CLC number: TU45

On-line Access: 2013-01-02

Received: 2012-07-01

Revision Accepted: 2012-11-15

Crosschecked: 2012-12-10

Cited: 7

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Zhen-dong Shan, Dao-sheng Ling, Hao-jiang Ding. Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition[J]. Journal of Zhejiang University Science A, 2013, 14(1): 61-70.

@article{title="Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition",

author="Zhen-dong Shan, Dao-sheng Ling, Hao-jiang Ding",

journal="Journal of Zhejiang University Science A",

volume="14",

number="1",

pages="61-70",

year="2013",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.A1200167"

}

%0 Journal Article

%T Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition

%A Zhen-dong Shan

%A Dao-sheng Ling

%A Hao-jiang Ding

%J Journal of Zhejiang University SCIENCE A

%V 14

%N 1

%P 61-70

%@ 1673-565X

%D 2013

%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.A1200167

TY - JOUR

T1 - Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition

A1 - Zhen-dong Shan

A1 - Dao-sheng Ling

A1 - Hao-jiang Ding

J0 - Journal of Zhejiang University Science A

VL - 14

IS - 1

SP - 61

EP - 70

%@ 1673-565X

Y1 - 2013

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.A1200167

**Abstract: **Based on consolidation equations proposed for unsaturated soil, an analytical solution for 1D consolidation of an unsaturated single-layer soil with nonhomogeneous mixed boundary condition is developed. The mixed boundary condition can be used for special applications, such as tests occur in laboratory. The analytical solution is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the nonhomogeneous boundary condition is first transformed into a homogeneous boundary condition. Then, the eigenfunction and eigenvalue are derived according to the consolidation equations and the new boundary condition. Finally, using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution for the new boundary condition is obtained. The present method is applicable to various types of boundary conditions. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with mixed boundary condition.

**
**

1. Introduction

The consolidation equation for saturated soil proposed by Terzaghi (

Using the simplified consolidation equations, several analytical solutions for the 1D consolidation of unsaturated soil have been published. Qin et al. (

These analytical solutions are for unsaturated single-layer soil with boundaries that are permeable or impermeable to both air and water phases. However, lab-testing equipment is commonly designed such that air flows upwards and water flows downwards during a test, i.e., the top boundary is permeable to air and impermeable to water, while the bottom boundary has the reverse condition. Fredlund and Rahardjo (

As an extension of Shan et al. (

2. Consolidation equations for unsaturated soil

The compression of unsaturated single-layer soil (Shan et al.,

3. Initial and boundary conditions

The following boundary condition is considered:

4. Derivation of the analytical solution

By substituting Eq. (

Thus, the nonhomogeneous boundary value problem (i.e., Eq. (

The solution of Eq. (

The solution of Eq. (

When

When

When

At this point, the eigenfunction of Eq. (

Using Eq. (

5. Numerical examples

It is assumed that the total stress

Step load:

Exponential load:

Fig.

Fig.

Fig.

Fig.

Fig.

6. Conclusions

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