Dynamic Response of Two-Dimensional Double-Layered Unsaturated Soil Foundations Under a Strip Load

Abstract

Purpose

In order to study the effect of shear modulus and saturation of topsoil on the dynamic response in two typical layered foundations (the upper stiff-layer and lower soft-layer ground/the upper soft-layer and lower stiff-layer ground).

Methods

First, based on the mixed theory of three-phase porous media, a double-layered unsaturated soil foundation model is established and the dynamic governing equations of unsaturated soils are established by using the Fourier transform and Helmholtz vector decomposition. Then, the dynamic response solution of double-layered unsaturated foundations in the transformed domain is obtained using the transfer matrix method and combining with the boundary conditions and continuity conditions of adjacent layers. The numerical solutions of displacement, stress, and pore pressure are obtained by using numerical inverse Fourier transformation. Finally, the influence of shear modulus and saturation of topsoil on the dynamic response of double-layered ground is addressed.

Results

With the increase of topsoil shear modulus, the vertical displacement, pore water pressure and pore air pressure in the upper soft-layer and lower stiff-layer ground and the upper stiff-layer and lower soft-layer ground decrease. With the increase of topsoil saturation, the vertical displacement and pore air pressure in the upper soft-layer and lower stiff-layer ground and the upper stiff-layer and lower soft-layer ground increase but the pore water pressure decreases significantly.

Conclusions

The results show that the shear modulus and saturation of topsoil have a significant effect on the vertical displacement, pore air pressure and pore water pressure of double-layered ground. The results will provide theoretical basis and guiding significance for foundation design.

Appendices

Appendix A

$N_{n} = 2\mu S_{n} i\xi \lambda_{n} (n = 1,2,3)$,$X = \frac{{i\mu \lambda_{0}^{2} }}{\xi } + i\mu \xi (n = 1,2,3)$,$L_{n} = \frac{{\omega^{2} \rho_{w} S_{n} \lambda_{n} - f_{wn} \lambda_{n} }}{{b^{w} \rho_{w} }}(n = 1,2,3)$,$K_{n} = \frac{{\omega^{2} \rho_{a} S_{n} \lambda_{n} - f_{an} \lambda_{n} }}{{b^{a} \rho_{a} }}(n = 1,2,3)$, $M_{n} = 2\mu S_{n} \lambda_{n}^{2} + \lambda - a\chi f_{wn} - a(1 - \chi )f_{an} (n = 1,2,3)$.

The elements in matrix $\left[ S \right]$ are

$S_{11} = S_{15} = 2\mu \lambda_{0}$,$S_{12} = S_{16} = M_{1}$, $S_{13} = S_{17} = M_{2}$, $S_{14} = S_{18} = M_{3}$, $S_{21} = - S_{25} = X$, $S_{22} = - S_{26} = N_{1}$, $S_{23} = - S_{27} = N_{2}$, $S_{24} = - S_{28} = N_{3}$, $S_{31} = S_{35} = 0$, $S_{32} = S_{36} = f_{w1}$, $S_{33} = S_{37} = f_{w2}$, $S_{34} = S_{38} = f_{w3}$, $S_{41} = S_{45} = 0$, $S_{42} = S_{46} = f_{a1}$, $S_{43} = S_{47} = f_{a2}$, $S_{44} = S_{48} = f_{a3}$, $S_{51} = S_{55} = \frac{{i\lambda_{0} }}{\xi }$, $S_{52} = S_{56} = S_{1} i\xi$, $S_{53} = S_{57} = S_{2} i\xi$, $S_{54} = S_{58} = S_{3} i\xi$, $S_{61} = - S_{65} = 1$, $S_{62} = - S_{66} = S_{1} \lambda_{1}$, $S_{63} = - S_{67} = S_{2} \lambda_{2}$, $S_{64} = - S_{68} = S_{3} \lambda_{3}$, $S_{71} = - S_{75} = \frac{{\omega^{2} }}{{b^{w} }}$, $S_{72} = - S_{76} = L_{1}$, $S_{73} = - S_{77} = L_{2}$, $S_{74} = - S_{78} = L_{3}$, $S_{81} = - S_{85} = \frac{{\omega^{2} }}{{b^{a} }}$, $S_{82} = - S_{86} = K_{1}$, $S_{83} = - S_{87} = K_{2}$,$S_{84} = - S_{88} = K_{3}$.

Appendix B

The elements in matrix $\left[ {U^{1} } \right]$ are

$U_{11}^{1} = U_{15}^{1} = 2\mu^{1} \lambda_{0}^{1}$, $U_{12}^{1} = U_{16}^{1} = M_{1}^{1}$, $U_{13}^{1} = U_{17}^{1} = M_{2}^{1}$, $U_{14}^{1} = U_{18}^{1} = M_{3}^{1}$; $U_{21}^{1} = - U_{25}^{1} = X^{1}$, $U_{22}^{1} = - U_{26}^{1} = N_{1}^{1}$, $U_{23}^{1} = - U_{27}^{1} = N_{2}^{1}$, $U_{24}^{1} = - U_{28}^{1} = N_{3}^{1}$; $U_{31}^{1} = U_{35}^{1} = 0$, $U_{32}^{1} = U_{36}^{1} = f_{w1}^{1}$, $U_{33}^{1} = U_{37}^{1} = f_{w2}^{1}$, $U_{34}^{1} = U_{38}^{1} = f_{w3}^{1}$;

$U_{41}^{1} = U_{45}^{1} = 0$, $U_{42}^{1} = U_{46}^{1} = f_{a1}^{1}$, $U_{43}^{1} = U_{47}^{1} = f_{a2}^{1}$, $U_{44}^{1} = U_{48}^{1} = f_{a3}^{1}$.

The elements in matrix $\left[ {U^{N} } \right]$ are

$U_{11}^{N} = \frac{{i\lambda_{0}^{N} }}{\xi }$, $U_{12}^{N} = S_{1}^{N}$, $U_{13}^{N} = S_{2}^{N} i\xi$, $U_{14}^{N} = S_{3}^{N}$, $U_{15}^{N} = \frac{{i\lambda_{0}^{N} }}{\xi }$, $U_{16}^{N} = S_{1}^{N} i\xi$, $U_{17}^{N} = S_{2}^{N} i\xi$, $U_{18}^{N} = S_{3}^{N} i\xi$;$U_{21}^{N} = 1$,$U_{22}^{N} = S_{1}^{N} \lambda_{1}^{N}$,$U_{23}^{N} = S_{2}^{N} \lambda_{2}^{N}$,$U_{24}^{N} = S_{3}^{N} \lambda_{3}^{N}$,$U_{25}^{N} = - 1$,$U_{26}^{N} = - S_{1}^{N} \lambda_{1}^{N}$,$U_{27}^{N} = - S_{2}^{N} \lambda_{2}^{N}$, $U_{28}^{N} = - S_{3}^{N} \lambda_{3}^{N}$;$U_{31}^{N} = \frac{{\omega^{2} }}{{b^{{w^{N} }} }}$,$U_{32}^{N} = L_{1}^{N}$, $U_{33}^{N} = L_{2}^{N}$,$U_{34}^{N} = L_{3}^{N}$,$U_{35}^{N} = - \frac{{\omega^{2} }}{{b^{{w^{N} }} }}$,$U_{36}^{N} = - L_{1}^{N}$,$U_{37}^{N} = - L_{2}^{N}$, $U_{38}^{N} = - L_{3}^{N}$;$U_{41}^{N} = \frac{{\omega^{2} }}{{b^{{a^{N} }} }}$,$U_{42}^{N} = K_{1}^{N}$, $U_{43}^{N} = K_{2}^{N}$,$U_{44}^{N} = K_{3}^{N}$,$U_{45}^{N} = - \frac{{\omega^{2} }}{{b^{{a^{N} }} }}$,$U_{46}^{N} = - K_{1}^{N}$,$U_{47}^{N} = - K_{2}^{N}$,$U_{48}^{N} = - K_{3}^{N}$.