μ νΈμ±
(Hosung Shin)
1β
μ€μΈλΆ
(Seboong Oh)
2
κΉμ¬λ―Ό
(Jae-min Kim)
3
-
μ’
μ νμβ€κ΅μ μ μβ€μΈμ°λνκ΅ κ±΄μ€ν경곡νλΆ κ΅μ
(Corresponding Authorβ€University of Ulsanβ€shingeo@ulsan.ac.kr)
-
μ’
μ νμβ€μλ¨λνκ΅ κ±΄μ€μμ€ν
곡νκ³Ό κ΅μ
(Yeungnam Universityβ€sebungoh@yu.ac.kr)
-
μ’
μ νμβ€μ λ¨λνκ΅ ν λͺ©κ³΅νκ³Ό κ΅μ
(Chonnam National Universityβ€jm4kim@jnu.ac.kr)
Copyright Β© 2021 by the Korean Society of Civil Engineers
ν€μλ
J2-κ²½κ³λ©΄ μμ±λͺ¨λΈ, λ°λ³΅νμ€, μμ μ곑μ λͺ¨λΈ, μ νμμλ²
Key words
J2-bounding surface plasticity model, Cyclic load, Modified hyperbolic model, FEM
1. μ λ‘
μ μ¬ μ°μ체(pseudo-continuum)μ ν΄λΉνλ ν μ¬μ§λ°μ κ°λ³μ
μμ μ§ν©μΌλ‘, λλΆλΆμ λ³νλ₯ λ²μμμ λΉμ νμ μΈ μλ ₯-λ³νλ₯ κ±°λμ 보μΈλ€(Kramer, 1996). μ§λ°μ μνμ κ±°λμ λͺ¨μ¬νκΈ° μν λ¨μΌ ν볡면(single yield surface) λͺ¨λΈμ μλ ₯ 곡κ°μ νμ±κ³Ό μμ±μμμ λΆλ¦¬νλ©°, κ³Όλν
νμ±μμκ³Ό νμμ± κ²½κ³μμ κΈκ²©ν κ°μ± λ³νλ₯Ό λ°μμν€λ λ¬Έμ κ° μλ€. λν μλ ₯ μ ν(stress reversal)μ΄ λ°μνλ λ°λ³΅νμ€(cyclic
load)μ λν μ§λ°μ 볡μ‘ν κ±°λμ λͺ¨λΈλ§νκΈ° μ΄λ ΅λ€(Yu, 2006).
μ§λ°μ λΉμ νμ μλ ₯-λ³νλ₯ κ΄κ³λ₯Ό λͺ¨μ¬νκΈ° μνμ¬ μ μλ κ²½κ³λ©΄ μμ±λͺ¨λΈ(bounding surface plasticity model)μ κ²½κ³λ©΄μ
λν ν볡면μ μλμ μΈ μμΉλ‘λΆν° κ²½νν¨μλ₯Ό μ°μ νμ¬ λ§€λλ¬μ΄(smooth) μλ ₯-λ³νλ₯ 곑μ μ μ¬νν μ μλ€(Dafalias and Popov, 1975; Dafalias and Taiebat, 2016; Yu, 2006). λ°λΌμ, κ΅ν΅νμ€, μ§μ§, λ°λ λ° νλ λ±κ³Ό κ°μ λ°λ³΅νμ€μ λν ν μ¬μ§λ°μ λΉμ νμ μΈ μλ ₯-λ³νλ₯ μ΄λ ₯(hysteretic) κ±°λμ μ¬νν
μ μλ€(Yu et al., 2007).
μ
μ(granular)μ μ§λ°μ κ·Ήλλ‘ μ νλ λ³νλ₯ λ²μμμλ§ νμ±κ±°λμ 보μ΄λ―λ‘, μλ ₯곡κ°μμ ν볡면μ ν¬κΈ°λ₯Ό 0μΌλ‘ μλ ΄μμΌ νμ±μμμ μ¬λΌμ§κ²
νλ©΄μ ν볡면μ νμ¬μ μλ ₯μ μΌλ‘ μΆμνλ μ°κ΅¬κ° μνλμλ€(Borja and Amies, 1994; Dafalias and Popov, 1977). νμ§λ§ λλΆλΆμ μ°κ΅¬λ μμ λ 벨μ λ¨μ μλ ₯곡κ°μμ μ λ¨λ³νμ λν μμνλ§μ μ μνκ³ , λ€μ°¨μμ μΈ μμΉν΄μμ μ μ©ν μ¬λ‘λ λΆμ‘±νλ€(Borja et al., 1999; PisanΓ² and JeremiΔ, 2014; Restrepo and Taborda, 2018).
λ³Έ λ
Όλ¬Έμμλ J2-μμ±λͺ¨λΈμ νμ±μμμ΄ μλ κ²½κ³λ©΄ κ°λ
μ λμ
νκ³ μ€μ©μ μΈ λ±λ°©μλ ₯ μμ‘΄μ μλ ₯-λ³νλ₯ μ¦λΆ κ΄κ³μμ μ μνλ€. κ·Έλ¦¬κ³ μ μλ μμ±λͺ¨λΈμ
μ΄λ‘ μμ λν κ²μ¦κ³Ό μμκΈ°μ΄μ μμ©νλ λ°λ³΅νμ€μ λν μμ ν΄μμ μννκ³ μ νλ€.
2. κ²½κ³λ©΄ μμ±λͺ¨λΈμ μμν
2.1 νμ±μμμ΄ μλ J2-κ²½κ³λ©΄ μμ±λͺ¨λΈ
J2 μμ±λͺ¨λΈμ νΈμ°¨μλ ₯ ν
μμ λν μλ ₯ λΆλ³λ(stress invariant) J2λ₯Ό λ³μλ‘ ν볡면μ μ μνλ€. κ²½κ³λ©΄ μμ±λͺ¨λΈμμ ν볡면(yield
surface)μ κ²½κ³λ©΄(bounding surface)λ νΈμ°¨μλ ₯μ λν p νλ©΄μμ μ(circle)μΌλ‘ ννλλ€(Fig. 1). κ·Έλ¦¬κ³ κ²½κ³λ©΄μ λ±λ°©μλ ₯ pμ λνμ¬ μ νμ μΌλ‘ μμ‘΄μ μ΄λ€.
μ¬κΈ°μ $\underline{\sigma}$λ μ ν¨μλ ₯(effective stress)μ΄κ³ , $\underline{1}=\delta_{ij}$λ
Kroneckerβs delta tensorμ΄λ€. $\underline{\sigma}'$λ νΈμ°¨μλ ₯(deviatoric stress) ν
μ(tensor)μ΄λ©°
$\underline{\sigma}'=\underline{\sigma}-p\underline{1}$μ΄λ€. p νλ©΄μμ $\underline{\alpha}$λ
ν볡면μ μ€μ¬μ λμνλ μλ ₯ ν
μμ΄λ©°, rμ ν볡면μ λ°κ²½μ΄λ€. J2 μλ ₯ λΆλ³λμ $\dfrac{1}{2}\sigma_{ij}'\sigma_{ij}'=\dfrac{1}{2}\underline{\sigma}':\underline{\sigma}'=\dfrac{1}{2}β₯\underline{\sigma'}β₯^{2}$μ΄λ©°,
qλ νΈμ°¨μλ ₯ λΆλ³λμΌλ‘ $q=\sqrt{3/2}β₯\underline{\sigma}'β₯ =\sqrt{3J_{2}}$κ³Ό κ°λ€. κ·Έλ¦¬κ³ κ²½κ³λ©΄μ λν
κ°λμ μ aμ bλ μ§λ°μ λ§μ°°κ°(f)κ³Ό μ μ°©λ ₯(c)μΌλ‘ $a=\dfrac{6\sin(\phi)}{3-\sin(\phi)}$, $b=\dfrac{6Β·
c Β·\sin(\phi)}{3-\sin(\phi)}$μ κ°μ΄ μ°μ ν μ μλ€.
ν볡면μ λ°κ²½ rμ 0μΌλ‘ μλ ΄μν€λ©΄, νμ±μμμ΄ μ¬λΌμ§κ³ νΈμ°¨μλ ₯κ³Ό ν볡면 μ€μ¬μλ ₯μ λ°©ν₯μ νΈμ°¨ λ¨μ ν
μ(deviatoric unit tensor)
$\underline{n}$μΌλ‘ λμΌνλ€(Borja and Amies, 1994).
Prager(1955)κ° μ μν μ΄λκ²½ν(kinematic hardening) κ·μΉμ Eq. (1)μ ν볡면μ μ μ©νλ©΄, νΈμ°¨μλ ₯ μ¦λΆμ λ°©ν₯ν
μλ νΈμ°¨ λ¨μν
μ $\underline{n}$κ³Ό μΌμΉνκ² λλ€.
μνμ μμ±λ³ν μ¦λΆμ μμ±νλ¦λ²μΉ(plastic flow rule)κ³Ό ν볡면μ λν μΌκ΄μ± 쑰건(consistency condition)μΌλ‘λΆν°
μ°μ ν μ μλ€.
μ¬κΈ°μ Hλ μμ±κ²½ν κ³μ(plastic hardening modulus)λ‘ μ μνλ€. μμ± ν¬ν
μ
ν¨μ(plastic potential function)
gλ g = q + d Β· p = 0μ΄λ©°, ν½μ°½λ₯ (dilatancy, d = tan(d))μ ν½μ°½κ°(dilatancy angle, d)μΌλ‘ ννλλ€.
μΈλ¦½ν μ ν½μ°½λ₯ μ νΈμ°¨μλ ₯λΉ(Roscoe and Schofield, 1963)λ‘λΆν° μ°μ νκ³ , 쑰립ν λ μνλ³μ(state parameter)λ‘λΆν° μ°μ ν μ μλ€(Been and Jefferies, 1985).
Fig. 1. J2-Bounding Surface and Yield Surface in Principal Stress Space and on the Ο-Plane
2.2 μλ ₯-λ³νλ₯ μ¦λΆ κ΄κ³
μνμ λ³νλ₯ μ¦λΆμ λ€μκ³Ό κ°μ΄ νμ±κ³Ό μμ± μ¦λΆμ ν©μΌλ‘ λνλΌ μ μλ€.
μ¬κΈ°μ νμ± κ°μ± ν
μ(elastic stiffness tensor)λ $\underline{D}^{e}= K\underline{1}\otimes\underline{1}$$+2G(\underline{I}-\dfrac{1}{3}\underline{1}\otimes\underline{1})$μ΄λ€.
κ·Έλ¦¬κ³ Kλ 체μ νμ±κ³μ(elastic bulk modulus)μ΄κ³ , Gλ μ λ¨νμ±κ³μ(elastic shear modulus)μ΄λ€.
Eq. (4)μ Eq. (5)μ λν μνμ λ³νμΌλ‘λΆν° λ€μμμ μ°μ ν μ μλ€.
νμ±μμμ΄ μλ J2-κ²½κ³λ©΄ μμ±λͺ¨λΈμ λν μλ ₯-λ³νλ₯ μ¦λΆμμ λ€μκ³Ό κ°λ€.
Eq. (7)μ νΈμ°¨μλ ₯κ³Ό λ³νλ₯ κ΄κ³μ λνμ¬ μ 리νλ©΄ λ€μκ³Ό κ°λ€.
μ¬κΈ°μ νΈμ°¨ λ³νλ₯ μ $\epsilon_{q =}\sqrt{\dfrac{2}{3}}DL\in E\underline{\epsilon'}DL\in E$μ΄λ€.
2.3 μ곑μ (hyperbolic) λͺ¨λΈμ λν μμ±κ²½ν κ³μ H
μ ν μ λͺ¨λμ λΉμ ν μλ ₯-λ³νλ₯ κ΄κ³μ μ¬μ©λλ μ곑μ (hyperbolic) λͺ¨λΈμμ λ€μκ³Ό κ°λ€(Hardin and Drnevich, 1972; Kondner, 1963).
μ¬κΈ°μ Gmaxλ μ΄κΈ° μ λ¨νμ±κ³μ(initial elastic shear modulus)μ΄κ³ , quλ μΆμ°¨ μ λ¨κ°λμ΄λ€.
Eq. (9)μ μ곑μ λͺ¨λΈμ μλ ₯-λ³νλ₯ κ΄κ³λ₯Ό λ€μκ³Ό κ°μ μ¦λΆννλ‘ ννν μ μλ€.
Eq. (8)κ³Ό Eq. (10)μΌλ‘λΆν° μ곑μ μλ ₯-λ³νλ₯ κ΄κ³μ λν μμ±κ²½ν κ³μλ₯Ό μ°μ ν μ μλ€.
κ°λ°λ νμ±μμμ΄ μλ J2-κ²½κ³λ©΄ μμ±λͺ¨λΈμ ν μ¬μ§λ°μ μ 리μλ°μ Thermo-Hydro-Mechanical μ°κ³ν΄μμ μνμ¬ κ°λ°λ Geo-COUS(Geo-COUpled
Simulator) μ νμμ νλ‘κ·Έλ¨κ³Ό κ²°ν©νμλ€(Shin, 2011; Shin and Santamarina, 2019).
3. κ°λ°λ μμ±λͺ¨λΈμ μμ ν΄μ
3.1 λ°λ³΅ μΌμΆμ€ν(cyclic tri-axial test)
λ¨μΌν볡면μ Drucker-Prager λͺ¨λΈ(βD-P λͺ¨λΈβ)κ³Ό κ°λ°λ νμ±μμμ΄ μλ Bounding Surface λͺ¨λΈ(βB-S λͺ¨λΈβ)μ λ°λ³΅μΌμΆμμΆ
쑰건μ λνμ¬ λΉκ΅νμλ€. D-P λͺ¨λΈμ ν볡면과 νκ΄΄λ©΄μ΄ λμΌνλ©°, B-S λͺ¨λΈμ κ²½κ³λ©΄μ ν볡면μΌλ‘ μ¬μ©νλ€. κ·Έλ¦¬κ³ D-P λͺ¨λΈμ μΌκ΄λ μ μ κ³μ(consistent
tangent modulus)μ μν λ΄μ¬μ μλ ₯μ λΆ(implicit stress integration)μ μ¬μ©νμ¬ 2μ°¨μλ ΄(quadratic convergence)νλ€(Simo and Taylor, 1985).
D-P λͺ¨λΈ(Fig. 2a)μ ν볡면(yield surface)κ³Ό νκ΄΄ν¬λ½μ (failure envelope)μ΄ λμΌνμ¬, ν볡면 λ΄μμ μ ννμ± κ±°λμ 보μΈλ€. κ·Έλ¦¬κ³ μμ¬ν/μ¬μ¬ν(unload-reload)μ
μλ ₯μ΄ ν볡면 λ΄μ μμΉνμ¬ νμ±λ³νλ§μ μ λ°νλ€. Eq. (9)μ μ곑μ λͺ¨λΈ(hyperbolic model)μ μ λ¨λ³νμ΄ λ¬΄νν 컀μ§λ©΄ μΆμ°¨μλ ₯μ μ λ¨ κ°λμ μλ ΄νλ κ²μ μ μ μλ€.
λ³Έ μ°κ΅¬μμ κ°λ°ν νμ±μμμ΄ μλ B-S λͺ¨λΈμ D-P λͺ¨λΈκ³Ό λμΌν λ¬Όμ±μΉλ₯Ό μ¬μ©νμλ€. Fig. 2bμμ μ΄κΈ°νμ€ μ¬νμ μλ ₯-λ³νλ₯ κ΄κ³λ μ곑μ λͺ¨λΈκ³Ό λμΌνλ©°, λ°λ³΅νμ€μ λνμ¬ μμ±λ³νμ΄ λμ λλ μ νμ μΈ μ΄λ ₯(hysteresis) 곑μ μ
보μΈλ€. κ°λ°λ λͺ¨λΈμ λ°λ³΅νμ€μ λν μ΄λ ₯(hysteresis) 곑μ μ μ곑μ λͺ¨λΈμ λν Masing rule(Masing, 1926)μ 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
3.2 λ°λ³΅νμ€μ λν μμκΈ°μ΄μ κ±°λ
κ°λ°λ λͺ¨λΈμ κ±°λνΉμ±μ λΆμνκΈ° μνμ¬ λ°λ³΅νμ€μ λ°λ μμκΈ°μ΄(shallow foundation)μ λν μμ ν΄μμ μννμλ€. μμΉν΄μμ 2μ°¨μ
νλ©΄λ³νλ₯ 쑰건μΌλ‘ 40,000κ°μ 8μ μ μμμ 121,001κ°μ μ μ μ μ΄μ©νμλ€. Table 1μ μμΉν΄μμ μ¬μ©λ μ§λ°μ λ¬Όμ±μΉμ νμ€ μ¬νν B = 10 mμ λν μμκΈ°μ΄μ κ·Ήν μ§μ§λ ₯(bearing capacity)μ λνλ΄κ³ μλ€(Terzaghi, 1943). D-Pλͺ¨λΈκ³Ό B-S λͺ¨λΈμ μ΄μ©νμ¬ μ λ¨κ°λκ° κ΅¬μμμ 무κ΄ν μ μ±ν μ§λ°κ³Ό ꡬμμμ λΉλ‘νμ¬ μ¦κ°νλ c-f μ§λ°μ λνμ¬ λΉκ΅ν΄μμ μννμλ€.
Fig. 3μ μ°μ§ λ°λ³΅νμ€μ μν μ°μ±κΈ°μ΄ μ€μμ λ³μλ₯Ό λνλ΄κ³ μλ€. Fig. 3a(μ μ±ν μ§λ°)μμ D-P λͺ¨λΈμ 50 kPaμ λ°λ³΅νμ€μ μν μλ ₯λ³νκ° ν볡면 λ΄λΆλ‘ κ·Ήνλμ΄ μ νμ μΈ μλ ₯-μΉ¨νλ κ±°λμ 보μΈλ€. λ°λ©΄, B-S
λͺ¨λΈμ λ°λ³΅νμ€μ μν μμ±μΉ¨νλμ λμ κ³Ό μμ¬νμ(unloading)μ μ¬μ¬νμ(reloading)λ³΄λ€ λ³μμ λ³νκ° μκ² λνλ¬λ€.
Fig. 3b(c-f μ§λ°)μμ μΌκ΄λ μ μ κ³μλ₯Ό μ¬μ©νλ D-P λͺ¨λΈμ μλ ΄μ±μ΄ μ νλμ΄ λ°μ°νμλ€. B-S λͺ¨λΈμ μ΄μ©νμ¬ ν½μ°½κ° d = 0o, 10oμ λνμ¬
100 kPaμ λ°λ³΅νμ€μ λν μμΉν΄μμ μννμλ€. ν½μ°½κ°μ΄ 컀μ§μλ‘, μμ±μ λ¨λ³νμ μν 체μ ν½μ°½μΌλ‘ μμκΈ°μ΄μ λ°μκ°μ±μ΄ 컀μ§λ©΄μ μμ±μΉ¨νλμ΄
κ°μνμλ€.
Fig. 4λ μμκΈ°μ΄μ λ°λ³΅ λͺ¨λ©νΈ μ¬νμ μ°μ±κΈ°μ΄μ νκ· νμ κ°μ λ³νλ₯Ό λνλ΄κ³ μλ€. Fig. 4a(μ μ±ν μ§λ°)λ μμκΈ°μ΄μ 150 kPaμ μ°μ§μλ ₯μ μ¬νν ν, λͺ¨λ©νΈ(800 kNΒ·m/m)λ₯Ό λ°λ³΅μ μΌλ‘ μ¬ννμλ€. λ°λ³΅νμ€λ¨κ³μμ D-P λͺ¨λΈμ
κΈ°μ΄μ μ ννμ±μ μΈ κ±°λμ 보μ΄κ³ , B-S λͺ¨λΈμ κΈ°μ΄μ νμ (rocking)μ λν κ°μ±μ΄ μ μ°¨μ μΌλ‘ μ¦κ°νλ©΄μ μλ ΄νμλ€. Fig. 4b(c-f μ§λ°)λ μμκΈ°μ΄μ 500 kPaμ μ°μ§μλ ₯μ μ¬νν ν, λ°λ³΅ λͺ¨λ©νΈ(1,250 kNΒ·m/m)λ₯Ό μμ©μμΌ°λ€. λ°λ³΅νμ€λ¨κ³μμ κΈ°μ΄μ νμ λ°μ
κ°μ±μ΄ μ μ°¨μ μΌλ‘ μ¦κ°νλ©΄μ μλ ΄νμλ€. ν½μ°½κ° dκ° μ¦κ°νλ©΄, μ λ¨λ³νμ μν 체μ ν½μ°½μ±μ μ¦κ°λ‘ μ°μ§μμ± μΉ¨νλμ΄ κ°μνκ³ , μμ κΈ°μ΄μ νμ λ°μ
κ°μ±μ΄ μ¦κ°νμλ€.
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
|
βcohβ
|
Gmax = 27 MPa, n = 0.3, gt = 15 kN/m3
c = 40 kPa, f = 0Β°, d = 0Β°
|
248 kPa
|
B-S model
|
βcohβ
|
βcoh-phi-dβ
|
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
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
3.3 λ°λ³΅νμ€μ λν λμ νΉμ± νκ°
μ§λ°μ§μ§κ³΅νμμ μ§λ°μ λμ νμ€μ λν κ±°λμ λͺ¨μ¬νλ μνμ λͺ¨λΈμ λ§€μ° μ€μνλ€. μ£Όλ‘ μ κ· μ λ¨ν μ κ³μ(G/Gmax, normalized secant
shear modulus)μ κ°μ λΉ(D, material or hysteretic damping ratio)μ μν 1μ°¨μ λ±κ°μ νλͺ¨λΈ(equivalent
linear model)μ μ΄μ©νμ¬ μ£Όνμμμ ν΄μμ μννκ³ μλ€. νμ§λ§, 1μ°¨μ λͺ¨λΈμ μ§λ°μ λΉκ°μμ (irreversible) μμ±λ³ν, κ°κ·Ήμμμ
μν₯, μΆμ°¨λ³νκ³Ό μ°κ³λ 체μ λ³ν, κ·Έλ¦¬κ³ λΉμ κ·μ μΈ(irregular) λμ νμ€ λ±μ κ³ λ €ν μ μλ€. λν λ°λ³΅ μ λ¨λ³νλ₯ μ λ°λ₯Έ μ κ· μ λ¨ν μ κ³μμ
κ°μ λΉλ‘ ννλλ 1μ°¨μ λ±κ°μ νλͺ¨λΈμ λ€μ°¨μ μκ°μ΄λ ₯ν΄μμ ν μ μλ€. λ°λΌμ, μΌλ°μ μΈ μλ ₯μ₯μμ μ μλλ νμμ± λͺ¨λΈμ λν μ°κ΅¬κ° μ§μμ μΌλ‘
μ§νλκ³ μμΌλ©°, μ€μ λ¬Έμ μ νμ©νλλ° μμ΄μ λ§μ κ°―μμ λ¬Όμ±μΉλ λ¬Έμ μ μΌλ‘ μΈμλκ³ μλ€(PisanΓ² and JeremiΔ, 2014).
Eq. (9)μ μ곑μ λͺ¨λΈμ Masing ruleμ μ μ©ν λ°λ³΅ μ λ¨λ³νλ₯ μ λν μλ ₯μ΄λ ₯곑μ μΌλ‘λΆν° μ λ¨ν μ κ³μ(G/Gmax)μ κ°μ λΉ(D)λ₯Ό μ°μ ν μ μλ€(Ishihara, 1996).
μ¬κΈ°μ grμ κΈ°μ€λ³νλ₯ (reference strain)μΌλ‘ tmax/Gmaxμ΄λ€.
Fig. 5λ λ€μν ν¬κΈ°μ λ°λ³΅ μ λ¨λ³νλ₯ μ λνμ¬ μ΄κΈ° νμ€μ¬ν ν 5λ²μ μμ¬ν-μ¬μ¬νμ λν μ λ¨μλ ₯-μ λ¨λ³νλ₯ μ κ²½λ‘λ₯Ό 보μ¬μ£Όκ³ μλ€. μ΄κΈ° κ²½λ‘λ
μ곑μ λͺ¨λΈ(hyperbolic model)κ³Ό μΌμΉν¨μ μ μ μλ€. μ λ¨ λ³νλ₯ μ΄ μμΌλ©΄, μ νμ κ·Όμ¬ν μλ ₯-λ³νλ₯ μ κ΄κ³λ₯Ό 보μ΄λ©΄μ G/Gmaxβ1.0,
κ°μ λΉ Dβ0μ κ·Όμ νλ€. μ€λ΄ μ€νκ²°κ³Όμμ λ§€μ° μμ μ λ¨ λ³νλ₯ μμ λ°μνλ μ΅μ κ°μ λΉ(Dmin)λ κ³΅κ·Ήλ΄ μ 체μ μ μ±(viscosity)μ
μ§λ°μ ν¬λ¦¬ν(creep)μ μνμ¬ λ°μνλ―λ‘(dβOnofrio et al., 1999), μκ°μ΄λ ₯ν΄μ(time history analysis)μμ μ΅μ κ°μ λΉ Dminμ κ°μ λ ₯μ μ§μ μ μ©ν μ μλ€. λ°λ³΅ μ λ¨λ³νλ₯ μ΄ μ»€μ§μλ‘, μ λ¨ν μ κ³μ
G/Gmaxλ κ°μνκ³ κ°μ λΉ Dλ μ μ°¨μ μΌλ‘ 컀μ§λ κ²μ μ μ μλ€(Kramer, 1996).
Fig. 6aμ 6bλ λ°λ³΅ μ λ¨λ³νλ₯ μ ν¬κΈ°μ λ°λ₯Έ G/Gmaxμ Dμ λ³νλ₯Ό λνλ΄κ³ μλ€. κ°λ°λ B-Sλͺ¨λΈμ λν μμΉν΄μ κ²°κ³Όλ Eq. (12)μ μ΄λ‘ μ κ³Ό λ§€μ° μΌμΉν¨μ μ μ μλ€. λ€λ§, λ°λ³΅ μ λ¨λ³νλ₯ μ΄ λ§€μ° ν° κ²½μ°μλ κ°μ λΉμ λν μμΉμ μΈ μ€μ°¨κ° λ°μν¨μ μ μ μλ€(Anastasopoulos et al., 2011).
μ€μΆκ³‘μ (backbone curve)μ Masing ruleμ μ μ©ν λ°λ³΅νμ€μ λν μ΄λ ₯(hysteric) μλ ₯κ²½λ‘λ μ€νκ²°κ³Όμ λΉνμ¬ ν° λ³νλ₯ μμ
κ°μ λΉλ₯Ό κ³Όλ μ°μ νμ¬ λμ νμ€μ λν μ§λ°μ λ°μμ κ³Όμνκ°νλ κ²½ν₯μ΄ μλ€(Stewart et al., 2008). μ΄λ¬ν λ¬Έμ μ μ ν΄κ²°νκΈ° μνμ¬, 1) μμ λ μ곑μ λͺ¨λΈ(Yang et al., 2022)μ μ€μΆκ³‘μ μΌλ‘ μ¬μ©νμ¬ μλ‘μ΄ μμ±κ²½ν κ³μλ₯Ό μ μνμλ€.
μ¬κΈ°μ c = 1μ΄λ©΄ κΈ°μ‘΄μ μ곑μ ν¨μμ μμ±κ²½ν κ³μμ λμΌνλ€.
2) μνμ€(unloading)μ μν μλ ₯μ ν(stress reversal)μ΄ μλ ₯ κ²½λ‘μ λ―ΈμΉλ μν₯μ κ³ λ €νμ¬ μΆκ°μ μΈ μμ±κ³μ H1μ λμ
νμλ€.
λ°λΌμ, μλ‘κ² μ μλλ μμ± κ²½νκ³μ H*μ μμ λ μ곑μ λͺ¨λΈμ λν κ²½νκ³μ(Eq. (13))μ μΆκ°λ‘ λμ
λ κ²½νκ³μ H1μ μ§λ ¬(serial) μ‘°ν©μΌλ‘ μ°μ ν μ μλ€.
μ¬κΈ°μ, q0λ μ΅μ’
μλ ₯μ ν(stress reversal)μμ μΆμ°¨μλ ₯μ΄λ€. H1μ΄ λ¬΄νν 컀μ§λ©΄, μλ‘μ΄ κ²½νκ³μ H*λ Hλ‘ μλ ΄νκ² λλ€.
1μ°¨μ λ±κ°μ νλͺ¨λΈμμ μ λ¨λ³νλ₯ gμ λν G/Gmaxμ Dμ κ΄κ³κ° μ£Όμ΄μ§ λ, Eq. (13)κ³Ό Eq. (14)μ λͺ¨λΈλ³μλ€μ κ²°μ νκΈ° μνμ¬ λ§€κ°λ³μ ν΄μμ μννμλ€. Fig. 7μ μ λ¨λ³νλ₯ g/gr = 0.5μμ λͺ¨λΈλ³μ mκ³Ό cμ λ°λ₯Έ G/Gmaxμ Dμ λ³νλ₯Ό 보μ¬μ£Όκ³ μλ€. κΈ°μ‘΄μ μ곑μ λͺ¨λΈμμ g/gr = 0.5μ
λν G/Gmax = 2/3, D = 8.6\%μ΄λ€(Eq. (12)). mκ³Ό cκ° μ¦κ°νλ©΄ λ°λ³΅μλ ₯κ²½λ‘μ G/Gmaxλ μ¦κ°νκ³ , κ°μ λΉλ cμ΄ μ¦κ°νλ©΄ μ μ°¨ κ°μνμλ€. λ€μν μ§λ°μ μ€λ΄μ€ν κ²°κ³Όμ λνμ¬ λͺ¨λΈλ³μλ€μ
κ²°μ νκΈ° μν μΆκ°μ μΈ μ°κ΅¬κ° νμνλ€.
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
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
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
4. κ²° λ‘
λ°λ³΅νμ€μ΄λ λμ νμ€μ λν μ§λ°μ μνμ μμ±λͺ¨λΈμ μ§λ°κ΅¬μ‘°λ¬Όμ λ€μ°¨μ λΉμ ν μμΉν΄μμμ λ§€μ° μ€μνλ€. λ¨μΌ ν볡면 μμ±λͺ¨λΈμ λ°λ³΅νμ€μ λν΄
μ νμ κ±°λμ 보μ΄κ³ , κΈ°μ‘΄μ κ²½κ³λ©΄ μμ±λͺ¨λΈμ 볡μ‘ν μλ ₯-λ³νλ₯ κ΄κ³λ₯Ό λͺ¨μ¬ν μ μμ§λ§ ꡬμ±λͺ¨λΈμ ꡬννκΈ° μνμ¬ λ§μ λ¬Όμ±μΉκ° νμνλ€. λ³Έ
λ
Όλ¬Έμ νμ±μμμ΄ μλ J2-κ²½κ³λ©΄ μμ±λͺ¨λΈμ λ¨μν λ¬Όμ±μΉλ‘ ν¨κ³Όμ μΌλ‘ μ§λ°μ λΉμ νμ±μ λͺ¨μ¬ν μ μλ€.
- κ°λ°λ κ²½κ³λ©΄ μμ±λͺ¨λΈμ p νλ©΄μμ ν볡면μ λ°κ²½μ 0μΌλ‘ μλ ΄μμΌ νμ±μμμ΄ μ¬λΌμ§λλ‘ μμννκ³ , μμ±κ²½ν κ³μκ³Ό ν½μ°½λ₯ μ μ΄μ©νμ¬ μμ±λ³ν
μ¦λΆμ μ μνμλ€. νμ±μμμ΄ μλ J2-κ²½κ³λ©΄ μμ±λͺ¨λΈμ μλ ₯-λ³νλ₯ μ¦λΆμμ μ μνκ³ , μ곑μ λͺ¨λΈμ λν μμ±κ²½ν κ³μλ₯Ό μ λνμλ€.
- λ°λ³΅ μΌμΆμ€ν쑰건μμ κΈ°μ‘΄μ λ¨μΌ ν볡면 μμ±λͺ¨λΈ(D-P λͺ¨λΈ)κ³Ό κ°λ°λ μμ±λͺ¨λΈ(B-S λͺ¨λΈ)μ κ±°λνΉμ±μ λΉκ΅ λΆμνμλ€. D-P λͺ¨λΈκ³Ό λμΌν
λ¬Όμ±μΉλ₯Ό μ¬μ©νλ B-S λͺ¨λΈμ μ΄κΈ°νμ€ μ¬νμ μ곑μ λͺ¨λΈκ³Ό μΌμΉνμμΌλ©°, λ°λ³΅νμ€μ λν μ§λ°μ μ΄λ ₯νμμ λͺ¨μ¬ν μ μμλ€. μμκΈ°μ΄μ κ±°λν΄μμμ
μΌκ΄λ μ μ κ³μλ₯Ό μ΄μ©νλ D-P λͺ¨λΈλ³΄λ€ μμ μ μΈ μλ ΄μ±μ 보μλ€. κ·Έλ¦¬κ³ ν½μ°½κ° dμ μ¦κ°λ μ°μ§λμ μΉ¨νλμ κ°μμν€κ³ νμ λ°μ κ°μ±μ μ¦κ°μμΌ°λ€.
- B-S λͺ¨λΈμ μ΄μ©ν λ°λ³΅ μ λ¨λ³νλ₯ μ μμΉν΄μ κ²°κ³Όλ μ곑μ λͺ¨λΈμ Masing ruleμ μ μ©ν μ΄λ‘ μκ³Ό μΌμΉνμλ€. λ€λ§, μ곑μ λͺ¨λΈμ
μν λ°λ³΅μ΄λ ₯κ²½λ‘λ ν° λ³νλ₯ μμ κ°μ λΉλ₯Ό κ³Όλνκ² μ°μ νλ€. λ°λΌμ, μμ λ μ곑μ ν¨μμ λν μμ±κ²½ν κ³μλ₯Ό μ μνκ³ , μνμ€ μλ ₯μ ν μλ ₯μ
κ³ λ €ν μλ‘μ΄ μμ±κ³μλ₯Ό μ μνμλ€. κ·Έλ¦¬κ³ μ μλ 1μ°¨μ λ±κ°μ νλͺ¨λΈμ λΆν©νλλ‘ λ€μ°¨μ κ±°λμ λͺ¨λΈλ§ν μ μλ λͺ¨λΈλ³μ μ°μ λ²μ μ μνμλ€.
κ°μ¬μ κΈ
λ³Έ μ°κ΅¬λ νκ΅μ°κ΅¬μ¬λ¨ μ€κ²¬μ°κ΅¬μμ§μμ¬μ
(NRF- 2022R1A2C200823612)κ³Ό νκ΅μλ ₯μμλ ₯(μ£Ό) ν΄μ€λ¦λλ§ΉμΆμ°μ¬μ
(μμ μ§μ νΉνμ°κ΅¬)μ
μ§μμΌλ‘ μνλμμΌλ©°, μ΄μ κΉμ κ°μ¬λ₯Ό λ립λλ€.β
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