮹᳕⦹ʑভྙᨱ⋉⚍⧕ᕾᔍᬊࡽ⧉ᙹ✚ᖒłᖁ᮹₉ᯕಽᯙ⧕
༉š⯂ᙹಆ᮹ qᗭĞ⨆ᯕ ᔢᯕ⦹í ӹ┡ӽ äᮝಽ ❱݉ࡽ݅.
Fig. 8ᮡᔍ໕ᔢᇡḡၹ᮹bᗭᨱᕽၽᔾ⦽↽᳦ᙹḢᄡ᭥
ෝࠥ⦽äᯕ݅. Type Aḡၹ᮹ᄡ᭥aType Bḡၹᅕ݅↽ݡ
8mmⓍíၽᔾ⦹ᩡᮝ໑, ᔍ໕⢽⊖ᨱᕽມᨕḩᙹಾᄡ᭥a᷾a
⦹۵ Ğ⨆ᮥ ӹ┡ԩ݅. ᄡ᭥۵ vᬑᨱ ᮹⧕ ⋉⚍a ၽᔾ⦹໕ᕽ
Ⅹʑeɚශᯕᄡ⦹íࡹŁeɚශ᮹ᄡ⪵۵ḡၹ᮹⧉ᙹ✚ᖒłᖁ ŝᇩ⡍⪵⚍ᙹĥᙹ॒ᮥᄡ⪵┅໑, ᗭ(element)᮹vᖒ⧪
(stiffness matrix)ĥᔑᨱᩢ⨆ᮥၙ⊹ʑভྙᨱḡၹ᳑Õᨱ
ᄡ᭥ၽᔾᱶࠥa݅íӹ┡ԍ݅. ᯕ᪡zᯕvᬑ⋉⚍ᨱ᮹⧕
☁พᯱॅeᨱᦶ⇶ᯕၽᔾ⦹Łᯱᵲᨱ᮹⦽ᩑḢႊ⨆ᄡ⩶ᯕၽᔾ
Fig. 9. Non-homogeneous soil slope mesh with two layers
Fig. 10. Variation of matric suction at A element (45°, 30mm/h)
(a) Surface layer: Type A and subsurface layer: Type B
(b) Surface layer: Type B and subsurface layer: Type A Fig. 11. Distribution of matric suction in two layered soil slope
(30mm/h)
⧉ᨱḡၹeɚශᯕqᗭ⦹ᩍᔍ໕ԕᇡᨱᕽၽᔾࡹ۵༉š⯂
ᙹಆᮡ Ⅹʑᨱ ᷾a⦹݅a qᗭ⦹۵ Ğ⨆ᮥ ᅕᯙ݅.
ḡɩʭḡ۵Ɂḩ⦽݉ᯝ⊖ᨱݡ⧕⋉⚍⧕ᕾᮥᙹ⧪⦹ᩡᮝӹ, ᝅᱽ⩥ᔢᨱᕽ۵ᖒ☁ӹᯱᩑ♕ᱢᨱ᮹⧕݅⊖ᮝಽḡၹᯕ᳑ᖒࡹ
ᨕᯩ۵Ğᬑa⮵⦹݅. ᕽᅙᩑǍᨱᕽࠥᔍ໕⢽⊖ᮥ
᧞2mࢱ̹อⓝḡၹ᮹᳦ඹෝǍᇥ⦹ᩍ⋉⚍⧕ᕾᮥᙹ⧪⦹ᩡ݅
(Fig. 9). ᔍ໕ԕᇡ(element A)ᨱᕽ᮹eᨱෙ༉š⯂ᙹಆ᮹
ᄡ⪵۵Fig. 10ᨱӹ┡ԕᨩᮝ໑, 2}᮹⊖ᮝಽ᳕ᰍ⦹۵ḡၹԕ᮹
༉š⯂ᙹಆᄡ⪵۵ᔍ໕ԕᇡ᮹ᬱḡၹᅕ݅⢽⊖ḡၹ᮹ᙹญ⦺ᱢ
✚ᖒᨱⓍí᳭ᬑࡹᨕ⢽⊖ᨱᇥ⡍⦹۵ḡၹ᮹⚍ᙹĥᙹa
ᬱḡၹᅕ᯲݅ᮥĞᬑᛞí⡍⪵ࡹ۵äᮥ⪶ᯙ⧁ᙹᯩ݅(Fig.
11). Fig. 11(a)ᨱᕽᅕᯕ۵ၵ᪡zᯕ, ᔍ໕⢽⊖ᇡᨱᕽ᧲᮹eɚᙹ ᦶᯕၽᔾ⦹۵äᮡ⢽⊖ᯕ⡍⪵ࡽ⬥(eɚᙹᦶ=0) bnodal point ᨱ ḡᗮᱢᮝಽ vᬑvࠥa ᱢᬊࡹŁ ᯩʑ ভྙᮝಽ ❱݉ࡽ݅.
5. đು
ᅙᩑǍᨱᕽ۵ᇩ⡍⪵⣮⪵☁ᔍ໕ᨱ⋉⚍⦹۵vᬑᨱݡ⦹ᩍ
ḡၹ✚ᖒ, vᬑ✚ᖒəญŁᔍ໕ĞᔍෝŁಅ⦽ᔍ໕᮹⋉⚍Ñ࠺
✚ᖒᮥᇥᕾ⦹ᩡ݅. ᯕᨱǎԕ᮹3aḡḡᩎᨱᕽ₥≉⦽⣮⪵☁ෝ
ݡᔢᮝಽ⧉ᙹ✚ᖒłᖁᮥđᱶ⦹Ł, ᮁ⦽ᗭ⧕ᕾ⥥ಽəఉᮥᯕ ᬊ⦹ᩍᔍ໕ԕ༉š⯂ᙹಆ᮹ᄡ⪵ᇥ⡍ෝš⊂⦹ᩡ݅. ᅙᩑǍđŝ
݅ᮭŝ zᮡ đುᮥ ᮥ ᙹ ᯩᨩ݅.
(1) vᬑ, ᔍ໕ԕᇡ᮹༉š⯂ᙹಆᮡvᬑvࠥa᷾a⧉ᨱ
qᗭ⦹۵እᮉᯕ᷾a⦹ᩍෙeᨱᔍ໕⢽⊖ḡၹᮥ⡍⪵
┅໑, ᔍ໕ᮥǍᖒ⦹۵ᇩ⡍⪵ḡၹ᮹ᙹญ⦺ᱢ✚ᖒᨱ
ᔍ໕ ԕᇡෝ ⡍⪵┅۵ eᮥ ḡᩑ┕ᮥ ᙹ ᯩᨩ݅.
(2) vᬑvࠥaḡၹ᮹⡍⪵⚍ᙹĥᙹᅕ݅ⓑĞᬑ, ᔍ໕᮹Ğᔍᨱ
ෙ༉š⯂ᙹಆ᮹ᄡ⪵۵ၙၙ⦹໑, ༉š⯂ᙹಆᮡĞᔍᨱ᮹
⦽ᩢ⨆ᅕ݅۵ḡၹ᮹ᙹญ⦺ᱢ✚ᖒŝvᬑvࠥa⮺᮹vࠥᨱ
ⓑ ᩢ⨆ᮥ ၙ⊽݅.
(3) ḡၹ᮹⡍⪵⚍ᙹĥᙹavᬑvࠥᅕ݅ⓑĞᬑᨱ۵༉š⯂ᙹಆ ᯕqᗭ⦹۵eᮡӹ, ⡍⪵⚍ᙹĥᙹavᬑvࠥᅕ᯲݅
ᮡĞᬑᨱ۵⢽⊖ḡၹ᮹⡍⪵aຝᱡၽᔾ⦽݅. đŝᱢᮝಽ,
⚍ᙹĥᙹa᯲ᮡḡၹᨱᕽ⡍⪵aᛞíᯝᨕӹ۵äᮥ᮹ၙ⦹
໑, ݅⊖ḡၹᨱᕽࠥ༉š⯂ᙹಆ᮹ᄡ⪵۵ᬱḡၹ᮹✚ᖒᅕ݅۵
ᔍ໕⢽⊖ ḡၹ ✚ᖒᨱ Ⓧí ᳭ᬑࡽ݅.
qᔍ᮹ɡ
ᯕםྙᮡ2011֥ࠥᱶᇡ(Ʊᮂŝ⦺ʑᚁᇡ)᮹ᰍᬱᮝಽ⦽ǎᩑ Ǎᰍ݉᮹ ḡᬱᮥ ၼᦥ ᙹ⧪ࡽ ᩑǍᯥ(No.2011-0030842).
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