竖向简谐荷载下二维层状饱和地基的解析层元解

基于Biot固结理论，运用解析层元方法求解竖向简谐荷载作用下二维层状饱和地基的动力响应问题。从直角坐标平面应变问题控制方程出发，通过Fourier-Laplace变换将偏微分方程组转化为常微分方程组，求解得到单层饱和地基的解析层元。结合层间连续条件和边界条件，组装得到多层饱和地基的总刚度矩阵方程，进而求得变换域内的解。借助Fourier-Laplace逆变换的数值积分方法，获得平面应变动力问题在物理域内的解，编制了相应的计算程序，其计算结果与已有文献结果吻合较好。通过算例分析了荷载圆频率、荷载作用深度及地基成层性对地基竖向位移的影响。计算结果表明：随荷载圆频率的增大，地基竖向位移先增加后减小；地基竖向位移在荷载作用点处呈现波峰，且受表层土性的影响较大。

Analytical layer element solution for two-dimensional multilayered saturated subsoils under a vertical time-harmonic load

This paper used the analytical layer element method to analyze two-dimensional dynamic response of multilayered saturated subsoils subjected to a vertical time-harmonic load based on Biot consolidation theory. Starting with the governing equations of plane strain problem in Cartesian coordinates, ordinary differential equations can be obtained from partial differential equations through the Fourier-Laplace transform, and the analytical layer element for a single saturated soil layer is established. The global stiffness matrix for multilayered saturated subsoils is assembled according to continuity conditions of adjacent layers and boundary conditions. Actual solutions are obtained by taking Fourier-Laplace inverse transform. The results of numerical calculation are in good agreement with the existing solution in literature. More numerical examples are designed to analyze the effects on vertical displacement due to the circular frequency, force depth and soil stratification. Numerical results show that the vertical displacement increases firstly and then decreases with the increases of circular frequency; the vertical displacement at the load point presents a crest and is influenced by the characteristic of topsoil greatly.