Simplified 2-D Model For Determination Of Free Field Seismic Motion Of Soil

This paper is based on 1-D shear-beam model, the simplified 2-D model for determination of free field seismic motion of soil strata is studied. The content is divided into 3 aspects as follows:(1) The derivation process of 1-D shear-beam model(single layer soil), including homogeneous soil and non-homogeneous soil with power function variation of shear modulus along the depth is Summarized. It is to provide the reference for the comparison with the subsequent simplified 2-D model.(2) The simplified 2-D model for determination of free field seismic motion of uniform soil strata is studied. Firstly, the two-dimensional wave equation of the soil is obtained by using the theory of elastic wave. The separation of variables method is used to solve the equation. The natural frequencies and mode shapes of vertical vibration of soil are determined by using the boundary conditions of the upper and lower soil interface. According to the wave attenuation properties of wave propagation to infinity the wave attenuation coefficient along the horizontal direction is determined. And the relative displacement analytical solution of free vibration is obtained. Then the forced vibration equation is solved by using the mode superposition method. As a result the analytical solution of the horizontal relative soil displacement is obtained. By the above procedure, the simplified 2-D model for determination of free field seismic motion of soil strata is obtained and analytical solution is obtained. Finally an numerical example is given to compare the 2-D model and 1-D model.(3) The simplified 2-D model for determination of free field seismic motion of non-homogeneous soil stratum is studied. The variation of shear modulus of non-homogeneous soil along the depth is taken in a power function. Firstly, the two-dimensional wave equation of the soil is obtained by using the theory of elastiodynamics. The variable separation method is used to solve the equation. According to the attenuation properties of wave propagation to infinity the wave attenuation coefficient along the horizontal direction is determined. And the analytical solution of horizontal component is obtained. Then the solution of horizontal component was substituted into the original two-dimensional motion equation and the two-dimensional motion equation is transformed into one-dimensional motion equation. The numerical solution is obtained by using MATLAB codes. Finally an numerical example is given to compare the 2-D model and 1-D model.According to the results of the study, the comparison shows that consideration of the horizontal dimension affects little on the seismic response of the homogeneous soil and non- homogeneous soil in the general earthquake. It is not to consider the effects of horizontal dimension in general engineering. But it is recommended that the 2-D model should be used to analyze the free field seismic motion of soil for small scale seismic events with the attenuation less than 300 meters.