Semi-analytical solution to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary under time-dependent loading

This paper presents semi-analytical solutions to Fredlund and Hasan's one-dimensional consolidation of unsaturated soils with semi-permeable drainage boundary under time-dependent loadings. Two variables are introduced to transform two coupled governing equations of pore-water and pore-air pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. The pore-water pressure, pore-air pressure and settlement are obtained in the Laplace domain. Crump's method is adopted to perform the inverse Laplace transform in order to obtain semi-analytical solutions in time domain. It is shown that the present solutions are more general and have a good agreement with the existing solutions from literatures. Furthermore, the current solutions can also be degenerated into conventional solutions to one-dimensional consolidation of unsaturated soils with homogeneous boundaries. Finally, several numerical examples are provided to illustrate consolidation behavior of unsaturated soils under four types of time-dependent loadings, including instantaneous loading, ramp loading, exponential loading and sinusoidal loading. Parametric studies are illustrated by variations of pore-air pressure, pore-water pressure and settlement at different values of the ratio of air–water permeability coefficient, depth and loading parameters.