Analytical solution to axisymmetric consolidation of unsaturated soil stratum under equal strain condition incorporating smear effects

Ho, Liem; Fatahi, Behzad

doi:10.1002/nag.2838

Cited by 37 publications

(54 citation statements)

References 43 publications

(69 reference statements)

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“…According to the equal‐strain assumption, 1,12,45 governing equations of excess pore pressures in unsaturated soils by vertical drain under the instantaneous loading are shown as follows:

\frac{\partial {\overset{false¯}{u}}_{a}}{\partial t} = - C_{a} \frac{\partial {\overset{false¯}{u}}_{w}}{\partial t} - C_{v}^{a} ((), \frac{\partial^{2} u_{a}}{\partial r^{2}} + \frac{1}{r} \frac{\partial u_{a}}{\partial r}),

\frac{\partial {\overset{false¯}{u}}_{w}}{\partial t} = - C_{w} \frac{\partial {\overset{false¯}{u}}_{a}}{\partial t} - C_{v}^{w} ((), \frac{\partial^{2} u_{w}}{\partial r^{2}} + \frac{1}{r} \frac{\partial u_{w}}{\partial r}),

where u a and u w , respectively, are excess pore‐air and water pressures; C a and C w , respectively, are the interaction constants of air and water phases;

C_{v}^{a}

and

C_{v}^{w}

are the consolidation coefficients of air and water phases; and

{\overset{false¯}{u}}_{a}

and

{\overset{false¯}{u}}_{w}

, respectively, are average excess pore‐air and water pressures defined as

{\overset{false¯}{u}}_{a} = \int_{r_{normalw}}^{r_{normals}} u_{{normala}_{normals}} 2 italicπrdr + \int_{r}

…”

Section: Mathematical Modelingmentioning

confidence: 99%

“…In addition, by applying the Fourier series expansion, analytical solutions for the two‐dimensional (2‐D) plane strain consolidation were presented with analytical methods by Conte, 38 Ho et al, 39 and Ho and Fatahi, 40 and semi‐analytical solutions were given by Wang et al 41–43 By considering the smear effect, the consolidation of drainage well foundation under free‐strain hypothesis was analyzed by Qin et al 44 with semi‐analytical procedure. Ho and Fatahi 45 obtained the analytical solutions under the equal‐strain assumption. In addition, drain resistance was introduced into the PVD foundation by Zhou et al, 46 whereas the factors of drain resistance were not presented in explicit forms.…”

Section: Introductionmentioning

confidence: 99%

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Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

Huang

et al. 2020

Num Anal Meth Geomechanics

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Summary Based on the consolidation theory raised by Fredlund, the solutions for the equal‐strain consolidation of unsaturated foundation with the prefabricated vertical drain considering smear effect and drain resistance are analytically formulated in this paper. Firstly, governing equations for excess pore pressures (i.e., excess pore‐air and pore‐water pressures) under the equal‐strain hypothesis are derived with the introduction of radial boundary conditions. Afterwards, the obtained coupled equations are solved by applying general integration, decoupling process, and Fourier sine series expansion. The smear coefficients and factors of drain resistance corresponding to air and water phases are both captured explicitly in the final solutions. Furthermore, the degenerated solutions are employed to verify the reliability of the current solutions. Finally, a parametric study is conducted to study the consolidation characteristics of the proposed foundation model against modeling sizes (S and N), smear coefficients (αa and αw), and drain resistance factors (Ga and Gw).

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“…According to the equal‐strain assumption, 1,12,45 governing equations of excess pore pressures in unsaturated soils by vertical drain under the instantaneous loading are shown as follows:

\frac{\partial {\overset{false¯}{u}}_{a}}{\partial t} = - C_{a} \frac{\partial {\overset{false¯}{u}}_{w}}{\partial t} - C_{v}^{a} ((), \frac{\partial^{2} u_{a}}{\partial r^{2}} + \frac{1}{r} \frac{\partial u_{a}}{\partial r}),

\frac{\partial {\overset{false¯}{u}}_{w}}{\partial t} = - C_{w} \frac{\partial {\overset{false¯}{u}}_{a}}{\partial t} - C_{v}^{w} ((), \frac{\partial^{2} u_{w}}{\partial r^{2}} + \frac{1}{r} \frac{\partial u_{w}}{\partial r}),

where u a and u w , respectively, are excess pore‐air and water pressures; C a and C w , respectively, are the interaction constants of air and water phases;

C_{v}^{a}

and

C_{v}^{w}

are the consolidation coefficients of air and water phases; and

{\overset{false¯}{u}}_{a}

and

{\overset{false¯}{u}}_{w}

, respectively, are average excess pore‐air and water pressures defined as

{\overset{false¯}{u}}_{a} = \int_{r_{normalw}}^{r_{normals}} u_{{normala}_{normals}} 2 italicπrdr + \int_{r}

…”

Section: Mathematical Modelingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

Huang

et al. 2020

Num Anal Meth Geomechanics

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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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“…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%

“…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.…”

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confidence: 99%

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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Analytical solution to axisymmetric consolidation of unsaturated soil stratum under equal strain condition incorporating smear effects

Cited by 37 publications

References 43 publications

Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

Analytical solutions for the consolidation of unsaturated foundation with prefabricated vertical drain

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

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