Research Article | Open Access

Volume 2015 |Article ID 618717 | 13 pages | https://doi.org/10.1155/2015/618717

# One-Dimensional Consolidation of Double-Layered Foundation with Depth-Dependent Initial Excess Pore Pressure and Additional Stress

Revised11 Sep 2015
Accepted14 Sep 2015
Published08 Oct 2015

#### 1. Introduction

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.

#### 2. Mathematical Modelling

A double-layered soil profile model is presented in Figure 1. The upper layer may be a natural or an artificial hard crust. The lower layer is a native soft soil layer. There are 2 kinds of drainage conditions in this model: (1) a single-drained condition, that is, only the top of upper layer is drained; (2) a double-drained condition, that is, both top and bottom of layers are freely drained. The soil properties of the th layer are the coefficient of consolidation , the coefficient of permeability , and the compression modulus . The compressible stratum has a total thickness of . A time-dependent loading, , is applied on the foundation surface, as shown in Figure 2.

When the time-dependent loading, , as shown in Figure 2 is applied on the foundation surface, the resulting additional stress along the depth inside the foundation, , can be expressed as follows:where is the initial loading; is duration and is the vertical coordinate; and are the additional stress coefficient and initial excess pore pressure, respectively; is the loading rate and

For a 1D consolidation problem, the assumptions in Terzaghi’s  consolidation theory are retained except for the depth-dependent initial pore pressure and additional stress and time-dependent loading. Then, let ; the partial differential equation for 1D consolidation of soils by vertical drainage is given as follows:where is the excess pore pressure.

Two commonly encountered drainage conditions are studied in this study:The former and the latter represent the drainage conditions (2) and (1) above, respectively.

The initial and continuity boundary conditions are given in the following forms:where and are the excess pore pressure and initial excess pore pressure in soil layer at depth , respectively (). Equation (4) is the initial condition. Equation (5) is the continuity condition of the two layers.

#### 3. General Analytical Consolidation Solutions

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.

##### 3.1. Single-Drained Condition

The following dimensionless parameters were defined to simplify the expression for consolidation:where is the compression coefficient of the layers ().

The excess pore pressure, , was described according to Terzaghi’s 1D consolidation solution :where and ; , , , , , and are undetermined coefficients, which can be derived according to (2)~(4). According to (5),

According to (8), can be gained:

Substituting (10) into (9), (11) is derived:

Substituting (7) into the consolidation equation (2), (12) and (13) can be gained:

is a time coefficient caused by . If , . So (12) is simplified:

Then, the coefficient can be determined as follows:

According to similar principle, another is determined based on (13):

So the parameter is derived:

Substituting (15) into (12) and (16) into (13), two equations are gained as follows:

The above two equations should satisfy

Using the following orthogonality relationship:

the coefficient is where is the additional stress coefficient in the soil layers at depth ().

According to the derivation processes of and (13), the coefficient can be given as follows:where is determined by eigen-equation .

Comparing (21) with (22), it is found that the expressions of and are similar. However, of (21) and of (22) represent the additional stresses coefficient and initial excess pore pressure in soil layer at depth .

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:

##### 3.2. Double-Drained Condition

According to the similar principle, for the consolidation problem of a double-layered foundation with a double-drained condition, the consolidation solution iswhere and are similar to (15)~(17); is determined by ;

#### 4. Special Cases

For the instantaneous loading, . So, (23)~(24) can be simplified as follows:

Assume the initial excess pore pressure has a bilinear distribution with depth; that is,where , , and are the initial excess pore pressure at the top of the upper layer, interface between the upper and lower layers, and bottom of the lower layer, respectively.

Substituting (27) into (22) and (25), the expression of is determined.

For the single-drained condition, For the double-drained condition, Then, the average excess pore pressure inside the upper and lower soils is derived as follows:So, the average consolidation degree inside the foundation defined by settlement iswhere is the compression of the two soil layers at time and is the final compression when the excess pore pressure is zero; is the average initial excess pore pressure in the layers (). From (31), it is evident that only relates to and when the parameters of , and are constant.

If a single-level uniform loading (dash line in Figure 2) is applied on the foundation surface, the following relationships can be gained:where is the time when the loading becomes a constant value,

When the self-weight consolidation is not considered, the initial excess pore pressure inside the foundation is equal to zero; that is, ; then . So, the consolidation solutions of the double-layered foundation with the depth-dependent additional stress are determined as follows.

When ,

When , where ; .

According to the research results by Zhang et al. , the distribution of the depth-dependent additional stress can be simplified to a bilinear line; that is,where , , and are the additional stress at the top of the upper layer, the interface between the upper and lower layers, and the bottom of the lower layer, respectively, as shown in Figure 1.

Substituting (35) into (21) and (25), the consolidation solutions with a single-level loading are determined as follows.

For the single-drained condition,

For the double-drained condition,

So, the average degree of consolidation inside double-layered foundation defined by settlement iswhere is the average coefficient of additional stress in the layers, respectively ().

#### 5. Consolidation Behaviour of the Soft Foundation with an Upper Crust

##### 5.1. Model Verification

A calculation program was developed to analyze the consolidation of the soft foundation with an upper curst in terms of (26)~(38). In order to verify the consolidation model above, the degree of consolidation of a double-layered foundation with a uniform additional stress with depth under a single-level uniform loading was calculated as shown in Figure 3; that is, , , , , and , and 10. In Figure 3, and denote the single-drained condition and double-drained condition. It is obvious from Figure 3 that the consolidation rate relates to the loading rates (the values of ) and drainage conditions, which is larger with the increasing loading rate (i.e., with a smaller value of ) and under the double-drained condition. Moreover, the degree of consolidation of a double-layered foundation with a uniform initial excess pore pressure with depth under an instantaneous loading was also obtained, as shown in Figure 4. For this calculation, , , , , and 10, for the single drainage condition, and for the double drainage condition. It can be seen from Figure 4 that the degree of consolidation relates to both the ratios of the modulus of the upper and lower layers (the values of ) and the drainage conditions. The consolidation rate becomes smaller with the increasing , which is larger under the double-drained condition than that under the single-drained condition. In addition, compared with the analytical results of Xie et al.  and Xie et al. , it can be found that the consolidation curves under the single and double drainage conditions in Figures 3 and 4 are perfectly similar to Figures and from Xie et al.  and Figures and from Xie et al. , respectively.

In addition, Pyrah  discussed the influence of soil parameters on the consolidation behaviour of a double-layered system through four idealized soil profiles using the finite element method. Zhu and Yin  gained the analytical solutions of the first three soil profiles and found that they are similar to the results of Pyrah . In order to further verify the consolidation model in this study, the degree of consolidation of the first three soil profiles (namely, ) was calculated with a uniform additional stress ( :  :  :  : ). The result with a depth-dependent additional stress ( :  :  :  : ), which decreases along the depth, also was gained to investigate the effects of the distribution of the additional stress. For all the calculations, and . Figure 5 presents the calculation results with the corresponding curves of Zhu and Yin . and denote the consolidation curves with the uniform and depth-dependent additional stress. denotes the curves of (1), (2), and (3) with in Figure from Zhu and Yin . It is evident from Figure 5 that curves are similar to curves, which indicates that the results in this study agree well with the analytical solutions of Zhu and Yin  and the finite element results of Pyrah . Meanwhile, the decreasing additional stress with depth ( :  :  :  : ) quickens the consolidation process of the double-layered foundation.

For the last comparison, the degree of consolidation was calculated for the single layer defined by Case in Example from literature , as shown in Figure 6. The analytical solution was gained by the model in this study and the numerical solution was calculated by the curve with the bilinear additional stress in Figure and the parameters in Table from literature . It can be concluded from Figure 6 that the analytical solution matches the numerical results well except a minor difference at the middle stage.

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.

##### 5.2. Effect of Distributions of Initial Excess Pore Pressure

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, will be smaller than 1 while will be larger than 1. So, assume , , and in the following analyses. In order to investigate the effect of the initial pore pressure on the consolidation of the double-layered foundation, three distributions of the initial pore pressure with depth were selected, namely,  :  :  :  : ,  :  : , and  :  : , which represent the uniform, gradually increasing, and decreasing distributions with depth, respectively.

Figure 7 shows the curves of degree of consolidation under different drainage conditions. It is evident from Figure 7 that the initial excess pore pressure distributions have a similar influence on the consolidation of the double-layered foundation under different drainage conditions. Comparing with the consolidation curve of a uniform initial excess pore pressure with depth ( :  :  :  : ), an increasing initial excess pore pressure with depth ( :  :  :  : ) slows down the consolidation process while the consolidation rate increases with a decreasing initial excess pore pressure with depth (). At the same time, the maximum differences of degree of consolidation between the uniform and increasing and decreasing initial excess pore pressure with depth under the single-drained condition (Curve 1 and Curve 2 in Figure 7(a)) are −9.9% and 20.8%, respectively, while the corresponding differences under the double-drained condition (Curve 1 and Curve 2 in Figure 7(b)) are −1.8% and 6.5%. Therefore, the initial excess pore pressure has a greater influence on the consolidation rate under the single-drained condition than that with the double-drained condition. For a practical project, the variation of initial excess pore pressure with depth should be considered, especially for the project with the single-drained condition.

##### 5.3. Effect of Depth-Dependent Additional Stress

Typically, is not equal to zero and 10% of the gravity stress is always used in settlement calculations. In order to compare the effect of different additional stress, two kinds of distributions were selected, namely,  :  :  :  :  and  :  : . In order to investigate the effect of loading rate on the consolidation of the soft foundation with an upper crust, different values of (i.e., ) were used.

Figure 9 gives the consolidation curves of different additional stresses with the double-drained conditions. It is clear from Figure 9 that the additional stress also affects the consolidation of the soft foundation with an upper crust under a double-drained condition and the maximum differences of the consolidation degree are 6%, 6%, 5%, and 2% for different . Comparing with the curves in Figure 8, it indicates that the effect of additional stress on the consolidation degree with the single-drained condition is much greater than that with the double-drained condition. At the same time, the effect of additional stress on the degree of consolidation with a double-drained condition also gradually weakens with the increase of .

#### Conflict of Interests

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

#### Acknowledgments

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

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