Research Article  Open Access
Qi Hu, "Retaining Structure ForceDeformation Analysis Model for an Ultradeep Foundation Pit", Mathematical Problems in Engineering, vol. 2013, Article ID 549491, 18 pages, 2013. https://doi.org/10.1155/2013/549491
Retaining Structure ForceDeformation Analysis Model for an Ultradeep Foundation Pit
Abstract
An ultradeep foundation pit is a complex system composed of a retaining structure, foundation soil, and groundwater. Design and construction of foundation pits for use at greater depths than in the past require continual improvement in the design methods and analysis methods applied. In this paper, a loaddeformation analysis model of a retaining structure based on a bearingmode analysis of an ultradeep foundation pit is proposed. A calculation method was theoretically derived for the horizontal foundation stiffness coefficient for this model, and the influences of factors such as space size, stress path, load level, and seepage were analyzed. A numerical example and a case study of an ultradeep foundation pit in the Hangzhou Metro Line number 1 test section are presented. The calculated results for deformation of the structure and for earth pressure were found to be similar to the results obtained from elasticplastic finite element analysis and similar to the measured results. The results of this study indicate that the proposed analysis model adequately reflects the forcedeformation characteristics of an ultradeep foundation pit and show that the proposed analysis model appropriately considers the influences of various factors.
1. Introduction
Numerous achievements have been reported in the study of deep foundation pits. Codes for the construction of foundation pits have been developed in numerous countries, [1, 2] and technical specifications have been prepared by local governments. Valuable experience in the design and construction of ultradeep foundation pits has been reported. Due to increasing depths of excavation, new support methods and construction technologies have been applied, and more stringent safety measures have been required. These advances require continuous improvement in the design methods and analysis theories for retaining structures.
The principle underlying current codes and technical specifications for the design of a foundation pit, which is a static design problem, is illustrated in Figure 1. The main steps in the design process can be summarized as follows. First, an initial state is considered in which the external earth pressure is equivalent to the active earth pressure, and this state is constant. Second, part of the initial internal earth pressure is offset by the external earth pressure, and the change in the internal earth pressure is represented by the soil spring force. The influence of seepage on the waterearth pressure and the horizontal foundation stiffness coefficient is not considered. This basic design approach has been proven to be applicable to shallow foundation pit excavation.
An ultradeep foundation pit is a complex system that is composed of a retaining structure, foundation soil, and groundwater. The following characteristics of an ultradeep foundation pit are illustrated in Figure 2. (1) The waterearth pressure on a retaining structure is large, and the earth pressure on a flexible retaining structure is closely related to the deformation of the retaining structure, which cannot be described by a single model. (2) Because the unloading of the soil inside the pit is large and the stress path is complex, the influences of the stress level and stress path on soil parameters must be considered. (3) The deformation mode of a retaining structure consists of the lateral deformation of the entire pile group. The magnitude of the lateral support that the soil inside the pit provides to the retaining structure is related not only to the properties of the soil but also to the space size, that is, the excavation widthand excavation depth. (4) In areas with high groundwater levels, the influence of seepage on the waterearth pressure and soil parameters is significant due to the large difference between the water level inside the pit and the water level outside the pit. Under the influence of seepage, the effective soil stress inside the pit is reduced and the deformation of the retaining structure is increased.
Many researchers have studied these problems. As Figure 3 shows, the displacementdependent earth pressure theory assumes that the earth pressure consists of active earth pressure and passive earth pressure. Xu [3] used trigonometric functions to describe the relationship between earth pressure and deformation. Chen et al. [4] and Zhao et al. [5] used exponential functions to describe the relationship between earth pressure and deformation. Bei and Zhao [6] analyzed the relationship between active earth pressure and the deformation of a retaining structure. Many researchers, such as Lade and Duncan [7], Yuan et al. [8], Liu and Hou [9], Liu [10], and Charles and Qun [11], have conducted soil stress path experiments on foundation pits. The current methods for determining horizontal foundation stiffness coefficient values can be classified into three categories. The first category encompasses empirical methods. The value of the horizontal foundation stiffness coefficient is selected on the basis of analyses of soil geological conditions, as well as experience with similar projects and codes [12, 13]. These methods are sometimes arbitrary. The second category encompasses field test methods, including horizontal static load tests, pressure meter tests, and flat dilatometer tests [14]. The third category encompasses laboratory test methods. According to certain theories, the relationships between the horizontal foundation stiffness coefficientand the soil modulusand shear strengthcan be expressed by the equations[15] and[16], respectively.
Although these three types of methods have yielded useful results, none of them reflects the forcedeformation behavior of an ultradeep foundation pit or considers the influence of various factors on that behavior. This paper proposes an analysis model for an ultradeep foundation pit, based on the forcedeformation behavior of a retaining structure and soil. A calculation method was theoretically derived for the horizontal foundation stiffness coefficient in this model. The influences of the space size, stress path, load level, and seepage on the forcedeformation behavior of a retaining structure were examined. The proposed method was verified using a numerical example and a case study of an ultradeep foundation pit in the Hangzhou Metro Line number 1 test section.
2. Analysis Model for an Ultradeep Foundation Pit
An analysis model framework for an ultradeep foundation pit, which can be decomposed into the processes described below, is shown in Figure 4. If deformation of a retaining structure does not occur after excavation, the earth pressure outside the pit maintains a static state. Unbalanced earth pressure is sustained by the support system, which is composed of soil springs inside and outside the pit. Due to the effect of the unbalanced earth pressure between the inside and the outside of the pit, the soil springs inside the pit are compressed, the earth pressure inside the pit increases, the soil springs outside the pit are stretched, and the earth pressure outside the pit decreases, until a new balanced state is achieved and formed. Three significant differences between this model and existing analysis models exist. (1) The initial loading state consists of static earth pressure without active earth pressure. (2) The change in earth pressure outside the pit is considered using the force of the soil springs outside the pit. (3) The soil spring stiffness or horizontal foundation stiffness coefficientis related not only to the soil properties but also to the loading modes of the retaining structure and the soil.
(a) Loading system
(b) Supporting system
As Figure 5 shows, according to the definition of a Winkler elastic foundation, the force on a soil spring is defined by, and the soil spring stiffness is defined by. The relationships among the force, the stiffness, and the horizontal displacement of the soil spring are described by the following equations: wheredenotes the horizontal strip load (in units of pressure),denotes the calculated horizontal width, anddenotes the calculated thickness.
As Figure 6 shows, the soil applies lateral pressuresandto the retaining structure, and the retaining structure applies lateral pressuresandto the soil. If the relationship between the change in the horizontal strip loadand the horizontal displacementis established, the value of the horizontal foundation stiffness coefficientfor any depth can be determined.
(a)
(b)
The solutions for the components of stress at any point in a semiinfinite elastic space due to a linear horizontal loadapplied at a depth(as illustrated in Figure 7) were proposed by Melan [17]. The solutions for stress at any point in a semiinfinite elastic space solution due to a uniform horizontal strip load(as illustrated in Figure 8), as well as the displacement solution and the horizontal foundation stiffness coefficient, can be obtained by integrating Melan’s solutions. These solutions for a uniform horizontal strip load reflect the actual forcedeformation pattern of an ultradeep foundation pit.
As mentioned above, the solutions for the components of stress at any point in a semiinfinite elastic space due to a linear horizontal loadapplied at a depth(as illustrated in Figure 7) were proposed by Melan [17]: wheredenotes the linear load,denotes Poisson’s ratio,denotes the depth of the linear load,denotes the horizontal coordinate of a point in the semiinfinite space, anddenotes the vertical coordinate of a point in the semiinfinite space.
The solutions for the components of stress at any point in a semiinfinite elastic space due to a uniform horizontal strip load(as illustrated in Figure 8) can be obtained by integrating Melan’s solutions: wheredenotes the uniform strip load, denotes the depth of the top of the uniform strip load, and denotes the depth of the bottom of the uniform strip load.
2.1. Basic Assumptions of the Analysis Model
A comparison of Figures 6 and 8 reveals certain differences between the physical model of an ultradeep foundation pit and a semiinfinite space. Therefore, certain assumptions can be made.
Assumption 1. The physical model of a strip foundation pit is usually described as a planestrain elastic problem in a semiinfinite space, as shown in Figure 9. The foundation pit is evenly divided into the left side and right side, regardless of the interaction between the left side and the right side (Figure 9(a)). Using a retaining structure as a border, the space inside and outside of the foundation pit is divided into two independent regions (Figure 9(b)). The two regions are asymmetric when subject to a lateral load and can be expanded into two separate semiinfinite elastic spaces (Figure 9(c)).
Assumption 2. Based on the provisions of settlement factors for a rigid base and a flexible load [18], if the width of the horizontal strip load is sufficiently small, the horizontal deformation of a rigid base is equivalent to the average deformation value of a flexible load,, as shown in Figure 10.
(a) Flexible load
(b) Rigid base
Assumption 3. For the purpose of calculating stresses, the soil is considered to be a singlephase, homogeneous, and isotropic material with a constant modulus.
2.2. Horizontal Foundation Stiffness Coefficient for the Analysis Model
According to the definition of the horizontal foundation stiffness coefficient, the horizontal deformation of the isotropic planestrain problem can be calculated from the following equations:
Using the integrals of the stress solutions in (3), the displacement in semiinfinite space due to a uniform horizontal strip loadcan be determined as follows:
The average displacement within the loading area is expressed as follows:
According to Assumption 1, the real load in a quarterinfinite space is, due to the operation of asymmetric mapping. When (4) and (5) are used to solve the horizontal deformation equation, the integral range shown in Figure 11 must be determined. In a symmetric excavation, the horizontal integral range is finite inside the pit:
The horizontal foundation coefficientof the soil springs inside the pit can be obtained from the average displacement within the loading area:
Determination of the theoretical influence zone, which is infinite outside the pit, is similar to the problem of determining the thickness of the underlying layer below a strip foundation. Based on the provisions of compression depth in calculating the foundation settlement [18], when the additional stress decreases to 10% of the gravity stress, the depth is defined as the compression depth. The influence zone of lateral soil deformation is assumed to satisfy the calculation when the lateral stress decreases to 10% of the horizontal load. According to the results obtained using the stress solutions in (3), when the additional horizontal stress is 10% of the horizontal load, the corresponding distance is approximately ten times the loading width. Thus, ten times the foundation pit depth was defined as the influence zone outside the pit:
The horizontal foundation coefficientof the soil springs outside the pit can be obtained from the average displacement within the loading area:
We define
Then, wheredenotes the coefficient of the horizontal foundation stiffness coefficient, which is related to the foundation pit space size and Poisson’s ratio (m^{−1}), anddenotes the elastic modulus of the soil.
3. Parametric Analysis
As shown in (8) and (10), the horizontal foundation stiffness coefficient can be described as. The factors that influenceinclude the space size of the foundation pit (a function of the pit width, the pit depth, and the influence zone outside the pit) and the soil parameters (elastic modulus and Poisson’s ratio).
3.1. Influence of Space Size of the Foundation Pit
The coefficientof the horizontal foundation stiffness coefficient for various foundation pit widths and depths is shown in Figures 12 and 13 for a Poisson’s ratio of soil of 0.3.
As Figures 12 and 13 show, the value of the coefficientdecreases when the foundation pit widthor the influence zoneincreases. Due to the low level of restraint on the surface soil, the value of the coefficientis also small. When the soil depthis half of the foundation pit width, the coefficientapproaches a constant value.
3.2. Influence of Poisson’s Ratio
As shown in Figure 14, the coefficientdecreases when Poisson’s ratioincreases. When the foundation pit is 20 meters wide,and.
3.3. Influence of Stress Path
The relationship between the soil modulus, the stress path, and the consolidation pressure is [19] where the coefficientdenotes the influence of the stress path;denotes the soil effective gravity; anddenotes the soil depth.
For the soil outside the pit, the lateral modulus coefficientmeans the lateral unloading stress path. For the soil inside the pit, the lateral modulus coefficientmeans the vertical unloading stress path.
Taking the influence of the stress path into consideration, the horizontal foundation coefficient can be expressed as follows:
The proportional coefficientof the horizontal foundation stiffness coefficient can be expressed as. The influence of the stress path on the coefficientof the initial tangent modulus, according to stress path tests on Hangzhou sandy silt, is indicated in the results shown in Table 1.

The average effective gravityof Hangzhou sandy silt is 9.0 kN/m^{3} [19]. According to the test results shown in Table 1, the lateral unloading stress path coefficientis approximately 80, and the vertical unloading stress path coefficientis approximately 470. The results for the proportional coefficientof sandy silt inside and outside the foundation pit, when Poisson’s ratiois 0.3, are shown in Figures 15 and 16 and Table 2.

As the values in Table 2 show, the variation in the proportional coefficientin the homogeneous foundation is similar to that of coefficient. The greater the width of the foundation pit is, the smaller the proportional coefficientis. The value of the proportional coefficientis smallest at the ground surface. When the soil depth is half of the foundation pit width, the proportional coefficientapproaches a constant value. In general, the proportional coefficientof the soil inside the pit is considerably larger than the proportional coefficientof the soil outside the pit.
3.4. Influence of Load Level
The stressstrain behavior of soil is nonlinear. As the load level increases, the rate of strain, the soil modulus, and the horizontal foundation stiffness coefficient decrease. Thus, the effect of load level should be considered. The secant modulus for the Duncan and Chang [21] hyperbolic equations, shown in Figure 17, can be expressed as follows: where.
Taking the influence of load level into consideration, the horizontal foundation stiffness coefficient can be expressed as follows:
As in the linear elastic model, the initial tangent modulus in (16) is unrelated to the stress levelsand, and the effect of load level is only related to the coefficient. Equation (16) is relatively complicated and must be solved by numerical integration. The influence of the load leveland the strength parameteron the coefficientwhen the foundation pit width is 20 m is shown in Figures 19 and 20. When the load levelincreases, the secant modulus of the soil and the coefficientdecrease. When the strength parameterincreases, the soil secant modulus and the coefficientdecrease.
According to the stress path test results for Hangzhou sandy silt, the strength parameterin the vertical unloading stressstrain curve which is illustrated in Figure 18 can be determined from the following equation: wherehas units of MPa^{−1} anddenotes the vertical consolidation pressure (kPa).
3.5. Influence of Seepage
As shown in (13) and (14), the horizontal foundation stiffness coefficient and the modulus of soil are related to the soil stress state. At a site with abundant groundwater, if seepage occurs in the foundation pit, the soil stress state will change. The watersoil pressure and horizontal foundation stiffness coefficient will also be affected.
Taking the influence of seepage into consideration, the horizontal foundation stiffness coefficient of the soil inside the pit can be expressed as follows: wheredenotes the average hydraulic gradient inside the pit.
Taking the influence of seepage into consideration, the horizontal foundation stiffness coefficient of the soil outside the pit can be expressed as follows: wheredenotes the average hydraulic gradient outside the pit.
Taking the influence of seepage into consideration, the static earth pressure outside the pit, the water pressure outside the pit, and the lateral pressure outside the pit can be expressed as follows: wheredenotes the static earth pressure coefficient.
Taking the influence of seepage into consideration, the static earth pressure inside the pit, the water pressure inside the pit, and the lateral pressure inside the pit can be expressed as follows:
4. Analysis Example
The parameters for the analysis example, illustrated in Figure 21, are as follows: an excavation depth of 20 m, five excavation steps, horizontal supporting structures consisting of 5 layers ofsteel pipes, a stiffness of, retaining structures consisting ofbored piles with embedded depths of 20 m, a concrete modulus GPa, excavation widths of 10 m and 40 m, a horizontal zone of influence of 100 m outside the pit, a water surface elevation equal to the ground surface elevation, and a water pressure unrelated to the earth pressure.
(a) Plane graph
(b) Sectional graph
The foundation soil consists of sandy silt with the following characteristics: saturated gravity kN/m^{3}, effective gravity kN/m^{3}, Poisson’s ratio, shear strength parameters kPa and, lateral unloading stress path coefficientof the initial tangent modulus of 80, and vertical unloading stress path coefficientof the initial tangent modulus of 470. The calculation methods and models are shown in Figures 22, 23, and 24, and the parameters are listed in Table 3.

4.1. Analysis Results for the Retaining Structures
Figures 25 to 28 illustrate the following points. (1) The influence of the foundation pit space size and seepage cannot be considered, and the earth pressure outside the pit and the proportional coefficientof the horizontal foundation stiffness coefficient are constant. Thus, the horizontal displacement and the bending moment determined by the code method are smaller than the horizontal displacement and the bending moment determined using the other two methods. The difference increases when the excavation width increases. (2) The method proposed in this study considers the influences of space size, stress path, load level, and seepage. The results obtained using this method are similar to those obtained for the continuum elasticplasticmedium finite element model.
Equations (18) to (21) illustrate the following points. First, due to the influence of seepage, the water pressure outside the pitdecreased, which caused a decrease in the horizontal displacement and bending moment of the retaining structures. Second, due to the influence of seepage, the horizontal foundation stiffness coefficient inside the pitdecreased, which increased the horizontal displacement and bending moment of the retaining structures. As Figures 25 and 26 show, when the width of the foundation pit is small, the hydraulic gradientinside the pit is large. The horizontal foundation stiffness coefficientinside the pit decreases rapidly under the influence of seepage, and the horizontal displacement and bending moment of the retaining structures increase significantly. As Figures 27 and 28 show, when the width of the foundation pit is large, the hydraulic gradientinside the pit is similar to the hydraulic gradientoutside the pit, and the horizontal foundation stiffness coefficientinside the pit and the water pressureoutside the pit decrease similarly under the influence of seepage. Thus, the horizontal displacement and the bending moment of the retaining structures vary slightly.
4.2. Analysis Results for Earth Pressure
In the model described in this paper, the initial state of the load consists of static earth pressure, and the earth pressure outside the pit changes when the soil springs are tensed. Figures 29, 30, 31, and 32 highlight the results obtained for earth pressure using the method proposed in this paper, which are similar to the results obtained for the continuum elasticplasticmedium finite element model. A comparison of the results indicates that the method proposed in this paper accurately simulates the distribution patterns and the changes in earth pressure both inside and outside the pit.
5. Case Study
5.1. Overview
As Figure 33 shows, the parameters of the ultradeep foundation pit in the Qiutao Road station of the Hangzhou Metro Line number 1 test section are as follows: a strip foundation pit, a 20 m excavation width, a 16.8 m excavation depth, a zone of influence of 200 m outside the pit, 6 excavation steps, horizontal supporting structures consisting of 5 layers ofsteel pipes with stiffnesses of, retaining structures consisting of 30 mlongbored piles, the concrete modulus GPa, and the water surface 2 m below the ground surface. The water inside the pit was pumped, and the water outside the pit was not pumped. As shown in Table 4 and Figure 34, the main soil layers are composed of permeable sandy silt. The soil layer 24 m below the ground surface is composed of impermeable muddy silty clay. Seepage cannot occur in muddy silty clay, so the influence of seepage can be disregarded.

The average effective gravity of sandy silt is kN/m^{3}. According to the measured results of the stress path tests shown in Table 1 and (14), the initial tangent modulusfor the lateral unloading soil outside the pit is MPa, and the initial tangent modulusfor the lateral unloading soil outside the pit is MPa. The values of the proportional coefficientfor the horizontal foundation stiffness coefficients, disregarding the influence of load level, are shown in Figure 35.
5.2. Application of the Calculation Method and Model
The calculation method and model, which are equivalent to the calculation methods and models used in the example analysis, are shown in Table 3 and Figures 21 through 24. The parameters are also similar to the parameters in the analysis example, with the exception that the proportional coefficientin the code method is 3,000 kN/m^{4}.
5.3. Analysis Results
As Figures 36 and 37 show, the results for the retaining structure deformation and earth pressure outside the pit are as follows.(1)The results obtained using the method proposed in this paper and using the continuum elasticplasticmedium finite element method are similar to the measured results, which indicates that the method and the model proposed in this paper can be used to accurately calculate the forces and deformations of the retaining structures of an ultradeep foundation pit.(2)The influences of the stress path, the size of the foundation pit, and the stress level on the horizontal foundation stiffness coefficient are not considered in the code method. The maximum horizontal displacements calculated using the code method ranged from 32 mm to 35 mm. The maximum horizontal displacement calculated using the method proposed in this paper was 48 mm, which was similar to the maximum horizontal displacement of 49 mm calculated using the continuum elasticplasticmedium finite element method. The measured values ranged from 46 mm to 51 mm.(3)The results obtained for earth pressure using the method proposed in this paper were similar to the results obtained with the continuum elasticplasticmedium finite element method and were similar to the measured results.
6. Conclusions
New loaddeformation model and method for analysis of retaining structures in ultradeep foundation pits are proposed in this paper. The horizontal foundation stiffness coefficient for this model can be expressed as. The coefficientis related to the size of the foundation pit, Poisson’s ratio, the stress path and the stress level. The soil modulusis also related to the stress path, and the stress level.(1)The value of the coefficientdecreases as the foundation pit width or zone of influence increases. The restraint applied to the surface of the soil is the smallest restraint; thus, coefficienthas the smallest value at the surface. When the soil depth is half of the depth of the foundation pit width, the coefficientapproaches a constant value. The value of the coefficientdecreases as Poisson’s ratio increases.(2)The proportional coefficientof the horizontal foundation stiffness coefficient reflects the effect of the stress path, which can be expressed as. The proportional coefficientat the ground surface exhibits the smallest influence. When the soil depth is half of the foundation pit width, the proportional coefficientapproaches a constant value. In general, the value ofof the soil inside the pit is significantly larger than the value ofof the soil outside the pit.(3)When the load levelincreases, the secant modulus of the soil and the value of the coefficientdecrease. When the strength parameterincreases, the soil secant modulus and the value of the coefficientdecrease.(4)Taking the influence of seepage into consideration, the horizontal foundation stiffness coefficientof the soil inside the pit can be expressed as, and the horizontal foundation stiffness coefficientof the soil outside the pit can be expressed as. Seepage will cause the value of the horizontal foundation stiffness coefficientof the soil inside the pit to decrease and the coefficientof the soil outside the pit to increase.(5)The results obtained for the example analysis and case study presented indicate that the model and method proposed in this paper yield results similar to measured results and similar to results obtained using a continuum elasticplasticmedium finite element model. The good agreement among the three types of results indicates that the method and model proposed in this paper are capable of accurately calculating the forces and deformations of retaining structures in an ultradeep foundation pit.
Acknowledgment
The author would like to acknowledge the financial support from the National Natural Science Foundation of China (NSFC Grant no. 51108417).
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Copyright
Copyright © 2013 Qi Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.