二维及轴对称条件下饱和层状地基的Lamb问题解答

Solution of Lamb′s problems of 2-D and axisymmetric saturated layered soil

在Biot饱和两相介质理论的基础上,对二维Biot波动方程进行水平坐标的Fourier变换和相应的变量代换,得到一组以土骨架位移分量与孔压为变量的二阶常微分方程。同时,对轴对称条件下的Biot波动方程进行径向坐标的Hankel变换,得到一组与二维情况相同的常微分方程。为求解该常微分方程组,运用薄层法原理在竖向进行离散求解,最终得到饱和层状地基的动力Lamb问题的基本解答,给出频域内土骨架位移和孔压的显示表达。此外,针对薄层法仅能在有限深度内求解的不足,根据旁轴近似原理,对饱和半空间频率-波数域内的刚度矩阵进行二阶Taylor级数展开,得到适合于薄层法底层边界的旁轴近似解答。最后,通过计算实例验证本文解答。

The dynamic response of saturated layered ground subjected to harmonic force is investigated.By using Fourier transform and variable substitutions,the 2-D Biot＇s dynamic equations in each layer are reduced into a series of ordinary differential equations.The axisymmetric Biot＇s dynamic equations are reduced to ordinary differential equations by performing Hankel transform.The equations can be solved by using Thin Layered Method（TLM）.The 2-D and axisymmetric solutions for displacement and pore pressure in frequency domain are derived by analytical Fourier transfrom and Hankel transform,respectively.In order to overcome the limitation that TLM only can work in finite depth,a solution of paraxial approximation,which can serve as the under-layer boundary of TLM,is obtained by using Taylor＇s series expansion with respect to the stiffness matrix of saturated half-space soil in wave-number domain.Several examples are investigated to validate the method.