تحلیل دینامیکی شمع اختلاط سیمان عمیق با هسته بتن تحت بارهای دینامیکی افقی

با ${ک}_{\mathrm{متر}}=\frac{\mathrm{۶}\left(\mathrm{۱}+n_{\mathrm{متر}}\right){}^{}}{}$

۱ + r c / r m ۲ ۲ ۷ + ۶ ν m ۱ + r c / r m ۲ ۲ + ۲۰ + ۱۲ ν m r c / r m ۲ , and $K_c=\frac{\mathrm{۶}\left(\mathrm{۱}+{\nu }_c\right)}{\mathrm{۷}+\mathrm{۶}{\nu }_m}$, where E_c and E_m are the elastic modulus of the concrete inner pile and cement–soil outer pile, respectively; I_c and I_m are the moment of inertia of the cross-section of the concrete inner pile and cement–soil outer pile, respectively; G_c and G_m are the shear modulus of the concrete inner pile and cement–soil outer pile, respectively; ν_c and ν_m are the Poisson’s ratio of the concrete inner pile and cement–soil outer pile, respectively; and K_c and K_m the shear correction factor of the concrete inner pile and cement–soil outer pile, respectively.

with

where λ = λ_s(1 + 2iβ_s) and μ = μ_s(1 + 2iβ_s) denote the complex moduli of soil, with λ_s and μ_s as the Lamé constant of soil, and β_s as the damping ratio of soil. u_r, u_θ, and u_z denote the radial, circumferential, and vertical displacement of soil, respectively. ρ_s denotes the mass density of soil. ∇^۲ is the Laplace operator. ε_s is the volume strain of soil and can be expressed as

where φ_s and ψ_s represent the scalar and vector potentials of soil displacement, respectively.

where F is the undetermined coefficient and $g_n=\left(\mathrm{۲}n-\mathrm{۱}\right)\pi \mathrm{i}/\left(\mathrm{۲}H\right)$ with $n=\mathrm{۱},\mathrm{۲},\mathrm{۳}\cdots$.

where B₁, B₂, C₁, C₂, D₁, D₂, F₁, and F₂ are undetermined coefficients. I_m(·) and K_m(·) represent the first and second modified m-order Bessel functions, respectively, with m as the natural number. ${\beta }_{\mathrm{۱}}^{\mathrm{۲}}=\left(-{\rho }_s{\omega }^{\mathrm{۲}}-\mu g_n^{\mathrm{۲}}\right)/\left(\lambda +\mathrm{۲}\mu \right)$ and ${\beta }_{\mathrm{۲}}^{\mathrm{۲}}=\left(-{\rho }_s{\omega }^{\mathrm{۲}}-\mu g_n^{\mathrm{۲}}\right)/\mu$.

with

with

with .

with

with

with .

with ${\zeta }_{\mathrm{۲}n}=H_{\mathrm{۵}}{g}_n{\xi }_{\mathrm{۳}n}/\left(g_n^{\mathrm{۴}}+H_{\mathrm{۱}}{g}_n^{\mathrm{۲}}+H_{\mathrm{۲}}\right)$.

with

with EI = E_mI_m + E_cI_c and AGK = A_mG_mK_m + A_cG_cK_c.

where the expressions of s₁~s₁₂ are detailed in Appendix A.

where the expressions of u₁~u₈ are detailed in Appendix B.

where Re(·) and Im(·) represent the real and imaginary parts, respectively. k_hh, k_rr, and k_hr represent the dynamic stiffness factor of horizontal, rocking, and horizontal–rocking dynamic impedances, respectively. c_hh, c_rr, and c_hr represent the dynamic damping factor of horizontal, rocking, and horizontal–rocking dynamic impedances, respectively.