State space solution to three-dimensional consolidation of multi-layered soils

From the governing equations of a saturated poro-elastic soil in the Cartesian coordinate system, the state space equations of Biot’s three-dimensional consolidation problems are obtained by the Laplace transform and the double Fourier transform. Transfer matrix describing the transfer relation between the state vectors for a finite layer is derived explicitly in the transform space. Based on the continuity conditions between adjacent layers and the boundary conditions, the solution for three-dimensional consolidation of multi-layered soils is derived in a transformed domain. This solution is then transferred to that in a physical domain by the inversion of the double Fourier transform and the Laplace transform. Numerical analysis is carried out for three-dimensional consolidation of single, two, and multi-layered soils. The results for single and two-layered soils are compared well with those by the finite layer method in the literature. Numerical results for three-dimensional consolidation of five-layered soils are presented in this paper as an example to illustrate the use of the solution in this paper for three-dimensional consolidation of more than two-layered soils.