A Simplified Theoretical Method for Estimating the Effect of Capsule Grouting Expansion Technology on Controlling Pile Deformation

Abstract

Deep excavation and shield tunneling processes often induce significant deformation in adjacent existing pile foundations, posing risks to the safety of superstructures. To address this issue, capsule grouting expansion technology has emerged as an efficient and economical technology to actively control pile deformation during such construction activities. This technology has been applied in some engineering projects aimed at safeguarding nearby buildings or facilities. However, theoretical research on capsule grouting expansion technology is limited and lags behind its practical engineering application. This paper proposed a simplified theoretical approach for estimating the effect of capsule grouting expansion technology on the deformation of neighboring pile foundations. Initially, virtual image technology was employed to obtain an analytical solution for the lateral displacement of soil resulting from the expansion of a cylindrical cavity with infinite height. The pile was considered as an Euler–Bernoulli beam on a Winkler foundation, allowing for the calculation of the loading response of the existing pile foundation. Utilizing the two-stage method, the theoretical analytical method of the lateral deformation of the pile shaft caused by capsule expansion was obtained. The reliability of the proposed theoretical method was verified by comparing the measured and calculated results. Further investigation into the controlling effect of the capsule grouting expansion technology on the deformation of the pile foundation was conducted through parametric analyses. These analyses encompass capsule radius, the distance between the capsule and pile foundation, the depth of the capsule, the length of the capsule, and the pile base constraints.

1. Introduction

With the continuous development of underground space and transportation facilities, underground construction processes will inevitably encounter existing pile foundations. Deep excavation and shield tunneling alter the initial stress state of soil, and the resulting lateral deformation of the soil will induce additional deformation and internal force on adjacent existing pile foundations. These will have serious consequences for engineering applications [1,2,3,4].

The additional deformation of the existing pile foundations induced by nearby construction will affect the safety performance of superstructures. Consequently, it is crucial to assess the impacts of soil deformation on existing pile foundations before construction. Many researchers have investigated the effects of deep excavation and shield tunneling on adjacent existing pile foundations through field tests, model tests, numerical simulations, and theoretical analyses [5,6,7,8,9,10,11,12,13]. For the deformation of existing pile foundations located outside the pit, this is significantly affected by the deformation of the retaining structure induced by pit excavation [10]. Controlling the deformation of retaining structures can effectively limit the additional deformation of piles located outside the pit. Hence, current engineering projects mainly focus on reducing the deformation of retaining structures to mitigate the construction effects of deep excavation [14]. This method to reduce the deformation of retaining structures mainly includes enhancing the stiffness of the supporting system [15,16], improving the soil properties [17], and excavating in subzones [18]. However, controlling the deformation of retaining structures depends heavily on the reasonableness of the design scheme and the accuracy of the preliminary evaluation of pile deformation, making it a passive control method [14]. Once the deformation of the pile exceeds the deformation control standard, the construction process must be interrupted, prolonging the construction period. For the deformation of existing pile foundations induced by shield tunneling, the soil loss during the tunneling process greatly impacts the deformation of pile foundations [19]. This soil loss can be effectively reduced by injecting cement paste into the soil layers (compensatory grouting technology), thereby controlling the deformation of neighboring pile foundations. Compensatory grouting technology has been widely used in building deflection correction [20,21] and for reducing tunnel settlement [22], lateral displacement [23], etc.

Compensatory grouting technology is a type of active control technology [14], mainly characterized by low construction investment, minimal impact on the construction period, a flexible control area, and a significant control effect. However, when directly grouted into soil, the cement paste is easy to split and infiltrate, resulting in a reduction in the grouting efficiency [24]. This makes it a challenge to accurately control the grouting area and achieve the expected effect. Zheng et al. [14] proposed a capsule grouting expansion technology, achieving the aim of a controllable grouting area through injecting cement paste in a capsule. The capsule is made of a puncture-resistant, high-strength, and non-ductile material, ensuring impermeability so that the injected cement paste does not penetrate the soil. Consequently, capsule grouting expansion technology can effectively control the volume and shape of the capsule.

For the application of capsule grouting expansion technology, the capsule is pre-sleeved onto a grouting pipe and vacuumed before being transported to the site. After drilling a hole at the desired location, the capsule and grouting pipe are inserted into the drilled hole to a predetermined depth. Fine sand is then used to backfill the void in the drilled hole. As shown in Figure 1, the capsule can be expanded through injecting the cement paste into it, and the resulting displacement of the soil will act on the deformed pile foundation, thereby reducing the pile deformation.

Capsule grouting expansion technology has been used in some engineering applications, with several field tests conducted to investigate its working mechanism [25]. However, there is a lack of theoretical research on capsule grouting expansion technology, warranting further investigation. The theoretical calculation method offers the advantages of a clear force mechanism and requires less time for calculation. Therefore, it is essential to analyze the control effect of capsule grouting expansion technology on the deformation of piles through theoretical research. In the theoretical calculations, the “two-stage” method is typically adopted [26,27,28]. Initially, the displacement field or stress field of soil is determined, disregarding the existence of pile foundations. Subsequently, the displacement field or stress field is applied to the neighboring pile foundation to obtain the force response of the pile foundation.

2. Predicting Soil Displacement Induced by Capsule Grouting

The expansion of the capsule is realized through grouting inside the capsule, and the mechanism of capsule expansion on the displacement field of the surrounding soil can be addressed utilizing the cavity expansion theory. However, the calculation method of cylindrical cavity expansion is based on the assumption of infinite cavity height, whereas the height of the capsule is finite in engineering applications. As a consequence, the results of calculating the cylindrical cavity expansion tend to overestimate the displacement field of the soil, but underestimate the dosage of the cement paste, making it impossible to achieve the desired deformation control effect. The image source method [29] provides a calculation method for the displacement field of soil induced by cavity expansion in a semi-infinite space, assuming that the soil is incompressible. Various analytical formulas derived from the image source method are widely adopted in the calculation of soil displacement fields caused by shield tunneling [30,31,32], deep excavation [33,34], etc. Although it is based on the assumption that the soil is incompressible, its practicality has been confirmed in a wide range of applications. Therefore, virtual image technology is employed to equate the expansion of a cylindrical cavity with finite height to the expansion of multiple spherical cavities. The displacement field of the surrounding soil caused by the expansion of the capsule is then determined by superposing the displacement field of the soil caused by the expansion of multiple spherical cavities.

The following assumptions are made to simplify the theoretical analysis:

• The pile is simplified as an Euler–Bernoulli beam acting on a Winkler foundation.
• The soil is assumed to be incompressible.

For saturated soil, the grouting period for the capsule is relatively short, allowing it to be considered under non-drainage conditions, which reach the incompressible conditions. Subsequently, soil displacement will recover to a certain extent with the dissipation of the pore water pressure. Hence, the proposed method in this paper is applicable to predicting the soil displacement at the initial stage after the completion of capsule grouting, while the calculation results will overestimate the soil displacement generated by the expansion of capsule grouting in the long-term prediction.

Firstly, the soil displacement field induced by the expansion of a cylindrical cavity with infinite height is derived, as shown in Figure 2. The lateral soil displacement δ can be calculated by the following equation as the radius of the cavity increases from 0 to R_c:

$$\pi {R_{\mathrm{c}}}^2=\pi {(R+\delta )}^2-\pi R^2$$

(1)

where R is the lateral distance from the calculation point to the axis of the capsule.

The lateral displacement (δ)_cy of soil can be calculated according to Equation (1), neglecting the higher-order infinitesimal quantities:

$${(\delta )}_{\mathrm{cy}}={R_{\mathrm{c}}}^2/2R$$

(2)

The soil displacement field due to the expansion of a cylindrical cavity with finite height is then derived based on Figure 3, assuming that the cylindrical cavity of height H consists of a series of spherical cavities with neighboring spherical cavities spaced at S.

In infinite space, for a single spherical cavity expansion, the lateral displacement of soil (δ)_sp is given as

$${(\delta )}_{\mathrm{sp}}={R_{\mathrm{c}}}^3/3{R_1}^2$$

(3)

where $R_1=\sqrt{{(x_1-x)}^2+{(y_1-y)}^2+{(z_1-z)}^2}$ is the distance from the cavity to the calculation point.

The image source method [29] was employed to solve the infinite half-space solution, considering the free surface, as depicted in Figure 4. Taking the free surface as the symmetry plane, a virtual expansion cavity is set up at the symmetry position of the expansion cavity. Positive stress will be generated on the ground surface; however, since its influence on the ground surface for the lateral displacement is limited and decays sharply with the depth [35], it is not considered in this research.

The x-direction displacement of soil at point T(x₁, y₁, z₁) caused by the expansion of positive image cavity at A(x, y, z) and the corresponding imaginary image cavity at A′(x, y, −z) can be calculated using the following equation:

$${\delta }_{ix}=\frac{{R_{\mathrm{c}}}^3}{3{R_1}^2}\frac{x_1-x}{R_1}+\frac{{R_{\mathrm{c}}}^3}{3{R_2}^2}\frac{x_1-x}{R_2}$$

(4)

where $R_2=\sqrt{{(x_1-x)}^2+{(y_1-y)}^2+{(z_1+z)}^2}$ is the distance from the virtual image cavity to the calculation point.

The displacement of soil in the x direction at point T(x₁, y₁, z₁) caused by a series of cavity expansions can be obtained by the superposition method with the integral expression:

$${\delta }_x=\frac{1}{S}{\displaystyle \underset{H_{\mathrm{t}}}{\overset{H_{\mathrm{b}}}{\int }}\frac{{R_{\mathrm{c}}}^3}{3{R_1}^2}\frac{x_1-x}{R_1}}+\frac{{R_{\mathrm{c}}}^3}{3{R_2}^2}\frac{x_1-x}{R_2}\mathrm{d}z$$

(5)

When the buried depth of the capsule head H_t = 0, and the buried depth of the capsule base H_b → +∞, the displacement of soil in the x-direction caused by the expansion of a series of cavities and their mirrors is equal to the displacement of soil in the x-direction caused by the expansion of an infinite-height cylindrical cavity:

$${\delta }_x\prime =\frac{2{R_{\mathrm{c}}}^3(x_1-x)}{3[{(x_1-x)}^2+{(y_1-y)}^2]S}={(\delta )}_{\mathrm{cy}}\frac{(x_1-x)}{R}=\frac{{R_{\mathrm{c}}}^2(x_1-x)}{2[{(x_1-x)}^2+{(y_1-y)}^2]}$$

(6)

Then, the space can be calculated using the following equation:

$$S=\frac{4}{3}{R}_{\mathrm{c}}$$

(7)

As described in Figure 5, the lateral displacement Δ(y) of soil at the axis of the pile (d₁, 0, z₁) caused by the expansion of the capsule is

$$\mathsf{\Delta }(y)=={\displaystyle \underset{H_{\mathrm{t}}}{\overset{H_{\mathrm{b}}}{\int }}\frac{{R_{\mathrm{c}}}^2}{4}}(\frac{d_1}{{r_1}^3}+\frac{d_1}{{r_2}^3})\mathrm{d}z$$

(8)

where $r_1=\sqrt{{d_1}^2+{(z-z_1)}^2}$, $r_2=\sqrt{{d_1}^2+{(z+z_1)}^2}$.

3. Pile Deformation Induced by Capsule Grouting

Considering the pile as an Euler–Bernoulli beam placed on a Winkler foundation, the differential control equations unfold as follows:

$$E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{{\mathrm{d}}^{4}w(y)}{ \mathrm{d}{z}^4}+k{D}_{\mathrm{p}}w(y)-q(y)D_{\mathrm{p}}=0$$

(9)

$$q(y)=k\mathsf{\Delta }(y)$$

(10)

where w(y) is the lateral deformation of the pile foundation, D_p is the diameter of the pile, E_p and I_p are the elasticity modulus and inertia moment of the pile foundation, respectively, k is the spring stiffness of the foundation, and q(y) is the additional load induced by the expansion of the capsule.

Vesic [36] obtained the spring stiffness of the foundation based on the surface long beam test with a comprehensive consideration of formation properties and beam stiffness. Additionally, Yu et al. [37] derived the calculation method for the foundation spring stiffness, taking into account the effect of the soil burial depth:

$$k=\frac{3.08E_{\mathrm{s}}}{\lambda D_{\mathrm{p}}(1-{\nu }^2)}\sqrt[8]{\frac{E_{\mathrm{s}}{D_{\mathrm{p}}}^4}{E_{\mathrm{p}}{I}_{\mathrm{p}}}}$$

(11)

$$\lambda =\left\{\begin{array}{l}2.18 z_1/D_{\mathrm{p}}\le 0.5\\ 1+\frac{1}{1.7z_1/D_{\mathrm{p}}} z_1/D_{\mathrm{p}}>0.5\end{array}\right.$$

(12)

where E_s is the modulus of elasticity of the foundation soil, ν is the Poisson ratio of the foundation soil, and λ is the spring stiffness depth coefficient.

The finite difference method is then employed to solve Equation (9), and the nodal unit is illustrated in Figure 6. The pile shaft is equally divided into n parts, each with length l. Two imaginary nodes are added at each end of pile, resulting in (n + 5) nodes totally. The finite difference format of Equation (9) is as follows:

$$E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{6w_i-4\left(w_{i+1}+w_{i-1}\right)+\left(w_{i+2}+w_{i-2}\right)}{l^4}+k_i{D}_{\mathrm{p}}{w}_i=D_{\mathrm{p}}{q}_i$$

(13)

When both ends of the pile are free, the bending moment M and shear force Q at both ends are 0:

$$\left\{\begin{array}{l}{M}_0=-E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{w_1-2w_0+w_{-1}}{l^2}=0\\ M_n=-E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{w_{n+1}-2w_n+w_{n-1}}{l^2}=0\\ Q_0=-E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{w_2-2w_1+2w_{-1}-w_{-2}}{2l^3}=0\\ Q_n=-E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{w_{n+2}-2w_{n+1}+2w_{n-1}-w_{n-2}}{2l^3}=0\end{array}\right.$$

(14)

The expressions for w₋₂, w₋₁, w_n+1, w_n+2 can be obtained by solving Equation (14), and the expressions are then substituted into Equation (13) and written in matrix form as

$$\left(K_1+K_2\right)w=P$$

(15)

where K₁ is the deformation stiffness matrix of the pile, K₂ is the stiffness matrix of the foundation, w is the lateral displacement column vector of the pile, P is the additional load column vector, and each matrix expression unfolds as follows:

$$K_1=\frac{E_{\mathrm{p}}{I}_{\mathrm{p}}}{l^4}{\left[\begin{array}{ccccccc}2& -4& 2& & & & \\ -2& 5& -4& 1& & 0& \\ 1& -4& 6& -4& 1& & \\ & \ddots & \ddots & \ddots & \ddots & \ddots & \\ & & 1& -4& 6& -4& 1\\ & 0& & 1& -4& 5& -2\\ & & & & 2& -4& 2\end{array}\right]}_{\left(n+1\right)\left(n+1\right)}$$

(16)

$$K_2=D_{\mathrm{p}}{\left[\begin{array}{ccccccc}{k}_0& & & & & & \\ & k_1& & & & 0& \\ & & k_2& & & & \\ & \ddots & \ddots & \ddots & \ddots & \ddots & \\ & & & & k_{n-2}& & \\ & 0& & & & k_{n-1}& \\ & & & & & & k_n\end{array}\right]}_{\left(n+1\right)\left(n+1\right)}$$

(17)

$$P=D_{\mathrm{p}}{{\left[\begin{array}{ccccccc}{q}_0& q_1& q_2& \cdots & q_{n-2}& q_{n-1}& q_n\end{array}\right]}^{\mathrm{T}}}_{\left(n+1\right)}$$

(18)

When both ends of the pile are fixed constraints, i.e., the angle of rotation θ and displacement w are 0 at both ends, the following is applied:

$$\left\{\begin{array}{l}{\theta }_{-1}=\frac{w_0-w_{-2}}{2l}=0\\ {\theta }_{n+1}=\frac{w_{n+2}-w_n}{2l}=0\\ w_{-1}=0\\ w_{n+1}=0\end{array}\right.$$

(19)

The matrix expression for the deformation of the pile with fixed constraints at both ends can also be obtained as follows:

$$\left({K_1}^{\prime }+K_2\right)w=P$$

(20)

where the matrix expression ${K_1}^{\prime }$ can be expressed as

$${K_1}^{\prime }=\frac{E_{\mathrm{p}}{I}_{\mathrm{p}}}{l^4}{\left[\begin{array}{ccccccc}7& -4& 1& & & & \\ -4& 6& -4& 1& & 0& \\ 1& -4& 6& -4& 1& & \\ & \ddots & \ddots & \ddots & \ddots & \ddots & \\ & & 1& -4& 6& -4& 1\\ & 0& & 1& -4& 6& -4\\ & & & & 1& -4& 7\end{array}\right]}_{\left(n+1\right)\left(n+1\right)}$$

(21)

4. Variation in the Proposed Method

4.1. Field Test One [38]

Zheng et al. [38] carried out a group of field tests on the effect of capsule grouting expansion on the lateral displacement of soil in Tianjin, China. The distribution of the soil layers is shown in Figure 7, and the water table is approximately 1.5 m below the ground surface. The capsule was located at depths ranging from 20 to 28 m, with an expansion diameter of 50 cm. The lateral soil displacement was monitored at distances of 3 m and 6 m from the capsule during the grouting process. Figure 7 depicts the comparison of the measured lateral displacement and the calculated displacement using the proposed method in this research. It is evident that the calculated lateral displacement at 6 m from the capsule fits well with the measured displacement. However, there are some differences between the calculated and measured results at 3 m from the capsule. Nevertheless, the calculated results are in close agreement with the measured results on the whole, indicating that the proposed method can effectively estimate the lateral displacement of soil induced by capsule grouting expansion.

4.2. Field Test Two [38]

Zheng et al. [38] also conducted a field test on the effect of capsule grouting expansion on the lateral displacement of the pile based on a deep excavation project in Tianjin, China. The capsule expansion area was mainly in silty clay layers (Figure 8). Similarly, the capsule expansion diameter was also 50 cm, and the capsule was buried at a depth of 12–16 m. The bored pile with a diameter of 1.2 m and a length of 50 m was selected as the test pile, with a net distance of 2 m between the test pile and the capsule. And the bored pile was made of C30 concrete with a Young’s modulus of 30 GPa. Two inclined pipes were buried inside the test pile to measure the lateral displacement of the pile shaft. The elasticity modulus of the silty soil was 40 MPa [39]. To solve the matrix equations, a man-made code is developed utilizing the commercial computing platform of MATLAB. Figure 8 presents the comparison between the measured and calculated lateral displacement. It can be seen that the calculated results align well with the measured results, further verifying the reliability of the proposed method.

5. Parameter Analysis

A basic arithmetic example model is firstly established, as shown in Figure 5. The depth of the capsule head H_t is 8 m, the length of the capsule is 4 m, and the expansion radius R_c is 0.20 m. The distance between the capsule and the test pile is 1.5 m. The test pile is a concrete bored pile with a diameter of 1.0 m, and the elastic modulus of the pile shaft is 30 GPa. The ends of the test pile are both freely constrained, and a thorough analysis of the effect of different parameters on the lateral displacement and bending moment of the pile caused by the capsule grouting expansion is conducted.

5.1. Effect of Capsule Expansion Radius

The effect of the capsule expansion radius R_c on the lateral displacement and bending moment of the pile is described in Figure 9. The radii of the capsule are 0.10 m, 0.15 m, 0.20 m, 0.25 m, and 0.30 m, respectively. It can be noticed that the lateral displacement of the pile clearly increases with the increase in the capsule expansion radius, with the maximum lateral displacement occurring at the center of the capsule (10 m). When the radius of the capsule is 0.30 m, the maximum lateral displacement of the pile reaches about 10.0 mm, accompanied by a maximum bending moment of approximately 1030 kN·m. Conversely, when the capsule radius reduces to 0.10 m, the maximum lateral displacement of the pile decreases to 1.1 mm, and the maximum bending moment declines to 114 kN·m. Notably, the maximum lateral displacement and bending moment of the pile both approximately increase by 9 times when the capsule expansion radius increases from 0.10 m to 0.30 m. It can be observed in Figure 10 that the maximum lateral displacement and bending moment of the pile are proportional to the square of the capsule expansion radius, i.e., proportional to the capsule grouting volume.

5.2. Effect of Distance between Capsule and Pile

The effect of distance between the pile and the capsule d on the lateral displacement and bending moment of the pile is presented in Figure 11 and Figure 12. The distance between the pile and the capsule is selected as 1.0 m, 1.5 m, 2.0 m, 2.5 m, and 3.0 m, respectively. Figure 11 depicts that both the lateral displacement and bending moment of the pile decrease with an increase in the distance between the pile and the capsule. For instance, when the distance is 1 m, the maximum lateral displacement of the pile is about 6.3 mm, and the maximum bending moment is 711 kN·m. Conversely, when the distance increases to 3 m, the maximum lateral displacement of the pile is about 2.1 mm, and the maximum bending moment is around 163 kN·m. Figure 12 demonstrates that the maximum lateral displacement and bending moment for the distance of 1.0 m are 3 times and 4.4 times, respectively, compared to a distance of 3.0 m. Moreover, the bending moment decays at a faster rate compared to the lateral displacement as the distance increases.

5.3. Effect of Pile Diameter

The pile diameter D_p is varied as follows: 0.6 m, 0.8 m, 1.0 m, 1.2 m, and 1.4 m, to investigate the effect on the lateral displacement and bending moment of the pile. It should be noted in Figure 13 and Figure 14 that the maximum lateral displacement and bending moment are 5.8 mm and 143 kN·m, respectively, when the pile diameter is 0.6 m. Additionally, the maximum lateral displacement and bending moment of the pile with a diameter of 0.6 m are 1.6 times and 0.16 times that of a pile with a diameter of 1.4 m. Therefore, with the increase in the pile diameter, the horizonal displacement of the pile gradually decreases, while the bending moment increases due to the increased bending stiffness of the pile with a larger diameter. Additionally, Figure 13 illustrates that with the increase in the pile diameter, the displacement at the pile head initially increases and then decreases. This is due to the fact that as the pile diameter increases, the lateral displacement decreases, leading to a reduction in the rotation angle of the pile. Conversely, a smaller pile diameter exhibits larger flexural deformation at the pile head. The jointed effect of these two aspects results in a variation in the lateral displacement and bending moment of the pile. It is notable from Figure 13 that the lateral displacement of the pile head is −0.9 mm when the pile diameter is 1.0 m, which is the largest among the calculated results.

5.4. Effect of Capsule Length

The effect of the capsule length H on the lateral displacement and bending moment of the pile is presented in Figure 15 and Figure 16. The depth of the middle part of the capsule is kept at 10 m, and the entire length of the capsule is varied as follows: 4 m, 6 m, 8 m, 10 m, and 12 m. It is evident that the lateral displacement of the pile increases with the increase in the capsule length, although the increasing rate decreases as the capsule length extends. The variation in the bending moment with the capsule length differs from that of the lateral displacement, and the maximum bending moment firstly increases with the increase in the capsule length and then decreases with the capsule length. This behavior can be attributed to the fact that the increase in the capsule length will increase the lateral displacement of the pile, which also increases the bending moment. In addition, the increase in the capsule length also increases the deformation area of the pile and makes the deformation more gentle, resulting in a decrease in the bending moment. Consequently, the bending moment of the pile increases initially and then decreases with the increase in the capsule length.

5.5. Effect of Capsule Depth

The buried depth of the capsule head H_t is set as 0 m, 4 m, 8 m, 12 m, and 16 m, respectively, to investigate its effect on the lateral displacement and bending moment of the pile. Figure 17 and Figure 18 show that the lateral displacement of the pile reaches the maximum value when the depth of the capsule head is 0 m, where the lateral displacement of the pile head is about 9.7 mm and the maximum bending moment is around −281 kN·m. The depth corresponding to the maximum lateral displacement also increases with the increase in the capsule depth. The lateral displacement of the pile head remains positive when the capsule depth is 4 m, and the maximum lateral displacement and bending moment are almost constant when the depth of the capsule is under 8 m. The value of the lateral displacement and bending moment are around 4.4 mm and 460 kN·m, respectively, when the depth of the capsule is under 8 m. Hence, it can be considered that when the depth of the capsule is under 8 m, the change in the capsule depth will only affect the locations of the maximum lateral displacement and bending moment rather than their values.

5.6. Effect of Pile End Constraint

The effect of the capsule depth on the lateral displacement and bending moment of the pile under the condition of fixing both ends of the pile are presented in Figure 19 and Figure 20. It is apparent that when the capsule depth exceeds 8 m, the maximum lateral displacement and bending moment of the pile remain almost constant, with values of around 4.4 mm and 470 kN·m, respectively. Moreover, both the maximum lateral displacement and bending moment occur in the middle of the capsule, resembling the condition of a pile with two free ends. This suggests that the loading and deformation characteristics of the pile are unaffected by the constraints at the pile ends when the capsule depth is above 8 m, and the influence of the capsule depth on the pile mirrors that of a pile with free ends. When the capsule depth is 0 m, the maximum lateral displacement of the pile is only 2.0 mm due to the strong constraint effect of the pile. Furthermore, the maximum lateral displacement occurs at 5 m below the pile head, which is also below the depth of the capsule base. Moreover, a large negative bending moment occurs at the pile head, with an absolute value of approximately 1200 kN·m. It can also be observed that the constraint effect of the pile head decreases with increasing depth, leading to an increase in the lateral displacement of the pile. However, the bending moment decreases with increasing depth. When the capsule depth is 4 m, the value of the maximum lateral displacement and bending moment are about 4.0 mm and 900 kN·m.

6. Conclusions

This paper proposes a simplified calculation method to estimate the effect of capsule grouting expansion on the deformation and internal force of neighboring piles, employing a two-stage method. The main conclusions drawn from this study are as follows:

(1)

The expression for the lateral displacement field of soil induced by capsule grouting is obtained utilizing the image source method, wherein the cylindrical cavity expansion with finite height is transferred to a series of spherical cavity expansions. Meanwhile, the pile is considered as an Euler–Bernoulli beam on a Winkler foundation to establish a theoretical analytical framework for controlling the lateral displacement of the pile resulting from capsule grouting expansion. The reliability of the proposed method is confirmed through comparisons, demonstrating a good fit between the calculated lateral displacement of the pile and soil and their measured values.

(2)

The lateral displacement of the pile can be controlled by enlarging the capsule radius, with the maximum lateral displacement of the pile being proportional to the cross-sectional area of the capsule grouting. Increasing the distance between the capsule and the pile diminishes the deformation of the pile, thereby compromising the effectiveness of capsule grouting in mitigating pile deformation. Additionally, expanding the pile diameter significantly curtails the lateral displacement of the pile, albeit leading to a gradual rise in the bending moment.

(3)

Increasing the capsule length results in heightened lateral displacement and a broader deformation range of the pile. Concurrently, the bending moment initially increases and then decreases with the augmentation of the capsule length. In scenarios where the depth of the capsule head is 0 m, substantial lateral displacement occurs at the pile head under free end constraints, while a significant bending moment manifests at the pile head under fixed constraints. However, when the capsule depth exceeds 8 times that of the pile diameter, the deformation and bending moment of the pile remain unaffected by the constraint at the pile head. The effect of parameter variations on the pile is consistent, regardless of the end constraints imposed on the pile.

(4)

The theoretical analysis method framework of capsule grouting expansion is effective. Nevertheless, the following limitations require attention in further research. (a) The soil is assumed to be incompressible; (b) the nonlinear behaviors of the soil–pile interaction are not taken into consideration.