Retaining Structure Force-Deformation 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 load-deformation analysis model of a retaining structure based on a bearing-mode analysis of an ultra-deep 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 ultra-deep 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 elastic-plastic finite element analysis and similar to the measured results. The results of this study indicate that the proposed analysis model adequately reflects the force-deformation characteristics of an ultra-deep 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 pressure𝑃_𝑎, 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 force𝐹_𝑖=𝐾_𝑖⋅𝛿_𝑖. The influence of seepage on the water-earth 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.

Figure 1 

Schematic of current design approach for a foundation pit.

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 water-earth 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 width𝑙and excavation depthℎ. (4) In areas with high groundwater levels, the influence of seepage on the water-earth 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.

Figure 2 

Schematic of an ultradeep foundation pit system.

Many researchers have studied these problems. As Figure 3 shows, the displacement-dependent 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 coefficient𝐾and the soil modulus𝐸_𝑠and shear strength𝐶_𝑢can be expressed by the equations𝐾=𝛼𝐸_𝑠[15] and𝐾=𝛽⋅𝐶_𝑢[16], respectively.

Figure 3 

Relationship between earth pressure and deformation.

Although these three types of methods have yielded useful results, none of them reflects the force-deformation 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 force-deformation 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 force-deformation 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 coefficient𝐾is 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

Figure 4 

Analysis model for an ultradeep foundation pit.

As Figure 5 shows, according to the definition of a Winkler elastic foundation, the force on a soil spring is defined by𝑞_𝑖⋅𝑤⋅Δℎ, and the soil spring stiffness is defined by𝐾_𝑖⋅𝑤⋅Δℎ. The relationships among the force, the stiffness, and the horizontal displacement of the soil spring are described by the following equations: 𝑞_𝑖⋅𝑤⋅Δℎ𝐾_𝑖=𝑞⋅𝑤⋅Δℎ𝐾^𝑖_𝑖=𝑢_𝑖𝑥or𝐾_𝑖=𝑞_𝑖𝑢_𝑖𝑥, (1) where𝑞_𝑖denotes the horizontal strip load (in units of pressure),𝑤denotes the calculated horizontal width, andΔℎdenotes the calculated thickness.

Figure 5 

Illustration of the concept of the horizontal foundation stiffness coefficient.

As Figure 6 shows, the soil applies lateral pressures𝑃_𝑜and𝑃_𝑖to the retaining structure, and the retaining structure applies lateral pressures𝑃^′_𝑜and𝑃^′_𝑖to the soil. If the relationship between the change in the horizontal strip loadΔ𝑞_𝑖and the horizontal displacement𝑢_𝑖𝑥is established, the value of the horizontal foundation stiffness coefficient𝐾_𝑖=Δ𝑞_𝑖/𝑢_𝑖𝑥for any depth can be determined.

(a)

(b)

Figure 6 

Force-deformation behavior of the retaining structure and the soil of an ultradeep foundation pit.

The solutions for the components of stress at any point in a semi-infinite elastic space due to a linear horizontal load𝑞applied at a depth𝑑(as illustrated in Figure 7) were proposed by Melan [17]. The solutions for stress at any point in a semi-infinite elastic space solution due to a uniform horizontal strip load𝑞(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 force-deformation pattern of an ultradeep foundation pit.

Figure 7 

Schematic diagram of a linear horizontal load.

Figure 8 

Schematic diagram of a horizontal uniform strip load.

As mentioned above, the solutions for the components of stress at any point in a semi-infinite elastic space due to a linear horizontal load𝑞applied at a depth𝑑(as illustrated in Figure 7) were proposed by Melan [17]: 𝜎_𝑥=𝑞𝑥𝑥2𝜋1−𝜇𝑟²⁴₁+𝑥²+8𝑑𝑧+6𝑑²𝑟⁴₂+8𝑑𝑧(𝑑+𝑧)²𝑟⁶₂+1−2𝜇21𝑟²₁+3𝑟²₂−4𝑧(𝑑+𝑧)𝑟⁴₂𝜎_𝑧=𝑞𝑥()2𝜋1−𝜇𝑧−𝑑²𝑟⁴₁−𝑑²−𝑧²+6𝑑𝑧𝑟⁴₂+8𝑑𝑧𝑥²𝑟⁶₂−1−2𝜇21𝑟²₁−1𝑟²₂−4𝑧(𝑑+𝑧)𝑟⁴₂,𝜎_𝑥+𝜎_𝑧=𝑞𝑥×12𝜋1−𝜇𝑟²₁+5𝑟²₂−4𝑧²+𝑥²𝑟⁴₂+21−2𝜇𝑟²₂, (2) where𝑞denotes the linear load,𝜇denotes Poisson’s ratio,𝑑denotes the depth of the linear load,𝑥denotes the horizontal coordinate of a point in the semi-infinite space, and𝑧denotes the vertical coordinate of a point in the semi-infinite space.

The solutions for the components of stress at any point in a semi-infinite elastic space due to a uniform horizontal strip load𝑞(as illustrated in Figure 8) can be obtained by integrating Melan’s solutions: 𝜎_𝑧=^𝑑2_𝑑1𝑞𝑥×()2𝜋1−𝜇𝑧−𝑑²𝑟⁴₁−𝑑²−𝑧²+6𝑑𝑧𝑟⁴₂+8𝑑𝑧𝑥²𝑟⁶₂−1−2𝜇21𝑟²₁−1𝑟²₂−4𝑧(𝑑+𝑧)𝑟⁴₂=d𝑑𝑞×12𝜋1−𝜇2arctan𝑑−𝑧𝑥−𝑥(𝑑−𝑧)2𝑟²₁−12arctan𝑑+𝑧𝑥+2𝑥𝑧𝑑(𝑑+𝑧)𝑟⁴₂+𝑥(𝑑+𝑧)2𝑟²₂−1−2𝜇2arctan𝑑−𝑧𝑥−arctan𝑑+𝑧𝑥+2𝑥𝑧𝑟²₂^𝑑=𝑑2_𝑑=𝑑1,𝜎_𝑥+𝜎_𝑧=^𝑑2_𝑑1𝑞𝑥12𝜋1−𝜇𝑟²₁+5𝑟²₂−4𝑧²+𝑥²𝑟⁴₂+21−2𝜇𝑟²₂=d𝑑𝑞2𝜋1−𝜇arctan𝑑−𝑧𝑥+5arctan𝑑+𝑧𝑥−2𝑧²+𝑥²×𝑑+𝑧𝑥𝑟²₂+1𝑥²arctan𝑑+𝑧𝑥+21−2𝜇arctan𝑑+𝑧𝑥^𝑑=𝑑2_𝑑=𝑑1, (3) where𝑞denotes the uniform strip load, 𝑑₁denotes the depth of the top of the uniform strip load, and 𝑑₂denotes 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 semi-infinite space. Therefore, certain assumptions can be made.

Assumption 1. The physical model of a strip foundation pit is usually described as a plane-strain elastic problem in a semi-infinite 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 semi-infinite elastic spaces (Figure 9(c)).

Figure 9 

Assumption 1 for the analysis model.

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,𝑢_𝑎=(𝑢_𝑡+2𝑢_𝑚+𝑢_𝑏)/4, as shown in Figure 10.

(a) Flexible load

(b) Rigid base

Figure 10 

Assumption 2 for the analysis model.

Assumption 3. For the purpose of calculating stresses, the soil is considered to be a single-phase, 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 coefficient𝐾=𝑞/𝑢_𝑥, the horizontal deformation of the isotropic plane-strain problem can be calculated from the following equations: 𝑢_𝑥=𝜀_𝑥𝜀d𝑥_𝑥=1−𝜇²𝐸_𝑠𝜎_𝑥−𝜇𝜎1−𝜇^𝑧𝑢or_𝑥=1−𝜇²𝐸_𝑠𝜎_𝑥𝜇d𝑥−𝜎1−𝜇^𝑧d𝑥. (4)

Using the integrals of the stress solutions in (3), the displacement in semi-infinite space due to a uniform horizontal strip load𝑞can be determined as follows: 𝜎_𝑧d𝑥=𝑞×−()2𝜋1−𝜇𝑧𝑑𝑑+𝑧𝑟²₂+𝜇𝑥arctan𝑑−𝑧𝑥−𝜇𝑥arctan𝑑+𝑧𝑥+1−2𝜇(𝑧−𝑑)4𝑟×ln¹𝑑−𝑧²+1−2𝜇(𝑑−𝑧)4𝑟×ln²𝑑+𝑧^2𝑑=𝑑2_𝑑=𝑑1,𝜎_𝑥d𝑥=𝑞×()2𝜋1−𝜇𝑧𝑑𝑑+𝑧𝑟²₂+1−𝜇𝑥arctan𝑑−𝑧𝑥+5𝑥−3𝜇𝑥+2𝑧²𝑥arctan𝑑+𝑧𝑥+3−2𝜇(𝑑−𝑧)4𝑟ln¹𝑑−𝑧²+𝑑5−6𝜇+𝑧7−10𝜇4𝑟×ln²𝑑+𝑧^2𝑑=𝑑2_𝑑=𝑑1. (5)

The average displacement within the loading area is expressed as follows: 𝑢_𝑥=𝑢_{𝑥,𝑧=𝑑1}+2𝑢_{𝑥,𝑧=(𝑑1+𝑑2)/2}+𝑢_{𝑥,𝑧=𝑑2}4. (6)

According to Assumption 1, the real load in a quarter-infinite space is𝑞=𝑞/2, 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: 𝑢_𝑥=₀^𝑙/2𝜀_𝑥𝑑𝑥−^𝑙_𝑙/2𝜀_𝑥𝑑𝑥. (7)

Figure 11 

Boundary conditions for a strip foundation pit.

The horizontal foundation coefficient𝐾_𝑖of the soil springs inside the pit can be obtained from the average displacement within the loading area:𝐾_𝑖=𝑞𝑢_𝑥=𝐸_𝑠1−𝜇²𝑞2₀^𝑙/2𝜎_𝑥d𝑥−^𝑙_𝑙/2𝜎_𝑥−𝜇d𝑥1−𝜇₀^𝑙/2𝜎_𝑧d𝑥−^𝑙_𝑙/2𝜎_𝑧d𝑥⁻¹. (8)

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 load𝑞, 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: 𝑢_𝑥=₀^10ℎ𝜀_𝑥𝑑𝑥. (9)

The horizontal foundation coefficient𝐾_𝑜of the soil springs outside the pit can be obtained from the average displacement within the loading area: 𝐾_𝑜=𝑞𝑢_𝑥=𝐸_𝑠1−𝜇²𝑞2₀^10ℎ𝜎_𝑥𝜇d𝑥−1−𝜇₀^10ℎ𝜎_𝑧d𝑥⁻¹. (10)

We define𝛼_𝑖=11−𝜇²𝑞2₀^𝑙/2𝜎_𝑥d𝑥−^𝑙_𝑙/2𝜎_𝑥−𝜇d𝑥1−𝜇₀^𝑙/2𝜎_𝑧d𝑥−^𝑙_𝑙/2𝜎_𝑧d𝑥⁻¹,𝛼_𝑜=11−𝜇²𝑞2₀^10ℎ𝜎_𝑥𝜇d𝑥−1−𝜇₀^10ℎ𝜎_𝑧d𝑥⁻¹. (11)

Then, 𝐾_𝑖=𝛼_𝑖𝐸_𝑠,𝐾_𝑜=𝛼_𝑜𝐸_𝑠, (12) where𝛼denotes the coefficient of the horizontal foundation stiffness coefficient𝐾, which is related to the foundation pit space size and Poisson’s ratio (m⁻¹), and𝐸_𝑠denotes the elastic modulus of the soil.

3. Parametric Analysis

As shown in (8) and (10), the horizontal foundation stiffness coefficient can be described as𝐾=𝛼𝐸_𝑠. The factors that influence𝐾include 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 coefficient𝛼_𝑖of 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.

Figure 12 

The coefficient𝛼𝑖of the horizontal foundation stiffness coefficient inside the pit.

Figure 13 

The coefficient𝛼𝑜of the horizontal foundation stiffness coefficient outside the pit.

As Figures 12 and 13 show, the value of the coefficient𝛼decreases when the foundation pit width𝑙or the influence zone𝐿increases. Due to the low level of restraint on the surface soil, the value of the coefficient𝛼is also small. When the soil depth𝑧_𝑖is half of the foundation pit width𝑙, the coefficient𝛼approaches a constant value.

3.2. Influence of Poisson’s Ratio

As shown in Figure 14, the coefficient𝛼_𝑖decreases when Poisson’s ratio𝜇increases. When the foundation pit is 20 meters wide,𝛼_{𝑖,𝜇=0.1}≈1.1𝛼_{𝑖,𝜇=0.3}and𝛼_{𝑖,𝜇=0.5}≈0.9𝛼_{𝑖,𝜇=0.3}.

Figure 14 

Influence of Poisson’s ratio on the coefficient𝛼𝑖(foundation pit width𝑙=20 m).

3.3. Influence of Stress Path

The relationship between the soil modulus, the stress path, and the consolidation pressure is [19] 𝐸_𝑠=𝜆⋅𝜎^′=𝜆⋅𝛾^′⋅𝑧, (13) where the coefficient𝜆denotes the influence of the stress path;𝛾^′denotes the soil effective gravity; and𝑧denotes the soil depth.

For the soil outside the pit, the lateral modulus coefficient𝜆means the lateral unloading stress path. For the soil inside the pit, the lateral modulus coefficient𝜆means the vertical unloading stress path.

Taking the influence of the stress path into consideration, the horizontal foundation coefficient can be expressed as follows: 𝐾=𝛼⋅𝐸_𝑠=𝛼⋅𝜆⋅𝛾^′⋅𝑧=𝑚⋅𝑧. (14)

The proportional coefficient𝑚of the horizontal foundation stiffness coefficient can be expressed as𝛼⋅𝜆⋅𝛾^′. The influence of the stress path on the coefficient𝜆of the initial tangent modulus𝐸_𝑖, according to stress path tests on Hangzhou sandy silt, is indicated in the results shown in Table 1.

Table 1 

Stress path coefficient 𝜆 of initial tangent modulus (Wei, 2006 [20]; Hu, 2008 [19]).

The average effective gravity𝛾^′of Hangzhou sandy silt is 9.0 kN/m³ [19]. According to the test results shown in Table 1, the lateral unloading stress path coefficient𝜆is approximately 80, and the vertical unloading stress path coefficient𝜆is approximately 470. The results for the proportional coefficient𝑚of sandy silt inside and outside the foundation pit, when Poisson’s ratio𝜇is 0.3, are shown in Figures 15 and 16 and Table 2.

Table 2 

Proportional coefficient 𝑚 of Hangzhou sandy silt.

Figure 15 

Proportional coefficient𝑚𝑖of sandy silt inside the pit.

Figure 16 

Proportional coefficient𝑚𝑜of sandy silt outside the pit.

As the values in Table 2 show, the variation in the proportional coefficient𝑚in the homogeneous foundation is similar to that of coefficient𝛼. The greater the width of the foundation pit is, the smaller the proportional coefficient𝑚is. The value of the proportional coefficient𝑚is smallest at the ground surface. When the soil depth is half of the foundation pit width, the proportional coefficient𝑚approaches a constant value. In general, the proportional coefficient𝑚of the soil inside the pit is considerably larger than the proportional coefficient𝑚of the soil outside the pit.

3.4. Influence of Load Level

The stress-strain 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: 𝐸_𝑞=𝐸_𝑖⋅𝜎1−¹−𝜎₃𝜎₁−𝜎_3ult=𝐸_𝑖⋅1−𝜎₁−𝜎₃⋅𝑏, (15) where𝑏=(𝜎₁−𝜎₃)⁻¹_ult.

Figure 17 

Nonlinear stress-strain relationship according to the Duncan-Chang model.

Taking the influence of load level into consideration, the horizontal foundation stiffness coefficient can be expressed as follows: 𝑞𝐾=𝑢_𝑥=𝑞∫𝜀_𝑥,𝜀d𝑥_𝑥=1−𝜇²𝐸_𝑞𝜎_𝑥−𝜇𝜎1−𝜇^𝑧=1−𝜇²𝐸_𝑖⋅1−𝜎_𝑥−𝜎_𝑧𝜎⋅𝑏^𝑥−𝜇𝜎1−𝜇^𝑧. (16)

As in the linear elastic model, the initial tangent modulus in (16) is unrelated to the stress levels𝜎_𝑥and𝜎_𝑧, and the effect of load level is only related to the coefficient𝛼. Equation (16) is relatively complicated and must be solved by numerical integration. The influence of the load level𝑞and the strength parameter𝑏on the coefficient𝛼when the foundation pit width is 20 m is shown in Figures 19 and 20. When the load level𝑞increases, the secant modulus of the soil and the coefficient𝛼decrease. When the strength parameter𝑏increases, the soil secant modulus and the coefficient𝛼decrease.

According to the stress path test results for Hangzhou sandy silt, the strength parameter𝑏in the vertical unloading stress-strain curve which is illustrated in Figure 18 can be determined from the following equation: 𝑏=4.6−0.0043𝜎^′_𝑧, (17) where𝑏has units of MPa⁻¹ and𝜎^′_𝑧denotes the vertical consolidation pressure (kPa).

Figure 18 

Relationship between𝑏and vertical consolidation pressure [19].

Figure 19 

Influence of the load level𝑞on the coefficient𝛼𝑖(𝑏=3 MPa−1).

Figure 20 

Influence of the strength parameter𝑏on the coefficient𝛼𝑖(𝑞=100 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 water-soil 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: 𝐾_𝑖=𝛼_𝑖⋅𝐸_𝑠=𝛼_𝑖⋅𝜆⋅𝜎^′_𝑧=𝛼_𝑖⋅𝜆⋅𝛾^′_𝑠−𝛾_𝑤⋅𝑖_𝑖⋅𝑧_𝑖, (18) where𝑖_𝑖denotes 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: 𝐾_𝑜=𝛼_𝑜⋅𝐸_𝑠=𝛼_𝑜⋅𝜆⋅𝜎^′_𝑧=𝛼_𝑜⋅𝜆⋅𝛾^′_𝑠+𝛾_𝑤⋅𝑖_𝑜⋅𝑧_𝑜, (19) where𝑖_𝑜denotes 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: 𝑃_𝑜,𝑠=𝛾^′_𝑠+𝛾_𝑤𝑖_𝑜⋅𝐾₀⋅𝑧_𝑜𝑃earthpressure,_𝑜,𝑤=𝛾_𝑤1−𝑖_𝑜⋅𝑧_𝑜𝑃waterpressure,_𝑜=𝛾^′_𝑠+𝛾_𝑤𝑖_𝑜⋅𝐾₀⋅𝑧_𝑜+𝛾_𝑤1−𝑖_𝑜⋅𝑧_𝑜lateralpressure, (20) where𝐾₀denotes 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: 𝑃_𝑖,𝑠=𝛾^′_𝑠−𝛾_𝑤𝑖_𝑖⋅𝐾₀⋅𝑧_𝑖𝑃earthpressure,_𝑖,𝑤=𝛾_𝑤1+𝑖_𝑖⋅𝑧_𝑖𝑃waterpressure,_𝑖=𝛾^′_𝑠−𝛾_𝑤𝑖_𝑖⋅𝐾₀⋅𝑧_𝑖+𝛾_𝑤1+𝑖_𝑖⋅𝑧_𝑖lateralpressure. (21)

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 of∅609×16@5000steel pipes, a stiffness of𝐸𝐴/𝐷/𝑙=1400/𝑙MN/m², retaining structures consisting of∅1000@1500bored piles with embedded depths of 20 m, a concrete modulus𝐸_𝑐=30 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

Figure 21 

Foundation pit layout for the analysis example.

The foundation soil consists of sandy silt with the following characteristics: saturated gravity𝛾_𝑠=19.0 kN/m³, effective gravity𝛾^′=9.0 kN/m³, Poisson’s ratio𝜇=0.3, shear strength parameters𝑐^′=5 kPa and𝜑^′=30^∘, lateral unloading stress path coefficient𝜆of the initial tangent modulus of 80, and vertical unloading stress path coefficient𝜆of 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.

Table 3 

Parameters according to different methods.

Figure 22 

Code analysis model.

Figure 23 

Analysis model proposed in this study.

Figure 24 

Continuum elastic-plastic-medium finite element model.

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 coefficient𝑚of 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 elastic-plastic-medium finite element model.

Figure 25 

Retaining structure deformation for 10 m wide pit.

Equations (18) to (21) illustrate the following points. First, due to the influence of seepage, the water pressure outside the pit𝑃_𝑜,𝑤decreased, 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 pit𝐾_𝑖decreased, 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 gradient𝑖_𝑖inside the pit is large. The horizontal foundation stiffness coefficient𝐾_𝑖inside 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 gradient𝑖_𝑖inside the pit is similar to the hydraulic gradient𝑖_𝑜outside the pit, and the horizontal foundation stiffness coefficient𝐾_𝑖inside the pit and the water pressure𝑃_𝑜,𝑤outside the pit decrease similarly under the influence of seepage. Thus, the horizontal displacement and the bending moment of the retaining structures vary slightly.

Figure 26 

Retaining structure bending moment for 10 m wide pit.

Figure 27 

Retaining structure deformation for 40 m wide pit.

Figure 28 

Retaining structure bending moment for 40 m wide pit.

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 elastic-plastic-medium 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.

Figure 29 

External earth pressure for 10 m wide pit.

Figure 30 

External earth pressure for 40 m wide pit.

Figure 31 

Earth pressure inside 10 m wide pit.

Figure 32 

Earth pressure inside 40 m wide 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 of∅609×16@4000steel pipes with stiffnesses of𝐸𝐴/𝐷/𝑙=1750/𝑙MN/m², retaining structures consisting of 30 m-long∅1000@1500bored piles, the concrete modulus𝐸_𝑐=30 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.

Table 4 

Physical and mechanical parameters of the soil.

Figure 33 

The ultradeep foundation pit in the Qiutao Road station of Hangzhou Metro Line number 1.

Figure 34 

Soil profile.

The average effective gravity of sandy silt is𝛾^′=9.1 kN/m³. According to the measured results of the stress path tests shown in Table 1 and (14), the initial tangent modulus𝐸_𝑖,𝑖for the lateral unloading soil outside the pit is0.73𝑧_𝑖 MPa, and the initial tangent modulus𝐸_𝑖,𝑜for the lateral unloading soil outside the pit is4.28𝑧_𝑜 MPa. The values of the proportional coefficient𝑚for the horizontal foundation stiffness coefficients, disregarding the influence of load level, are shown in Figure 35.

Figure 35 

Horizontal foundation stiffness coefficients for each of six excavation steps.

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 coefficient𝑚in the code method is 3,000 kN/m⁴.

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 elastic-plastic-medium 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 elastic-plastic-medium 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 elastic-plastic-medium finite element method and were similar to the measured results.

Figure 36 

Deformation in retaining structures.

Figure 37 

Earth pressure outside the pit.

6. Conclusions

New load-deformation 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 as𝐾=𝛼𝐸_𝑠. The coefficient𝛼is related to the size of the foundation pit, Poisson’s ratio, the stress path and the stress level. The soil modulus𝐸_𝑠is also related to the stress path, and the stress level.(1)The value of the coefficient𝛼decreases as the foundation pit width or zone of influence increases. The restraint applied to the surface of the soil is the smallest restraint; thus, coefficient𝛼has the smallest value at the surface. When the soil depth is half of the depth of the foundation pit width, the coefficient𝛼approaches a constant value. The value of the coefficient𝛼decreases as Poisson’s ratio increases.(2)The proportional coefficient𝑚of the horizontal foundation stiffness coefficient reflects the effect of the stress path, which can be expressed as𝛼⋅𝜆⋅𝛾^′. The proportional coefficient𝑚at the ground surface exhibits the smallest influence. When the soil depth is half of the foundation pit width, the proportional coefficient𝑚approaches a constant value. In general, the value of𝑚of the soil inside the pit is significantly larger than the value of𝑚of the soil outside the pit.(3)When the load level𝑞increases, the secant modulus of the soil and the value of the coefficient𝛼decrease. When the strength parameter𝑏increases, the soil secant modulus and the value of the coefficient𝛼decrease.(4)Taking the influence of seepage into consideration, the horizontal foundation stiffness coefficient𝐾_𝑖of the soil inside the pit can be expressed as𝛼⋅𝜆⋅(𝛾^′_𝑠−𝛾_𝑤⋅𝑖_𝑖)⋅𝑧_𝑖, and the horizontal foundation stiffness coefficient𝐾_𝑜of the soil outside the pit can be expressed as𝛼⋅𝜆⋅(𝛾^′_𝑠+𝛾_𝑤⋅𝑖_𝑜)⋅𝑧_𝑜. Seepage will cause the value of the horizontal foundation stiffness coefficient𝐾_𝑖of the soil inside the pit to decrease and the coefficient𝐾_𝑜of 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 elastic-plastic-medium 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).