An explicit one‐dimensional consolidation solution with semi‐permeable drainage boundary for unsaturated soil

Abstract

Existing solutions for analyzing one-dimensional (1-D) consolidation of unsaturated soil are only derived to cater to two extreme drainage conditions (fully drained and undrained). This study presents a new explicit solution for 1-D consolidation of unsaturated soil with semi-permeable drainage boundary. Based on the assumptions of two independent stress variables and the governing equations proposed by Fredlund, the eigenfunction expansion method is adopted to develop an explicit analytical solution to calculate excess pore-water and pore-air pressures in an unsaturated soil when it is subjected to external loads. The developed general solutions are expressed in terms of depth, z, and time, t. For the semi-permeable drainage boundary, eigenvalues and eigenfunctions in the space domain are developed. The technique of Laplace transform is used to solve the coupled ordinary differential equations in the time domain. The newly derived explicit solution is verified with the existing semi-analytical method in the literature, and an excellent agreement is obtained. Compared with the semi-analytical solution, the newly derived analytical solution is more straightforward and explicit so that this solution is relatively easier to be implemented into a computer program to carry out a preliminary assessment of 1-D consolidation of unsaturated soil.

Solutions for one‐dimensional consolidation of unsaturated soil with general boundary conditions subjected to time‐dependent load

“…Figure 5 presents the dissipation of excess pore water pressure and excess pore air pressure with time with same impeded drainages on air and water phases at the boundary. The results obtained from the solutions in this paper are compared with results obtained by the eigenfunction expansion method proposed by Zhao et al, 25 and the accuracy of the series solutions in this research can be concluded again. The larger parameters mean better permeability for air and water phases across the boundary.…”

“…when R tw = R ta at the top surface and R bw = R ba at the bottom surface, the assumption of same impeded drainage on air and water phases at the boundary adopted by Wang et al 21 and Zhao et al 25 can be reproduced. Otherwise, different impeded drainages at the boundary can be presented.…”

“…The consolidation theory of unsaturated soil proposed by Fredlund and Hasan, 6 which will be adopted in this paper as well as widely adopted by many researchers before, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] is based on the following assumptions. for pore air and pore water, and the coefficients for volume change resulted from change of net normal stress and matric suction are supposed to remain constant during the consolidation process.…”

“…Wang et al [21][22][23][24] adopted the Laplace transform and its numerical inversion to derive semi-analytical solutions for consolidation of unsaturated soil with semi-permeable boundary conditions under static or varying load. Zhao et al 25 also analyzed the consolidation of unsaturated soil with semi-permeable boundary conditions by the Laplace transform and eigenfunction expansion method. Huang and Zhao 26,27 proposed general analytical solutions for one-dimensional and two-dimensional consolidation of unsaturated soil incorporating impeded drainage boundaries, when the load is exerted instantaneously on the soil.…”

Analysis of One‐Dimensional Consolidation for Unsaturated Soils under Piecewise Cyclic Loading

“…e authors of [21][22][23] investigated a series of semianalytical solutions for 1D consolidation of unsaturated soils with semipermeable drainage boundary and exponentially time-growing drainage boundary. Zhao et al [24] investigated the 1D consolidation characteristics of unsaturated foundation considering the semipermeable boundary condition with explicit solutions. By taking the lateral and vertical drainage, Huang et al [25] presented the general analytical solutions of tow-dimensional consolidation of unsaturated soils.…”