三维层状饱和地基动力Green函数的薄层法求解

Dynamic Green function solution of three-dimensional layered saturated ground using thin layer element method

建立了三维层状饱和地基模型并在模型底部施加无反射边界条件,引入Fourier变换和正交变换将饱和地基Biot动力方程化为Love和Rayleigh模态方程,将地基沿深度方向划分为数个薄层单元并对单元位移场引入线性插值函数,从而将模态方程离散化,通过求解本征值问题最终给出了三维层状饱和地基的频域动力Green函数。分析了均质饱和地基的Love和Rayleigh模态的弥散特性,求解了均质和成层饱和半空间的Lamb问题,并与Philippacopoulos的数值积分解对比,说明了所求Green函数的正确性。所得Green函数无需关于Hankel函数进行数值逆变换且能直接考虑地基的成层性,提高了计算效率。

A model consisting of three-dimensional （3D） layered saturated ground and absorbing boundary applied at the bottom was established in this paper. By introducing the Fourier and orthogonal transformations, Blot＇s equations for the saturated soil are generated into equations governing Love and Rayleigh modes. By dividing the ground into several thin layer elements along the vertical direction and introducing linear interpolation function for the elemental displacement fields, the equations governing Love and Rayleigh modes for a single thin layer element are derived and then assembled to obtain global modal equations. After solving the homogenous global modal equations, the eigenvalues and corresponding eigenvectors of the saturated ground are obtained for Love and Rayleigh modes, separately. Eventually, the Green function for the 3D saturated ground is obtained for point loads by mode superposition method. Numerical results are presented to portray the evolution processes of eigenvalues of Love and Rayleigh modes with respect to the increasing excitation frequency and soil permeability. The Lamb problem of a homogeneous and layered saturated half-space is solved and compared with the numerical integration solutions given by Philippacopoulos. Through the comparisons, the obtained Green function is proven to be valid and accurate. Meanwhile, it does not require numerical integration with respect to Hankel functions and can take the soil stratification into consideration in a natural way, so as to improve the computation efficiency.