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  1. μ’…μ‹ νšŒμ›β€€κ΅μ‹ μ €μžβ€€μšΈμ‚°λŒ€ν•™κ΅ κ±΄μ„€ν™˜κ²½κ³΅ν•™λΆ€ ꡐ수 (Corresponding Authorβ€€University of Ulsanβ€€shingeo@ulsan.ac.kr)
  2. μ’…μ‹ νšŒμ›β€€μ˜λ‚¨λŒ€ν•™κ΅ κ±΄μ„€μ‹œμŠ€ν…œκ³΅ν•™κ³Ό ꡐ수 (Yeungnam Universityβ€€sebungoh@yu.ac.kr)
  3. μ’…μ‹ νšŒμ›β€€μ „λ‚¨λŒ€ν•™κ΅ ν† λͺ©κ³΅ν•™κ³Ό ꡐ수 (Chonnam National Universityβ€€jm4kim@jnu.ac.kr)



J2-경계면 μ†Œμ„±λͺ¨λΈ, λ°˜λ³΅ν•˜μ€‘, μˆ˜μ •μŒκ³‘μ„  λͺ¨λΈ, μœ ν•œμš”μ†Œλ²•
J2-bounding surface plasticity model, Cyclic load, Modified hyperbolic model, FEM

1. μ„œ λ‘ 

μœ μ‚¬ 연속체(pseudo-continuum)에 ν•΄λ‹Ήν•˜λŠ” ν† μ‚¬μ§€λ°˜μ€ κ°œλ³„μž…μžμ˜ μ§‘ν•©μœΌλ‘œ, λŒ€λΆ€λΆ„μ˜ λ³€ν˜•λ₯  λ²”μœ„μ—μ„œ λΉ„μ„ ν˜•μ μΈ 응λ ₯-λ³€ν˜•λ₯  거동을 보인닀(Kramer, 1996). μ§€λ°˜μ˜ 역학적 거동을 λͺ¨μ‚¬ν•˜κΈ° μœ„ν•œ 단일 항볡면(single yield surface) λͺ¨λΈμ€ 응λ ₯ 곡간을 탄성과 μ†Œμ„±μ˜μ—­μ„ λΆ„λ¦¬ν•˜λ©°, κ³Όλ„ν•œ νƒ„μ„±μ˜μ—­κ³Ό νƒ„μ†Œμ„± κ²½κ³„μ—μ„œ κΈ‰κ²©ν•œ κ°•μ„± λ³€ν•˜λ₯Ό λ°œμƒμ‹œν‚€λŠ” λ¬Έμ œκ°€ μžˆλ‹€. λ˜ν•œ 응λ ₯ μ „ν™˜(stress reversal)이 λ°œμƒν•˜λŠ” λ°˜λ³΅ν•˜μ€‘(cyclic load)에 λŒ€ν•œ μ§€λ°˜μ˜ λ³΅μž‘ν•œ 거동을 λͺ¨λΈλ§ν•˜κΈ° μ–΄λ ΅λ‹€(Yu, 2006).

μ§€λ°˜μ˜ λΉ„μ„ ν˜•μ  응λ ₯-λ³€ν˜•λ₯  관계λ₯Ό λͺ¨μ‚¬ν•˜κΈ° μœ„ν•˜μ—¬ μ œμ•ˆλœ 경계면 μ†Œμ„±λͺ¨λΈ(bounding surface plasticity model)은 경계면에 λŒ€ν•œ ν•­λ³΅λ©΄μ˜ μƒλŒ€μ μΈ μœ„μΉ˜λ‘œλΆ€ν„° κ²½ν™”ν•¨μˆ˜λ₯Ό μ‚°μ •ν•˜μ—¬ λ§€λ„λŸ¬μš΄(smooth) 응λ ₯-λ³€ν˜•λ₯  곑선을 μž¬ν˜„ν•  수 μžˆλ‹€(Dafalias and Popov, 1975; Dafalias and Taiebat, 2016; Yu, 2006). λ”°λΌμ„œ, κ΅ν†΅ν•˜μ€‘, 지진, λ°”λžŒ 및 νŒŒλ™ λ“±κ³Ό 같은 λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ ν† μ‚¬μ§€λ°˜μ˜ λΉ„μ„ ν˜•μ μΈ 응λ ₯-λ³€ν˜•λ₯  이λ ₯(hysteretic) 거동을 μž¬ν˜„ν•  수 μžˆλ‹€(Yu et al., 2007).

μž…μƒ(granular)의 μ§€λ°˜μ€ κ·Ήλ„λ‘œ μ œν•œλœ λ³€ν˜•λ₯  λ²”μœ„μ—μ„œλ§Œ 탄성거동을 λ³΄μ΄λ―€λ‘œ, 응λ ₯κ³΅κ°„μ—μ„œ ν•­λ³΅λ©΄μ˜ 크기λ₯Ό 0으둜 μˆ˜λ ΄μ‹œμΌœ νƒ„μ„±μ˜μ—­μ„ μ‚¬λΌμ§€κ²Œ ν•˜λ©΄μ„œ 항볡면을 ν˜„μž¬μ˜ 응λ ₯점으둜 μΆ•μ†Œν•˜λŠ” 연ꡬ가 μˆ˜ν–‰λ˜μ—ˆλ‹€(Borja and Amies, 1994; Dafalias and Popov, 1977). ν•˜μ§€λ§Œ λŒ€λΆ€λΆ„μ˜ μ—°κ΅¬λŠ” μš”μ†Œ 레벨의 λ‹¨μˆœ 응λ ₯κ³΅κ°„μ—μ„œ μ „λ‹¨λ³€ν˜•μ— λŒ€ν•œ μˆ˜μ‹ν™”λ§Œμ„ μ œμ‹œν•˜κ³ , 닀차원적인 μˆ˜μΉ˜ν•΄μ„μ— μ μš©ν•œ μ‚¬λ‘€λŠ” λΆ€μ‘±ν•˜λ‹€(Borja et al., 1999; PisanΓ² and JeremiΔ‡, 2014; Restrepo and Taborda, 2018).

λ³Έ λ…Όλ¬Έμ—μ„œλŠ” J2-μ†Œμ„±λͺ¨λΈμ— νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” 경계면 κ°œλ…μ„ λ„μž…ν•˜κ³  μ‹€μš©μ μΈ 등방응λ ₯ μ˜μ‘΄μ μ‘λ ₯-λ³€ν˜•λ₯  증뢄 관계식을 μ œμ‹œν•œλ‹€. 그리고 μ œμ•ˆλœ μ†Œμ„±λͺ¨λΈμ˜ 이둠식에 λŒ€ν•œ 검증과 μ–•μ€κΈ°μ΄ˆμ— μž‘μš©ν•˜λŠ” λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ μ˜ˆμ œν•΄μ„μ„ μˆ˜ν–‰ν•˜κ³ μž ν•œλ‹€.

2. 경계면 μ†Œμ„±λͺ¨λΈμ˜ μˆ˜μ‹ν™”

2.1 νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” J2-경계면 μ†Œμ„±λͺ¨λΈ

J2 μ†Œμ„±λͺ¨λΈμ€ νŽΈμ°¨μ‘λ ₯ ν…μ„œμ— λŒ€ν•œ 응λ ₯ λΆˆλ³€λŸ‰(stress invariant) J2λ₯Ό λ³€μˆ˜λ‘œ 항볡면을 μ •μ˜ν•œλ‹€. 경계면 μ†Œμ„±λͺ¨λΈμ—μ„œ 항볡면(yield surface)와 경계면(bounding surface)λŠ” νŽΈμ°¨μ‘λ ₯에 λŒ€ν•œ p ν‰λ©΄μ—μ„œ 원(circle)으둜 ν‘œν˜„λœλ‹€(Fig. 1). 그리고 경계면은 등방응λ ₯ p에 λŒ€ν•˜μ—¬ μ„ ν˜•μ μœΌλ‘œ μ˜μ‘΄μ μ΄λ‹€.

(1)

Yield surface $f= βˆ₯\underline{\sigma}'-\underline{\alpha}βˆ₯ -r=0$

Bounding surface $B = q-(a Β· p+b)=0$

μ—¬κΈ°μ„œ $\underline{\sigma}$λŠ” μœ νš¨μ‘λ ₯(effective stress)이고, $\underline{1}=\delta_{ij}$λŠ” Kronecker’s delta tensor이닀. $\underline{\sigma}'$λŠ” νŽΈμ°¨μ‘λ ₯(deviatoric stress) ν…μ„œ(tensor)이며 $\underline{\sigma}'=\underline{\sigma}-p\underline{1}$이닀. p ν‰λ©΄μ—μ„œ $\underline{\alpha}$λŠ” ν•­λ³΅λ©΄μ˜ 쀑심에 λŒ€μ‘ν•˜λŠ” 응λ ₯ ν…μ„œμ΄λ©°, r은 ν•­λ³΅λ©΄μ˜ λ°˜κ²½μ΄λ‹€. J2 응λ ₯ λΆˆλ³€λŸ‰μ€ $\dfrac{1}{2}\sigma_{ij}'\sigma_{ij}'=\dfrac{1}{2}\underline{\sigma}':\underline{\sigma}'=\dfrac{1}{2}βˆ₯\underline{\sigma'}βˆ₯^{2}$이며, qλŠ” νŽΈμ°¨μ‘λ ₯ λΆˆλ³€λŸ‰μœΌλ‘œ $q=\sqrt{3/2}βˆ₯\underline{\sigma}'βˆ₯ =\sqrt{3J_{2}}$κ³Ό κ°™λ‹€. 그리고 경계면에 λŒ€ν•œ κ°•λ„μ •μˆ˜ a와 bλŠ” μ§€λ°˜μ˜ 마찰각(f)κ³Ό 점착λ ₯(c)으둜 $a=\dfrac{6\sin(\phi)}{3-\sin(\phi)}$, $b=\dfrac{6Β· c Β·\sin(\phi)}{3-\sin(\phi)}$와 같이 μ‚°μ •ν•  수 μžˆλ‹€.

ν•­λ³΅λ©΄μ˜ 반경 r을 0으둜 μˆ˜λ ΄μ‹œν‚€λ©΄, νƒ„μ„±μ˜μ—­μ΄ 사라지고 νŽΈμ°¨μ‘λ ₯κ³Ό 항볡면 쀑심응λ ₯의 λ°©ν–₯은 편차 λ‹¨μœ„ ν…μ„œ(deviatoric unit tensor) $\underline{n}$으둜 λ™μΌν•˜λ‹€(Borja and Amies, 1994).

(2)
$\dfrac{\underline{\sigma'}}{βˆ₯\underline{\underline{\sigma'}}βˆ₯}=\dfrac{\underline{\alpha}}{βˆ₯\underline{\underline{\alpha}}βˆ₯}=\underline{n}$

Prager(1955)κ°€ μ œμ•ˆν•œ μš΄λ™κ²½ν™”(kinematic hardening) κ·œμΉ™μ„ Eq. (1)의 항볡면에 μ μš©ν•˜λ©΄, νŽΈμ°¨μ‘λ ₯ μ¦λΆ„μ˜ λ°©ν–₯ν…μ„œλŠ” 편차 λ‹¨μœ„ν…μ„œ $\underline{n}$κ³Ό μΌμΉ˜ν•˜κ²Œ λœλ‹€.

(3)
$\dfrac{\underline{\dot{\sigma}}'}{βˆ₯\underline{\dot{\sigma}}'βˆ₯}=\underline{n}$

역학적 μ†Œμ„±λ³€ν˜• 증뢄은 μ†Œμ„±νλ¦„λ²•μΉ™(plastic flow rule)κ³Ό 항볡면에 λŒ€ν•œ 일관성 쑰건(consistency condition)μœΌλ‘œλΆ€ν„° μ‚°μ •ν•  수 μžˆλ‹€.

(4)
$\dot{\underline{\epsilon}^{p}}=\dfrac{1}{H}(\underline{n}:\dot{\underline{\sigma}})\dfrac{\partial g}{\partial\underline{\sigma}}$

μ—¬κΈ°μ„œ HλŠ” μ†Œμ„±κ²½ν™” κ³„μˆ˜(plastic hardening modulus)둜 μ •μ˜ν•œλ‹€. μ†Œμ„± ν¬ν…μ…œ ν•¨μˆ˜(plastic potential function) gλŠ” g = q + d Β· p = 0이며, 팽창λ₯ (dilatancy, d = tan(d))은 팽창각(dilatancy angle, d)으둜 ν‘œν˜„λœλ‹€. μ„Έλ¦½ν† μ˜ 팽창λ₯ μ€ νŽΈμ°¨μ‘λ ₯λΉ„(Roscoe and Schofield, 1963)λ‘œλΆ€ν„° μ‚°μ •ν•˜κ³ , μ‘°λ¦½ν† λŠ” μƒνƒœλ³€μˆ˜(state parameter)λ‘œλΆ€ν„° μ‚°μ •ν•  수 μžˆλ‹€(Been and Jefferies, 1985).

Fig. 1. J2-Bounding Surface and Yield Surface in Principal Stress Space and on the Ο€-Plane
../../Resources/KSCE/Ksce.2023.43.4.0469/fig1.png

2.2 응λ ₯-λ³€ν˜•λ₯  증뢄 관계

역학적 λ³€ν˜•λ₯  증뢄은 λ‹€μŒκ³Ό 같이 탄성과 μ†Œμ„± μ¦λΆ„μ˜ ν•©μœΌλ‘œ λ‚˜νƒ€λ‚Ό 수 μžˆλ‹€.

(5)
$\dot{\underline{\epsilon}}=\dot{\underline{\epsilon}^{e}}+\dot{\underline{\epsilon}^{p}}=\underline{D}^{e^{-1}}:\dot{\underline{\sigma}}+\dot{\underline{\epsilon^{p}}}$

μ—¬κΈ°μ„œ 탄성 κ°•μ„± ν…μ„œ(elastic stiffness tensor)λŠ” $\underline{D}^{e}= K\underline{1}\otimes\underline{1}$$+2G(\underline{I}-\dfrac{1}{3}\underline{1}\otimes\underline{1})$이닀. 그리고 KλŠ” μ²΄μ νƒ„μ„±κ³„μˆ˜(elastic bulk modulus)이고, GλŠ” μ „λ‹¨νƒ„μ„±κ³„μˆ˜(elastic shear modulus)이닀.

Eq. (4)와 Eq. (5)에 λŒ€ν•œ μˆ˜ν•™μ  λ³€ν˜•μœΌλ‘œλΆ€ν„° λ‹€μŒμ‹μ„ μ‚°μ •ν•  수 μžˆλ‹€.

(6)
$\underline{n}:\underline{\dot{\sigma}}=\dfrac{2G\underline{n}:\underline{\dot{\epsilon}}}{1+\sqrt{6}G/H}$

νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” J2-경계면 μ†Œμ„±λͺ¨λΈμ— λŒ€ν•œ 응λ ₯-λ³€ν˜•λ₯  증뢄식은 λ‹€μŒκ³Ό κ°™λ‹€.

(7)
$\dot{\underline{\sigma}}=\left[\underline{D}^{e}-\dfrac{2G}{H+\sqrt{6}G}(Kd\underline{1}\otimes\underline{1}+\sqrt{6}G\underline{n}\otimes\underline{n})\right]:\dot{\underline{\epsilon}}$

Eq. (7)을 νŽΈμ°¨μ‘λ ₯κ³Ό λ³€ν˜•λ₯  관계에 λŒ€ν•˜μ—¬ μ •λ¦¬ν•˜λ©΄ λ‹€μŒκ³Ό κ°™λ‹€.

(8)
$\dot{q}=\dfrac{3G}{1+\dfrac{\sqrt{6}G}{H}}\dot{\epsilon_{q}}$

μ—¬κΈ°μ„œ 편차 λ³€ν˜•λ₯ μ€ $\epsilon_{q =}\sqrt{\dfrac{2}{3}}DL\in E\underline{\epsilon'}DL\in E$이닀.

2.3 μŒκ³‘μ„ (hyperbolic) λͺ¨λΈμ— λŒ€ν•œ μ†Œμ„±κ²½ν™” κ³„μˆ˜ H

점토와 λͺ¨λž˜μ˜ λΉ„μ„ ν˜• 응λ ₯-λ³€ν˜•λ₯  관계에 μ‚¬μš©λ˜λŠ” μŒκ³‘μ„ (hyperbolic) λͺ¨λΈμ‹μ€ λ‹€μŒκ³Ό κ°™λ‹€(Hardin and Drnevich, 1972; Kondner, 1963).

(9)
$q=\dfrac{\epsilon_{q}}{\dfrac{1}{3G_{\max}}+\dfrac{\epsilon_{q}}{q_{u}}}$

μ—¬κΈ°μ„œ GmaxλŠ” 초기 μ „λ‹¨νƒ„μ„±κ³„μˆ˜(initial elastic shear modulus)이고, quλŠ” μΆ•μ°¨ 전단강도이닀.

Eq. (9)의 μŒκ³‘μ„  λͺ¨λΈμ„ 응λ ₯-λ³€ν˜•λ₯  관계λ₯Ό λ‹€μŒκ³Ό 같은 μ¦λΆ„ν˜•νƒœλ‘œ ν‘œν˜„ν•  수 μžˆλ‹€.

(10)
$\dot{q}= 3G_{\max}\left(1-\dfrac{q}{q_{u}}\right)^{2}\dot{\epsilon_{q}}$

Eq. (8)κ³Ό Eq. (10)μœΌλ‘œλΆ€ν„° μŒκ³‘μ„  응λ ₯-λ³€ν˜•λ₯  관계에 λŒ€ν•œ μ†Œμ„±κ²½ν™” κ³„μˆ˜λ₯Ό μ‚°μ •ν•  수 μžˆλ‹€.

(11)
$H=\sqrt{6}G_{\max}\dfrac{(q_{u}- q)^{2}}{q(2q_{u}-q)}$

개발된 νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” J2-경계면 μ†Œμ„±λͺ¨λΈμ€ ν† μ‚¬μ§€λ°˜μ™€ μ ˆλ¦¬μ•”λ°˜μ˜ Thermo-Hydro-Mechanical 연계해석을 μœ„ν•˜μ—¬ 개발된 Geo-COUS(Geo-COUpled Simulator) μœ ν•œμš”μ†Œ ν”„λ‘œκ·Έλž¨κ³Ό κ²°ν•©ν•˜μ˜€λ‹€(Shin, 2011; Shin and Santamarina, 2019).

3. 개발된 μ†Œμ„±λͺ¨λΈμ˜ 예제 해석

3.1 반볡 μ‚ΌμΆ•μ‹€ν—˜(cyclic tri-axial test)

λ‹¨μΌν•­λ³΅λ©΄μ˜ Drucker-Prager λͺ¨λΈ(β€œD-P λͺ¨λΈβ€)κ³Ό 개발된 νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” Bounding Surface λͺ¨λΈ(β€œB-S λͺ¨λΈβ€)을 λ°˜λ³΅μ‚ΌμΆ•μ••μΆ• 쑰건에 λŒ€ν•˜μ—¬ λΉ„κ΅ν•˜μ˜€λ‹€. D-P λͺ¨λΈμ€ 항볡면과 파괴면이 λ™μΌν•˜λ©°, B-S λͺ¨λΈμ˜ 경계면을 ν•­λ³΅λ©΄μœΌλ‘œ μ‚¬μš©ν•œλ‹€. 그리고 D-P λͺ¨λΈμ€ μΌκ΄€λœ μ ‘μ„ κ³„μˆ˜(consistent tangent modulus)에 μ˜ν•œ λ‚΄μž¬μ  응λ ₯적뢄(implicit stress integration)을 μ‚¬μš©ν•˜μ—¬ 2차수렴(quadratic convergence)ν•œλ‹€(Simo and Taylor, 1985).

D-P λͺ¨λΈ(Fig. 2a)은 항볡면(yield surface)κ³Ό νŒŒκ΄΄ν¬λ½μ„ (failure envelope)이 λ™μΌν•˜μ—¬, 항볡면 λ‚΄μ—μ„œ μ„ ν˜•νƒ„μ„± 거동을 보인닀. 그리고 μ—­μž¬ν•˜/μž¬μž¬ν•˜(unload-reload)μ‹œ 응λ ₯이 항볡면 내에 μœ„μΉ˜ν•˜μ—¬ νƒ„μ„±λ³€ν˜•λ§Œμ„ μœ λ°œν•œλ‹€. Eq. (9)의 μŒκ³‘μ„  λͺ¨λΈ(hyperbolic model)은 μ „λ‹¨λ³€ν˜•μ΄ λ¬΄ν•œνžˆ 컀지면 좕차응λ ₯은 전단 강도에 μˆ˜λ ΄ν•˜λŠ” 것을 μ•Œ 수 μžˆλ‹€.

λ³Έ μ—°κ΅¬μ—μ„œ κ°œλ°œν•œ νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” B-S λͺ¨λΈμ€ D-P λͺ¨λΈκ³Ό λ™μΌν•œ λ¬Όμ„±μΉ˜λ₯Ό μ‚¬μš©ν•˜μ˜€λ‹€. Fig. 2bμ—μ„œ μ΄ˆκΈ°ν•˜μ€‘ μž¬ν•˜μ‹œ 응λ ₯-λ³€ν˜•λ₯  κ΄€κ³„λŠ” μŒκ³‘μ„  λͺ¨λΈκ³Ό λ™μΌν•˜λ©°, λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•˜μ—¬ μ†Œμ„±λ³€ν˜•μ΄ λˆ„μ λ˜λŠ” μ „ν˜•μ μΈ 이λ ₯(hysteresis) 곑선을 보인닀. 개발된 λͺ¨λΈμ˜ λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ 이λ ₯(hysteresis) 곑선은 μŒκ³‘μ„  λͺ¨λΈμ— λŒ€ν•œ Masing rule(Masing, 1926)을 λ”°λ₯Έλ‹€.

Fig. 2. Cyclic Behavior under Tri-axial Condition: (a) Drucker-Prager Model with Hyperbolic Model, (b) Bounding-Surface with Hyperbolic Model. Material Properties: Gmax = 27 MPa (VS = 200 m/s, gt=1.8 g/cm3), n = 0.3, tmax= cu = 40 kPa(qu= 80 kPa), Dilatancy Angle d = 0o
../../Resources/KSCE/Ksce.2023.43.4.0469/fig2.png

3.2 λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ μ–•μ€κΈ°μ΄ˆμ˜ 거동

개발된 λͺ¨λΈμ˜ κ±°λ™νŠΉμ„±μ„ λΆ„μ„ν•˜κΈ° μœ„ν•˜μ—¬ λ°˜λ³΅ν•˜μ€‘μ„ λ°›λŠ” μ–•μ€κΈ°μ΄ˆ(shallow foundation)에 λŒ€ν•œ μ˜ˆμ œν•΄μ„μ„ μˆ˜ν–‰ν•˜μ˜€λ‹€. μˆ˜μΉ˜ν•΄μ„μ€ 2차원 ν‰λ©΄λ³€ν˜•λ₯  쑰건으둜 40,000개의 8절점 μš”μ†Œμ™€ 121,001개의 μ ˆμ μ„ μ΄μš©ν•˜μ˜€λ‹€. Table 1은 μˆ˜μΉ˜ν•΄μ„μ— μ‚¬μš©λœ μ§€λ°˜μ˜ λ¬Όμ„±μΉ˜μ™€ ν•˜μ€‘ μž¬ν•˜ν­ B = 10 m에 λŒ€ν•œ μ–•μ€κΈ°μ΄ˆμ˜ κ·Ήν•œ 지지λ ₯(bearing capacity)을 λ‚˜νƒ€λ‚΄κ³  μžˆλ‹€(Terzaghi, 1943). D-Pλͺ¨λΈκ³Ό B-S λͺ¨λΈμ„ μ΄μš©ν•˜μ—¬ 전단강도가 ꡬ속압에 λ¬΄κ΄€ν•œ 점성토 μ§€λ°˜κ³Ό ꡬ속압에 λΉ„λ‘€ν•˜μ—¬ μ¦κ°€ν•˜λŠ” c-f μ§€λ°˜μ— λŒ€ν•˜μ—¬ 비ꡐ해석을 μˆ˜ν–‰ν•˜μ˜€λ‹€.

Fig. 3은 연직 λ°˜λ³΅ν•˜μ€‘μ— μ˜ν•œ μ—°μ„±κΈ°μ΄ˆ μ€‘μ•™μ˜ λ³€μœ„λ₯Ό λ‚˜νƒ€λ‚΄κ³  μžˆλ‹€. Fig. 3a(점성토 μ§€λ°˜)μ—μ„œ D-P λͺ¨λΈμ€ 50 kPa의 λ°˜λ³΅ν•˜μ€‘μ— μ˜ν•œ 응λ ₯λ³€ν™”κ°€ 항볡면 λ‚΄λΆ€λ‘œ κ·Ήν•œλ˜μ–΄ μ„ ν˜•μ μΈ 응λ ₯-μΉ¨ν•˜λŸ‰ 거동을 보인닀. 반면, B-S λͺ¨λΈμ€ λ°˜λ³΅ν•˜μ€‘μ— μ˜ν•œ μ†Œμ„±μΉ¨ν•˜λŸ‰μ˜ λˆ„μ κ³Ό μ—­μž¬ν•˜μ‹œ(unloading)에 μž¬μž¬ν•˜μ‹œ(reloading)보닀 λ³€μœ„μ˜ λ³€ν™”κ°€ μž‘κ²Œ λ‚˜νƒ€λ‚¬λ‹€.

Fig. 3b(c-f μ§€λ°˜)μ—μ„œ μΌκ΄€λœ μ ‘μ„ κ³„μˆ˜λ₯Ό μ‚¬μš©ν•˜λŠ” D-P λͺ¨λΈμ€ μˆ˜λ ΄μ„±μ΄ μ €ν•˜λ˜μ–΄ λ°œμ‚°ν•˜μ˜€λ‹€. B-S λͺ¨λΈμ„ μ΄μš©ν•˜μ—¬ 팽창각 d = 0o, 10o에 λŒ€ν•˜μ—¬ 100 kPa의 λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ μˆ˜μΉ˜ν•΄μ„μ„ μˆ˜ν–‰ν•˜μ˜€λ‹€. 팽창각이 컀질수둝, μ†Œμ„±μ „λ‹¨λ³€ν˜•μ— μ˜ν•œ 체적팽창으둜 μ–•μ€κΈ°μ΄ˆμ˜ λ°˜μ‘κ°•μ„±μ΄ μ»€μ§€λ©΄μ„œ μ†Œμ„±μΉ¨ν•˜λŸ‰μ΄ κ°μ†Œν•˜μ˜€λ‹€.

Fig. 4λŠ” μ–•μ€κΈ°μ΄ˆμ— 반볡 λͺ¨λ©˜νŠΈ μž¬ν•˜μ‹œ μ—°μ„±κΈ°μ΄ˆμ˜ 평균 νšŒμ „κ°μ˜ λ³€ν™”λ₯Ό λ‚˜νƒ€λ‚΄κ³  μžˆλ‹€. Fig. 4a(점성토 μ§€λ°˜)λŠ” μ–•μ€κΈ°μ΄ˆμ— 150 kPa의 연직응λ ₯을 μž¬ν•˜ν•œ ν›„, λͺ¨λ©˜νŠΈ(800 kNΒ·m/m)λ₯Ό 반볡적으둜 μž¬ν•˜ν•˜μ˜€λ‹€. λ°˜λ³΅ν•˜μ€‘λ‹¨κ³„μ—μ„œ D-P λͺ¨λΈμ€ 기초의 μ„ ν˜•νƒ„μ„±μ μΈ 거동을 보이고, B-S λͺ¨λΈμ€ 기초의 νšŒμ „(rocking)에 λŒ€ν•œ 강성이 점차적으둜 μ¦κ°€ν•˜λ©΄μ„œ μˆ˜λ ΄ν•˜μ˜€λ‹€. Fig. 4b(c-f μ§€λ°˜)λŠ” μ–•μ€κΈ°μ΄ˆμ— 500 kPa의 연직응λ ₯을 μž¬ν•˜ν•œ ν›„, 반볡 λͺ¨λ©˜νŠΈ(1,250 kNΒ·m/m)λ₯Ό μž‘μš©μ‹œμΌ°λ‹€. λ°˜λ³΅ν•˜μ€‘λ‹¨κ³„μ—μ„œ 기초의 νšŒμ „λ°˜μ‘ 강성이 점차적으둜 μ¦κ°€ν•˜λ©΄μ„œ μˆ˜λ ΄ν•˜μ˜€λ‹€. 팽창각 dκ°€ μ¦κ°€ν•˜λ©΄, μ „λ‹¨λ³€ν˜•μ— μ˜ν•œ μ²΄μ νŒ½μ°½μ„±μ˜ μ¦κ°€λ‘œ μ—°μ§μ†Œμ„± μΉ¨ν•˜λŸ‰μ΄ κ°μ†Œν•˜κ³ , 얕은 기초의 νšŒμ „λ°˜μ‘ 강성이 μ¦κ°€ν•˜μ˜€λ‹€.

Table 1. Material Properties and Bearing Capacity of Shallow Foundation for Fig. 3 and 4

Model

Material properties

Bearing capacity(B = 5m)

D-P model

β€œcoh”

Gmax = 27 MPa, n = 0.3, gt = 15 kN/m3

c = 40 kPa, f = 0Β°, d = 0Β°

248 kPa

B-S model

β€œcoh”

β€œcoh-phi-d”

Gmax = 27 MPa, n = 0.3, gt = 15 kN/m3

c = 10 kPa, f = 30Β°, d = 0Β° or 10Β°

2,228 kPa

Fig. 3. Behavior of Shallow Foundation under Vertical Cyclic Load with Drucker-Prager and Developed Bounding-Surface Model: (a) Vertical Load-Settlement of the Foundation on Cohesive Soil, (b) Vertical Load-Settlement of the Foundation on Cohesive-Frictional Soil under 5 Cyclic Loads
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Fig. 4. Behavior of Shallow Foundation under 5 Cyclic Moments with Drucker-Prager and Developed Bounding-Surface Model: (a) Moment- Rotation Curve of the Foundation on Cohesive Soil, (b) Moment-Rotation Curve of the Foundation on Cohesive-Frictional Soil
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3.3 λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ λ™μ νŠΉμ„± 평가

μ§€λ°˜μ§€μ§„κ³΅ν•™μ—μ„œ μ§€λ°˜μ˜ λ™μ ν•˜μ€‘μ— λŒ€ν•œ 거동을 λͺ¨μ‚¬ν•˜λŠ” 역학적 λͺ¨λΈμ€ 맀우 μ€‘μš”ν•˜λ‹€. 주둜 μ •κ·œ μ „λ‹¨ν• μ„ κ³„μˆ˜(G/Gmax, normalized secant shear modulus)와 감쇠비(D, material or hysteretic damping ratio)에 μ˜ν•œ 1차원 λ“±κ°€μ„ ν˜•λͺ¨λΈ(equivalent linear model)을 μ΄μš©ν•˜μ—¬ μ£ΌνŒŒμˆ˜μ˜μ—­ 해석을 μˆ˜ν–‰ν•˜κ³  μžˆλ‹€. ν•˜μ§€λ§Œ, 1차원 λͺ¨λΈμ€ μ§€λ°˜μ˜ 비가역적(irreversible) μ†Œμ„±λ³€ν˜•, κ°„κ·Ήμˆ˜μ••μ˜ 영ν–₯, μΆ•μ°¨λ³€ν˜•κ³Ό μ—°κ³„λœ μ²΄μ λ³€ν˜•, 그리고 λΉ„μ •κ·œμ μΈ(irregular) λ™μ ν•˜μ€‘ 등을 κ³ λ €ν•  수 μ—†λ‹€. λ˜ν•œ 반볡 μ „λ‹¨λ³€ν˜•λ₯ μ— λ”°λ₯Έ μ •κ·œ μ „λ‹¨ν• μ„ κ³„μˆ˜μ™€ κ°μ‡ λΉ„λ‘œ ν‘œν˜„λ˜λŠ” 1차원 λ“±κ°€μ„ ν˜•λͺ¨λΈμ€ 닀차원 μ‹œκ°„μ΄λ ₯해석을 ν•  수 μ—†λ‹€. λ”°λΌμ„œ, 일반적인 응λ ₯μž₯μ—μ„œ μ •μ˜λ˜λŠ” νƒ„μ†Œμ„± λͺ¨λΈμ— λŒ€ν•œ 연ꡬ가 μ§€μ†μ μœΌλ‘œ μ§„ν–‰λ˜κ³  있으며, μ‹€μ œ λ¬Έμ œμ— ν™œμš©ν•˜λŠ”λ° μžˆμ–΄μ„œ λ§Žμ€ 갯수의 λ¬Όμ„±μΉ˜λŠ” 문제점으둜 μΈμ‹λ˜κ³  μžˆλ‹€(PisanΓ² and JeremiΔ‡, 2014).

Eq. (9)의 μŒκ³‘μ„  λͺ¨λΈμ— Masing rule을 μ μš©ν•œ 반볡 μ „λ‹¨λ³€ν˜•λ₯ μ— λŒ€ν•œ 응λ ₯이λ ₯κ³‘μ„ μœΌλ‘œλΆ€ν„° μ „λ‹¨ν• μ„ κ³„μˆ˜(G/Gmax)와 감쇠비(D)λ₯Ό μ‚°μ •ν•  수 μžˆλ‹€(Ishihara, 1996).

(12)

$\dfrac{G}{G_{\max}}=\dfrac{1}{1+\gamma /\gamma_{r}}$

$D=\dfrac{4}{\pi}\left(1+\dfrac{1}{\gamma /\gamma_{r}}\right)\left[1-\dfrac{\ln(1+\gamma /\gamma_{r})}{\gamma /\gamma_{r}}\right]-\dfrac{2}{\pi}$

μ—¬κΈ°μ„œ gr은 κΈ°μ€€λ³€ν˜•λ₯ (reference strain)으둜 tmax/Gmax이닀.

Fig. 5λŠ” λ‹€μ–‘ν•œ 크기의 반볡 μ „λ‹¨λ³€ν˜•λ₯ μ— λŒ€ν•˜μ—¬ 초기 ν•˜μ€‘μž¬ν•˜ ν›„ 5번의 μ—­μž¬ν•˜-μž¬μž¬ν•˜μ— λŒ€ν•œ 전단응λ ₯-μ „λ‹¨λ³€ν˜•λ₯ μ˜ 경둜λ₯Ό 보여주고 μžˆλ‹€. 초기 κ²½λ‘œλŠ” μŒκ³‘μ„  λͺ¨λΈ(hyperbolic model)κ³Ό μΌμΉ˜ν•¨μ„ μ•Œ 수 μžˆλ‹€. 전단 λ³€ν˜•λ₯ μ΄ μž‘μœΌλ©΄, μ„ ν˜•μ— κ·Όμ‚¬ν•œ 응λ ₯-λ³€ν˜•λ₯ μ˜ 관계λ₯Ό λ³΄μ΄λ©΄μ„œ G/Gmaxβ‰ˆ1.0, 감쇠비 Dβ‰ˆ0에 κ·Όμ ‘ν•œλ‹€. μ‹€λ‚΄ μ‹€ν—˜κ²°κ³Όμ—μ„œ 맀우 μž‘μ€ 전단 λ³€ν˜•λ₯ μ—μ„œ λ°œμƒν•˜λŠ” μ΅œμ†Œ 감쇠비(Dmin)λŠ” 곡극내 유체의 점성(viscosity)와 μ§€λ°˜μ˜ 크리프(creep)에 μ˜ν•˜μ—¬ λ°œμƒν•˜λ―€λ‘œ(d’Onofrio et al., 1999), μ‹œκ°„μ΄λ ₯해석(time history analysis)μ—μ„œ μ΅œμ†Œ 감쇠비 Dmin을 감쇠λ ₯에 직접 μ μš©ν•  수 μžˆλ‹€. 반볡 μ „λ‹¨λ³€ν˜•λ₯ μ΄ 컀질수둝, μ „λ‹¨ν• μ„ κ³„μˆ˜ G/GmaxλŠ” κ°μ†Œν•˜κ³  감쇠비 DλŠ” 점차적으둜 μ»€μ§€λŠ” 것을 μ•Œ 수 μžˆλ‹€(Kramer, 1996).

Fig. 6a와 6bλŠ” 반볡 μ „λ‹¨λ³€ν˜•λ₯ μ˜ 크기에 λ”°λ₯Έ G/Gmax와 D의 λ³€ν™”λ₯Ό λ‚˜νƒ€λ‚΄κ³  μžˆλ‹€. 개발된 B-Sλͺ¨λΈμ— λŒ€ν•œ μˆ˜μΉ˜ν•΄μ„ κ²°κ³ΌλŠ” Eq. (12)의 이둠적과 맀우 μΌμΉ˜ν•¨μ„ μ•Œ 수 μžˆλ‹€. λ‹€λ§Œ, 반볡 μ „λ‹¨λ³€ν˜•λ₯ μ΄ 맀우 큰 κ²½μš°μ—λŠ” 감쇠비에 λŒ€ν•œ 수치적인 μ˜€μ°¨κ°€ λ°œμƒν•¨μ„ μ•Œ 수 μžˆλ‹€(Anastasopoulos et al., 2011).

쀑좔곑선(backbone curve)에 Masing rule을 μ μš©ν•œ λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ 이λ ₯(hysteric) 응λ ₯κ²½λ‘œλŠ” μ‹€ν—˜κ²°κ³Όμ— λΉ„ν•˜μ—¬ 큰 λ³€ν˜•λ₯ μ—μ„œ 감쇠비λ₯Ό κ³ΌλŒ€ μ‚°μ •ν•˜μ—¬ λ™μ ν•˜μ€‘μ— λŒ€ν•œ μ§€λ°˜μ˜ λ°˜μ‘μ„ κ³Όμ†Œν‰κ°€ν•˜λŠ” κ²½ν–₯이 μžˆλ‹€(Stewart et al., 2008). μ΄λŸ¬ν•œ λ¬Έμ œμ μ„ ν•΄κ²°ν•˜κΈ° μœ„ν•˜μ—¬, 1) μˆ˜μ •λœ μŒκ³‘μ„  λͺ¨λΈ(Yang et al., 2022)을 μ€‘μΆ”κ³‘μ„ μœΌλ‘œ μ‚¬μš©ν•˜μ—¬ μƒˆλ‘œμš΄ μ†Œμ„±κ²½ν™” κ³„μˆ˜λ₯Ό μ œμ‹œν•˜μ˜€λ‹€.

(13)

$q=\dfrac{\epsilon_{q}}{\left[\left(\dfrac{1}{3G_{\max}}\right)^{\chi}+\left(\dfrac{\epsilon_{q}}{q_{u}}\right)^{\chi}\right]^{1/\chi}}$

$H=\sqrt{6}G_{\max}\dfrac{\left(1-\left(q/q_{u}\right)^{\chi}\right)^{1+1/\chi}}{\left(1-\left(1-(q/q_{u}\right)^{\chi}\right)^{1+1/\chi}}$

μ—¬κΈ°μ„œ c = 1이면 기쑴의 μŒκ³‘μ„  ν•¨μˆ˜μ™€ μ†Œμ„±κ²½ν™” κ³„μˆ˜μ™€ λ™μΌν•˜λ‹€.

2) μ—­ν•˜μ€‘(unloading)에 μ˜ν•œ 응λ ₯μ „ν™˜(stress reversal)이 응λ ₯ κ²½λ‘œμ— λ―ΈμΉ˜λŠ” 영ν–₯을 κ³ λ €ν•˜μ—¬ 좔가적인 μ†Œμ„±κ³„μˆ˜ H1을 λ„μž…ν•˜μ˜€λ‹€. λ”°λΌμ„œ, μƒˆλ‘­κ²Œ μ •μ˜λ˜λŠ” μ†Œμ„± κ²½ν™”κ³„μˆ˜ H*은 μˆ˜μ •λœ μŒκ³‘μ„  λͺ¨λΈμ— λŒ€ν•œ κ²½ν™”κ³„μˆ˜(Eq. (13))와 μΆ”κ°€λ‘œ λ„μž…λœ κ²½ν™”κ³„μˆ˜ H1의 직렬(serial) μ‘°ν•©μœΌλ‘œ μ‚°μ •ν•  수 μžˆλ‹€.

(14)

$H_{1}= m\sqrt{6}G\dfrac{1}{(q_{0}/q_{u})^{n}}$

$H *=\dfrac{1}{\dfrac{1}{H}+\dfrac{1}{H_{1}}}$

μ—¬κΈ°μ„œ, q0λŠ” μ΅œμ’… 응λ ₯μ „ν™˜(stress reversal)μ‹œμ˜ 좕차응λ ₯이닀. H1이 λ¬΄ν•œνžˆ 컀지면, μƒˆλ‘œμš΄ κ²½ν™”κ³„μˆ˜ H*λŠ” H둜 μˆ˜λ ΄ν•˜κ²Œ λœλ‹€.

1차원 λ“±κ°€μ„ ν˜•λͺ¨λΈμ—μ„œ μ „λ‹¨λ³€ν˜•λ₯  g에 λŒ€ν•œ G/Gmax와 D의 관계가 μ£Όμ–΄μ§ˆ λ•Œ, Eq. (13)κ³Ό Eq. (14)의 λͺ¨λΈλ³€μˆ˜λ“€μ„ κ²°μ •ν•˜κΈ° μœ„ν•˜μ—¬ λ§€κ°œλ³€μˆ˜ 해석을 μˆ˜ν–‰ν•˜μ˜€λ‹€. Fig. 7은 μ „λ‹¨λ³€ν˜•λ₯  g/gr = 0.5μ—μ„œ λͺ¨λΈλ³€μˆ˜ mκ³Ό c에 λ”°λ₯Έ G/Gmax와 D의 λ³€ν™”λ₯Ό 보여주고 μžˆλ‹€. 기쑴의 μŒκ³‘μ„  λͺ¨λΈμ—μ„œ g/gr = 0.5에 λŒ€ν•œ G/Gmax = 2/3, D = 8.6\%이닀(Eq. (12)). mκ³Ό cκ°€ μ¦κ°€ν•˜λ©΄ λ°˜λ³΅μ‘λ ₯경둜의 G/GmaxλŠ” μ¦κ°€ν•˜κ³ , κ°μ‡ λΉ„λŠ” c이 μ¦κ°€ν•˜λ©΄ 점차 κ°μ†Œν•˜μ˜€λ‹€. λ‹€μ–‘ν•œ μ§€λ°˜μ˜ μ‹€λ‚΄μ‹€ν—˜ 결과에 λŒ€ν•˜μ—¬ λͺ¨λΈλ³€μˆ˜λ“€μ„ κ²°μ •ν•˜κΈ° μœ„ν•œ 좔가적인 연ꡬ가 ν•„μš”ν•˜λ‹€.

Fig. 5. Cyclic Stress-Strain Paths at Different Shear Stress Levels. Material Properties: Gmax = 27 MPa, n = 0.3, tmax = 40 kPa, Dilatancy Angle d = 0o
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Fig. 6. (a) Comparison of G/Gmax from B-S model and Hyperbolic Model, (b) Comparison of Damping Ratio from B-S Model and Hyperbolic Model. Material Properties: Gmax = 27 MPa, n = 0.3, tmax = 40 kPa, Dilatancy Angle d = 0 o
../../Resources/KSCE/Ksce.2023.43.4.0469/fig6.png
Fig. 7. Parametric Chart on Model Parameters (m and c, Fixed n = 0.5) at Deviatoric Strain Level g/gr = 0.5: (a) Normalized Secant Shear Modulus, G/Gmax, (b) Damping Ratio, D
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4. κ²° λ‘ 

λ°˜λ³΅ν•˜μ€‘μ΄λ‚˜ λ™μ ν•˜μ€‘μ— λŒ€ν•œ μ§€λ°˜μ˜ 역학적 μ†Œμ„±λͺ¨λΈμ€ μ§€λ°˜κ΅¬μ‘°λ¬Όμ˜ 닀차원 λΉ„μ„ ν˜• μˆ˜μΉ˜ν•΄μ„μ—μ„œ 맀우 μ€‘μš”ν•˜λ‹€. 단일 항볡면 μ†Œμ„±λͺ¨λΈμ€ λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•΄ μ„ ν˜•μ  거동을 보이고, 기쑴의 경계면 μ†Œμ„±λͺ¨λΈμ€ λ³΅μž‘ν•œ 응λ ₯-λ³€ν˜•λ₯  관계λ₯Ό λͺ¨μ‚¬ν•  수 μžˆμ§€λ§Œ ꡬ성λͺ¨λΈμ„ κ΅¬ν˜„ν•˜κΈ° μœ„ν•˜μ—¬ λ§Žμ€ λ¬Όμ„±μΉ˜κ°€ ν•„μš”ν•˜λ‹€. λ³Έ λ…Όλ¬Έμ˜ νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” J2-경계면 μ†Œμ„±λͺ¨λΈμ€ λ‹¨μˆœν•œ λ¬Όμ„±μΉ˜λ‘œ 효과적으둜 μ§€λ°˜μ˜ λΉ„μ„ ν˜•μ„±μ„ λͺ¨μ‚¬ν•  수 μžˆλ‹€.

- 개발된 경계면 μ†Œμ„±λͺ¨λΈμ€ p ν‰λ©΄μ—μ„œ ν•­λ³΅λ©΄μ˜ λ°˜κ²½μ„ 0으둜 μˆ˜λ ΄μ‹œμΌœ νƒ„μ„±μ˜μ—­μ΄ 사라지도둝 μˆ˜μ‹ν™”ν•˜κ³ , μ†Œμ„±κ²½ν™” κ³„μˆ˜κ³Ό 팽창λ₯ μ„ μ΄μš©ν•˜μ—¬ μ†Œμ„±λ³€ν˜• 증뢄을 μ •μ˜ν•˜μ˜€λ‹€. νƒ„μ„±μ˜μ—­μ΄ μ—†λŠ” J2-경계면 μ†Œμ„±λͺ¨λΈμ˜ 응λ ₯-λ³€ν˜•λ₯  증뢄식을 μ œμ‹œν•˜κ³ , μŒκ³‘μ„  λͺ¨λΈμ— λŒ€ν•œ μ†Œμ„±κ²½ν™” κ³„μˆ˜λ₯Ό μœ λ„ν•˜μ˜€λ‹€.

- 반볡 μ‚ΌμΆ•μ‹€ν—˜μ‘°κ±΄μ—μ„œ 기쑴의 단일 항볡면 μ†Œμ„±λͺ¨λΈ(D-P λͺ¨λΈ)κ³Ό 개발된 μ†Œμ„±λͺ¨λΈ(B-S λͺ¨λΈ)의 κ±°λ™νŠΉμ„±μ„ 비ꡐ λΆ„μ„ν•˜μ˜€λ‹€. D-P λͺ¨λΈκ³Ό λ™μΌν•œ λ¬Όμ„±μΉ˜λ₯Ό μ‚¬μš©ν•˜λŠ” B-S λͺ¨λΈμ€ μ΄ˆκΈ°ν•˜μ€‘ μž¬ν•˜μ‹œ μŒκ³‘μ„  λͺ¨λΈκ³Ό μΌμΉ˜ν•˜μ˜€μœΌλ©°, λ°˜λ³΅ν•˜μ€‘μ— λŒ€ν•œ μ§€λ°˜μ˜ 이λ ₯ν˜„μƒμ„ λͺ¨μ‚¬ν•  수 μžˆμ—ˆλ‹€. μ–•μ€κΈ°μ΄ˆμ˜ κ±°λ™ν•΄μ„μ—μ„œ μΌκ΄€λœ μ ‘μ„ κ³„μˆ˜λ₯Ό μ΄μš©ν•˜λŠ” D-P λͺ¨λΈλ³΄λ‹€ μ•ˆμ •μ μΈ μˆ˜λ ΄μ„±μ„ λ³΄μ˜€λ‹€. 그리고 팽창각 d의 μ¦κ°€λŠ” μ—°μ§λˆ„μ  μΉ¨ν•˜λŸ‰μ„ κ°μ†Œμ‹œν‚€κ³  νšŒμ „λ°˜μ‘ 강성을 μ¦κ°€μ‹œμΌ°λ‹€.

- B-S λͺ¨λΈμ„ μ΄μš©ν•œ 반볡 μ „λ‹¨λ³€ν˜•λ₯ μ— μˆ˜μΉ˜ν•΄μ„ κ²°κ³ΌλŠ” μŒκ³‘μ„  λͺ¨λΈμ— Masing rule을 μ μš©ν•œ 이둠식과 μΌμΉ˜ν•˜μ˜€λ‹€. λ‹€λ§Œ, μŒκ³‘μ„  λͺ¨λΈμ— μ˜ν•œ 반볡이λ ₯κ²½λ‘œλŠ” 큰 λ³€ν˜•λ₯ μ—μ„œ 감쇠비λ₯Ό κ³ΌλŒ€ν•˜κ²Œ μ‚°μ •ν•œλ‹€. λ”°λΌμ„œ, μˆ˜μ •λœ μŒκ³‘μ„ ν•¨μˆ˜μ— λŒ€ν•œ μ†Œμ„±κ²½ν™” κ³„μˆ˜λ₯Ό μ œμ‹œν•˜κ³ , μ—­ν•˜μ€‘ 응λ ₯μ „ν™˜ 응λ ₯을 κ³ λ €ν•œ μƒˆλ‘œμš΄ μ†Œμ„±κ³„μˆ˜λ₯Ό μ œμ•ˆν•˜μ˜€λ‹€. 그리고 μ œμ‹œλœ 1차원 λ“±κ°€μ„ ν˜•λͺ¨λΈμ— λΆ€ν•©ν•˜λ„λ‘ 닀차원 거동을 λͺ¨λΈλ§ν•  수 μžˆλŠ” λͺ¨λΈλ³€μˆ˜ 산정법을 μ œμ•ˆν•˜μ˜€λ‹€.

κ°μ‚¬μ˜ κΈ€

λ³Έ μ—°κ΅¬λŠ” ν•œκ΅­μ—°κ΅¬μž¬λ‹¨ μ€‘κ²¬μ—°κ΅¬μžμ§€μ›μ‚¬μ—…(NRF- 2022R1A2C200823612)κ³Ό ν•œκ΅­μˆ˜λ ₯μ›μžλ ₯(μ£Ό) ν•΄μ˜€λ¦„λ™λ§ΉμΆœμ—°μ‚¬μ—…(원전지역 νŠΉν™”μ—°κ΅¬)의 μ§€μ›μœΌλ‘œ μˆ˜ν–‰λ˜μ—ˆμœΌλ©°, 이에 κΉŠμ€ 감사λ₯Ό λ“œλ¦½λ‹ˆλ‹€. 

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