Analytical Solutions to the Axisymmetrical Consolidation of Unsaturated Soils

Zhou, Wan-Huan; Zhao, Lin‐Shuang; Lok, T. M. H.; Mei, Guoxiong; Li, Xiaobo

doi:10.1061/(asce)em.1943-7889.0001382

Cited by 32 publications

(45 citation statements)

References 39 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

One-Dimensional Consolidation Analysis of Unsaturated Soils under Cyclic Loadings

Kim¹,

Ri²,

Kim³

et al. 2020

Shock and Vibration

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This paper presents analytical solutions to Fredlund and Hasan’s one-dimensional consolidation equations for unsaturated soils subjected to various cyclic loadings. Two new variables are introduced so that the governing equations for excess pore air and water pressures can be transformed to a set of conventional diffusion equations. Based on the general solutions for two introduced variables, the analytical solutions are derived for one-dimensional consolidation of unsaturated soils under trapezoidal, rectangular, triangular, and haversine cyclic loadings. It shows through the degeneration into the existing solutions for unsaturated and saturated soils that the proposed solutions are more general ones for one-dimensional consolidation of soils from unsaturated to saturated states. A comprehensive parametric study is conducted to investigate the effects of different parameters on one-dimensional consolidation of unsaturated soils under various cyclic loadings. The proposed solutions can be effectively utilized in the analysis of consolidation of unsaturated soils subjected to various cyclic loadings.

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Section: Introductionmentioning

confidence: 99%

One-Dimensional Consolidation Analysis of Unsaturated Soils under Cyclic Loadings

Kim¹,

Ri²,

Kim³

et al. 2020

Shock and Vibration

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show abstract

“…By applying the separation variable method, Zhou et al presented a set of simple analytical solutions for the homogeneous and partially permeable boundaries. Lately, Zhou et al obtained the analytical solutions for the axisymmetric consolidation in an unsaturated soil stratum on the basis of the assumption of free and equal strain, respectively. And Shan et al gave implicit analytical solutions of the 1D consolidation equations for the single layer and multilayer unsaturated soils, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In general, the SAS had been developed considering all kinds of boundaries and time‐dependent loadings for 1D dimensional consolidation problem in unsaturated soils . The literature about the 2D plane strain consolidation in unsaturated soils are applicable to the homogeneous and mixed drainage boundary condition subjected to four time‐dependent loadings, and there are only analytical solutions for the axisymmetric consolidation in unsaturated soils with the homogeneous boundary and considering the smear effect and time‐dependent loading . However, there are no SAS of the 2D plane strain consolidation considering the LSDB.…”

Section: Introductionmentioning

confidence: 99%

“…Notations: C a , interactive constant with respect to air phase; C w , interactive constant with respect to water phase; C a vx , coefficient of volume change with respect to air phase in x-direction; C a vz , coefficient of volume change with respect to air phase in z-direction; C w vx , coefficient of volume change with respect to water phase in x-direction; C w vz , coefficient of volume change with respect to water phase in z-direction; g, gravitational acceleration; h, thickness of soil layer; k ax , coefficient of air permeability in x-direction; k az , coefficient of air permeability in z-direction; k wx , coefficient of water permeability in x-direction; k wz , coefficient of water permeability in z-direction; l, length of soil layer; M, molecular mass of air; m a 1 , coefficient of air volume change with respect to a change in σ − u a ; m a 2 , coefficient of air volume change with respect to a change in u a − u w ; m s 1 , coefficient of volume change with respect to a change in σ − u a ; m s 2 , coefficient of volume change with respect to a change in u a − u w ; m w 1 , coefficient of water volume change with respect to a change in σ − u a ; m w 2 , coefficient of water volume change with respect to a change in u a − u w ; n 0 , initial porosity; q 0 , initial surcharge; R, universal gas constant; S r0 , initial degree of saturation; s, complex number; t, time; u a , excess pore-air pressure; u atm , atmospheric pressure; u 0 a , initial excess pore-air pressure; u w , pore-water pressure; u 0 w , initial excess pore-water pressure; w, settlement; w*, normalized settlement; x, investigated distance; z, investigated depth; γ w , unit weight of water; ε v , total volumetric strain; Θ, absolute temperature boundaries. Lately, Zhou et al 28 obtained the analytical solutions for the axisymmetric consolidation in an unsaturated soil stratum on the basis of the assumption of free and equal strain, respectively. And Shan et al 29,30 gave implicit analytical solutions of the 1D consolidation equations for the single layer and multilayer unsaturated soils, respectively.…”

mentioning

confidence: 99%

“…21,23,38 The literature about the 2D plane strain consolidation in unsaturated soils are applicable to the homogeneous and mixed drainage boundary condition subjected to four time-dependent loadings, 25,34 and there are only analytical solutions for the axisymmetric consolidation in unsaturated soils with the homogeneous boundary and considering the smear effect and time-dependent loading. 28,[35][36][37] However, there are no SAS of the 2D plane strain consolidation considering the LSDB. In addition, the consolidation property is significantly impacted by the lateral boundary for the 2D plane strain condition.…”

mentioning

confidence: 99%

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

Xia

2019

Num Anal Meth Geomechanics

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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.

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

Zhao

ZHANG

et al. 2020

Num Anal Meth Geomechanics

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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.

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Analytical Solutions to the Axisymmetrical Consolidation of Unsaturated Soils

Cited by 32 publications

References 39 publications

One-Dimensional Consolidation Analysis of Unsaturated Soils under Cyclic Loadings

One-Dimensional Consolidation Analysis of Unsaturated Soils under Cyclic Loadings

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

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

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