MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 937421 10.1155/2013/937421 937421 Research Article Analytical Solution for Stress and Displacement after X-Section Cast-in-Place Pile Installation Zhou Hang Liu Han Long Kong Gangqiang Cao Zhaohu Liu Yuji College of Civil and Transportation Engineering Hohai University Nanjing Jiangsu 210098 China hhu.edu.cn 2013 9 12 2013 2013 13 10 2013 01 11 2013 2013 Copyright © 2013 Hang Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Driven cast-in-place pile, which belongs to displacement pile type, is widely used in China . 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) , strain path method (SPM) , 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)) . 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)  and modified by Sagaseta et al. (1997) . 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 . 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 strength  Su, and the pile hole pressure  P  were discussed.

2. Mathematical Model 2.1. Definition of the Problem and Basic Assumption

Figure 1 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 as  Plim=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, the  x-axis and  y-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.

2.2. Basic Governing Equations

According to the elasticity , 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=2Uy2,σy=2Ux2,τxy=-2Uxy.

According to the complex variable elasticity , the stress around the X-shaped cavity can be expressed with two stress functions φ1(z) and  ψ1(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-),where  φ1(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 the  φ1(z)  and ψ1(z). u and v are the displacement component acting in  x-axis and  y-axis directions, respectively.  E  is the Young’s modulus of the soil.  μ  is 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 the z-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=7  is 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 . 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 pressure  P  at 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.

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-),where  φ(ξ)=φ1(w(ξ)),  ψ(ξ)=ψ1(w(ξ)).

The two stress functions φ1(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 functions  φ(ξ)  and  ψ(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),where  F-x  and  F-y  are the composite surface force in  x  and  y  direction on the X-shaped cavity boundary, respectively. One has B=(σ1+σ2)/4  (σ1  and σ2  are the principal stress at infinity) and B+C=-(1/2)(σ1-σ2)e-2iα (α is the angle between the principal stress  σ1  and ox-axis).

From the mechanics model in Figure 2, the following equations are established:(8a)F-x=F-y=0,(8b)B=σ0,(8c)B=C=0.

Additionally, the two stress functions  φ0(ξ) and  ψ0(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 boundary  s  at both sides:(13a)12πiφ0(σ)dσ+12πiw(σ)w(σ-)φ0(σ-)dσ+12πiψ0(σ-)dσ=12πi-(P+σ0)w(σ)dσ,(13b)12πiφ0(σ-)dσ+12πiw(σ-)w(σ)φ0(σ)dσ+12πiψ0(σ)dσ=12πi-(P+σ0)w(σ-)dσ.

According to the principle of the series expansion, the terms of  w(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-1b1b3b5b2n-3]T,C=[c1c3c5c7c2n-1]T.

After the coefficients b-1 and b2k-3  (k=1,2,3,n) are determined by solving (15), substituting the expression of  w(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-30b33b50000b2n-30000000],(18b)α=[α1α3α2n-1α-1α-3α-2n-1]T,(18c)N=-(P+σ0)[c1c3c2n-1c-1c-3c-2n-1]T,M-  is the conjugate matrix of M, and  En  is n-dimension unit matrix.

Similarly, the coefficient  β2k-1  of the stress function  ψ0(ξ)  can be calculated like  α2k-1. Thus, the stress functions  φ0(ξ) and  ψ0(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 distance  ρ from the center of the cavity, the equation of equilibrium in the phase plane can be expressed as follows: (19)Δσρρ+Δσρ-Δσθρ=0, where  Δσρ and  Δσθ  are radial and circumference stress increment, respectively and  ρ  is the radial position of the soil particle.

Note that the Tresca yield criteria has the following form (20)Δσρ-Δσθ=2Su, where  Su  is the undrained strength of the soil.

The stress boundary conditions in the phase plane are(21a)Δσρ=Pat  ρ=1,(21b)Δσρ=0at  ρ.

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,where  λ  is 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,where  ρb  is 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 pressure  Plim  into (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 zone  Rb(θ)  is the function of the polar angle.

According to the above analysis, EP boundary is circle curve with radius equal to ρb  in 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)|), where  Rb  is 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 coefficient λ into the elastic analysis. In other words, the new stress functions λφ(ξ) and λψ(ξ) instead of  φ(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.

The ub2 is higher order driblet and can be ignored and thus (28) can be expressed as (31)ub=Ax2πRb.

The stress redistribution factor λ 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 Table 1. 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).

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.

Geometry of XCC pile cross-section.

The arrangement of the test equipment and measuring points are shown in Figure 4. 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.

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 Figure 5. The radial stress and displacement are plotted against the normalized radius, r/R, where the variable  r  is the radial position and  R  is 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.

Figure 5 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 pressure  P, and the undrained strength  Su. The stress, displacement along  x-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 pressure  Plim, the undrained strength  Plim/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 radius  Rp  is plotted against the variable of  Plim/Su  with different parameters of the XCC pile cross-section in Figure 6. As shown in Figures 6(a), 6(b), and 6(c), the higher  Plim/Su  develops the larger plastic zone radius  Rp. From Figure 6(a), it can be seen that the outsourcing diameter  a  increases with the increasing plastic zone radius Rp, provided that all other factors are held constants. With the variable of  a  range from 500 mm to 1000 mm, it is found that the increasing amplitude of the  Rp  increases with the increasing  Plim/Su, provided that the variables of  b  and θ are constant. Figure 6(b) shows that the open arc distance  b  has similar characteristics as the outsourcing diameter  a. 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 distance  b  and open arc angle θ. In all, the outsourcing diameter  a  is 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 radius Rp 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°.

4.2. Stress and Displacement Distribution Analysis

Based on the three geometric parameters of the XCC pile cross-section and the undrained strength  Su, the influence characteristics of the stress changes and displacement in the elastic zone are obtained. The limit pressure  Plim  is assumed to be 10 kPa.

As shown in Figure 7, with the outsourcing diameter range from 330 mm to 730 mm, the radial displacement increases with the increasing outsourcing diameter  a  in 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 diameter  a  along the radial direction (b = 110 mm, θ = 90°,  Su  = 10 kPa): (a) radial stress, (b) radial displacement.

Radial stress and displacement distribution of different open arc angle  b  along the radial direction (a = 530 mm, θ = 90°,  Su= 10 kPa): (a) radial stress, (b) radial displacement.

Radial stress and displacement distribution of different open arc angle θ along the radial direction (a = 530 mm, b = 110 mm,  Su= 10 kPa): (a) radial stress, (b) radial displacement.

Figure 10(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 different  Su  along the radial direction (a = 530 mm, b = 110 mm, θ = 90°): (a) radial stress, (b) radial displacement.

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.

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.

The radius of the plastic zone caused by the XCC pile installation can be calculated conveniently by this study. The  Rp  increased with the increasing of  Plim/Su, outsourcing diameter  a, and open arc distance  b  while it decreases with the increasing of open arc distance θ. The outsourcing diameter  a  is 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.

The radial displacement increases with the increasing of outsourcing diameter  a  in the elastic zone, and the outsourcing diameter  a  has little influence on the horizontal stress in the elastic zone. The stress and displacement increased with the increasing of undrained strength  Su  obviously in the elastic zone. Both of the open arc distance  b  and 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.

Acknowledgments

The authors wish to thank the National Science Foundation of China (nos. 51278170 and U1134207), Program for Changjiang Scholars and Innovative Research Team in Hohai University (no. IRT1125), and 111 Project (no. B13024) for financial support.

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