# Retaining Structure Force-Deformation Analysis Model for an Ultradeep Foundation Pit

Authors: Qi Hu
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Figure1. 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 pressurePa, 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 forceFi=Ki·δi. 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 Figure2. (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 Figure3 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 modulusEsand shear strengthCucan be expressed by the equationsK=αEs[15] andK=β·Cu[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 Figure4. 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.Analysis model for an ultradeep foundation pit.
(a)
Loading system
(b)
Supporting systemAs Figure5 shows, according to the definition of a Winkler elastic foundation, the force on a soil spring is defined byqi·w·Δh, and the soil spring stiffness is defined byKi·w·Δh. The relationships among the force, the stiffness, and the horizontal displacement of the soil spring are described by the following equations:
(1)(qi·w·Δh)(Ki·w·Δh)=qiKi=uixorKi=qiuix,
whereqidenotes 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 Figure6 shows, the soil applies lateral pressuresPoandPito the retaining structure, and the retaining structure applies lateral pressuresPo′andPi′to the soil. If the relationship between the change in the horizontal strip loadΔqiand the horizontal displacementuixis established, the value of the horizontal foundation stiffness coefficientKi=Δqi/uixfor any depth can be determined.Force-deformation behavior of the retaining structure and the soil of an ultradeep foundation pit.
(a)
(b)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]:
(2)σx=qx2π(1-μ){x2r14+x2+8dz+6d2r24+8dz(d+z)2r26hhhhhhhhhh+1-2μ2[1r12+3r22-4z(d+z)r24]}σz=qx2π(1-μ){(z-d)2r14-d2-z2+6dzr24+8dzx2r26hhhhhhhhhh-1-2μ2[1r12-1r22-4z(d+z)r24]},σx+σz=qx2π(1-μ)×{1r12+5r22-4(z2+x2)r24+2(1-2μ)r22},
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:
(3)σz=∫d1d2q¯x2π(1-μ)×{(z-d)2r14-d2-z2+6dzr24+8dzx2r26hhhhhh-1-2μ2[1r12-1r22-4z(d+z)r24]}dd=q¯2π(1-μ)×{2xzr2212arctand-zx-x(d-z)2r12hhhh-12arctand+zx+2xzd(d+z)r24hhhh+x(d+z)2r22hhhh-1-2μ2(2xzr22arctand-zxhhhhhhhhhhhhhh-arctand+zx+2xzr22)}|d=d1d=d2,σx+σz=∫d1d2q¯x2π(1-μ){1r12+5r22-4(z2+x2)r24+2(1-2μ)r22}dd=q¯2π(1-μ){arctand-zx+5arctand+zxhhhhhhhhhh-2(z2+x2)hhhhhhhhhh×(d+zxr22+1x2arctand+zx)hhhhhhhhhh+2(1-2μ)arctand+zx}|d=d1d=d2,
whereq¯denotes 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 Figures6 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 Figure9. 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
Assumption1 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,ua=(ut+2um+ub)/4, as shown in Figure 10.Assumption2 for the analysis model.
(a)
Flexible load
(b)
Rigid baseAssumption 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/ux, the horizontal deformation of the isotropic plane-strain problem can be calculated from the following equations:
(4)ux=∫εxdxεx=1-μ2Es(σx-μ1-μσz)orux=1-μ2Es(∫σxdx-μ1-μ∫σzdx).Using the integrals of the stress solutions in (3), the displacement in semi-infinite space due to a uniform horizontal strip loadq¯can be determined as follows:
(5)∫σzdx=q¯2π(1-μ)×{-zd(d+z)r22+μxarctand-zx-μxarctand+zx+(1-2μ)(z-d)4×ln(r1d-z)2+(1-2μ)(d-z)4×ln(r2d+z)2zd(d+z)r22}|d=d1d=d2,∫σxdx=q¯2π(1-μ)×{zd(d+z)r22+(1-μ)xarctand-zx+(5x-3μx+2z2x)arctand+zx+(3-2μ)(d-z)4ln(r1d-z)2+d(5-6μ)+z(7-10μ)4×ln(r2d+z)2zd(d+z)r22}|d=d1d=d2.The average displacement within the loading area is expressed as follows:(6)u¯x=(ux,z=d1+2ux,z=(d1+d2)/2+ux,z=d2)4.According to Assumption1, 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:
(7)ux=∫0l/2εxdx-∫l/2lεxdx.Figure 11
Boundary conditions for a strip foundation pit.The horizontal foundation coefficientKiof the soil springs inside the pit can be obtained from the average displacement within the loading area:(8)Ki=qu¯x=Es1-μ2q¯2[(∫0l/2σxdx-∫l/2lσxdx)-μ1-μ(∫0l/2σzdx-∫l/2lσzdx)-]-1.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:
(9)ux=∫010hεxdx.The horizontal foundation coefficientKoof the soil springs outside the pit can be obtained from the average displacement within the loading area:
(10)Ko=qu¯x=Es1-μ2q¯2(∫010hσxdx-μ1-μ∫010hσzdx-)-1.We define(11)αi=11-μ2q¯2[(∫0l/2σxdx-∫l/2lσxdx)-μ1-μ(∫0l/2σzdx-∫l/2lσzdx)-]-1,αo=11-μ2q¯2(∫010hσxdx-μ1-μ∫010hσzdx-)-1.Then,(12)Ki=αiEs,Ko=αoEs,
whereαdenotes the coefficient of the horizontal foundation stiffness coefficientK, which is related to the foundation pit space size and Poisson’s ratio (m−1), andEsdenotes the elastic modulus of the soil.


## 2.1. Basic Assumptions of the Analysis Model

A comparison of Figures6 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 Figure9. 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
Assumption1 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,ua=(ut+2um+ub)/4, as shown in Figure 10.Assumption2 for the analysis model.
(a)
Flexible load
(b)
Rigid baseAssumption 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/ux, the horizontal deformation of the isotropic plane-strain problem can be calculated from the following equations:
(4)ux=∫εxdxεx=1-μ2Es(σx-μ1-μσz)orux=1-μ2Es(∫σxdx-μ1-μ∫σzdx).Using the integrals of the stress solutions in (3), the displacement in semi-infinite space due to a uniform horizontal strip loadq¯can be determined as follows:
(5)∫σzdx=q¯2π(1-μ)×{-zd(d+z)r22+μxarctand-zx-μxarctand+zx+(1-2μ)(z-d)4×ln(r1d-z)2+(1-2μ)(d-z)4×ln(r2d+z)2zd(d+z)r22}|d=d1d=d2,∫σxdx=q¯2π(1-μ)×{zd(d+z)r22+(1-μ)xarctand-zx+(5x-3μx+2z2x)arctand+zx+(3-2μ)(d-z)4ln(r1d-z)2+d(5-6μ)+z(7-10μ)4×ln(r2d+z)2zd(d+z)r22}|d=d1d=d2.The average displacement within the loading area is expressed as follows:(6)u¯x=(ux,z=d1+2ux,z=(d1+d2)/2+ux,z=d2)4.According to Assumption1, 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:
(7)ux=∫0l/2εxdx-∫l/2lεxdx.Figure 11
Boundary conditions for a strip foundation pit.The horizontal foundation coefficientKiof the soil springs inside the pit can be obtained from the average displacement within the loading area:(8)Ki=qu¯x=Es1-μ2q¯2[(∫0l/2σxdx-∫l/2lσxdx)-μ1-μ(∫0l/2σzdx-∫l/2lσzdx)-]-1.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:
(9)ux=∫010hεxdx.The horizontal foundation coefficientKoof the soil springs outside the pit can be obtained from the average displacement within the loading area:
(10)Ko=qu¯x=Es1-μ2q¯2(∫010hσxdx-μ1-μ∫010hσzdx-)-1.We define(11)αi=11-μ2q¯2[(∫0l/2σxdx-∫l/2lσxdx)-μ1-μ(∫0l/2σzdx-∫l/2lσzdx)-]-1,αo=11-μ2q¯2(∫010hσxdx-μ1-μ∫010hσzdx-)-1.Then,(12)Ki=αiEs,Ko=αoEs,
whereαdenotes the coefficient of the horizontal foundation stiffness coefficientK, which is related to the foundation pit space size and Poisson’s ratio (m−1), andEsdenotes the elastic modulus of the soil.


## 3. Parametric Analysis

As shown in (8) and (10), the horizontal foundation stiffness coefficient can be described asK=αEs. 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 coefficientαiof 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 Figures12 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 depthziis half of the foundation pit widthl, the coefficientαapproaches a constant value.


### 3.2. Influence of Poisson’s Ratio

As shown in Figure14, the coefficientαidecreases when Poisson’s ratioμincreases. When the foundation pit is 20 meters wide,αi,μ=0.1≈1.1αi,μ=0.3andαi,μ=0.5≈0.9αi,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]
(13)Es=λ·σ′=λ·γ′·z,
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:(14)K=α·Es=α·λ·γ′·z=m·z.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 modulusEi, according to stress path tests on Hangzhou sandy silt, is indicated in the results shown in Table 1.Table 1
Stress path coefficientλ of initial tangent modulus (Wei, 2006 [20]; Hu, 2008 [19]).
Soil layer
Stress path
Initial tangent modulusEi
Stress path coefficientλ
2-3
Vertical compression
Vertical tangent modulus
158
2-3
Lateral unloading
Lateral tangent modulus
83
2-3
Vertical unloading
469
2-6
Vertical unloading
511The 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 coefficientm of Hangzhou sandy silt.
Location
Soil inside pit
Soil outside pit
Foundation pit width L/m
10
20
40
100
Horizontal influence scope L/m
100
200
Ground surface
m
i/kN m−4
1,689
1,248
990
778
m
o/kN m−4
100
89
Deep inside
4,230
3,170
2,598
1,879
186
150Figure 15
Proportional coefficientmiof sandy silt inside the pit.Figure 16
Proportional coefficientmoof sandy silt outside the pit.As the values in Table2 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:
(15)Eq=Ei·(1-σ1-σ3(σ1-σ3)ult)=Ei·(1-(σ1-σ3)·b),
whereb=(σ1-σ3)ult-1.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:(16)K=qu¯x=q(∫εxdx¯),εx=1-μ2Eq(σx-μ1-μσz)=1-μ2Ei·[1-(σx-σz)·b](σx-μ1-μσz).As in the linear elastic model, the initial tangent modulus in (16) is unrelated to the stress levelsσxandσz, 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:
(17)b=4.6-0.0043σz′,
wherebhas units of MPa−1 andσz′denotes 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:(18)Ki=αi·Es=αi·λ·σz′=αi·λ·(γs′-γw·i-i)·zi,
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:(19)Ko=αo·Es=αo·λ·σz′=αo·λ·(γs′+γw·i-o)·zo,
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:(20)Po,s=(γs′+γwi-o)·K0·zoearthpressure,Po,w=γw(1-i-o)·zowaterpressure,Po=(γs′+γwi-o)·K0·zo+γw(1-i-o)·zolateralpressure,
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:(21)Pi,s=(γs′-γwi-i)·K0·ziearthpressure,Pi,w=γw(1+i-i)·ziwaterpressure,Pi=(γs′-γwi-i)·K0·zi+γw(1+i-i)·zilateralpressure.


## 3.1. Influence of Space Size of the Foundation Pit

The coefficientαiof 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 Figures12 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 depthziis half of the foundation pit widthl, the coefficientαapproaches a constant value.


## 3.2. Influence of Poisson’s Ratio

As shown in Figure14, the coefficientαidecreases when Poisson’s ratioμincreases. When the foundation pit is 20 meters wide,αi,μ=0.1≈1.1αi,μ=0.3andαi,μ=0.5≈0.9αi,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]
(13)Es=λ·σ′=λ·γ′·z,
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:(14)K=α·Es=α·λ·γ′·z=m·z.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 modulusEi, according to stress path tests on Hangzhou sandy silt, is indicated in the results shown in Table 1.Table 1
Stress path coefficientλ of initial tangent modulus (Wei, 2006 [20]; Hu, 2008 [19]).
Soil layer
Stress path
Initial tangent modulusEi
Stress path coefficientλ
2-3
Vertical compression
Vertical tangent modulus
158
2-3
Lateral unloading
Lateral tangent modulus
83
2-3
Vertical unloading
469
2-6
Vertical unloading
511The 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 coefficientm of Hangzhou sandy silt.
Location
Soil inside pit
Soil outside pit
Foundation pit width L/m
10
20
40
100
Horizontal influence scope L/m
100
200
Ground surface
m
i/kN m−4
1,689
1,248
990
778
m
o/kN m−4
100
89
Deep inside
4,230
3,170
2,598
1,879
186
150Figure 15
Proportional coefficientmiof sandy silt inside the pit.Figure 16
Proportional coefficientmoof sandy silt outside the pit.As the values in Table2 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:
(15)Eq=Ei·(1-σ1-σ3(σ1-σ3)ult)=Ei·(1-(σ1-σ3)·b),
whereb=(σ1-σ3)ult-1.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:(16)K=qu¯x=q(∫εxdx¯),εx=1-μ2Eq(σx-μ1-μσz)=1-μ2Ei·[1-(σx-σz)·b](σx-μ1-μσz).As in the linear elastic model, the initial tangent modulus in (16) is unrelated to the stress levelsσxandσz, 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:
(17)b=4.6-0.0043σz′,
wherebhas units of MPa−1 andσz′denotes 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:(18)Ki=αi·Es=αi·λ·σz′=αi·λ·(γs′-γw·i-i)·zi,
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:(19)Ko=αo·Es=αo·λ·σz′=αo·λ·(γs′+γw·i-o)·zo,
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:(20)Po,s=(γs′+γwi-o)·K0·zoearthpressure,Po,w=γw(1-i-o)·zowaterpressure,Po=(γs′+γwi-o)·K0·zo+γw(1-i-o)·zolateralpressure,
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:(21)Pi,s=(γs′-γwi-i)·K0·ziearthpressure,Pi,w=γw(1+i-i)·ziwaterpressure,Pi=(γs′-γwi-i)·K0·zi+γw(1+i-i)·zilateralpressure.


## 4. Analysis Example

The parameters for the analysis example, illustrated in Figure21, are as follows: an excavation depth of 20 m, five excavation steps, horizontal supporting structures consisting of 5 layers ofØ609×16@5000steel pipes, a stiffness ofEA/D/l=1400/lMN/m2, retaining structures consisting ofØ1000@1500bored piles with embedded depths of 20 m, a concrete modulusEc=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.Foundation pit layout for the analysis example.
(a)
Plane graph
(b)
Sectional graphThe foundation soil consists of sandy silt with the following characteristics: saturated gravityγs=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.
Methods
Code method [1]
Method proposed in this study
Continuum medium finiteelement method
Parameters
Horizontal foundation stiffness coefficient
Earth pressure
Horizontal foundation coefficient
Earth pressure
Soil modulus
Model
Proportional coefficient of horizontal foundation stiffness coefficientm = 4,000
Active earth pressure coefficientKa=0.33
Considering the influence of space size, stress path, load level, and seepage
Static earth pressure coefficient considering the influence of seepageKo=0.5
Initial tangent modulusEi considering the influence of stress path
Mohr-Coulomb elastic-plastic modelFigure 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

Figures25 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 pitPo,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 pitKidecreased, 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 coefficientKiinside 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 coefficientKiinside the pit and the water pressurePo,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. Figures29, 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.


## 4.1. Analysis Results for the Retaining Structures

Figures25 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 pitPo,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 pitKidecreased, 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 coefficientKiinside 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 coefficientKiinside the pit and the water pressurePo,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. Figures29, 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 Figure33 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 ofØ609×16@4000steel pipes with stiffnesses ofEA/D/l=1750/lMN/m2, retaining structures consisting of 30 m-longØ1000@1500bored piles, the concrete modulusEc=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.
Layer number
Soil name
γ
s/kN m−3
Void ratioe
Shear strength parameters
Permeability coefficient/*10−4m s−1
c/kPa
φ/°
K
V
K
H
2-1
Sandy silt
18.9
0.853
7.6
28.5
3.39
2.41
2-3
Sandy silt
19.2
0.788
5.6
31.2
2.39
2.04
2-4
Sandy silt
19.0
0.858
6.1
30.8
2.42
1.82
2-5
Sandy silt with sand
19.3
0.772
4.7
31.5
2.65
3.00
2-6
Sandy silt
18.7
0.916
7.9
29.3
0.13
2-7
Sandy silt with sand
19.3
0.775
5.5
31.2
1.20
2.98
5
Muddy silty clay
18.3
1.067
19.3
11.7
(10−5 to 10−6 cm/s)
6-2
Silty clay
19.2
0.866
50.6
15.7
6-3
Silty clay with silt
20.0
0.69
34.9
19.6
8-1
Sand
18.7
0.829
1.6
32.9Figure 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 modulusEi,ifor the lateral unloading soil outside the pit is0.73zi MPa, and the initial tangent modulusEi,ofor the lateral unloading soil outside the pit is4.28zo 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 Table3 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 Figures36 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.


## 5.1. Overview

As Figure33 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 ofØ609×16@4000steel pipes with stiffnesses ofEA/D/l=1750/lMN/m2, retaining structures consisting of 30 m-longØ1000@1500bored piles, the concrete modulusEc=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.
Layer number
Soil name
γ
s/kN m−3
Void ratioe
Shear strength parameters
Permeability coefficient/*10−4m s−1
c/kPa
φ/°
K
V
K
H
2-1
Sandy silt
18.9
0.853
7.6
28.5
3.39
2.41
2-3
Sandy silt
19.2
0.788
5.6
31.2
2.39
2.04
2-4
Sandy silt
19.0
0.858
6.1
30.8
2.42
1.82
2-5
Sandy silt with sand
19.3
0.772
4.7
31.5
2.65
3.00
2-6
Sandy silt
18.7
0.916
7.9
29.3
0.13
2-7
Sandy silt with sand
19.3
0.775
5.5
31.2
1.20
2.98
5
Muddy silty clay
18.3
1.067
19.3
11.7
(10−5 to 10−6 cm/s)
6-2
Silty clay
19.2
0.866
50.6
15.7
6-3
Silty clay with silt
20.0
0.69
34.9
19.6
8-1
Sand
18.7
0.829
1.6
32.9Figure 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 modulusEi,ifor the lateral unloading soil outside the pit is0.73zi MPa, and the initial tangent modulusEi,ofor the lateral unloading soil outside the pit is4.28zo 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 Table3 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 Figures36 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=αEs. The coefficientαis related to the size of the foundation pit, Poisson’s ratio, the stress path and the stress level. The soil modulusEsis 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 coefficientKiof the soil inside the pit can be expressed asα·λ·(γs′-γw·i-i)·zi, and the horizontal foundation stiffness coefficientKoof the soil outside the pit can be expressed asα·λ·(γs′+γw·i-o)·zo. Seepage will cause the value of the horizontal foundation stiffness coefficientKiof the soil inside the pit to decrease and the coefficientKoof 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.



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*Source: 549491-2013-12-01.xml*