An Analytical Solution for Seismic Response of Saturated Soil Layer to Vertical Earthquake Excitation

Several analytical and numerical techniques have been developed for solving poroelastic governing equations; however, no closed-form solution in time domain for the general material case, even in simple one-dimensional geometry, has been yet introduced. Analytical solution for wave propagation in saturated porous media is limited and cannot be easily obtained for earthquake loading. The existence of such analytical solutions to simplified problems of seismic wave propagation is essential. In the present paper, a closed-form solution in time domain is obtained for saturated soil layer subjected to vertical component of earthquake acceleration. Saturated soil is assumed as a saturated poroelastic media and corresponding governing differential equations for earthquake loading are derived. In a poroelastic medium under the effect of seismic waves, solid phase displacement and pore pressure are coupledand interact with each other. If the ground surface and the boundaries between soil layers are horizontal, the lateral extent of the deposit has no influence on the response, and hence, the deposit can be considered as a one-dimensional confined column. The vibration modes, the modal shapes and their corresponding frequencies are obtained from the free vibration condition of the governing equation. By applying the method of separation of variables, the governing equation, which is a second order hyperbolicpartial differential equation, is separated into a Bessel equation in space and a single-degree-of-freedom vibration equation in time. The Bessel equation and the single-degree-of-freedom vibration equation are solved using the Bessel functions and Newmark's direct integration method, respectively. In order to examine the accuracy of the analytical response presented in this paper, acceleration values recorded in a saturated alluvial during one of the previous earthquakes are compared with the calculated values by the analytical method. A numerical example is presented to further analyze the analytical solution. The numerical example shows that a decrease in permeability has a damping effect on acceleration, whereas, amplifies the excessive pore pressure. The suggested solution can be used for dynamic analysis of wave propagation in saturated soils during earthquakes.