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A model for one-dimensional consolidation of a double-layered foundation considering the depth-dependent initial excess pore pressure and additional stress and time-dependent loading under different drainage conditions was presented in this study and its general analytical solution was deduced. The consolidation solutions of several special cases of single-drained and double-drained conditions under an instantaneous loading and a single-level uniform loading were derived. Then, the average degree of consolidation of the double-layered foundation defined by settlement was gained and verified. Finally, the effects of the initial excess pore pressure distributions, depth-dependent additional stress, and loading modes on the consolidation of the soft foundation with an upper crust with different drainage conditions were revealed. The results show that the distributions of initial excess pore pressure and additional stress with depth and loading rates have a significant influence on the consolidation process of the soft foundation with an upper crust. This influence is larger with the single-drained condition than that with the double-drained condition. Comparing the consolidation rate with a uniform initial pore pressure and additional stress, their decreasing distribution with depth quickens the consolidation at the former and middle stages. Moreover, the larger the loading rate is, the quicker the consolidation of the soft foundation with an upper crust is.

Soft soil foundations are widely distributed in coastal and inland lake areas. Because of some natural actions or long-term engineering practices, a kind of natural or artificial hard crust with a thickness of a few meters on the soft substratum forms. The consolidation behavior of this kind of double-layered foundation with an upper crust is quite different from that of a homogeneous foundation.

Since Gray [

For a double-layered foundation, if the loading, which is always time-dependent, applied on the foundation surface is large enough, such as a high embankment or a high building, its influence region inside the foundation is deep and the vertical additional stress may vary with depth. At the same time, there may exist different distributions of initial excess pore pressure for a practical project. Therefore, a consolidation model of the double-layered foundation which takes account of the depth-dependent initial excess pore pressure and additional stress, time-dependent loading needs to be built. Based on the preceding discussions, the current researchers are not able to consider the above three factors in one consolidation model. Though Zhu and Yin [

The paper is organized as follows. The forthcoming section introduces the mathematical model used in this study. Subsequently, the general analytical consolidation solutions of the double-layered foundation considering the depth-dependent initial excess pore pressure and additional stress, time-dependent loading under different drainage conditions are derived. The next section deduces the consolidation solutions of some several cases of a bilinear initial excess pore pressure and additional stress with depth under an instantaneous loading and a single-level uniform loading. After that, the average degree of consolidation defined by settlement of the special cases is verified and the effects of the depth-dependent initial excess pore pressure and additional stress as well as loading rate on the consolidation of the soft foundation with an upper crust under different drainage conditions are discussed. The final section summarizes the major findings of this study.

A double-layered soil profile model is presented in Figure

Consolidation problem considered.

Time-dependent loading.

When the time-dependent loading,

For a 1D consolidation problem, the assumptions in Terzaghi’s [

Two commonly encountered drainage conditions are studied in this study:

The initial and continuity boundary conditions are given in the following forms:

The general consolidation solutions to consider depth-dependent initial excess pore pressure and additional stress, time-dependent loading, and different drainage conditions are deduced as follows.

The following dimensionless parameters were defined to simplify the expression for consolidation:

The excess pore pressure,

According to (

Substituting (

Substituting (

Then, the coefficient

According to similar principle, another

So the parameter

Substituting (

The above two equations should satisfy

Using the following orthogonality relationship:

the coefficient

According to the derivation processes of

Comparing (

So, for the consolidation problem of a double-layered foundation with the pervious top surface of the upper layer, the 1D consolidation solution is gained as follows:

According to the similar principle, for the consolidation problem of a double-layered foundation with a double-drained condition, the consolidation solution is

Based on the above solutions, results for two kinds of loading conditions, (1) instantaneous loading and (2) single-level uniform loading, are given. The bilinear distributions with depth of initial excess pore pressure and additional stress are considered.

For the instantaneous loading,

Assume the initial excess pore pressure has a bilinear distribution with depth; that is,

Substituting (

For the single-drained condition,

If a single-level uniform loading (dash line in Figure

When the self-weight consolidation is not considered, the initial excess pore pressure inside the foundation is equal to zero; that is,

When

When

According to the research results by Zhang et al. [

Substituting (

For the single-drained condition,

For the double-drained condition,

So, the average degree of consolidation inside double-layered foundation defined by settlement is

A calculation program was developed to analyze the consolidation of the soft foundation with an upper curst in terms of (

Consolidation curves with a uniform vertical additional stress.

Consolidation curves with a uniform vertical additional stress initial excess pore pressure.

In addition, Pyrah [

Degree of consolidation for different soil profiles.

For the last comparison, the degree of consolidation was calculated for the single layer defined by Case

Degree of consolidation for a single layer foundation.

Therefore, the consolidation model in this study is rational by several comparisons above with other analytical and numerical results, which can be further used to analyze the effects of distributions of initial excess pore pressure and depth-dependent additional stress on the consolidation of the double-layered foundation.

For the soft foundation with an upper crust, the upper layer always forms due to the soil sedimentation or some long-term engineering practices; it has the same ingredient and smaller compressibility. Then,

Figure

Consolidation curves with different drainage conditions.

Curves with a single-drained condition

Curves with a double-drained condition

Typically,

The consolidation curves of different additional stress with a pervious top of the upper layer are shown in Figure

Consolidation curves with a single-drained condition.

Figure

Consolidation curves with a double-drained condition.

The 1D consolidation model of the double-layered foundation proposed in this study can account for the depth-dependent initial excess pore pressure and additional stress and time-dependent loading under different drainage conditions, which can be utilized to comprehensively investigate the consolidation behavior of the double-layered foundation. The general solutions of the model under different drainage conditions were gained.

The consolidation solutions of special cases, which consider two loading modes, that is, instantaneous loading and single-level uniform loading, and the bilinear distributions with depth of the initial excess pore pressure and additional stress, were derived. Then the average degree of consolidation defined by settlement was deduced and verified.

Regardless of the drainage conditions, both the distribution of initial excess pore pressure and additional stress with depth and the loading rate have a great influence on the consolidation process of the soft foundation with an upper crust. This influence is larger with the single-drained condition than that with the double-drained condition. With the decrease of the initial excess pore pressure or the additional stress with depth, the consolidation rate increases. The larger the loading rate is, the quicker the consolidation process of the soft foundation with an upper crust is.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors acknowledge the National Natural Science Foundation of China (51108048, 51208517, and 51478054) and Jiangxi Communications Department Program (2013C0011) for the financial support.