Semi‐analytical solutions of two‐dimensional plane strain consolidation in unsaturated soils subjected to the lateral semipermeable drainage boundary

Abstract

Summary The study presents semi‐analytical solutions of two‐dimensional plane strain consolidation problem in unsaturated soils incorporating the lateral semipermeable drainage boundary by adopting Fourier sine series and Laplace transform. The two‐dimensional plane strain consolidation equations in the form of two‐order partial differential equations with three variables are firstly converted to two‐order partial differential equations with two variables, which are similar to those of one‐dimensional consolidation problem. The four‐order ordinary differential equations about excess pore‐air and excess pore‐water pressures are got by applying Laplace transform and the substitution method. Then, the solutions of excess pore pressures and settlement are achieved in the Laplace transform domain. Afterwards, on the basis of Crump's method, the inverse Laplace transform is conducted to obtain the analytical solutions in time domain. The comparison is conducted to verify the exactness of the obtained solutions, and the two‐dimensional plane strain consolidation property with the lateral semipermeable drainage boundary is illustrated and discussed. Parametric studies are demonstrated for the excess pore pressures and normalized settlement with the change of the boundary parameters, air‐water and lateral‐vertical permeability coefficients, and the distance and depth. It can be found that the lateral semipermeable drainage boundary impedes the consolidation rate obviously, and when different investigated parameters are adopted, the consolidation property is similar to each other under the later permeable and semipermeable drainage boundary conditions.

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

“…However, these assumptions are essential and widely used in the analytical analysis of unsaturated soil to deliver a prediction of dissipation for the excess pore pressures. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][26][27][28] Based on these assumptions, the governing equations can be formulated as follows 1 :…”

“…For instance, in the preloading project of soft clay ground, when the drainage capability of the sand cushion is not very effective, there will be a partially drained boundary on the top of the adjacent soft clay layer just beneath it. 23 Wang et al [26][27][28] first incorporated the semi-permeable boundary conditions into the consolidation theory of unsaturated soil. In these studies, Wang et al [26][27][28] transformed PDEs into an equivalent set of partial differential equations by introducing two new variables.…”

“…23 Wang et al [26][27][28] first incorporated the semi-permeable boundary conditions into the consolidation theory of unsaturated soil. In these studies, Wang et al [26][27][28] transformed PDEs into an equivalent set of partial differential equations by introducing two new variables. Then, the solutions in the frequency domain could be easily derived using the method of Laplace transform.…”

“…However, it was quite challenging to transform these solutions from the frequency domain to the time domain analytically. Wang et al [26][27][28] used the numerical inversion of Laplace transform and obtained the semianalytical solutions for excess pore-water and pore-air pressures. Although this set of semi-analytical solutions was correct, these solutions were not explicit in the time domain and derived from a complicated mathematical process, which made these formulas hard to use.…”

One-Dimensional Consolidation Analysis of Unsaturated Soils under Cyclic Loadings

“…Although coupled models are theoretically more rigorous than uncoupled ones, the uncoupled theory proposed by Fredlund and Hasan [20] is now widely accepted because it is generally simple and sufficiently accurate for practical projects [24,25]. On the basis of the consolidation theory proposed by Fredlund and Hasan [20], many researchers have investigated consolidation of unsaturated soil under different time-dependent loadings [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. Among them, Feng et al [30] presented an analytical solution for one-dimensional consolidation of unsaturated soil under sinusoidal cyclic loading.…”