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Developing an analytical solution for the consolidation of unsaturated soils remains a challenging task due to the complexity of coupled governing equations for air and water phases. This paper presents an equal-strain model for the radial consolidation of unsaturated soils by vertical drains, and the effect of drain resistance is also considered. Simplified governing equations are established, and an analytical solution to calculate the excess pore-air and pore-water pressures is derived by using the methods of matrix analysis and eigenfunction expansion. The average degrees of consolidation for air and water phases and the ground surface settlement are also given. The solutions of the equal-strain model are verified by comparing the proposed free-strain model with the equal-strain model, and reasonably good agreement is obtained. Moreover, parametric studies regarding the drain resistance effect are graphically presented.

As a common phenomenon of civil engineering, consolidation is a process of decreasing soil volume when soil is subjected to an increased stress. Understanding of this phenomenon is vital to designs of soft soil foundations, pavements, and other engineering structures. Therefore, Terzaghi [

Since the inception of the one-dimensional (1D) consolidation theory of unsaturated soft soils proposed by Fredlund and Hasan [

In subgrade and pavement engineering, vertical drains are one of the most commonly used techniques to accelerate the consolidation process and increase the bearing capacity of the subgrade. It is essential to expand the consolidation theory of unsaturated soft soils from one-dimensional vertical consolidation to radial consolidation. For this purpose, Conte [

However, a lot of attempts have been made to solve the axisymmetric consolidation model of unsaturated soft soils particularly with analytical approaches. Among the pioneered studies, Conte [

Most of the models mentioned above were obtained based on the free-strain assumption. However, the solutions of the free-strain model are complex and inconvenient to use in practice. In the research of saturated soft soils, Barron [

At present, there is no reference about the equal-strain model in the research of unsaturated soft soils. The objective of this paper is to propose an equal-strain consolidation model to investigate the consolidation behavior of unsaturated soft soils. In addition, drain resistance will also be considered in this study, and the accuracy of the equal-strain model will be verified by comparing with the free-strain model proposed by Qin et al. [

Figure

Calculation diagram of vertical drain in an unsaturated soil stratum.

The basic assumptions are the same as those of Fredlund’s consolidation theory for unsaturated soft soils and Barron’s [

All the soil parameters are constants during consolidation.

The strains at the same depth of the foundation are equal (equal-strain assumption).

The distribution of excess pore-air pressure in unsaturated soft soils is uniform.

The governing equations for the consolidation of unsaturated soft soils with vertical drains consist of water phase and air phase equations under equal-strain conditions. In the polar coordinate system, considering a representative element of unsaturated soft soil foundation, with water and air flow in and out in the radial direction during consolidation, the derivation process is given as follows.

The net flux of water through the element is computed from the volume of water entering and leaving the element within a period of time. By utilizing Darcy’s law, the net flux of water per unit volume of the soil can be expressed in the polar coordinate system as

The net flux of water per unit volume of the soil can be obtained by differentiating the water phase constitutive relation with respect to time:

Substituting (

The net flux of air through the element is computed from the volume of air entering and leaving the element within a period of time. The flux of air per unit volume of the soil can be obtained by Fick’s law as

Based on assumption 3, the distribution of excess pore-air pressure in unsaturated soft soils is uniform. The density of air is a function of air pressure in accordance with the ideal gas law. It can be expressed as

Replacing the air density,

The flux of air per unit volume of the soil due to changes in the net normal stress,

By substituting (

By rearranging the governing equations of water and air phases, the consolidation equations for water and air phases can be written in matrix form as

The external radius,

Excess pore pressures at the interface of soil stratum and vertical drain satisfy flow continuity condition. The following boundary condition is considered:

The vertical boundary conditions are

The initial excess pore pressures are uniform throughout the soil mass when

According to the boundary condition (

Then, an integration of (

Incorporating (

The boundary condition of

Substituting (

General solution for

The average excess pore pressure

By substituting (

Based on the orthogonality of the sine function, the initial condition of

According to the initial condition (

Combining (

After solving

Based on Fredlund’s theory, the volume change is given by the following constitutive equation for unsaturated soft soils [

The ground surface settlement

The normalized settlement, denoted as

When the effect of drain resistance is neglected, the permeability of drain well is endless, videlicet

Based on the theory of series, the following can be obtained:

By substituting (

The equations from (

In this study, the accuracy of the proposed equal-strain consolidation model in the unsaturated soft soils is investigated by comparing with the free-strain model. The consolidation behavior and influence factors are also discussed in this section. Following Fredlund and Rahardjo [

Mechanical parameters

Fundamental constants of physics

Geometrical parameters

Other parameters

The above parameters are assumed to be constant during the consolidation process. An instantaneous compression induced by the external applied load

The validity of the equal-strain model had been verified in the saturated soft soils. In this section, the validity of the proposed equal-strain model in the unsaturated soft soils will be verified as well. For this purpose, the special consolidation case without considering drain resistance is analyzed by using the proposed equal-strain solution and one available free-strain solution [

Figures

Average degrees of consolidation of (a) air phase and (b) water phase varying with different ratios of

Figures

Difference of degrees of consolidation in (a) air phase and (b) water phase varying with different ratios of

Figure

Degrees of consolidation of (a) air phase, (b) water phase, and (c) normalized settlement varying with different values of

Figure

Degrees of consolidation of (a) air phase, (b) water phase, and (c) normalized settlement varying with different ratios of

By comparing Figure

The equal-strain model established in this paper has a negligible difference for average degree of consolidation compared with the free-strain model. Parametric studies regarding well-resistance effect are graphically presented and discussed, and the key findings are summarized as follows:

The solutions calculated from the equal-strain model show good agreement with that obtained from the free-strain model. The main difference between the two models is mainly found at the earlier stage of the consolidation.

The higher the ratio of

With the increase of

Both the water and air phases of consolidation proceed more quickly with the increase of

The authors declare that they have no conflicts of interest.