Analytical solution for one-dimensional consolidation of unsaturated soils using eigenfunction expansion method

This paper introduces an exact analytical solution for governing flow equations for one-dimensional consolidation in unsaturated soil stratum using the techniques of eigenfunction expansion and Laplace transformation. The homogeneous boundary conditions adopted in this study are as follows: (i) a one-way drainage system of homogenous soils, in which the top surface is considered as permeable to air and water, whereas the base is an impervious bedrock; and (ii) a two-way drainage system where both soil ends allow free dissipation of pore-air and pore-water pressures. In addition, the analytical development adopts initial conditions capturing both uniform and linear distributions of the initial excess pore pressures within the soil stratum. Eigenfunctions and eigenvalues are parts of the general solution and can be obtained based on the proposed boundary conditions. Besides, the Laplace transform method is adopted to solve the first-order differential equations. Once equations with transformed domain are all obtained, the final solutions, which are proposed to be functions of time and depth, can be achieved by taking an inverse Laplace transform. To verify the proposed solution, two worked examples are provided to present the consolidation characteristics of unsaturated soils based on the proposed method. The validation of the recent results against other existing analytical solutions is graphically demonstrated. Copyright (c) 2013 John Wiley & Sons, Ltd.