Lateral Deformation Response of an Adjacent Passive Pile under the Combined Action of Surcharge Loading and Foundation Excavation

Abstract

With the increasing development of civil engineering in large cities, more and more excavations and surcharge loadings are being constructed or planned adjacent to existing building piles in crowded urban areas. Previous study on pile deformation has primarily focused on surcharge loading or foundation excavation and given little concern to the combined action of surcharge loading and foundation excavation. The article develops a two-stage process to assess the lateral displacement of nearby pile foundations induced by the combined action of surcharge loading and excavation. Firstly, the local plastic deformation theory and Boussinesq solution are used to accurately predict the passive loading of adjacent pile foundations caused by surcharge loading; Mindlin formulas are adopted to predict the passive pile’s additional lateral stress applied by excavation. Secondly, Pasternak models are adopted and the finite difference method is used to establish the deflection differential formula of the single passive pile. Last but not least, a parametric study is conducted to investigate the influence of the loading dimensions, loading magnitudes, and three-dimensional excavation dimensions. The findings of the calculations reveal that the loading magnitudes have a more significant impact on the lateral displacement of the pile compared to the loading dimensions. Therefore, a concentrated surcharge loading should be avoided. Additionally, the excavation depth has a greater influence on the lateral displacement of the pile compared to the excavation area. In order to mitigate this situation, a step excavation should be implemented for each layer of soil, with the soil excavated away from the pile foundation first.

1. Introduction

The rise of city transport hubs has led to an increase in engineering constructions built on soft soil, including pile foundations, tunnels, and pipelines, which are increasingly affecting each other. As the density of excavation engineering increases, the overlapping factors affecting existing underground buildings gradually increase as a result. The lateral movement of the surrounding soil caused by excavation exerts a lateral thrust on the pile foundations of adjacent buildings, thereby affecting the pile foundations within the scope of influence. The loading on the surrounding pile foundations results in lateral pressure on the pile foundations and leads to pile deformation. Previous research on the analysis approach for the response of passive piles has primarily focused on single factors such as surcharge loading or excavation [1,2,3,4,5,6,7,8]. However, insufficient attention has been given to the combined effects of surcharge loading and foundation pit excavation, which may prove harmful for the adjacent buildings. For example, in the case of the collapse of the Lotus Riverside Residential Building in Shanghai [9,10,11,12,13], the second rapid surcharge loading caused the foundation soil to reach its limit state, resulting in significant lateral deformation of the soil and excessive lateral displacement of the pile, leading to pile eccentric compression failure. Meanwhile, the unfavorable combination of heavy rainfall and over-excavation of the foundation pit on the west end of the south side of Building 7 resulted in soil sliding, leading to the destruction of the southwest pile foundation of Building 7 and then causing a chain reaction to the south pile foundation. As a result, Building 7 eventually tilted to the south under the combined action of surcharge loading and foundation excavation. Consequently, research on the deformation of nearby pile foundations under the combined action of surcharge loading and foundation excavation has become significant and crucial in geotechnical engineering [14].

Numerous experts and scholars in geotechnical engineering have conducted extensive research on the impact of surcharge loading or foundation pit excavation on adjacent pile foundations [15,16,17,18,19,20,21,22]. Li et al. (2007) [15] and Zhu et al. (2016) [16] derived the flexural differential equation of a passive pile by changing the initial conditions of the local plastic deformation theory under the surcharge loading. Deng et al. (2021) [17] proposed a method of analyzing the effects of surcharge loading on passive pile loading by taking into account the time factor, based on the Boussinesq plastic deformation theory combined with the elastic foundation beam model. Poulos et al. (1997) [18] attempted to understand the factors that influence the force deformation of adjacent pile foundations during foundation pit excavation and produced diagrams to estimate the lateral deformation and bending moment of pile foundations, using a combination of two-dimensional finite element and boundary element methods. Taking into account the Winkler elastic foundation model and coordinated deformation conditions of pile–soil interaction, Liang et al. (2010, 2013) [19,20] derived a simple formula of single pile and pile group due to excavation using a process in two stages. Using a process in two stages, Zhang et al. (2013) [21] deduced an analytical solution for the pile response based on the image source method due to excavation. Qiu et al. (2020) [22] solved the lateral displacement of the pile using the stress release method and the Pasternak elastic foundation model.

The present study utilizes the two-stage passive pile analysis approach to investigate the deformation behavior of adjacent piles subjected to a combination of surcharge loading and excavation. First of all, a method to identify the time-dependent passive pile loading is proposed based on the balance point position of the thrust generated by the local plastic deformation theory and the Boussinesq additional stress plastic deformation theory in the plastic zone of the soil arching. Moreover, Mindlin formulas are adopted to predict the passive pile’s additional lateral stress applied by excavation. Furthermore, Pasternak foundation models are adopted to establish the deflection differential formula of the single pile. Finally, the method in this paper is validated by analyzing two published measurements and a 3D numerical simulation. Furthermore, a sensitivity analysis of the loading dimensions, loading magnitudes, and excavation three-dimensional dimensions is carried out.

2. Analysis Method

In this analysis, a two-step analysis approach is selected and applied to the excavation–pile–loading interaction, commonly adopted for structure–ground interaction issues. The analysis method is divided into two stages but interconnected. In the first stage, the surcharge loading triggers lateral passive loading acting on the pile, which is calculated using the local plastic deformation theory and Boussinesq solution. The excavation resulting from the additional total lateral stress applied to the single pile is analyzed using the double Gauss–Legendre formula, based on the Mindlin solution. In the second stage, the Pasternak model and finite difference method are adopted to solve the pile response to the corresponding lateral passive loading and additional lateral stress.

2.1. The Lateral Passive Loading Cause by the Adjacent Surcharge Loading

Soft soil creep can be characterized by exponential variation in the critical earth pressure with time. It is expressed as:

$${\sigma }_{h,t}={\sigma }_{h0}+e^{-at}({\sigma }_{ha}-{\sigma }_{h0})$$

(1)

where t denotes the start loading time; σ_h,t represents the critical earth pressure at any time; σ_ha and σ_h0 denote the active and static earth pressure, respectively; a represents the time factor, which can be determined by back-analysis based on the maximum pile displacement or earth pressure test results.

Based on the local plastic deformation theory (Figure 1) [23], the formula of lateral stress on the AA′ plane per unit area is provided below:

$${\sigma }_{hA{A}^{\prime }}={\sigma }_{x,}{}_{z=0}=\{\begin{array}{cc}{\sigma }_{ha}=\gamma z{K}_a-2c\sqrt{K_a}& \mathrm{Active} \mathrm{earth} \mathrm{pressure} \mathrm{state}\\ {\sigma }_{h,t}={\sigma }_{h0}+e^{-at}({\sigma }_{ha}-{\sigma }_{h0})& \mathrm{Critical} \mathrm{earth} \mathrm{pressure} \mathrm{at} \mathrm{any} \mathrm{moment}\\ {\sigma }_{h0}=\gamma z{K}_0& \mathrm{Static} \mathrm{earth} \mathrm{pressure} \mathrm{state}\end{array}$$

(2)

where γ denotes soil weight; K_a represents the active earth pressure coefficient, K_a = tan2(π/4 − φ/2); K₀ represents the static earth pressure coefficient.

Similarly, according to the Boussinesq formulas, the formula of lateral stress on the BEE′B′ plane per unit area is expressed as follows:

$${\sigma }_{x,BE{E}^{‘}{B}^{‘}}=\frac{1}{{}^{N_{\phi }+N_{\phi }^{1/2}\tan \phi -1}}\left\{\begin{array}{l}{(\frac{D}{D_1})}^{{}^{N_{\phi }+N_{\phi }^{1/2}\tan \phi -1}}\left[p_{\mathrm{sd}}\left({}^{N_{\phi }+N_{\phi }^{1/2}\tan \phi -1}\right)+c\left(2\tan \phi +2N_{\phi }^{1/2}+N_{\phi }^{-1/2}\right)\right]\\ -c\left(2\tan \phi +2N_{\phi }^{1/2}+N_{\phi }^{-1/2}\right)\end{array}\right\}$$

(3)

where p_sd denotes the formula of lateral stress on the BB′ plane per unit area, p_sd = p_s + σ_hAA′, p_s represents the additional lateral stress caused by surcharge loading, calculated by using a modified solution of the Boussinesq theory of elasticity; N_φ = tan²(π/4 + φ/2).

The formula of lateral stress on the AEE′A′ plane per unit area is provided below:

$$\begin{array}{l}{\sigma }_{x,AE{E}^{‘}{A}^{‘}}=\{\frac{N_{\phi }\tan \phi }{\left(N_{\phi }+N_{\phi }^{1/2}\tan \phi -1\right)\exp \left(N_{\phi }\tan \phi \frac{D_1-D_2}{D_2}\tan \left(\frac{\pi }{8}+\frac{\phi }{4}\right)\right)}\\ \left\{\begin{array}{l}{\left(\frac{D}{D_1}\right)}^{{}^{N_{\phi }+N_{\phi }^{1/2}\tan \phi -1}}\left[\begin{array}{l}{p}_{\mathrm{sd}}\left(N_{\phi }+N_{\phi }^{1/2}\tan \phi -1\right)\\ +c\left(\begin{array}{l}2\tan \phi +2\\ N_{\phi }^{1/2}+N_{\phi }^{-1/2}\end{array}\right)\end{array}\right]-\\ c\left(2\tan \phi +2N_{\phi }^{1/2}+N_{\phi }^{-1/2}\right)\end{array}\right\}+\frac{c\left(1+2N_{\phi }^{1/2}\tan \phi \right)}{\exp \left(N_{\phi }\tan \phi \frac{D_1-D_2}{D_2}\tan \left(\frac{\pi }{8}+\frac{\phi }{4}\right)\right)}\}\\ \frac{\exp \left(\frac{2N_{\phi }\tan \phi }{D_2}x\right)}{N_{\phi }\tan \phi }-\frac{c\left(1+2N_{\phi }^{1/2}\tan \phi \right)}{N_{\phi }\tan \phi }\end{array}$$

(4)

where c and φ are taking into account the degree of consolidation at any given time.

To obtain the passive loading of pile foundations, it is crucial to establish the division of the critical earth pressure state. Assuming that the AA′ side is the starting point of the x-axis, denoted as x₀; the BB′ side is the end point of the x-axis, denoted as x₂; and the EE′ side is x₁, which can be expressed as follows:

$$\{\begin{array}{l}{x}_0=0\\ x_1=\frac{D_1-D_2}{2}\tan \left(\frac{\pi }{8}+\frac{\phi }{4}\right)\\ x_2=\frac{D_1-D_2}{2}\left[\tan \left(\frac{\pi }{8}+\frac{\phi }{4}\right)+\tan \left(\frac{\pi }{4}-\frac{\phi }{2}\right)\right]\end{array}$$

(5)

The vertical length D(x) of the BEE′B′ area at x∈[x1, x2],

$$D\left(x\right)=D_2+2\left(x-x_1\right)\tan \left(\frac{\pi }{4}+\frac{\phi }{2}\right)\left(x\in \left[x_1,x_2\right]\right)$$

(6)

To determine the passive loading on the pile, the following steps need to be taken:

(1)

Utilize Equation (4) to calculate the lateral stress σ_1x and the corresponding lateral pushing force p_1x = σ_1xD₂ in the area AEE′A′;

(2)

Employ Equation (2) to determine the lateral stress σ₁₀ = σ_hAA′ and the corresponding lateral pushing force p₁₀ = σ₁₀D₂ = σ_AA′D₂ at any given time during the critical earth pressure state in the area AEE′A′;

(3)

Utilize Equation (3) to calculate the lateral stress σ_2x and the corresponding lateral pushing force p_2x = σ_2xD(x) in the area BEE′B′;

(4)

Employ Equation (2) to determine the lateral stress σ₂₀ = σ_hAA′ and the corresponding lateral pushing force q₂₀ = σ₂₀D(x) = σ_hAA′D(x) at any given time during the critical earth pressure state in the area BEE‘B′;

(5)

Assuming that the depth at the calculation point is z₀, the lateral force acting on the BB′ plane is p_BB′ = p_sdD₁;

(6)

First, the initial step in this analysis entails verifying the transfer of the lateral pushing force p_BB′ from the surcharge loading on the BB′ side to the area AEE′A′. To do this, one should equate p_1x to p₁₀ and solve for the corresponding x_s1. The next step involves determining whether x_s1 lies within the range [x₀, x₁]. If it does, x_s1 should be substituted into Equation (4) to obtain the passive loading at the corresponding depth, which is calculated as q_real = p_BB′ − p_1x(x_s1);

(7)

If it does not, x_s2 should be solved for the area BEE′B′ by equating p_2x to q₂₀. Once x_s2 is determined, one should verify whether x_s2 falls within the range [x₁, x₂]. If it does, x_s2 should be substituted into Equation (3) to obtain the passive loading at the corresponding depth, which is calculated as q_real = p_BB′ − p_1x(x_s2);

(8)

If neither x_s1 nor x_s2 fall within [x₀, x₂], then the p_AA′ derived from the Boussinesq solution will exceed the p_AA′ derived from the local plastic deformation theory.

From the above steps, the lateral passive loading cause by the adjacent surcharge loading was calculated using the local plastic deformation theory and Boussinesq solution.

2.2. The Additional Lateral Stress Cause by the Adjacent Excavation

A number of researchers have studied the pile response caused by single excavation, as shown in Figure 2. The computational model of the pile–soil induced by excavation is easily established within the co-ordinate system. The basic assumptions of the modeling are as follows: (a) a semi-space, homogeneous, and isotropic soil; (b) ignore the sequence and rainfall of the foundation excavation; (c) the lateral soil unloading at the sidewalls due to excavation is up to applying a triangular load distribution; (d) the perpendicular unloading at the bottom of the foundation excavation is equal to applying a rectangular load distribution; (e) ignore the lateral soil unloading at the ② sidewall; (f) ignore the influence of the presence of the pile.

According to Mindlin stress solutions [25], the vertical unloading stress σ_x^d and the horizontal unloading stress σ_x¹, σ_x³, and σ_x⁴ of the sidewalls along the pile centerline are obtained:

$${\sigma }_x^d=-\frac{\sigma }{8\pi \left(1-\mu \right)}{\displaystyle {\int }_{-B/2}^{B/2}{\displaystyle {\int }_{L/2}^{L/2}\left(\begin{array}{l}-\frac{\left(1-2\mu \right)\left(z-H\right)}{T_1^3}-\frac{\left(1-2\mu \right)\left[3\left(z-H\right)-4\mu \left(z+H\right)\right]}{T_2^3}+\frac{3{\left(x-\xi \right)}^2\left(z-H\right)}{T_1^5}\\ +\left\{\begin{array}{l}3\left(3-4\mu \right){\left(x-\xi \right)}^2\left(z-H\right)-\\ 6H\left(z+H\right)\left[\left(1-2\mu \right)z-2\mu H\right]\end{array}\right\}/T_2^5+\frac{4\left(1-\mu \right)\left(1-2\mu \right)}{T_2\left(T_2+z+H\right)}\\ \left[1-\frac{{\left(x-\xi \right)}^2}{T_2\left(T_2+z+H\right)}-\frac{{\left(x-\xi \right)}^2}{T_2{}^2}\right]+\frac{30{\left(x-\xi \right)}^{2}zH\left(z+H\right)}{T_2{}^7}\end{array}\right)d\xi d\eta }}$$

(7)

where T₁² = (x − ξ)² + (y − η)² + (z − H)²; T₂² = (x − ξ)² + (y − η)² + (z + H)².

$${\sigma }_x{}^1=-\frac{\beta K_0\gamma \left(x-B/2\right)}{8\pi \left(1-\mu \right)}{\displaystyle {\int }_{-L/2}^{L/2}{\displaystyle {\int }_0^d\left[\begin{array}{l}\frac{\left(1-2\mu \right)}{R_1^3}-\frac{\left(1-2\mu \right)\left(5-4\mu \right)}{R_2^3}+\frac{3{\left(x-B/2\right)}^2}{R_1^5}+\frac{3{\left(x-B/2\right)}^2\left(3-4\mu \right)}{R_2^5}\\ +\frac{4\left(1-\mu \right)\left(1-2\mu \right)}{R_2{\left(R_2+z+\tau \right)}^2}\left(3-\frac{{\left(x-B/2\right)}^2\left(3R_2+z+\tau \right)}{R_2{}^2\left(R_2+z+\tau \right)}\right)-\frac{6\tau }{R_2{}^5}\\ \left(3\tau -\left(3-2\mu \right)\left(z+\tau \right)+\frac{5{\left(x-B/2\right)}^{2}z}{R_2{}^2}\right)\end{array}\right]}}\tau d\eta d\tau$$

(8)

where K₀ = μ/(1 − μ); β represents the stress loss rate of the retaining wall [26]; R₁² = (x −B/2)²+(y − η)²+(z − τ)²; R₂² = (x − B/2)² + (y − η)² + (z + τ)².

$${\sigma }_x{}^4=\frac{\beta K_0\gamma \left(y+L/2\right)}{8\pi (1-\mu )}{\displaystyle {\int }_{-B/2}^{B/2}{\displaystyle {\int }_0^d\left[\begin{array}{l}\frac{\left(1-2\mu \right)}{R_3^3}+\frac{\left(1-2\mu \right)\left(3-4\mu \right)}{R_4^3}-\frac{3{\left(y+L/2\right)}^2}{R_3^5}-\frac{3{\left(y+L/2\right)}^2\left(3-4\mu \right)}{R_4^5}-\\ \frac{4\left(1-\mu \right)\left(1-2\mu \right)}{R_4{\left(R_4+z+\tau \right)}^2}\left(1-\frac{{\left(y+L/2\right)}^2\left(3R_4+z+\tau \right)}{R_4{}^2\left(R_4+z+\tau \right)}\right)+\\ \frac{6\tau }{R_4{}^5}\left(\tau -\left(1-2\mu \right)\left(z+\tau \right)+\frac{5{\left(y+L/2\right)}^{2}z}{R_4{}^2}\right)\end{array}\right]\tau d\eta d\tau }}$$

(9)

where R₃² = (x − ξ)² + (y + L/2)² + (z − τ)²; R₄² = (x − ξ)² + (y + L/2)²+(z + τ)².

$${\sigma }_x{}^3=-\frac{\beta K_0\gamma \left(y-L/2\right)}{8\pi (1-\mu )}{\displaystyle {\int }_{-B/2}^{B/2}{\displaystyle {\int }_0^d\left[\begin{array}{l}\frac{\left(1-2\mu \right)}{R_5^3}+\frac{\left(1-2\mu \right)\left(3-4\mu \right)}{R_6^3}-\frac{3{\left(y-L/2\right)}^2}{R_5^5}-\frac{3{\left(y-L/2\right)}^2\left(3-4\mu \right)}{R_6^5}-\\ \frac{4\left(1-\mu \right)\left(1-2\mu \right)}{R_6{\left(R_6+z+\tau \right)}^2}\left(1-\frac{{\left(y-L/2\right)}^2\left(3R_6+z+\tau \right)}{R_6{}^2\left(R_6+z+\tau \right)}\right)+\\ \frac{6\tau }{R_6{}^5}\left(\tau -\left(1-2\mu \right)\left(z+\tau \right)+\frac{5{\left(y-L/2\right)}^{2}z}{R_6{}^2}\right)\end{array}\right]\tau d\eta d\tau }}$$

(10)

where R₅² = (x − ξ)²+(y − L/2)²+(z − τ)²; R₆² = (x − ξ)²+(y − L/2)² + (z + τ)².

In order to predict the additional lateral stress applied to a single pile, a double Gauss–Legendre formula was applied in the single excavation analysis using MATLAB R2016a, and the principle of superposition of the total additional lateral stress caused by adjacent excavation can be described as follows:

$${\sigma }_x={\sigma }_x^d+{\sigma }_x^1+{\sigma }_x{}^3+{\sigma }_x{}^4$$

(11)

2.3. Analysis Method

2.3.1. Pasternak Foundation Model

The Pasternak foundation model [27], a two-parameter model, is presented for the lateral deformation and internal force of nearby piles, which adds a shear layer on the spring side to considering the shearing effect, and the combined action of surcharge loading and excavation denoted as:

$$p=ky-G\frac{d^{2}y}{d{x}^2}$$

(12)

The governing equilibrium differential equation for the deformation of passive piles lying on the Pasternak elastic foundation, suffering from lateral passive loading and additional lateral stress under the combined action of surcharge loading and excavation, can be expressed as follows:

$$EI\frac{d^{4}y}{d{x}^4}+Q\frac{d^{2}y}{d{x}^2}-GD\frac{d^{2}y}{d{x}^2}+kDy=q+{\sigma }_{x}D$$

(13)

where Q represents the vertical load at the top of the pile; q denotes the lateral passive loading cause by the adjacent surcharge loading; σ_x represents the total additional lateral stress on the pile. Note that G is equal to zero; the governing equation degenerates into the Winkler foundation model.

To solve the direct solution for the fourth-order homogenous difference equation, the finite difference method is adopted. Correspondingly, Equation (13) can be written:

$$\alpha y_{i-2}+\beta y_{i-1}+\lambda y_i+\beta y_{i+1}+\alpha y_{i+2}={\sigma }_x{}_i+q_i/D$$

(14)

where σ_xi denotes the additional lateral stress of node i on the pile; q_i represents the lateral passive loading of node i on the pile.

$$\left[\begin{array}{c}\alpha \\ \beta \\ \lambda \end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ -4& 1& 0\\ 6& -2& 1\end{array}\right]\left\{B\right\}$$

(15)

$$\left\{B\right\}={\left\{\begin{array}{ccc}\frac{EI}{D{h}^4}& \frac{Q-GD}{D{h}^2}& k\end{array}\right\}}^T$$

(16)

The finite difference method was adopted in the pile combined action of surcharge loading and excavation.

The following parameters, such as the rotations, bending moment, lateral force, and horizontal pressure of the pile, are shown in below:

$${\theta }_i=\frac{1}{2h}\left(y_{i+1}-y_{i-1}\right)$$

(17)

$$M_i=\frac{EI}{h^2}\left(y_{i+1}-2y_i+y_{i-1}\right)$$

(18)

$$F_i=\frac{EI}{2h^3}\left(y_{i+2}-2y_{i+1}+2y_{i-1}-y_{i-2}\right)+\frac{Q-GD}{2h}\left(y_{i+1}-y_{i-1}\right)$$

(19)

$${\left(q+{\sigma }_{x}D-pD\right)}_i=\frac{EI}{h^4}\left(y_{i+2}-4y_{i+1}+6y_i-4y_{i-1}+y_{i-2}\right)+\frac{Q-GD}{h^2}\left(y_{i+1}-2y_i+y_{i-1}\right)$$

(20)

By assuming that the pile is free up and down, the boundary conditions can be presented:

$$\{\begin{array}{c}{y}_1-2y_0+y_{-1}=0\\ \frac{EI}{2h^3}\left(y_2-2y_1+2y_{-1}-y_{-2}\right)+\frac{Q-GD}{2h}\left(y_1-y_{-1}\right)=0\\ y_{n+1}-2y_n+y_{n-1}=0\\ \frac{EI}{2h^3}\left(y_{n+2}-2y_{n+1}+2y_{n-1}-y_{n-2}\right)+\frac{Q-GD}{2h}\left(y_{n+1}-y_{n-1}\right)=0\end{array}$$

(21)

Combining Equations (14) and (21), the pile displacement expression of the combined action of surcharge loading and excavation is given:

$${\left\{y\right\}}_{\left(n+1\right)\times 1}={\left[K\right]}_{\left(n+1\right)\times \left(n+1\right)}^{-1}{\left({\sigma }_x+q/D\right)}_{\left(n+1\right)\times 1}$$

(22)

where [K]⁻¹ is the inverse matrix of [K].

$$\left[K\right]={\left[\begin{array}{ccccccc}\lambda +2\beta +4\alpha & -4\alpha & 2\alpha & & & & 0\\ \beta +2\alpha & \lambda -\alpha & \beta & \alpha & & & \\ \alpha & \beta & \lambda & \beta & \alpha & & \\ & \cdots & \cdots & \cdots & \cdots & & \\ & & \alpha & \beta & \lambda & \beta & \alpha \\ & & & \alpha & \beta & \lambda -\alpha & \beta +2\alpha \\ 0& & & & 2\alpha & -4\alpha & \lambda +2\beta +4\alpha \end{array}\right]}_{\left(n+1\right)\times \left(n+1\right)}$$

(23)

2.3.2. Determination of Parameters

G and k are the two parameters in the Pasternak model. And a solution for the shear stiffness G was presented by Tanahashi et al. [28], as below:

$$G=\frac{E_{s}t}{6(1+\mu )}$$

(24)

where t is equal to 11D, suggested by Shi et al. [29].

A lateral infinite beam lying on a flexible foundation using the modified Boit’s expression has been previously proposed [30]. An empirical formula k, namely the coefficient of the subgrade modulus, has been adopted by numerous academics [14,20,31].

$$k=\frac{0.65E_s}{\left(1-{\mu }^2\right)}{\left(\frac{E_s{D}^4}{EI}\right)}^{1/12}$$

(25)

3. Verification

3.1. Yan et al.’s In Situ Test

Yan et al. (1983) [32] conducted an in situ test to evaluate the reaction of pile foundations in soft soil to adjacent surcharge loading. The in situ test was conducted at the Baosteel construction site, which was characterized by a 20 m thick clay layer of high compressibility, overlain with a crust of firm clay layer of 2.5 m. The in situ test experiment comprised the construction of a 30 m × 22 m (length × width) embankment in close proximity to the pile groups, which were composed of four steel open tube piles with a diameter of 609 mm, 11 mm wall thickness, and a concrete cap measuring 5.4 m × 5.4 m × 2.45 m (length × width × height). The spacing between piles was 4.2 m, they had a length of 60 m, and a bending rigidity EI of 2.1 × 10⁸ kN·m² was calibrated. The load increments applied during the experiment produced a surcharge loading of 60, 90, 120, and 150 kPa, within a period of 417 days.

Figure 3 shows the difference between the measured and estimated lateral displacements of the front pile of Baosteel’s entire pile loading test. The results indicate that the proposed method is in good agreement with the measured values of the front row piles (PI1 and PI3). Additionally, the study compares the calculation results of several researchers [15,17,33,34]. Shao et al. (1994) [33] applied the analytical solution method and showed good agreement within the range of 20 m, but the deformation between 20 m and 30 m exceeded the measured values. Luan et al. (2004) [34] employed the non-linear p-curve method, with a surcharge loading equivalent to 112 kPa and an average soil density of 17.9 kN/m³, and the lateral displacement of the pile was closer to the measured value of the pile PI3. Li et al. (2007) [15] calculated the lateral displacement of the pile using Ito’s plasticity theory and the m-method, which exceeded the measured value. Deng et al. (2021) [17] estimated the lateral displacement of the pile using Boussinesq’s plastic deformation theory with time effects and the elastic–plastic model, which exhibited a good agreement, but the deformation in the range of 15 m–25 m was larger than the measured value.

3.2. Ong et al.’s Single Pile Centrifuge Model Test

A set of centrifuge experiments from Ong et al. (2006) [35] were adopted in kaolin clay to predict the deformation on an adjacent single pile due to excavation and the result has already been verified [24]. The results showed the lateral displacement of the free head pile versus that measured for a 1 m, 3 m, 5 m, and 7 m spacing from the excavation face to the pile centerline; the proposed method fits the measured results well.

3.3. Validation of Numerical Simulation

To validate the proposed approach, the lateral displacement of an adjacent pile induced by the combined action of surcharge loading and foundation excavation was simulated using PLAXIS 3D. Figure 4 shows the calculated model dimensions and plan view. Specifically, a single pile with a free-head length of 12.5 m and a bending stiffness of 2.2 × 10⁵ kNm², with a diameter of 600 mm, was positioned 3 m away from both the excavation pit, 10 m × 10 m × 1.2 m (length × width × depth), and the surcharge loading, 5 m × 5 m and 10 kPa (length × width and loading magnitudes). The retaining wall structure, which was 10 m in depth, was constructed to the left of the pile. The top soft clay layer depth was 6.5 m and the underlying sand layer depth was 6 m. A numerical simulation was carried out employing direct excavation and surcharge loading, with an elastic model for the retaining wall structure and the single pile: the Mohr–Coulomb model for soils. The structural parameters are given in

Table 1. Parameters of the structures.

Structure Type	Young’s Modulus/MPa	Poisson Ratio	Unit Weight/kN·m3	c/kPa	φ/Degrees
Soft clay	150cu *	0.4	16.5	1	23
Sand	6z **	0.3	20	1	43
Retaining wall	20,000	0.3	27	-	-
Pile	40,000	0.3	27	-	-

* c_u denotes the undrained shear strength from Ong et al. (2006) [35]; ** z represents the depth.

.

Figure 5 presents a comparison between the results obtained from the analytical method proposed in the paper and numerical simulation. The lateral displacement profiles of the pile were analyzed, and the figures in the paper demonstrate an agreement between the proposed method and the numerical modelling results. Although certain differences were observed, they were attributed to the model size and the boundary effect in the numerical simulation.

From the above discussion, the proposed method was validated with two published tests, which examined the issue of the pile foundation induced by adjacent surcharge loading and excavation, respectively. The method was also validated with a numerical simulation for pile deformation resulting from the combined action of surcharge loading and foundation excavation. The comparison showed that the proposed method is rapid and effective in evaluating pile responses subjected to such conditions during the preliminary design stage.

4. Parametric Study and Results of Analysis

In this part, a series of parametric investigations were conducted, aimed at comprehensively understanding the influence of various factors on the lateral displacement of the pile due to the combined action of surcharge loading and foundation excavation, including the loading dimensions, loading magnitudes, and three-dimensional excavation dimensions. To facilitate direct comparison between the different factors, this study employs the aforementioned numerical simulation example. In general, the basic condition is as follows: the length Le, width Be, and depth d of the excavation are defined as 10 m, 10 m, and 1.2 m, respectively; the length L, width B, and loading magnitude p of the surcharge loading are given as 5 m, 5 m, and 10 kPa, respectively.

4.1. Influence of Loading Dimensions and Loading Magnitudes

The findings presented in Figure 6 demonstrate a relationship between the horizontal deformation of a pile and the surcharge loading conditions. The results indicate that increasing loading dimensions and magnitudes lead to a corresponding increase in the induced lateral displacement of the pile. Specifically, the maximum horizontal deformation of the pile increases by 33.9% as the surcharge loading length is increased from 5 m to 15 m, as depicted in Figure 6a. Additionally, Figure 6b illustrates that increasing the surcharge loading width from 1 m to 10 m results in a 72.3% increase in the maximum horizontal deformation of the pile. Moreover, Figure 6c shows that as the loading magnitudes increase from 5 kPa to 20 kPa; and the maximum lateral deformation of the pile increases by 285.3%, from 19 mm to 73.2 mm. Based on these results, it can be concluded that loading magnitudes have the greatest impact on the sensitivity to lateral displacement of the pile under different loading conditions, followed by the loading width and loading length, respectively. Therefore, it is recommended to avoid concentrated loading and to divide the loading area into long strips as much as possible in engineering design.

4.2. Influence of Three-Dimensional Excavation Dimensions

The lateral displacement of a pile with three-dimensional excavation dimensions is presented in Figure 7. The maximum horizontal deformation of the pile increases by 78.6% as the excavation length increases from 10 m to 20 m, as shown in Figure 7a. Similarly, an increase of 43.9% in the maximum lateral displacement of the pile is observed in Figure 7b as the excavation width increases from 1 m to 10 m. Moreover, Figure 7c illustrates that an increase of 143.9% in the maximum lateral displacement of the pile is observed as the excavation depth increases from 1.2 m to 5 m. Consequently, the sensitivity to lateral displacement of the pile under different three-dimensional excavation dimensions is ranked as follows, from largest to smallest: excavation depth, excavation length, excavation width. Therefore, it is recommended to conduct a step excavation for each layer of soil, and for the soil to be excavated away from the pile foundation first in engineering design.

5. Conclusions

This paper presents a simplified analytical approach to assess the pile deformation response due to the combined action of surcharge loading and foundation excavation. The following conclusions can be shown:

(1)

The proposed method utilizes a two-stage process for predicting the pile deformation. In the first stage, the passive loading caused by surcharge loading and the additional lateral stress acting on the centerline of the passive pile due to excavation are calculated. Furthermore, the exponential variation in the critical earth pressure with time is also taken into account. In the second stage, the Pasternak foundation model and finite difference method are used to solve the pile deformation.

(2)

The feasibility of the suggested approach is validated through two published measurements of surcharge loading and excavation, respectively, and a 3D numerical simulation under the combined action of surcharge loading and foundation excavation. The predictions of the suggested approach are in general agreement with the results of two published measurements and a 3D numerical simulation.

(3)

Parametric analyses are performed to explore the influence of several factors on the lateral displacement of a pile in association with the combined action of surcharge loading and foundation excavation, including the loading dimensions, loading magnitudes, and excavation three-dimensional dimensions.

(4)

Increasing loading magnitudes will significantly increase the adverse effects on the pile, making it imperative to avoid concentrated loading and instead divide the loading area into long strips in engineering design.

(5)

Increasing excavation depth will clearly improve the negative impacts on the pile, prompting the recommendation that a step excavation be conducted for each soil layer and that soil be excavated away from the pile foundation in engineering design.

(6)

For further research, the response of passive pile groups under the combined action of pile loading and the excavation of foundation pits [36], the time-dependent aspects of passive pile groups, and environmental factors will require future research.