1 Introduction

A new type of displacement pile, an X-section cast-in-place concrete (XCC) pile, has recently been developed for soft ground improvement during highway and railway construction in China ^[1-4]. The XCC pile has a special cross section, namely, an X-shaped cross section, which has a larger perimeter (usually 50% larger) than conventional circular cross sections with the same area. As a cast-in-place pile, the installation procedure for XCC piles can be divided into three steps: (a) driving the steel pile mold with the X-shaped cross section into the soil to create an X-shaped cavity, (b) pouring the concrete into the cavity to form the XCC pile body, and (c) extracting the pile mold. In practice, the produced X-shaped cavity undergoes the deformation due to the pressure induced by the self-weight of the concrete and the vibration of the pile mold, and the non-uniform deformation effect (NUDE) occurs around the X-shaped cavity. This deformation is unfavorable to the XCC pile because it can decrease the cambered section of the XCC pile cross section, resulting in a reduction in the perimeter of the XCC pile and further leading to a decreased XCC pile shaft capacity. However, for a conventional circular cross section cast-in-place concrete pile, the deformation of the circular cavity is uniformly subject to an inner pressure, which means that the circular cavity after expansion continues to have a circular shape but with an increased radius. The increase in the circular radius increases the side surface area of the circular pile and the pile shaft capacity. In this case, the deformation improves the circular pile shaft capacity ^[5-6]. Therefore, evaluating the NUDE of the new XCC pile during the installation process serves an important purpose of providing a set of modifications for the design of XCC piles in engineering practice.

The evaluation of the NUDE can be related to the topic of an X-shaped cavity subject to an inner pressure in an infinite elastic medium, which is a fundamental boundary value problem in elasticity ^[7-13]. A review of previous work on this topic indicated that systematic work on these problems had been conducted by Savin ^[10]. By adopting the complex variable theory and the conformal transformation technique developed by Muskhelishvili ^[9], Savin ^[10] obtained elastic solutions of the stress distribution for cavities of various shapes. These solutions were obtained using a conformal mapping function that transforms the area outside a cavity of a specified shape into the region either outside or inside a unit circle. Given that the conformal mapping function is known, the stress functions can be derived. Hence, the stress and displacement distributions around the cavity can be obtained. The key to solving such a boundary value problem is to find the conformal mapping function. Savin ^[10] presented conformal mapping functions for various cavities, such as a square cavity and a triangular cavity. Lei et al. ^[14] provided an approximate conformal mapping function for a rectangular cavity. Guo and Liu ^[15] presented a mapping function that can be used to map the exterior of elliptic hole with two straight cracks onto unit disk for a semi-infinite crack in a piezoelectric strip. Wang and Gao ^[16] introduced an analytical mapping function, provided by Bowie ^[17], to investigate the problem of double cracks at the edge of a circle hole in an infinite piezoelectric solid. Lu et al. ^[18] presented an analytical expression of the conformal mapping function for a problem of two semi-infinite collinear cracks in a piezoelectric strip. Guo and Lu ^[19] and Guo et al. ^[20] investigated the problem of elastic-plastic fields around an anti-plane elliptic hole using the classical mapping function of elliptical cavity. Yu et al. ^[21] developed a complex variable method for an anti-plane elliptical cavity in one-dimensional hexagonal piezoelectric quasicrystals using the mapping function of elliptical cavity. More recently, Liu and Guo ^[22] introduced the Schwartz-Christopher conformal mapping to investigate the interaction between a screw dislocation and an oblique edge crack in a half-infinite magnetoelectroelastic solid. In addition, Zhou et al. ^[23-26] and Liu et al. ^[27] investigated the problem of elliptical cavity expansion. Though numerous mapping functions have been presented in the past, no work on the conformal mapping function of X-shaped cavities has been reported.

2 Objectives

In this paper, a theoretical model of the X-cavity boundary value problem is proposed based on the framework of complex variable plane elasticity. Using the least squares method, a new conformal mapping function is provided to map the X-shaped cavity in the original physical plane onto the unit circle in the phase plane. Such a mathematical transformation can transform the non-cylindrically symmetric boundary value problem into a cylindrically symmetric one, thereby providing a solution to the difficulty in solving non-cylindrically symmetric boundary value problems. Then, the stress function and hence the stress and displacement distributions around an X-shaped cavity are readily derived using the conformation mapping function and the complex variable theory developed by Muskhelishvili. Subsequently, the derived analytical solution is used to evaluate the NUDE, the concrete filling index (CFI), and the reduction coefficient of the side surface of the XCC pile. The calculated NUDE, CFI, and reduction coefficient of the side surface of the XCC pile are then compared with the field test results provided by Liu et al. ^[1]. Closed-form solutions for the CFI and reduction coefficient of the side surface of the XCC pile are presented for use in practical engineering design.

3 Problem definition and basic assumptions 3.1 Description of X-shaped cavity

The X-shaped cavity (XCC pile cross section) contains four flat sections and four cambered sections ^[1]. The center of the cambered section is located at the point O. Three parameters, a, b, and θ, are used to control the size and shape of the XCC pile cross section. The parameter a describes the distance between the two flat sections in the diagonal direction. In addition, the parameter b is the length of the flat section, and the variable θ defines the angle of the cambered section. Once the three parameters are selected, the XCC pile cross section is determined. Two cases (i.e., Case 1: a=611 mm, b=120 mm, θ =90°; Case 2: a=611 mm, b=120 mm, θ =130°) are commonly used for XCC piles in practical engineering.

3.2 Description of boundary value problem

Figure 1(a) shows the initial stress state before the X-shaped cavity is created in an infinite and homogeneous elastic soil. The soil is everywhere subjected to an initial stress σ₀. Figure 1(b) demonstrates that an X-shaped cavity is created in the soil and subjected to a uniform in-plane inner pressure p around the cavity wall and initial stress σ₀ at infinity. Both Cartesian and polar coordinates systems are used in the analysis. The origins of the two coordinate systems are located at the center of the initial X-shaped cavity. For the Cartesian coordinate system, the x-and y-axes are along the horizontal and vertical directions in the cross section plane, respectively. However, the positive r-axis points toward infinity in the polar coordinate system. Additionally, the developed analytical solutions are based on the following three assumptions:

(a) Hooke's elasticity law governs the elastic behavior of the soil.

(b) A plane strain condition is assumed in the direction perpendicular to the X-shaped cavity plane.

(c) The "penetration effect" (the pile mold is forced into the soil to create an X-shaped cavity) of the pile mold during installation is neglected when deriving the analytical solution.

However, a slight modification of the analytical solution obtained by considering this effect will be given when it is used to evaluate the CFI and reduction coefficient of the side surface of the XCC pile.

4 Basic governing equations

According to Ref. [9], the governing equations for the plane problem in the theory of elasticity can be expressed using two stress functions, φ₁(z) and ψ₁(z), as

(1a)

(1b)

(1c)

where σ_x, σ_y, and τ_xy are stress components in the rectangular coordinate system. u and v are horizontal and vertical displacement components, respectively. G is the shear modulus of the soil. μ is Poisson's ratio of the soil. z=x+iy is a complex variable, where . Re represents the real part of a complex variable. An overscored symbol denotes the conjugate of that parameter, and the prime represents differentiation with respect to the corresponding independent variable.

5 Conformal mapping 5.1 General expression for conformal mapping function

Considering the non-cylindrically symmetric characteristics of the initial X-shaped cavity wall, it is difficult to address the X-shaped boundary value problem in the original physical plane (z-plane). As a powerful mathematical tool, the conformal mapping technique can deal with boundary value problems with complex boundary conditions. This technique can transform the complex boundary conditions in the physical plane into simple conditions in the phase plane. Hence, in this paper, the conformal mapping technique is used to map the outer region of the X-shaped cavity wall boundary in the physical plane (z-plane) onto the inner region of a unit circle in the phase plane (ζ-plane) (see Fig. 5). Thus, a non-cylindrically symmetric problem in the physical plane can be transformed into a cylindrically symmetric problem in the phase plane. The general expression for a conformal mapping equation can be expressed as an infinite series ^[9],

(2)

where

in which z is a complex variable in the z-plane, ζ is a complex variable in the ζ-plane, x and y are the variables in the Cartesian coordinate system of the z-plane, and r and θ are the variables in the polar coordinate system of the z-plane. ζ and η are the variables in the Cartesian coordinate system of the ζ-plane, ρ and are the variables in the polar coordinate system of the ζ-plane, and c₀ and c_2n-1 are constant real-valued coefficients.

5.2 Derivation of constant coefficients of conformal mapping function

Equation (2) presents an infinite series for the conformal mapping function, while a finite series provides sufficient accuracy for describing an X-shaped cavity. In this section, the well-known least squares method is used to derive the constant coefficients of the conformal mapping function ^[28]. Based on this method, the algorithm for the computation of the constant coefficients is developed in this work as follows.

(ⅰ) First, the curve of the X-shaped cavity is divided into (m+1) sampling points in a counter clockwise manner. Then, the first approximation calculation is performed based on the assumption that these sampling points in the physical plane correspond to equidistant points on the boundary of the unit circle in the phase plane with a counterclockwise manner. The approximate calculation obeys the following overdetermined linear system of m equations with n unknowns:

(3)

where

According to the least squares method, Eq. (3) can be rearranged as

(4)

From Eq. (4), the constant coefficients can be obtained as the first-order approximate solution.

(ⅱ) Substituting the obtained constant coefficients into Eq. (2) leads to the conformal mapping function. In addition, the new points at the X-shaped cavity boundary in the z-plane can be calculated using the conformal mapping function.

(ⅲ) The intersections of the line that connects the new points and the center of the coordinates with the prescribed physical X-shaped cavity boundary are used as the new values of z.

(ⅳ) Substituting the new values of z into Eq. (4) results in the improved values of the constant coefficients.

(ⅴ) This iterative procedure is continued until a prescribed small error, which is equal to the sum of the squares of the differences between the actual and predicted boundary points of the X-shaped cavity, is achieved.

Liu et al. ^[1] presented the approximations of the boundary of the X-shaped cavity using the proposed algorithm for the two cases, as it is commonly used for the XCC pile in practice. n=4 and n=6 result in sufficient accuracy and provide adequate approximations of the X-shaped cavity boundary for Cases 1 and 2, respectively. Thus, n=4 and n=6 are used in the conformal mapping function for the two cases in the following calculations. The subsequent analysis is derived based on the general expressions of the conformal mapping function. Thus, the derived solutions may be suitable for arbitrary cavities and not only for the X-shaped cavity.

6 Transformation of governing equations

The governing equation (see Eq. (1)) is written as a function of the complex variable z, which should be transformed into the complex variable ζ in the phase plane. Hence, by introducing the conformal mapping equation (Eq. (2)), the governing equations can be transformed as

(5a)

(5b)

(5c)

where φ (ζ) and ψ (ζ) are the two stress functions of the complex variable ζ in the phase plane, and G and v are the shear modulus and Poisson's ratio of the soil, respectively.

7 Boundary conditions

As shown by Muskhelishvili ^[9], the general expression for the stress boundary condition in the phase plane can be written as

(6)

where f_x and f_y are the surface stresses acting on the X-shaped cavity boundary in the x-and y-directions, respectively, and s represents the arc length at the X-shaped cavity boundary.

Consider that a uniform pressure p is applied at the X-shaped cavity wall, and Eq. (6) becomes

(7)

Note that ζ=σ at the X-shaped cavity wall. Thus, Eq. (7) is transformed into

(8)

8 Solution of stress functions

To solve the stress and displacement distributions around the X-shaped cavity, the key problem is to obtain the two stress functions in Eq. (5). In this paper, the same procedures and similar symbols for obtaining the elastic solutions for the X-shaped cavity presented by Muskhelishvili ^[9] and Savin ^[10] are followed. The stress functions φ (ζ) and ψ (ζ) can be expressed as ^[9]

(9a)

(9b)

where F_x and F_y represent the resultants at the X-shaped cavity wall in the x-and y-directions, respectively. In this paper, the two variables F_x and F_y are equal to zero due to the boundary conditions of a uniform pressure applied at the X-shaped cavity wall. In addition, the variables B, B'+iC', and B'-iC' can be written as

(10)

(11)

(12)

where σ₁ and σ₂ are the principal stresses at infinity, and α is the angle between the maximum principal direction and the x-direction. According to the defined problem, the stress functions φ₀ (ζ) and ψ₀ (ζ) in Eqs. (9a) and (9b) are analytic at infinity and can be written as a Laurent series, thus Eqs. (9a) and (9b) can be simplified as

(13a)

(13b)

Substituting the stress functions into the stress boundary conditions (see Eq. (8)) leads to

(14)

where

Solving Eq. (14) with Muskhelishvili's method, one obtains

(15a)

(15b)

The coefficients in Eqs. (15a) and (15b) can be determined using the following two equations:

(16a)

(16b)

where I_n-1 is an (n-1)-dimensional unit matrix,

(16c)

(16d)

(16e)

(16f)

(16g)

(16h)

(16i)

Once the stress functions are determined by Eq. (16), the stress can be derived from the governing Eq. (5). The stress solution of Eq. (5) is expressed as a function of the complex variable ζ, which is in the phase plane. To obtain the stress solutions in the original physical plane, the complex variable ζ should be transformed back to the physical plane via the conformal mapping equation. Note that the explicit analytical expression for ζ is difficult to be obtained and thus the value of ζ should be determined numerically through Eq. (2).

9 Model validation

To validate the proposed analytical solution, it is not sufficient to compare the degenerate solution with Timoshenko's classical circular cavity elastic solution, which is a special case of the X-shaped cavity expansion problem. Hence, the FEM in the ABAQUS software is used to verify the analytical solution. Figure 2 shows the finite element mesh of the model of the X-shaped cavity boundary value problem. Only a quarter of the model is built due to the symmetry of the problem. The plain strain is adopted in the model. The initial stress around the X-shaped cavity is assumed to be zero. The symmetry boundary is restrained from the direction perpendicular to the boundary in the plane. To avoid the influence of the boundary, the outer boundary of the model is approximately 50 times the XCC pile cross section parameter a. Only Case 1 for the XCC pile cross section (a=0.611 m, b=0.12 m, θ =90°) is used for comparison in this section. Moreover, Poisson's ratio and the elastic modulus of the surrounding soil are set as 0.3 and 5 MPa, respectively, and a uniform inner pressure of 100 kPa is used. Two typical radial paths (θ =0° and 45°) around the X-shaped cavity are selected for the comparison analysis. The normalized stress components (σ_r/p, σ_θ/p) and the radial displacement u_rG/(pr_{b, θ =0° or 45°})(r_{b, θ =0° or 45°} denotes the radial position of the X-shaped cavity boundary along the two typical paths) predicted by the analytical solution and FEM are plotted against the normalized radius r/r_{b, θ =0° or 45°} in Figs. 3 and 4. The predictions of the three normalized stress components along the radial direction are in reasonable agreement with the FEM results, as this is true for both of the two paths. This indicates that the analytical solution presented in this paper is reliable.

10 Distributions of stress and displacement around X-shaped cavity

The distributions of the stress and displacement around an X-shaped cavity are critical because they are useful in understanding the mechanism of the X-shaped cavity and may provide guidelines for practicing engineers. Therefore, the distributions of the stress and displacement around an X-shaped cavity are presented in the form of contour plots in this section. A quarter of the model is selected for analysis due to the symmetry of the problem. All of the parameters required for the calculations are the same as above. Figure 5 plots the contour of the normalized radial stress components σ_r/p around the X-shaped cavity. A stress bubble (stress concentration) is found adjacent to the corner of the X-shaped cavity boundary. This phenomenon is similar to the one found in the classic problem of a vertical point load on a half elastic space in soil mechanics. Another interesting phenomenon is that the contours of the radial stress surrounding the cambered section of the X-shaped cavity boundary transit from a "positive arc", which is consistent with the cambered section of the X-shaped cavity boundary, to a "negative arc", the outer normal direction of which is contrary to the "positive arc". Furthermore, the radial stress adjacent to the cambered section clearly vanishes rapidly along the radial direction, whereas it slowly surrounds the flat section. In other words, the pressure applied at the X-shaped cavity boundary may have a larger zone of influence along the direction perpendicular to the flat section compared with that to the cambered section. Similarly, the contour of the normalized circumferential stress components σ_θ/p around the X-shaped cavity is shown in Fig. 6. The phenomenon (the transition from the "positive arc" to the "negative arc" around the X-shaped cavity boundary) also appears near the cambered section of the X-shaped cavity boundary. In addition, the contour of the normalized radial displacement components, u_rG/(pa), around the X-shaped cavity is plotted in Fig. 7. An interesting phenomenon whereby the contours of the radial displacement tend to be circles when the radial distance exceeds 0.55 m can be observed. This finding can be explained by the well-known Saint-Venant's principle in the classical elasticity theory. Moreover, the middle of the cambered section of the X-shaped cavity boundary is more easily expanded, which means that the middle of the cambered section of the X-shaped cavity boundary may exhibit a larger radial displacement compared with the cambered section near the corner of the X-shaped cavity boundary under a uniform inner pressure. In practice, this phenomenon is the previously mentioned "NUDE". This effect may reduce the length of the cambered section of the X-shaped cavity boundary and the XCC pile shaft capacity.

11 Potential yielding mechanisms around X-shaped cavity

In this section, the potential yielding mechanism around an X-shaped cavity is presented. Investigating the plastic behavior of the surrounding soil is extremely useful. The soil is assumed to be an elastic-perfectly plastic material following the Tresca yield criterion, which is a commonly used constitutive model for clay in practice. The contour of the maximum shear stress can then be observed as the dividing line between the elastic and plastic zones. According to the Tresca yield criterion, the maximum shear stress is defined as follows:

(17)

where σ₁ and σ₃ are the maximum and minimum principal stresses, respectively.

Therefore, the key to understanding the yield mechanism is to obtain the contour of the maximum shear stress around the X-shaped cavity. Figure 8 plots the contour of the normalized maximum shear stress τ_max/p around the X-shaped cavity. The surrounding soil and geometric parameters of the X-shaped cavity are consistent with the previous analysis. Because the stress distributions near the X-shaped cavity are of greater concern than the far-field stress distributions, a relatively small zone with dimensions of 0.6 m×0.6 m around the X-shaped cavity is investigated. The soil is initially yielded at the corner of the X-shaped cavity wall where a stress concentration phenomenon occurs, and then the plastic zone extends to the middle of the cambered section with the increasing uniform pressure p. Furthermore, the contour of the maximum shear stress tends to be a circle with the increasing radial distance. In other words, beyond a specified radial distance, the plastic zone boundary is a circular curve. These findings concerning the yielding mechanism may provide guidance into the elastic-plastic X-shaped cavity boundary value problem in the future.

12 Application of proposed analytical solution 12.1 Evaluation of NUDE of XCC pile 12.1.1 Derivation for deformation around X-shaped cavity wall

Although the simple derived solution in this paper is for a plate with an X-shaped cavity subjected to an inner pressure, it can be readily adopted to the evaluation of the NUDE of XCC piles during installation. If the effects of the installation on stress changes and deformations at any depth are considered and idealized as a plane train problem, the derived analytical solution in this paper can be readily used to calculate the displacement around the X-shaped cavity and evaluate the NUDE. In this section, the NUDE of XCC piles during installation is discussed based on the proposed analytical solution. Before evaluating the NUDE, it is necessary to further simplify the analytical solution and present an expression for the deformation around the X-shaped cavity. At the X-shaped cavity wall, ρ=1, and . Thus, the stress functions can be transformed into

(18a)

(18b)

Substituting Eq. (18) into Eq. (5c) leads to

(19)

where u_b and v_b are the lateral and vertical displacements in the cross section plane at the X-shaped cavity boundary, respectively.

Equation (19) provides an explicit expression for predicting the deformation around the X-shaped cavity. Based on this expression, the following evaluations of the NUDE are conducted.

12.1.2 Comparison of predicted deformation around X-shaped cavity wall with field test results

To validate the suitability of Eq. (19) in evaluating the non-uniform deformation of the XCC pile cross section (namely, the X-shaped cavity) during installation, a field test was conducted near the fourth bridge of Nanjing. The XCC pile technique was used to improve the soft ground. Thus, monitoring data for the XCC pile, including the real size of the XCC pile cross section (using a ruler to measure the real size), were obtained. These data are used here to verify the suitability of the proposed analytical solution. Detailed descriptions, including the soil stratigraphy and cone penetration test (CPT) results, for the field test were provided by Liu et al. ^[1], who investigated the change in the displacement and stress induced by the XCC pile mold penetration into soil to create an X-shaped cavity. However, the deformation of the X-shaped cavity due to the pressure induced by the self-weight of the concrete or the vibration of the pile mold is of concern in this paper. Case 2 (a=611 mm, b=120 mm, θ =130°) for the XCC pile cross section is used in this project. The shear modulus G of the soil profile is obtained from the laboratory test (compression test) and ranges from 0.75 MPa to 1.25 MPa. Thus, an average value of the soil shear modulus of 1 MPa is adopted in the calculation. For soft clay, Poisson's ratio is set as 0.5, which represents the typical undrained behavior. Moreover, three XCC piles (Cases a, b, and c) in the field are measured. The surrounding soil around the three XCC piles is excavated to a depth of approximately 2 m (see Fig. 9), and the measured cross section of the XCC pile is located at a depth of 1 m.

In evaluating the deformation of the XCC pile after installation, the uniform inner pressure p is another important input parameter. As described above, the uniform inner pressure p contains two components: the contribution of the self-weight of the concrete and the vibration of the pile mold during installation. The pressure induced by the self-weight of the concrete can be easily evaluated. For instance, the pressure induced by the self-weight of the concrete at a depth of 1 m is 25 kPa because the unit weight of concrete is 25 kN/m³. However, it is difficult to evaluate the pressure induced by the vibration of the pile mold. Therefore, the input parameter for the pressure p varies from 25 kPa to 250 kPa, which provides a reasonable estimate of the uniform inner pressure p. Figure 9 plots the predicted and measured geometric size of a quarter of the XCC pile cross section. The predicted values of the XCC pile cross section after installation using values of p=150 kPa to p=250 kPa are in agreement with the three field test results, except for the overestimate of the flat section of the XCC pile cross section. This overestimate occurs because of the penetration effect of the pile mold, which is neglected when deriving the analytical solution. Due to this effect, the pressure applied at the X-shaped cavity wall may be not uniform. Thus, the pressure applied at the flat section of the X-shaped cavity is less than that at the cambered section. For simplicity, the non-uniform pressure is assumed as uniform in the present analytical solution. This assumption has a small effect on the prediction of the deformation of the cambered section, because the calculated deformation of the cambered section is in agreement with the measured value. However, it may produce a notable effect on the flat section, where the predicted deformation is not consistent with the field test results. In this case, the present analytical solution used to evaluate the NUDE must be modified and will be given in the next section. In addition, the pile mold vibration contributes to the primary component of the inner pressure p (25 kPa induced by the self-weight of the concrete). In addition, the deformation of the flat section is relatively small and can even be neglected compared with the cambered section. In other words, the NUDE of the XCC pile is mainly attributed to the deformation of the cambered section.

12.2 Modification of proposed analytical solution

As described above, the proposed analytical solution may overestimate the deformation of the flat section of the XCC pile cross section. Thus, various modifications should be made before they are used to predict the CFI and the reduction coefficient of the side surface of the XCC pile. The predictions for the deformation of the cambered section using the analytical solution are consistent with the field test results, particularly at the middle of the cambered section. In addition, the deformation of the flat section of the XCC pile cross section is relatively small according to the field test results. Based on these two findings, an assumption is made whereby the deformation of the flat section is neglected, and the deformation of the cambered section is predicted by the analytical solution. Such an assumption is reasonable because the reduction in the side surface of XCC pile is mainly produced by the reduction in the cambered section. Figure 10 presents the modified analytical model used in evaluating the NUDE of the XCC pile. The cambered section exhibits a non-uniform deformation, and the radial displacement at the middle of the cambered section is defined as u_{c, r}, which can be calculated using Eq. (19). The radii of the two cambered sections (before and after deformation) are defined as R₁ and R₂, and the centers are O₁ and O₂, respectively. Following this model, the length of the cambered section before deformation can be easily obtained as

(20)

Then, the expression for the length of the cambered section after deformation must be derived. The length of can be expressed as

(21)

where

Furthermore, using the law of cosines, one obtains

(22)

In addition,

(23)

where θ₁ =θ/2.

Substituting Eq. (23) into Eq. (22) yields

(24)

which can be simplified as

(25)

Thus, the variable of R₂ can be expressed as

(26)

Then, the length of the cambered section after deformation can be written as

(27)

12.3 Evaluation of CFI for XCC pile 12.3.1 Expression for CFI

The CFI is an important parameter for the new XCC pile in practical engineering. The CFI represents the ratio of the volume of concrete actually used for each XCC pile and the designed value. This parameter is more useful for XCC piles than that for conventional circular cast-in-place piles. Because there was no theoretical method for evaluating the reduction in the side surface of the XCC pile in the past, the CFI measured in the field provides an important reference index for engineers when empirically assessing the reduction in the side surface of the XCC pile. In this section, a theoretical assessment of the CFI is presented based on the modified analytical solution. Although the CFI can be measured in the field, a theoretical assessment of the CFI is also useful. Because a comparison analysis of the measured CFI with the theoretical assessment may provide powerful support for the application of the proposed analytical solution, a simple expression for the CFI is derived for the XCC pile, and the suitability of the expression is subsequently verified via comparison with field test results. Before deriving the solution for the CFI, one should make an assumption. The uniform inner pressure p in the analytical solution is mainly generated by the vibration of the pile mold, which has been previously demonstrated. In addition, the pressure induced by the vibration of the pile mold remains constant if the power of the pile installation equipment is fixed. The pressure p can be observed as constant along the pile depth, although the pressure induced by the self-weight of the concrete increases linearly along the pile depth. According to the definition of the CFI, the following relation is established:

(28)

where

in which V_actual and V_designed are the actual and designed concrete volumes for each XCC pile, respectively, and A_{a, XCC} and A_{d, XCC} are the actual and designed areas of the XCC pile cross section, respectively, which can be calculated using the proposed modified analytical solution.

12.3.2 Comparison of calculated CFI using analytical solution with field test results

The deformation of the X-shaped cavity can be obtained from the analytical solution. Hence, the CFI is affected by the pressure p and soil shear modulus G. Then, the variation in the CFI with the normalized pressure p/G is plotted in Fig. 11. The field test result for the CFI is also selected for comparison. The pressure p in practice may vary from 150 kPa to 250 kPa, because the calculated values of the XCC pile cross section size using values of p= 150 kPa to p= 250 kPa are consistent with the field test results. Hence, the normalized value of p/G, which ranges from 0.15 to 0.25 (G=1 MPa in practice), is selected to represent the actual pressure in the field test. Figure 11 shows that the value of the CFI increases linearly with increase in p/G. A convenient linear fit equation is presented to describe the relation between p/G and CFI. The equation can be expressed as

(29)

which is a simple closed-form expression and may be more convenient to be used compared with Eq. (28). However, the simple expression of Eq. (29) does not reduce the accuracy when evaluating the relation between p/G and CFI, because the calculated values of the CFI using Eq. (28) are consistent with those of Eq. (29). In addition, the measured CFIs (1.32, 1.16, and 1.32 for Cases a, b, and c, respectively) from the field test are in reasonable agreement with those predicted by Eqs. (28) and (29).

12.4 Evaluation of reduction in side surface of XCC pile

Due to the pressure induced by the self-weight of the concrete or the vibration of the pile mold, the NUDE will occur around the X-shaped cavity. In addition, the NUDE will reduce the XCC pile shaft capacity because it will decrease the side surface of the XCC pile, particularly reducing the cambered section of the XCC pile cross section. Thus, a modification of the side surface of the XCC pile obtained by considering the pile installation process is necessary when evaluating the XCC pile shaft capacity. Based on the modification of the proposed analytical solution described above, a reduction coefficient for the side surface of the XCC pile is defined as

(30)

where C_{a, XCC} and C_{d, XCC} are the actual and designed perimeters of the XCC pile cross section, respectively.

Figure 12 plots the reduction coefficient λ as a function of the normalized pressure p/G. The relationship between the reduction coefficient λ and normalized pressure p/G can be expressed by the following closed-form non-linear equation:

(31)

The result calculated using this equation is consistent with the prediction based on the analytical solution. In addition, the field test result for the reduction coefficient λ is also selected for comparison in Fig. 12. The normalized value of p/G also ranges from 0.15 to 0.25. The prediction obtained by the non-linear equation is slightly smaller than that by the field test. However, such an underestimate of the reduction coefficient λ may increase safety in practical engineering design, because it will underestimate the XCC pile shaft capacity. The reserve for the XCC pile shaft capacity is reasonable in practice.

13 Limitations

The analysis presented in this paper is based on the elasticity framework. The analytical solutions for the stress and displacement around a pressurized X-shaped cavity are obtained. Since no analytical solution for the X-shaped cavity boundary value problem has been reported, the present analysis serves the important purpose of providing a set of benchmark results that can underpin further, more sophisticated analyses. Currently, there is no quantitative design guidance for evaluating the CFI and the reduction in the side surface of the XCC pile. Therefore, the present closed-form solutions for the CFI and the reduction coefficient provide a simple and improved approach for engineers for assessing the XCC pile shaft capacity. A precise solution for the problem of the non-uniform pressure applied to an X-shaped cavity obtained by considering the pile mold penetration effect would recreate the true NUDE of the XCC pile. Furthermore, the disturbance effect of the soil (the degradation of the soil shear modulus) induced by the pile mold penetration is not considered though one could use a reduction shear modulus in the calculation.

14 Conclusions

This paper presents an analytical solution for the X-cavity boundary value problem to capture the NUDE of an XCC pile. The stress and displacement around a pressurized X-shaped cavity are obtained using a conformal mapping technique and a complex variable theory. The analytical solution is verified by comparison with the FEM. Subsequently, the analytical solution is used to evaluate the NUDE of the XCC pile, and the results are compared with those of the field test provided by Liu et al. ^[1]. These solutions are also slightly modified and subsequently used to calculate the CFI and reduction coefficient of the side surface of the XCC pile. The calculated NUDE and CFI are also compared with the field test results. Reasonable agreement is achieved. Based on the comparison analysis, two closed-form expressions for evaluating the CFI and reduction coefficient are presented for practical engineering design. The present analytical solution provides a general solution procedure for solving the X-shaped cavity boundary value problem and can be extended to arbitrary cavity boundary value problems if an appropriate conformal mapping function is used. A brief application of the evaluation of the NUDE, CFI, and reduction coefficient of the side surface of the XCC pile confirms their potential application in geotechnical engineering.