Dynamic analysis of rigid cantilever wall retaining linear elastic granular soil with microstructure under time harmonic seismic motion

The plane strain problem of the seismic behavior of a rigid cantilever wall retaining a linear elastic granular soil layer over bedrock is solved analytically. The horizontal seismic motion is assumed to be time harmonic, and thus, the problem is solved in the frequency domain. The granular soil is simulated by a linear elastic solid with microstructure due to Mindlin characterized by both micro-stiffness and micro-inertia. The problem is solved first for the case of two rigid walls, and the solution for one rigid wall considered here is obtained as the special case with the wall separation distance to be large enough. The equations of motion for the soil are two partial differential equations with two unknowns (the horizontal and vertical displacements) but of the 4th order instead of the 2nd one, which is the case of classical elasticity. Thus, the boundary conditions here are 8 instead of 4 in classical elasticity. The two displacements are expanded in Fourier sine and cosine series along the horizontal direction, and thus, the two partial differential equations of the problem become ordinary differential equations, which can be easily solved analytically. Finally, the resultant seismic pressure on the wall, the seismic base shear and moment as well as the location of the point of application of that resultant force on the wall are all determined in closed form. The solution is verified against the classical elastic solution by using it as a special case with microstructural effects going to zero. Parametric studies are also performed in order to assess the effects of microstructure on the seismic response of the system.