A simple analytical solution to one‐dimensional consolidation for unsaturated soils

Abstract

SUMMARY This paper presents a simple analytical solution to Fredlund and Hasan's one‐dimensional (1‐D) consolidation theory for unsaturated soils. The coefficients of permeability and volume change for unsaturated soils are assumed to remain constant throughout the consolidation process. The mathematical expression of the present solution is much simpler compared with the previous available solutions in the literature. Two new variables are introduced to transform the two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved with standard mathematical formulas. It is shown that the present analytical solution can be degenerated into that of Terzaghi consolidation for fully saturated condition. The analytical solutions to 1‐D consolidation of an unsaturated soil subjected to instantaneous loading, ramp loading, and exponential loading, for different drainage conditions and initial pore pressure conditions, are summarized in tables for ease of use by practical engineers. In the case studies, the analytical results show good agreement with the available analytical solution in the literature. The consolidation behaviors of unsaturated soils are investigated. The average degree of consolidation at different loading patterns and drainage conditions is presented. The pore‐water pressure isochrones for two different drainage conditions and three initial pore pressure distributions are presented and discussed. Copyright © 2013 John Wiley & Sons, Ltd.

Analytical solution for the two-dimensional plane strain consolidation of an unsaturated soil stratum subjected to time-dependent loading

“…3a and b illustrates the excess pore-air and pore-water pressure dissipation patterns varying with k a =k w , respectively. Considering a logarithmic time-scale, when the loading time is short, the dissipation of excess pore pressures induced by ramped loading is similar to that induced by constant loading [15,17,18]. After a considerable duration (i.e., 10 5 sÞ, when k a =k w > 1, the excess pore-air pressure has almost diminished (Fig.…”

“…However, to achieve a closed-form solution in this study, it is necessary to assume that these soil properties remain constant during the loading process. Moreover, constant properties have been already adopted in several existing analytical methods proposed by Qin et al [15,16], Shan et al [17], and Zhou et al [18] in consolidation studies for unsaturated soils.…”

“…5, the validation is conducted against the existing analytical solution adapted for ramped loading given by Zhou et al [18]. In this comparison, the parameters q 0 ¼ 0 and a ¼ 0:41 kPa=s are adopted as equivalent to the normalized loading parameters demonstrated by Zhou et al [18]. As observed, the analytical predictions obtained from Eq.…”

“…The entire mathematical procedure was conducted under a constant surcharge [15] and an exponentially time-dependent surcharge [16]. On the other hand, Shan et al [17] and Zhou et al [18] introduced alternative terms, w i and / i , respectively, which consist of excess pore-air and pore-water pressures, to convert the nonlinear inhomogeneous PDEs into traditional homogeneous PDEs and finally obtain solutions. In these studies, the unsaturated soil stratum is subjected to exponential loading [17,18] and sine wave loading [17].…”

“…On the other hand, Shan et al [17] and Zhou et al [18] introduced alternative terms, w i and / i , respectively, which consist of excess pore-air and pore-water pressures, to convert the nonlinear inhomogeneous PDEs into traditional homogeneous PDEs and finally obtain solutions. In these studies, the unsaturated soil stratum is subjected to exponential loading [17,18] and sine wave loading [17]. The mentioned analytical methods help to validate the numerical results for the 1D consolidation of unsaturated soils under a variety of time-dependent loadings.…”