Seismic pressures on rigid cantilever walls retaining elastic continuously non-homogeneous soil: An exact solution

• Seismic pressures on rigid walls retaining non-homogeneous soil are obtained analytically.
• Solution combines Fourier series in the horizontal direction and power series for the resulting ODEs.
• Non-homogeneity reduces seismic pressures on the walls by 10–20%.
• Use of the average shear modulus over the soil depth does not provide satisfactory results.

The dynamic response of an elastic continuously nonhomogeneous soil layer over bedrock retained by a pair of rigid cantilever walls to a horizontal seismic motion and the associated seismic pressure acting on these walls are determined analytically–numerically. The soil non-homogeneity is described by a shear modulus increasing nonlinearly with depth. The problem is solved in the frequency domain under conditions of plane strain and its exact solution is obtained analytically. This is accomplished with the aid of Fourier series along the horizontal direction and solution of the resulting system of two ordinary differential equations with variable coefficients by the method of Frobenius in power series. Due to the complexity of the various analytical expressions, the final results are determined numerically. These results include seismic pressures, resultant horizontal forces and bending moments acting on the walls. The solution of the problem involving a single retaining wall can be obtained as a special case by assuming the distance between the two walls to be very large. Results are presented in terms of numerical values and graphs using suitable dimensionless quantities. The effect of soil non-homogeneity on the system response is assessed through comparisons for typical sets of the parameters involved.