# Analytical Solution for Stress and Displacement after X-Section Cast-in-Place Pile Installation

Authors: Hang Zhou; Han Long Liu; Gangqiang Kong; Zhaohu Cao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

X-section cast-in-place (referred to as XCC) pile, which is one of new pile types developed by Hohai University, is widely used for pile foundation and pile-supported embankment over soft ground in China. However, little research has been carried out on this new type pile, especially the surrounding soil disturbance under XCC pile installation. This paper presents an analytical solution for estimating the horizontal stress and displacement of surrounding soil of XCC pile after XCC pile installation. The reliability and accuracy of the present solution are verified by comparing them with the field test results. Then, parametric studies, such as outsourcing diameter(a), open arc distance (b), open arc angle (θ), the undrained strength (Su), the limit pile cavity pressure (Plim), and the radius of the plastic zone (Rp), are discussed for the practice engineering design. The results show that the stress and displacement distributions of surrounding soil calculated by this paper are in agreement with field test results.


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## Body

## 1. Introduction

Driven cast-in-place pile, which belongs to displacement pile type, is widely used in China [1]. Surrounding environment will be influenced by displacement pile installation obviously. If handed improperly, it will cause the uplift or subsidence of the ground and even engineering accidents. Therefore, it is essential to predict the horizontal stress and displacement induced by the pile installation. Various approaches have been used to study the horizontal stress and displacement including cavity expansion method (CEM) [2–7], strain path method (SPM) [8], and modified SPM (SSPM) [9, 10]. CEM was proposed by Bishop et al. firstly and was used to solve the metal indentation problems. Subsequently, CEM was applied to solve the geotechnical problems such as the pile penetration and the bearing capacity of deep foundation (Vesic (1972)) [2]. In this method, a cylindrical (or spherical) cavity of zero radius was assumed in soil located near the tip of pile. The pressure around the tip of a pile to cause penetration is the limit pressure required to expand the cavity from an initial radius to the radius of the pile. The limit pressure for the expansion of the cavity is a function of the shear strength and compressibility of the soil. Based on the fluid mechanics, the strain path method (SPM) was proposed by Baligh (1985) [8] and modified by Sagaseta et al. (1997) [9]. The process of the pile penetration is assumed as a steady flow of soil around the pile rather than an expansion of a cavity in soil. Although the CEM and SPM are simple and easy to use, they can only solve the axisymmetric or spherical symmetric problem. However, for the special shaped pile installation, such as XCC pile, rectangular cross-section pile (barrette), they are unavailable.As a new immersed tube pile, XCC pile is developed by Hohai University, China [11]. The pile is formed by installation of the X cross-section steel mold which is protected by valve pile shoe or precast pile tip. The installation procedure includes immersing the tube, pouring concrete, vibratory extubation, and curing the concrete. XCC pile is one of new displacement piles, and widely used in practice engineering. However, the theoretical research are far behind the application, especially the horizontal stress and displacement induced by pile installation. In this paper, an analytical solution is provided to study the horizontal stress and displacement distribution of surrounding soil. Then, based on the Fourth Yangtze River Bridge’s north-line soft soil treatment engineering in Nanjing, the analytical solution was compared with the field test results. Finally, the geometric parameters of XCC pile (outsourcing diameter a, open arc distance b, and open arc angle θ), the undrained strengthSu, and the pile hole pressurePwere discussed.


## 2. Mathematical Model

### 2.1. Definition of the Problem and Basic Assumption

Figure1 shows that the elastoplastic soil, which is described by the Tresca model, is under initial stress σ0 before the XCC pile installation. Then, the XCC pile installation progress is simplified as the expansion of a cavity from zero to the X-shaped cavity of the XCC pile cross-section. The soil around the pile enter into the passive limit balance state after the XCC pile installation. Thus, the X-shaped cavity internal pressure after XCC pile installation can be assumed asPlim=σ0tan2(45°+φ/2)+2ctan(45°+φ/2), where c and φ are the soil cohesion and the internal friction angle, respectively. Additionally, the plain strain condition is assumed in the model. The Cartesian coordinates system is selected for the analysis. The origin of the coordinates is located at the center of the cavity. For the Cartesian coordinates system, thex-axis andy-axis are in the horizontal and vertical direction, respectively. The stress and strain are taken as positive in the positive direction of the coordinates system.Mechanics model: (a) passive limit balance state after the pile mold installation; (b) XCC pile cross-section.
(a)
(b)


### 2.2. Basic Governing Equations

According to the elasticity [12], the stress around the X-shaped cavity should obey the fourth-order partial differential equation as follows:
(1)∇4U=0,
where∇is Laplace operator and U is stress function in the plastic zone.The three stress components in Cartesian coordinates system,σx,σy and τxy, around the X-shaped cavity can be determined by (1):
(2)σx=∂2U∂y2,σy=∂2U∂x2,τxy=-∂2U∂x∂y.According to the complex variable elasticity [13–17], the stress around the X-shaped cavity can be expressed with two stress functions φ1(z) andψ1(z)as follows:(3a)σx+σy=4Re[φ1′(z)],(3b)σy-σx+2iτxy=2[z-φ1′′(z)+ψ1′(z)],(3c)E1+μ(u+iv)=(3-4μ)φ1(z)-zφ1′(z-)-ψ1(z-),whereφ1′(z) and φ1′′(z)are the first and second derivative of the function φ1(z), respectively.ψ1′(z) is the first derivative of the function ψ1(z). φ1′(z-) and ψ1(z-)are the conjugate complex functions of theφ1′(z)and ψ1(z). u and v are the displacement component acting inx-axis andy-axis directions, respectively.Eis the Young’s modulus of the soil.μis the Poisson ratio of the soil.For obtaining the solution to calculate the stress and displacement distributions, a conformal mapping function is provided to map the outside of the X-shaped cavity in thez-plane onto the outside of the unit circle in the phase plane, namely, ξ-plane (ξ=ζ+iη=ρeiθ) in Figure 2. The conformal mapping function can be expressed in a series as follows:
(4)z=w(ξ)=c0ξ+∑k=1nc2k-1ξ1-2k,|ξ|≤1,
where n is integral number (in this paper, n=7is selected for analysis and it can give enough accuracy). The constant coefficients c0 and c2k-1(k=1,2,…,n) are real numbers and can be obtained by the iterative technique [18]. z=x+iy(x and y are the variables in the Cartesian coordinates system, i=-1) is complex variable. ξ is complex variable in the phase plane, ξ=ρeiθ.(a)z-plane containing X-shaped cavity subjected uniform pressurePat the cavity and isotropic initial stress at infinity; (b) transformed ξ-plane containing a unit circle subjected uniform pressure at the cavity and isotropic initial stress at infinity.
(a)
(b)Substituting the conformal mapping function (4) into (3a), (3b), and (3c) leads to the following equations:(5a)σx+σy=4Re[φ′(ξ)w′(ξ)],(5b)σy-σx+2iτxy=2w′(ξ)[w(ξ)-(φ′(ξ)w′(ξ))′+ψ′(ξ)],(5c)E1+μ(u+iv)=χφ(ξ)-w(ξ)w′(ξ)φ′(ξ-)-ψ(ξ-),whereφ(ξ)=φ1(w(ξ)),ψ(ξ)=ψ1(w(ξ)).The two stress functionsφ1(z) and ψ1(z) are transformed as φ(ξ) and ψ(ξ). To solve the stress functions φ(ξ) and ψ(ξ), the stress boundary conditions should be considered. From Figure 2, it can be seen that the stress boundary condition can be expressed as
(6)[φ(ξ)+w(ξ)w′(ξ-)φ′(ξ-)+ψ′(ξ)]s=-pw(ξ),
where s is the X-shaped cavity boundary curve and P is the pressure at the X-shaped cavity.By the complex elasticity [10, 11], the stress functionsφ(ξ)andψ(ξ)can be written as follows:(7a)φ(ξ)=18π(1-μ)(F-x+iF-y)lnξ+Bw(ξ)+φ0(ξ),(7b)ψ(ξ)=-3-4μ8π(1-μ)(F-x-iF-y)lnξ+(B′+iC′)w(ξ)+ψ0(ξ),whereF-xandF-yare the composite surface force inxandydirection on the X-shaped cavity boundary, respectively. One has B=(σ1+σ2)/4(σ1and σ2are the principal stress at infinity) and B′+C′=-(1/2)(σ1-σ2)e-2iα (α is the angle between the principal stressσ1andox-axis).From the mechanics model in Figure2, the following equations are established:(8a)F-x=F-y=0,(8b)B=σ0,(8c)B′=C′=0.Additionally, the two stress functionsφ0(ξ) andψ0(ξ) can be expressed as the series:(9a)φ0(ξ)=∑k=0nα2k-1ξ2k-1,(9b)ψ0(ξ)=∑k=1nβ2k-1ξ2k-1,where α2k-1 and β2k-1 are the coefficients of the complex functions φ0(ξ) and ψ0(ξ). They can be determined by the boundary conditions.Thus (7a) and (7b) can be simplified by substituting (8a), (8b), (8c), (9a), and (9b) into (7a) and (7b) as(10a)φ(ξ)=σ0w(ξ)+∑k=0nα2k-1ξ2k-1,(10b)ψ(ξ)=∑k=1nβ2k-1ξ2k-1.Then, (10a) and (10b) are substituted into the stress boundary condition (6), and (6) can be transformed as follows:
(11)φ0(σ)+w(σ)w′(σ)-φ0′(σ-)+ψ0(σ-)=-(P+σ0)z,
where,σ=eiθ.Equations (10a) and (10b) are conjugated at both sides:
(12)φ0(σ-)+w(σ-)w′(σ)φ0′(σ)+ψ0(σ)=-(P+σ0)z-.Equations (10a), (10b), and (11) are multiplied by(1/2πi)(dσ/(σ-ξ))and integrated along the cavity boundarysat both sides:(13a)12πi∫φ0(σ)dσ+12πi∫w(σ)w′(σ-)φ0′(σ-)dσ+12πi∫ψ0(σ-)dσ=12πi∫-(P+σ0)w(σ)dσ,(13b)12πi∫φ0(σ-)dσ+12πi∫w(σ-)w′(σ)φ0′(σ)dσ+12πi∫ψ0(σ)dσ=12πi∫-(P+σ0)w(σ-)dσ.According to the principle of the series expansion, the terms ofw(σ)/w′(σ-)can be expressed as follows:
(14)w(σ)w′(σ-)=b2k-3σ2k-3+b2k-5σ2k-5+⋯+b1σ1+b-1σ-1+O(1σ3).Equation (4) is substituted into (14) and can be written in matrix form as follows:
(15)AB=C,
where
(16)A=[-c-0c-13c-35c-5⋯⋯(2n-3)c-2n-3-c-0c-13c-3⋯⋯(2n-5)c-2n-5-c-0c-1⋯⋯(2n-7)c-2n-7⋱⋯⋯⋮⋱⋯⋮-c-0c-1-c-0],B=[b-1b1b3b5⋯b2n-3]T,C=[c1c3c5c7⋯c2n-1]T.After the coefficientsb-1 and b2k-3(k=1,2,3…,n) are determined by solving (15), substituting the expression ofw(σ)/w′(σ-)into (13a), the equation for calculating the coefficients of the stress functions can be obtained as follows:
(17)[EnMM-En]α=N,
where(18a)M=[b13b3⋯(2n-3)b2n-30b33b5⋯00⋮⋯⋯00b2n-30⋯0000⋯00],(18b)α=[α1α3⋯α2n-1α-1α-3⋯α-2n-1]T,(18c)N=-(P+σ0)[c1c3⋯c2n-1c-1c-3⋯c-2n-1]T,M-is the conjugate matrix of M, andEnis n-dimension unit matrix.Similarly, the coefficientβ2k-1of the stress functionψ0(ξ)can be calculated likeα2k-1. Thus, the stress functionsφ0(ξ) andψ0(ξ)are completely determined, and the horizontal stress change and displacement are obtained by solving (5a), (5b), and (5c).


### 2.3. Elastoplastic Boundary (EP Boundary)

Normally, the surrounding soil will enter into plastic stage after XCC pile installation and lead to a formation of a plastic zone around the X-shaped cavity wall. Therefore an elastoplastic analysis is necessary. The nonaxisymmetric problem in the original plane can be transformed into axisymmetric problem in the phase plane by the conformal mapping technique. Thus, it can be easily processed in the phase plane for the axisymmetric characteristics. Considering an element at a radial distanceρ from the center of the cavity, the equation of equilibrium in the phase plane can be expressed as follows:
(19)∂Δσρ∂ρ+Δσρ-Δσθρ=0,
whereΔσρ andΔσθare radial and circumference stress increment, respectively andρis the radial position of the soil particle.Note that the Tresca yield criteria has the following form(20)Δσρ-Δσθ=2Su,
whereSuis the undrained strength of the soil.The stress boundary conditions in the phase plane are(21a)Δσρ=Patρ=1,(21b)Δσρ=0atρ⟶∞.Combining (19), (20), and the stress boundary conditions, the stress in the plastic zone can be obtained:(22a)Δσρp=-2Sulnρ-P,(22b)Δσθp=-2Su(lnρ+1)-P.In the elastic zone, the stress can be written as(23a)Δσρe=λPρ2,(23b)Δσθe=λPρ2,whereλis the stress redistribution coefficient in the elastic zone.At the EP boundary, the stress in the elastic zone should also obey the Tresca yield criteria. Thus, the stress redistribution coefficient in the elastic zone can be expressed as follows by substituting (23a) and (23b) into (22a) and (22b):(24a)σρ=Suρb2ρ2,(24b)σθ=-Suρb2ρ2,whereρbis the radius of the plastic zone in the phase plane.At the EP boundary, the stress in the plastic zone should be equal to that in the elastic zone. Therefore, combining (22a), (22b) and (24a), (24b) the relationship of the pressure-plastic zone radius can be expressed as follows:
(25)ρb=e1+P/Su/2.Substituting the limit cavity pressurePliminto (25), the radius of the plastic zone in ξ-plane after the XCC pile installation can be obtained as follows:
(26)ρb=e1+Plim/Su/2.The radius of the plastic zone in the physical plane can be obtained by combining (4) and (25):
(27)Rb(θ)=|w(ρbeiθ)|,
where the plastic zoneRb(θ)is the function of the polar angle.According to the above analysis, EP boundary is circle curve with radius equal toρbin the phase plane. The real EP boundary in the physical plane is not circular curve and it can be calculated by (27). However, the EP boundary in the physical plane is closed to circular curve far away from the X-shaped cavity from (27). Therefore, the radius of the plastic zone can be assumed as follows:
(28)Rb=max(|w(ρbeiθ)|),
whereRbis the maximum radius of the plastic zone in the physical plane.


### 2.4. The Horizontal Stress and Displacement Solutions in the Elastic Zone

After the soil around the cavity wall enters yield state, the stress in the elastic zone has a redistribution effect and the stress in the elastic zone cannot be calculated by the elastic analysis directly. However, the stress redistribution effect can be considered by introducing a coefficientλ into the elastic analysis. In other words, the new stress functions λφ(ξ) and λψ(ξ) instead ofφ(ξ) and ψ(ξ)are introduced into the governing equations. Thus the governing equations (5a), (5b), and (5c) of the elastic zone can be expressed as(29a)σx+σy=4Re[λφ′(ξ)w′(ξ)],(29b)σy-σx+2iτxy=2w′(ξ)[λw(ξ-)(φ′(ξ)w′(ξ))′+λψ′(ξ)],(29c)2G(ux+iuy)=(3-4μ)λφ(ξ)-λw(ξ)w′(ξ-)φ′(ξ-)-λψ(ξ-).Under the undrained condition, the volume change of the X-shaped cavity induced by the XCC pile installation is equivalent to the change in position of the EP boundary. The mathematical relation can be expressed as follows:(30)Ax=πRb2-π(Rb-ub)2,
where Ax is the area of X-shaped cavity, Rb is the radius of the plastic zone, and up is the radial displacement at the EP boundary.Theub2 is higher order driblet and can be ignored and thus (28) can be expressed as
(31)ub=Ax2πRb.The stress redistribution factorλ can be obtained by solving the coupled equations (28), (29c), and (31); then the new stress functions λφ(ξ) and λψ(ξ) can be obtained. Substituting the new stress functions into the governing equation of the elastic zone (29a), (29b), and (29c), the horizontal stress and displacement in the elastic zone can be determined.


## 2.1. Definition of the Problem and Basic Assumption

Figure1 shows that the elastoplastic soil, which is described by the Tresca model, is under initial stress σ0 before the XCC pile installation. Then, the XCC pile installation progress is simplified as the expansion of a cavity from zero to the X-shaped cavity of the XCC pile cross-section. The soil around the pile enter into the passive limit balance state after the XCC pile installation. Thus, the X-shaped cavity internal pressure after XCC pile installation can be assumed asPlim=σ0tan2(45°+φ/2)+2ctan(45°+φ/2), where c and φ are the soil cohesion and the internal friction angle, respectively. Additionally, the plain strain condition is assumed in the model. The Cartesian coordinates system is selected for the analysis. The origin of the coordinates is located at the center of the cavity. For the Cartesian coordinates system, thex-axis andy-axis are in the horizontal and vertical direction, respectively. The stress and strain are taken as positive in the positive direction of the coordinates system.Mechanics model: (a) passive limit balance state after the pile mold installation; (b) XCC pile cross-section.
(a)
(b)


## 2.2. Basic Governing Equations

According to the elasticity [12], the stress around the X-shaped cavity should obey the fourth-order partial differential equation as follows:
(1)∇4U=0,
where∇is Laplace operator and U is stress function in the plastic zone.The three stress components in Cartesian coordinates system,σx,σy and τxy, around the X-shaped cavity can be determined by (1):
(2)σx=∂2U∂y2,σy=∂2U∂x2,τxy=-∂2U∂x∂y.According to the complex variable elasticity [13–17], the stress around the X-shaped cavity can be expressed with two stress functions φ1(z) andψ1(z)as follows:(3a)σx+σy=4Re[φ1′(z)],(3b)σy-σx+2iτxy=2[z-φ1′′(z)+ψ1′(z)],(3c)E1+μ(u+iv)=(3-4μ)φ1(z)-zφ1′(z-)-ψ1(z-),whereφ1′(z) and φ1′′(z)are the first and second derivative of the function φ1(z), respectively.ψ1′(z) is the first derivative of the function ψ1(z). φ1′(z-) and ψ1(z-)are the conjugate complex functions of theφ1′(z)and ψ1(z). u and v are the displacement component acting inx-axis andy-axis directions, respectively.Eis the Young’s modulus of the soil.μis the Poisson ratio of the soil.For obtaining the solution to calculate the stress and displacement distributions, a conformal mapping function is provided to map the outside of the X-shaped cavity in thez-plane onto the outside of the unit circle in the phase plane, namely, ξ-plane (ξ=ζ+iη=ρeiθ) in Figure 2. The conformal mapping function can be expressed in a series as follows:
(4)z=w(ξ)=c0ξ+∑k=1nc2k-1ξ1-2k,|ξ|≤1,
where n is integral number (in this paper, n=7is selected for analysis and it can give enough accuracy). The constant coefficients c0 and c2k-1(k=1,2,…,n) are real numbers and can be obtained by the iterative technique [18]. z=x+iy(x and y are the variables in the Cartesian coordinates system, i=-1) is complex variable. ξ is complex variable in the phase plane, ξ=ρeiθ.(a)z-plane containing X-shaped cavity subjected uniform pressurePat the cavity and isotropic initial stress at infinity; (b) transformed ξ-plane containing a unit circle subjected uniform pressure at the cavity and isotropic initial stress at infinity.
(a)
(b)Substituting the conformal mapping function (4) into (3a), (3b), and (3c) leads to the following equations:(5a)σx+σy=4Re[φ′(ξ)w′(ξ)],(5b)σy-σx+2iτxy=2w′(ξ)[w(ξ)-(φ′(ξ)w′(ξ))′+ψ′(ξ)],(5c)E1+μ(u+iv)=χφ(ξ)-w(ξ)w′(ξ)φ′(ξ-)-ψ(ξ-),whereφ(ξ)=φ1(w(ξ)),ψ(ξ)=ψ1(w(ξ)).The two stress functionsφ1(z) and ψ1(z) are transformed as φ(ξ) and ψ(ξ). To solve the stress functions φ(ξ) and ψ(ξ), the stress boundary conditions should be considered. From Figure 2, it can be seen that the stress boundary condition can be expressed as
(6)[φ(ξ)+w(ξ)w′(ξ-)φ′(ξ-)+ψ′(ξ)]s=-pw(ξ),
where s is the X-shaped cavity boundary curve and P is the pressure at the X-shaped cavity.By the complex elasticity [10, 11], the stress functionsφ(ξ)andψ(ξ)can be written as follows:(7a)φ(ξ)=18π(1-μ)(F-x+iF-y)lnξ+Bw(ξ)+φ0(ξ),(7b)ψ(ξ)=-3-4μ8π(1-μ)(F-x-iF-y)lnξ+(B′+iC′)w(ξ)+ψ0(ξ),whereF-xandF-yare the composite surface force inxandydirection on the X-shaped cavity boundary, respectively. One has B=(σ1+σ2)/4(σ1and σ2are the principal stress at infinity) and B′+C′=-(1/2)(σ1-σ2)e-2iα (α is the angle between the principal stressσ1andox-axis).From the mechanics model in Figure2, the following equations are established:(8a)F-x=F-y=0,(8b)B=σ0,(8c)B′=C′=0.Additionally, the two stress functionsφ0(ξ) andψ0(ξ) can be expressed as the series:(9a)φ0(ξ)=∑k=0nα2k-1ξ2k-1,(9b)ψ0(ξ)=∑k=1nβ2k-1ξ2k-1,where α2k-1 and β2k-1 are the coefficients of the complex functions φ0(ξ) and ψ0(ξ). They can be determined by the boundary conditions.Thus (7a) and (7b) can be simplified by substituting (8a), (8b), (8c), (9a), and (9b) into (7a) and (7b) as(10a)φ(ξ)=σ0w(ξ)+∑k=0nα2k-1ξ2k-1,(10b)ψ(ξ)=∑k=1nβ2k-1ξ2k-1.Then, (10a) and (10b) are substituted into the stress boundary condition (6), and (6) can be transformed as follows:
(11)φ0(σ)+w(σ)w′(σ)-φ0′(σ-)+ψ0(σ-)=-(P+σ0)z,
where,σ=eiθ.Equations (10a) and (10b) are conjugated at both sides:
(12)φ0(σ-)+w(σ-)w′(σ)φ0′(σ)+ψ0(σ)=-(P+σ0)z-.Equations (10a), (10b), and (11) are multiplied by(1/2πi)(dσ/(σ-ξ))and integrated along the cavity boundarysat both sides:(13a)12πi∫φ0(σ)dσ+12πi∫w(σ)w′(σ-)φ0′(σ-)dσ+12πi∫ψ0(σ-)dσ=12πi∫-(P+σ0)w(σ)dσ,(13b)12πi∫φ0(σ-)dσ+12πi∫w(σ-)w′(σ)φ0′(σ)dσ+12πi∫ψ0(σ)dσ=12πi∫-(P+σ0)w(σ-)dσ.According to the principle of the series expansion, the terms ofw(σ)/w′(σ-)can be expressed as follows:
(14)w(σ)w′(σ-)=b2k-3σ2k-3+b2k-5σ2k-5+⋯+b1σ1+b-1σ-1+O(1σ3).Equation (4) is substituted into (14) and can be written in matrix form as follows:
(15)AB=C,
where
(16)A=[-c-0c-13c-35c-5⋯⋯(2n-3)c-2n-3-c-0c-13c-3⋯⋯(2n-5)c-2n-5-c-0c-1⋯⋯(2n-7)c-2n-7⋱⋯⋯⋮⋱⋯⋮-c-0c-1-c-0],B=[b-1b1b3b5⋯b2n-3]T,C=[c1c3c5c7⋯c2n-1]T.After the coefficientsb-1 and b2k-3(k=1,2,3…,n) are determined by solving (15), substituting the expression ofw(σ)/w′(σ-)into (13a), the equation for calculating the coefficients of the stress functions can be obtained as follows:
(17)[EnMM-En]α=N,
where(18a)M=[b13b3⋯(2n-3)b2n-30b33b5⋯00⋮⋯⋯00b2n-30⋯0000⋯00],(18b)α=[α1α3⋯α2n-1α-1α-3⋯α-2n-1]T,(18c)N=-(P+σ0)[c1c3⋯c2n-1c-1c-3⋯c-2n-1]T,M-is the conjugate matrix of M, andEnis n-dimension unit matrix.Similarly, the coefficientβ2k-1of the stress functionψ0(ξ)can be calculated likeα2k-1. Thus, the stress functionsφ0(ξ) andψ0(ξ)are completely determined, and the horizontal stress change and displacement are obtained by solving (5a), (5b), and (5c).


## 2.3. Elastoplastic Boundary (EP Boundary)

Normally, the surrounding soil will enter into plastic stage after XCC pile installation and lead to a formation of a plastic zone around the X-shaped cavity wall. Therefore an elastoplastic analysis is necessary. The nonaxisymmetric problem in the original plane can be transformed into axisymmetric problem in the phase plane by the conformal mapping technique. Thus, it can be easily processed in the phase plane for the axisymmetric characteristics. Considering an element at a radial distanceρ from the center of the cavity, the equation of equilibrium in the phase plane can be expressed as follows:
(19)∂Δσρ∂ρ+Δσρ-Δσθρ=0,
whereΔσρ andΔσθare radial and circumference stress increment, respectively andρis the radial position of the soil particle.Note that the Tresca yield criteria has the following form(20)Δσρ-Δσθ=2Su,
whereSuis the undrained strength of the soil.The stress boundary conditions in the phase plane are(21a)Δσρ=Patρ=1,(21b)Δσρ=0atρ⟶∞.Combining (19), (20), and the stress boundary conditions, the stress in the plastic zone can be obtained:(22a)Δσρp=-2Sulnρ-P,(22b)Δσθp=-2Su(lnρ+1)-P.In the elastic zone, the stress can be written as(23a)Δσρe=λPρ2,(23b)Δσθe=λPρ2,whereλis the stress redistribution coefficient in the elastic zone.At the EP boundary, the stress in the elastic zone should also obey the Tresca yield criteria. Thus, the stress redistribution coefficient in the elastic zone can be expressed as follows by substituting (23a) and (23b) into (22a) and (22b):(24a)σρ=Suρb2ρ2,(24b)σθ=-Suρb2ρ2,whereρbis the radius of the plastic zone in the phase plane.At the EP boundary, the stress in the plastic zone should be equal to that in the elastic zone. Therefore, combining (22a), (22b) and (24a), (24b) the relationship of the pressure-plastic zone radius can be expressed as follows:
(25)ρb=e1+P/Su/2.Substituting the limit cavity pressurePliminto (25), the radius of the plastic zone in ξ-plane after the XCC pile installation can be obtained as follows:
(26)ρb=e1+Plim/Su/2.The radius of the plastic zone in the physical plane can be obtained by combining (4) and (25):
(27)Rb(θ)=|w(ρbeiθ)|,
where the plastic zoneRb(θ)is the function of the polar angle.According to the above analysis, EP boundary is circle curve with radius equal toρbin the phase plane. The real EP boundary in the physical plane is not circular curve and it can be calculated by (27). However, the EP boundary in the physical plane is closed to circular curve far away from the X-shaped cavity from (27). Therefore, the radius of the plastic zone can be assumed as follows:
(28)Rb=max(|w(ρbeiθ)|),
whereRbis the maximum radius of the plastic zone in the physical plane.


## 2.4. The Horizontal Stress and Displacement Solutions in the Elastic Zone

After the soil around the cavity wall enters yield state, the stress in the elastic zone has a redistribution effect and the stress in the elastic zone cannot be calculated by the elastic analysis directly. However, the stress redistribution effect can be considered by introducing a coefficientλ into the elastic analysis. In other words, the new stress functions λφ(ξ) and λψ(ξ) instead ofφ(ξ) and ψ(ξ)are introduced into the governing equations. Thus the governing equations (5a), (5b), and (5c) of the elastic zone can be expressed as(29a)σx+σy=4Re[λφ′(ξ)w′(ξ)],(29b)σy-σx+2iτxy=2w′(ξ)[λw(ξ-)(φ′(ξ)w′(ξ))′+λψ′(ξ)],(29c)2G(ux+iuy)=(3-4μ)λφ(ξ)-λw(ξ)w′(ξ-)φ′(ξ-)-λψ(ξ-).Under the undrained condition, the volume change of the X-shaped cavity induced by the XCC pile installation is equivalent to the change in position of the EP boundary. The mathematical relation can be expressed as follows:(30)Ax=πRb2-π(Rb-ub)2,
where Ax is the area of X-shaped cavity, Rb is the radius of the plastic zone, and up is the radial displacement at the EP boundary.Theub2 is higher order driblet and can be ignored and thus (28) can be expressed as
(31)ub=Ax2πRb.The stress redistribution factorλ can be obtained by solving the coupled equations (28), (29c), and (31); then the new stress functions λφ(ξ) and λψ(ξ) can be obtained. Substituting the new stress functions into the governing equation of the elastic zone (29a), (29b), and (29c), the horizontal stress and displacement in the elastic zone can be determined.


## 3. Verification

### 3.1. Engineering Description

The Fourth Yangtze River Bridge’s north-line soft soil treatment field is located in Nanjing, China. The total length of the soft ground improvement engineering is 29.0 km. Physical-mechanical properties of soils on site are shown in Table1. The form of plum-shaped layout is carried out in the engineering. The pile spacing and length are 2.2 m and 12 m, respectively. The three parameters of the XCC pile cross-section the outsourcing diameter (parameter a), the open arc distance (parameter b), and the open arc angle (parameter θ) are 611 mm, 120 mm, and 130°, respectively (see Figure 3).Table 1
Physical-mechanical properties of soils on site.
Soil name
h (m)
w (%)
γ (kN/m3)
υ
E
s (MPa)
e
c (kPa)
φ (°)
Silt clay
2.00
30.5
18.50
0.3
5.49
0.913
26.4
15.1
Silt clay
4.60
41.4
17.90
0.3
2.97
1.159
10.8
3.4
Silt soil
3.40
30.3
18.60
0.3
11.68
0.897
12.7
26.4
Silt clay
3.70
41.4
17.90
0.3
2.97
1.159
10.8
3.4
Silt soil
2.00
30.3
18.60
0.3
11.68
0.897
12.7
26.4
Silt clay
1.60
32.9
18.80
0.3
4.41
0.915
26.4
16.1
Silt soil
0.30
30.3
18.60
0.3
11.68
0.897
12.7
26.4
Notes:h: the thickness of the soil layer; w: the moisture content; γ: the bulk density; υ: the Poisson ratio; Es: the compression modulus; e: the void ratio;c: the cohesion; φ: the internal friction angle.Figure 3
Geometry of XCC pile cross-section.The arrangement of the test equipment and measuring points are shown in Figure4. The location of the test instrument is concluded as follows. (1) Inclinometer tubes were buried at the distance from the XCC pile center: 1 m, 2 m, and 3.5 m, respectively. (2) Pore water pressure gauges were buried at the depth of 6 m and 9 m, and the distances from the pile center equal 1 m, 2 m, and 3.5 m, respectively. (3) Earth pressure cells were buried at the depth of 3 m and 6 m, and the distances from the pile center equal 1 m, 2 m, and 3.5 m, respectively.Figure 4
Equipment arrangement and measuring points on site.


### 3.2. Comparison on the Theoretical Calculated Results with Field Test Results

The radial stress and displacement at the depth of 3 meters are selected for comparison, which are shown in Figure5. The radial stress and displacement are plotted against the normalized radius, r/R, where the variableris the radial position andRis the radius of the outsourcing round of XCC pile cross-section. The stress and displacement of soil around the XCC pile calculated by this study are similar to those of the measured results on site. Therefore, this study can simulate the stress and displacement induced by XCC pile installation well. Additionally, the stress and radial displacement decrease rapidly with the distance from the pile center. The radius of the influence zone induced by the XCC pile installation is about 12 R.Comparison between this study and measured data: (a) radial displacement, (b) radial stress.
(a)
(b)Figure5 also gives the comparison between the XCC pile, the circular pile A (the same area with XCC pile cross-section), and the circular pile B (outsourcing round of XCC pile cross-section). The results show that the displacements for circular pile B, which is calculated by cylindrical cavity expansion method (CEM), are larger than those of the circular pile A and XCC pile. Additionally, the displacement of circular pile A and that of XCC pile are almost the same. In other words, the area of the pile cross-section governs the displacement induced by the pile installation. Therefore, it is reasonable to calculate displacement caused by XCC pile installation with circular pile A instead of XCC pile. As shown in Figure 5(b), the stress of circular pile B is also larger than those of the circular pile A and XCC pile. However, the stress of circular pile A and XCC pile is different which means the stress is related to the pile cross-section. Thus, it is not accurate to calculate the stress or excess pore pressure with circular pile A instead of XCC pile. When it refers to the stress or excess pore pressure, this study should be used.


## 3.1. Engineering Description

The Fourth Yangtze River Bridge’s north-line soft soil treatment field is located in Nanjing, China. The total length of the soft ground improvement engineering is 29.0 km. Physical-mechanical properties of soils on site are shown in Table1. The form of plum-shaped layout is carried out in the engineering. The pile spacing and length are 2.2 m and 12 m, respectively. The three parameters of the XCC pile cross-section the outsourcing diameter (parameter a), the open arc distance (parameter b), and the open arc angle (parameter θ) are 611 mm, 120 mm, and 130°, respectively (see Figure 3).Table 1
Physical-mechanical properties of soils on site.
Soil name
h (m)
w (%)
γ (kN/m3)
υ
E
s (MPa)
e
c (kPa)
φ (°)
Silt clay
2.00
30.5
18.50
0.3
5.49
0.913
26.4
15.1
Silt clay
4.60
41.4
17.90
0.3
2.97
1.159
10.8
3.4
Silt soil
3.40
30.3
18.60
0.3
11.68
0.897
12.7
26.4
Silt clay
3.70
41.4
17.90
0.3
2.97
1.159
10.8
3.4
Silt soil
2.00
30.3
18.60
0.3
11.68
0.897
12.7
26.4
Silt clay
1.60
32.9
18.80
0.3
4.41
0.915
26.4
16.1
Silt soil
0.30
30.3
18.60
0.3
11.68
0.897
12.7
26.4
Notes:h: the thickness of the soil layer; w: the moisture content; γ: the bulk density; υ: the Poisson ratio; Es: the compression modulus; e: the void ratio;c: the cohesion; φ: the internal friction angle.Figure 3
Geometry of XCC pile cross-section.The arrangement of the test equipment and measuring points are shown in Figure4. The location of the test instrument is concluded as follows. (1) Inclinometer tubes were buried at the distance from the XCC pile center: 1 m, 2 m, and 3.5 m, respectively. (2) Pore water pressure gauges were buried at the depth of 6 m and 9 m, and the distances from the pile center equal 1 m, 2 m, and 3.5 m, respectively. (3) Earth pressure cells were buried at the depth of 3 m and 6 m, and the distances from the pile center equal 1 m, 2 m, and 3.5 m, respectively.Figure 4
Equipment arrangement and measuring points on site.


## 3.2. Comparison on the Theoretical Calculated Results with Field Test Results

The radial stress and displacement at the depth of 3 meters are selected for comparison, which are shown in Figure5. The radial stress and displacement are plotted against the normalized radius, r/R, where the variableris the radial position andRis the radius of the outsourcing round of XCC pile cross-section. The stress and displacement of soil around the XCC pile calculated by this study are similar to those of the measured results on site. Therefore, this study can simulate the stress and displacement induced by XCC pile installation well. Additionally, the stress and radial displacement decrease rapidly with the distance from the pile center. The radius of the influence zone induced by the XCC pile installation is about 12 R.Comparison between this study and measured data: (a) radial displacement, (b) radial stress.
(a)
(b)Figure5 also gives the comparison between the XCC pile, the circular pile A (the same area with XCC pile cross-section), and the circular pile B (outsourcing round of XCC pile cross-section). The results show that the displacements for circular pile B, which is calculated by cylindrical cavity expansion method (CEM), are larger than those of the circular pile A and XCC pile. Additionally, the displacement of circular pile A and that of XCC pile are almost the same. In other words, the area of the pile cross-section governs the displacement induced by the pile installation. Therefore, it is reasonable to calculate displacement caused by XCC pile installation with circular pile A instead of XCC pile. As shown in Figure 5(b), the stress of circular pile B is also larger than those of the circular pile A and XCC pile. However, the stress of circular pile A and XCC pile is different which means the stress is related to the pile cross-section. Thus, it is not accurate to calculate the stress or excess pore pressure with circular pile A instead of XCC pile. When it refers to the stress or excess pore pressure, this study should be used.


## 4. Parametric Studies

In order to provide engineers and researchers with calculation charts and tables for estimating horizontal stress, displacements, and the radius of the plastic zone induced by XCC pile installation, a parametric study is carried out. The stress, displacement, and the radius of the plastic zone have many influence factors. This paper focus on the factors of outsourcing diameter, open arc distance, open arc angle, the pile hole pressureP, and the undrained strengthSu. The stress, displacement alongx-axis in the elastic zone, and the radius of the plastic zone are analyzed. The Young’s modulus of the soil is selected for 5 MPa and the Poisson’s ratio is 0.3. The influence characteristics of the parameters on the stress, displacement of the soil around the pile, and the radius of the plastic zone are obtained by the parametric study.


### 4.1. The Radius of the Plastic Zone Analysis

From (26) and (28), the radius of the plastic zone is the function of the ratio of the limit pressurePlim, the undrained strengthPlim/Su, and the parameters of the XCC pile cross-section (the outsourcing diameter, the open arc angle, and the open distance). Thus, the four parameters are selected for the parametric studies.The plastic zone radiusRpis plotted against the variable ofPlim/Suwith different parameters of the XCC pile cross-section in Figure 6. As shown in Figures 6(a), 6(b), and 6(c), the higherPlim/Sudevelops the larger plastic zone radiusRp. From Figure 6(a), it can be seen that the outsourcing diameteraincreases with the increasing plastic zone radius Rp, provided that all other factors are held constants. With the variable ofarange from 500 mm to 1000 mm, it is found that the increasing amplitude of theRpincreases with the increasingPlim/Su, provided that the variables ofband θ are constant. Figure 6(b) shows that the open arc distancebhas similar characteristics as the outsourcing diametera. Figure 6(c) shows that the open arc angle θ reduces with the increasing plastic zone radius Rp. However, the plastic zone radius Rp is not sensitive to the open arc distanceband open arc angle θ. In all, the outsourcing diameterais the most obvious influence parameter of the radius of the plastic zone among the three geometric parameters of XCC pile cross-section.Variation of the plastic radiusRp with different geometric parameters (a,b,θ) and Plim/Su: (a) a = 500 mm to 1000 mm, b = 120 mm, θ=90°; (b) a = 600 mm, b = 120 mm to 200 mm, θ = 90°; (c) a = 600 mm, b = 120 mm, θ = 80° to 130°.
(a)
(b)
(c)


### 4.2. Stress and Displacement Distribution Analysis

Based on the three geometric parameters of the XCC pile cross-section and the undrained strengthSu, the influence characteristics of the stress changes and displacement in the elastic zone are obtained. The limit pressurePlimis assumed to be 10 kPa.As shown in Figure7, with the outsourcing diameter range from 330 mm to 730 mm, the radial displacement increases with the increasing outsourcing diameterain the elastic zone, provided that all other factors are held constant. However, the stresses do not almost change. It can be concluded that the outsourcing diameter has little influence on the horizontal stress in the elastic zone. It can be seen that the radius of stress influence zone is about 17 Rp, which is less than that of the stress influence zone (more than 17 Rp) by comparing Figure 7(a) with Figure 7(b). From Figures 8 and 9, it can be observed that both of the open arc distance and open arc angle have little influence on the horizontal stress and displacement.Radial stress and displacement distribution of different outsourcing diameteraalong the radial direction (b = 110 mm, θ = 90°,Su= 10 kPa): (a) radial stress, (b) radial displacement.
(a)
(b)Radial stress and displacement distribution of different open arc anglebalong the radial direction (a = 530 mm, θ = 90°,Su= 10 kPa): (a) radial stress, (b) radial displacement.
(a)
(b)Radial stress and displacement distribution of different open arc angleθ along the radial direction (a = 530 mm, b = 110 mm,Su= 10 kPa): (a) radial stress, (b) radial displacement.
(a)
(b)Figure10(a) shows that the stress increased with the increasing of undrained strength Su in the elastic zone. From Figure 10(b), it can be seen that the larger the undrained strength Su is, the larger the displacement will be. It is because the radius of the plastic zone Rp reduces with the increasing Su. However, the volume of the plastic zone is constant under Tresca condition, and the volume change induced by the XCC pile installation can only be manifested in the elastic zone. Thus, the volume change in the elastic zone increases with the reducing Rp and the displacement in the elastic zone will increase.Radial stress and displacement distribution of differentSualong the radial direction (a = 530 mm, b = 110 mm, θ = 90°): (a) radial stress, (b) radial displacement.
(a)
(b)


## 4.1. The Radius of the Plastic Zone Analysis

From (26) and (28), the radius of the plastic zone is the function of the ratio of the limit pressurePlim, the undrained strengthPlim/Su, and the parameters of the XCC pile cross-section (the outsourcing diameter, the open arc angle, and the open distance). Thus, the four parameters are selected for the parametric studies.The plastic zone radiusRpis plotted against the variable ofPlim/Suwith different parameters of the XCC pile cross-section in Figure 6. As shown in Figures 6(a), 6(b), and 6(c), the higherPlim/Sudevelops the larger plastic zone radiusRp. From Figure 6(a), it can be seen that the outsourcing diameteraincreases with the increasing plastic zone radius Rp, provided that all other factors are held constants. With the variable ofarange from 500 mm to 1000 mm, it is found that the increasing amplitude of theRpincreases with the increasingPlim/Su, provided that the variables ofband θ are constant. Figure 6(b) shows that the open arc distancebhas similar characteristics as the outsourcing diametera. Figure 6(c) shows that the open arc angle θ reduces with the increasing plastic zone radius Rp. However, the plastic zone radius Rp is not sensitive to the open arc distanceband open arc angle θ. In all, the outsourcing diameterais the most obvious influence parameter of the radius of the plastic zone among the three geometric parameters of XCC pile cross-section.Variation of the plastic radiusRp with different geometric parameters (a,b,θ) and Plim/Su: (a) a = 500 mm to 1000 mm, b = 120 mm, θ=90°; (b) a = 600 mm, b = 120 mm to 200 mm, θ = 90°; (c) a = 600 mm, b = 120 mm, θ = 80° to 130°.
(a)
(b)
(c)


## 4.2. Stress and Displacement Distribution Analysis

Based on the three geometric parameters of the XCC pile cross-section and the undrained strengthSu, the influence characteristics of the stress changes and displacement in the elastic zone are obtained. The limit pressurePlimis assumed to be 10 kPa.As shown in Figure7, with the outsourcing diameter range from 330 mm to 730 mm, the radial displacement increases with the increasing outsourcing diameterain the elastic zone, provided that all other factors are held constant. However, the stresses do not almost change. It can be concluded that the outsourcing diameter has little influence on the horizontal stress in the elastic zone. It can be seen that the radius of stress influence zone is about 17 Rp, which is less than that of the stress influence zone (more than 17 Rp) by comparing Figure 7(a) with Figure 7(b). From Figures 8 and 9, it can be observed that both of the open arc distance and open arc angle have little influence on the horizontal stress and displacement.Radial stress and displacement distribution of different outsourcing diameteraalong the radial direction (b = 110 mm, θ = 90°,Su= 10 kPa): (a) radial stress, (b) radial displacement.
(a)
(b)Radial stress and displacement distribution of different open arc anglebalong the radial direction (a = 530 mm, θ = 90°,Su= 10 kPa): (a) radial stress, (b) radial displacement.
(a)
(b)Radial stress and displacement distribution of different open arc angleθ along the radial direction (a = 530 mm, b = 110 mm,Su= 10 kPa): (a) radial stress, (b) radial displacement.
(a)
(b)Figure10(a) shows that the stress increased with the increasing of undrained strength Su in the elastic zone. From Figure 10(b), it can be seen that the larger the undrained strength Su is, the larger the displacement will be. It is because the radius of the plastic zone Rp reduces with the increasing Su. However, the volume of the plastic zone is constant under Tresca condition, and the volume change induced by the XCC pile installation can only be manifested in the elastic zone. Thus, the volume change in the elastic zone increases with the reducing Rp and the displacement in the elastic zone will increase.Radial stress and displacement distribution of differentSualong the radial direction (a = 530 mm, b = 110 mm, θ = 90°): (a) radial stress, (b) radial displacement.
(a)
(b)


## 5. Conclusions

An analytical solution considering the pile cross-section shape for the horizontal stress and displacement of the soil around the XCC pile after installation is presented in this study. An elastoplastic model for calculating the horizontal stress and displacement is established by complex variables. Some main results can be concluded as follows.(1)
Compared with the data of the field test, it can be seen that the elastoplastic model calculation results on the horizontal stress and displacement of the soil around the XCC pile after installation are in agreement with those of field results. A theoretical method for studying the special shaped piles installation is provided in this paper.(2)
The radius of the plastic zone caused by the XCC pile installation can be calculated conveniently by this study. TheRpincreased with the increasing ofPlim/Su, outsourcing diametera, and open arc distancebwhile it decreases with the increasing of open arc distance θ. The outsourcing diameterais the most obvious influence parameter of the radius of the plastic zone among the three geometric parameters (a, b, and θ) of XCC pile cross-section.(3)
The radial displacement increases with the increasing of outsourcing diameterain the elastic zone, and the outsourcing diameterahas little influence on the horizontal stress in the elastic zone. The stress and displacement increased with the increasing of undrained strengthSuobviously in the elastic zone. Both of the open arc distanceband the open arc angle θ have little influence on the horizontal stress and displacement. The extent of the displacement influence zone is larger than that of the stress influence zone.



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*Source: 937421-2013-12-09.xml*