Analytical and Numerical Investigation of Internal Force Distribution in “Pile-Wall” Structures Based on Finite Difference Method

Abstract

The “pile-wall” structural system involves the use of conventional temporary retaining piles as part of the underground structure during its operational phase. While this approach has been implemented in engineering practice, there is a significant research gap regarding the theoretical and numerical understanding of the internal force distribution and load-sharing characteristics of the “pile-wall” system, especially in relation to the reinforcement bars used as connection nodes at the interface between the retaining piles and basement walls. This study addresses this gap by proposing a composite structural model that integrates an elastic foundation beam with a continuous beam to simulate the mechanical behavior of the “pile-wall” system. A finite difference equation and computational method are developed to analyze the internal forces and displacements in the system during two key stages: excavation and completion of the basement wall construction. The analytical solutions are validated through comparison with finite element simulations, which show a high degree of consistency between the results. The main contributions of this study include the development of a reliable calculation method for analyzing “pile-wall” internal forces and providing new insights into the load-sharing mechanisms of the system. These findings contribute to a deeper understanding of the mechanical behavior of the “pile-wall” structure and offer a solid theoretical foundation for its broader application in engineering design.

1. Introduction

With the rapid development of the social economy and the continuous advancement of urbanization, the construction of underground engineering projects is increasing, such as deep excavation for subway stations, high-rise buildings, large pipeline trenches, and other engineering construction. Engineering the surrounding environment has become increasingly complex. At the same time, excavation support technology is facing new challenges; the development of underground engineering construction needs to constantly update the support technology to protect the safety and stability of the excavation process. The technology of “pile-wall” structure is a new type of support system that has gradually emerged in the field of excavation engineering in recent years. When designing the underground permanent structure, the retaining pile in the excavation construction period is considered to be a part of the permanent structure in the use phase, and the underground structure wall is loaded together, and the effective combination of the pile and the wall is realized through the setting of force-transmitting members or connecting nodes to achieve the purpose of reducing the thickness of the underground structure wall and thus reducing the project cost. The “pile-wall” structure can be divided into different structural types: (1) a separate “pile-wall” structure involves setting up force-transmitting members in the horizontal direction parallel to the underground layered beam plate and transmit part of the earth pressure acting on the supporting piles to the beam plate of each basement floor; (2) a tight “pile-wall” structure involves connecting the supporting piles with the basement wall by connecting nodes (such as reinforcement bars, etc.) so that the retaining piles and the basement wall can be connected as a whole, so as to achieve the purpose of increasing the bending rigidity of the cross-section.

Currently, the separated “pile-wall” structure has been widely applied in practical engineering projects, such as the Jinmen Foundation Excavation Project in Tianjin and the Shanghai Chest Hospital Pulmonary Oncology Clinical Medicine Center Ward Project. Numerous scholars have conducted extensive research on the separated “pile-wall” structure, accumulating valuable experience and producing significant research findings [1,2,3,4,5,6].

However, theoretical studies on the closely attached “pile-wall” structure remain relatively scarce, and its engineering applications are also limited. In terms of theoretical derivation, Zheng et al. [7] established a calculation formula for the maximum crack width under bending effects based on the cross-sectional characteristics of the closely attached “pile-wall” structure. Through model tests, they revealed the crack development and the distribution of crack widths at different positions under failure loads. Zuo [8] investigated the internal force calculation method for closely attached “pile-wall” structures, proposed a theoretical analytical solution, and performed finite element model simulations. In terms of numerical simulation, Liu et al. [9] systematically analyzed the design aspects of closely attached “pile-wall” retaining structures, established a corresponding geotechnical calculation model, and conducted geotechnical–structural cooperative simulation analysis. Tao et al. [10] studied the internal force distribution of “pile-wall” structures through finite element analysis and verified the accuracy of the finite element simulation results using the Lizheng Deep Excavation software (v7.0). In terms of engineering and technical analysis, Lou et al. [11] successfully promoted and applied the closely attached “pile-wall” technology in an excavation project near the river in Lujiazui, Shanghai. It is evident that current research on closely attached “pile-wall” structures primarily focuses on establishing computational models or analyzing specific engineering applications, lacking comprehensive investigations into their bearing and deformation characteristics.

To address this gap, this study focuses on the excavation engineering of the Optics Valley Central City Distributed Energy Project in Wuhan. One main research question guides this study: How can the bearing and deformation characteristics of closely attached “pile-wall” structures be systematically analyzed, and what is the validity of simplified computational models in simulating these characteristics? Using the Plaxis 3D (v2024) finite element platform, the study aims to develop a simplified computational model for the closely attached “pile-wall” structure. The model separately analyzes the excavation and construction stages through displacement matrix calculations, deriving the internal force matrix accordingly. Additionally, numerical simulations are conducted to verify the correctness and applicability of the simplified model, providing insights into the stress and deformation behaviors throughout the entire process. By addressing the identified research gap, this study contributes to advancing the understanding and application of “pile-wall” technology in excavation engineering.

2. Theoretical Derivation of the “Pile-Wall” Condition

2.1. Calculation Assumption

Based on the above analysis, the following basic assumptions are made [12]:

(1)

The deformation between the soil and the retaining pile is coordinated without relative displacement;

(2)

The displacement of the inner support and the retaining pile is the same, and the deformation is coordinated;

(3)

The interaction between the retaining pile and its surrounding soil is replaced by the equivalent of soil spring.

2.2. Calculation of Excavation Conditions

For a pile-supported support structure, the influence of lateral displacement generated by soil behind the retaining pile on the earth pressure around the pile should not be neglected. In this paper, with the help of the Winkler foundation model, the soil is regarded as the soil spring, whose stiffness varies with depth, and the stiffness coefficient of the soil spring is expressed by k_i and i denotes the number of soil springs so that, according to the Hooke’s law, the stiffness coefficient k_i of the i-th soil spring can be obtained in the process of pressure:

$$k_i={\displaystyle \frac{F_i}{{\delta }_i}}$$

(1)

where δ_i denotes the deformation of the i-th soil spring, which can be solved by the Boussinesq equation [13]; F_i is the force applied to the corresponding i-th soil spring.

When the soil spring above the excavation surface is tensile, the tensile soil spring loses its effect due to the inability of the soil to withstand the tensile force, i.e., k_i is 0. Similarly, the inner support is also replaced by the equivalent of the spring of the corresponding stiffness, whose stiffness is expressed in terms of k_T, and the simplified model is built as shown in Figure 1. The spring is also used in the literature [14] to simulate the inner support and the soil in the excavation project, and the rationality of the simplified structure shown in Figure 1 can be strengthened by the literature [14].

As the number of the inner supports set up in this actual foundation excavation project is two, this paper takes the support form of two inner supports as an example, which can be seen in Figure 2.

In Figure 2, H₁ is the distance from the first inner support to the ground surface, H₂ is the distance from the second inner support to the ground surface, and H₃ is the distance from the ground surface of the excavation to the bottom of the excavation. The inner support can be simulated by the corresponding stiffness of the spring, the stiffness coefficient of the inner support spring is denoted by k_T, and the force exerted on the retaining piles by the first and the second inner supports is denoted by T₁ and T₂.

Due to the excavation, the earth pressure from the soil on the retaining pile becomes active earth pressure; the distribution is shown in Figure 2. Kunlin Lu [15] obtained the approximate relationship between active earth pressure and the displacement of the retaining structure by fitting data from the literature [16,17,18] and analyzing it, assuming that the relationship between active earth pressure and the displacement of the retaining structure satisfies the hyperbolic function:

$${E_a}^{*}={\displaystyle \frac{\exp \left[{\tan }^2({45}^{\circ }-\phi /2)\right]\cdot s}{\left\{\exp \left[{\tan }^2({45}^{\circ }-\phi /2)\right]-1\right\}\cdot s_a+s}}\cdot (E_0-E_a)$$

(2)

$$E_0=K_0\gamma z$$

(3)

$$E_a=K_a\gamma z-2c\sqrt{K_a}$$

(4)

where E₀ is the static earth pressure; E_a is the Rankine active earth pressure; E_a* is the active earth pressure considering the lateral displacement of the retaining pile; φ is the friction angle within the soil; s_a is the displacement of the active earth pressure, which can be obtained from the “Handbook of Foundation Pit Engineering” [19]; γ is the gravity of the soil; c is the cohesion of the soil; K₀ is the static earth pressure coefficient; K_a is the Rankine active earth pressure coefficient, K_a = tan²(45 − φ/2); z is the height of the calculation point from the ground; and s is the displacement of the retaining pile.

At this time, the passive earth pressure below the excavation surface is modeled as soil spring, and the soil spring force is replaced by the equivalent of soil spring force, while, according to the model study of the effect of excavation unloading on the soil spring parameters in the literature [20], the distribution of the soil spring parameters in the absence of excavation is k_h0 = mz, where z is the distance of the calculation point to the excavation surface. If the same distribution pattern with the depth of excavation is used, then the depth of excavation is h; the same distribution pattern with the depth of burial is used to determine the soil spring parameters for the range of z > h. The distribution of soil spring parameters can be obtained by k_h1 = m(z − h), which is shown in Figure 3, where m is the horizontal resistance ratio coefficient of the soil, which can be calculated according to the following formula:

$$m={\displaystyle \frac{0.2{\phi }^2-\phi +c}{{\nu }_b}}$$

(5)

where c is the soil cohesion, φ is the friction angle within the soil, and ν_b is taken with reference to the current “Technical Specification for Retaining and Protection of Building Foundation Excavations” (JGJ120-2012) [21].

In summary, with the aid of Hooke’s law, the soil spring reaction force below the excavation surface after excavation to the bottom of the excavation can be found to be:

$$f=m\left(z-H_3\right)y$$

(6)

For the inner support spring T₁, T₁ = k_Ty, which can be found with the help of Hooke’s law, where k_T can be solved by the formula in the literature [22]:

$$k_T={\displaystyle \frac{{\alpha }_{R}EA{b}_a}{\lambda l_{0}S}}$$

(7)

where λ is the adjustment coefficient of immovable point of support, taking λ = 0.5; α_R is the relaxation coefficient of support, taking α_R = 1.0 for concrete support and steel support with preloaded axial force; A is the cross-sectional area of inner support; l₀ is the length of compressed inner support member (m); S is the horizontal spacing of the support members (m); b_a is the distance of the retaining pile; and E is the modulus of elasticity of the inner support material.

In the process of the excavation, the essence of the elastic foundation beam method is to solve the force equilibrium equation of the supporting structure (retaining pile). The external force on the retaining pile mainly comes from the active soil pressure outside the excavation, the support force acting on the retaining pile by the inner support, and the soil pressure released by the soil excavation inside the excavation. Assuming that the number of inner supports set up in the excavation is n, with the help of the inner supports, the retaining piles are divided into n + 1 units and two virtual nodes are set up at each end of the retaining piles; therefore, the elastic foundation beam formed by the retaining piles is divided into n + 5 units, in which the deflection equation of the retaining piles under the excavation condition is shown in Equation (8).

$$E{I}_1{\displaystyle \frac{d^4\omega (z)}{d{z}^4}}+(k_{T,i}+k_{f,i})b_{0}z=E_a$$

(8)

Here, E is the modulus of elasticity of the retaining pile (kPa); I₁ is the moment of inertia of the cross-section of the retaining pile (m⁴); k_T,i is the stiffness of the inner support (kN/m/m); k_f,i is the horizontal resistance coefficient of the soil in the excavation; and E_a is the active earth pressure on the outside of the excavation, which can be seen in Equations (2)–(4).

According to the finite difference principle, the differential term in Equation (8) can be transformed as:

$${\displaystyle \frac{d^4\omega (z)}{d{z}^4}}={\displaystyle \frac{6{\omega }_i-4({\omega }_{i+1}+{\omega }_{i-1})+({\omega }_{i+2}+{\omega }_{i-2})}{h^4}}$$

(9)

where h is the length of each nodal.

Assuming that the bending moment M and shear force Q end of the retaining pile are both 0, Equation (10) can be obtained:

$$\left\{\begin{array}{l}{M}_0=E{I}_1\cdot {\displaystyle \frac{d^2\omega (z)}{d{z}^2}}z=0\\ Q_0=E{I}_1\cdot {\displaystyle \frac{d^3\omega (z)}{d{z}^3}}z=0\end{array}\right.$$

(10)

Using the finite difference method, the second- and third-order differential terms of Equation (10) can be expressed as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^2\omega (z)}{d{z}^2}}={\displaystyle \frac{{\omega }_{i+1}-2{\omega }_i+{\omega }_{i-1}}{h^2}}\\ {\displaystyle \frac{d^3\omega (z)}{d{z}^3}}={\displaystyle \frac{{\omega }_{i+2}-2{\omega }_{i+1}+2{\omega }_{i-1}-{\omega }_{i-2}}{2h^3}}\end{array}\right.$$

(11)

Combining Equation (10) with Equation (11), the virtual nodal horizontal displacement at the end of the retaining pile can be obtained:

$$\left\{\begin{array}{l}{\omega }_{-2}=4{\omega }_0-4{\omega }_1+{\omega }_2\\ {\omega }_{-1}=2{\omega }_0-{\omega }_1\end{array}\right.$$

(12)

Substituting Equation (12) into Equation (9), each virtual node displacement can be obtained, so the displacement stiffness matrix of the retaining pile is expressed as:

$$\left[K\right]={\displaystyle \frac{E{I}_1}{h^4}}\cdot \left[\begin{array}{l}2 -4 2 0 0\\ -2 5 -4 1\\ 1 -4 6 -4 1\\ 1 -4 6 -4 1\\ \cdots \cdots \cdots \cdots \cdots \\ 1 -4 6 -4 1\\ 0 1 -4 6 -4 1\end{array}\right]$$

(13)

From the assumptions of the elastic foundation beam method, the inner support is equated to a spring, and the spring stiffness can be obtained by referring to Equation (7). Therefore, we can obtain the total stiffness matrix of the internal support [K_T]:

$$\left[K_T\right]=b{k}_T\cdot \left[\begin{array}{l}1 0\\ 1\\ \cdots \cdots \cdots \\ 0 1\end{array}\right]$$

(14)

where b is the calculated width of a single retaining pile and k_T is the spring stiffness factor of the inner support.

The excavation of the soil in the excavation is carried out by the “m” method, and the compression stiffness of the soil spring increases linearly with the increase in the burial depth, as shown in Figure 3. The soil spring stiffness of each structural unit corresponding to the depth of burial can be obtained with the help of Equation (5). The calculated individual soil spring stiffnesses are then integrated into the overall soil spring stiffness matrix [K_f]:

$$\left[K_f\right]=m(z-h)b\cdot \left[\begin{array}{l}1 0\\ 1\\ \cdots \cdots \cdots \\ 0 1\end{array}\right]$$

(15)

where b is the calculated width of a single retaining pile; z is the depth of the calculation point from the ground; h is the length of each nodal unit of the retaining pile; m is the scaling factor of the horizontal soil resistance coefficient increasing with depth, with the value shown in Equation (5).

Ultimately, the force balance Equation (8) for the retaining pile can be written in matrix form as follows:

$$\left[E_a\right]=\left(\left[K\right]+\left[K_f\right]+\left[K_T\right]\right)\cdot \left[\Delta \right]$$

(16)

$$\left[E_a\right]=\left[E_{a0} E_{a1} E_{a2}\cdots \cdots E_{an}\right]$$

(17)

$$\left[\Delta \right]=\left[{\Delta }_0 {\Delta }_1 {\Delta }_2\cdots \cdots {\Delta }_n\right]$$

(18)

where [E_a] is the active earth pressure matrix outside the excavation, E_ai is the active earth pressure on the “ith” unit of the retaining pile, [K] is the overall stiffness matrix of the retaining pile, [Δ] is the deformation matrix of the retaining pile, Δ_i is the displacement of each unit of the retaining pile, [K_f] is the soil spring stiffness matrix of the excavated soil, and [K_T] is the total stiffness matrix of the inner support.

Equation (16) can be further simplified as:

$$\left[\Delta \right]={\left(\left[K\right]+\left[K_f\right]+\left[K_T\right]\right)}^{-1}\cdot \left[E_a\right]$$

(19)

The iterative process is shown in Figure 4, substituting the solved [Δ_n¹] into the active earth pressure Formula (2), calculating the corresponding active earth pressure value [E_an¹] under this displacement, and then substituting it into the above displacement solving Formula (19) to obtain the corresponding retaining pile displacement [Δ_n²]. Repeat the iterative solution until the deformation of the retaining pile [Δ_n^p−1] and [Δ_n^p] of the two solutions before and after satisfying the error requirements which we set.

2.3. Calculation of Construction Conditions

For the basement wall construction conditions, due to the excavation conditions of the external earth pressure, the main earth pressure is E_a, and the basement wall construction is completed after the operation of the external earth pressure for the use of the operating conditions of the static earth pressure E₀. Therefore, the basement wall construction conditions of the external earth pressure change process of the active earth pressure to the static earth pressure changes in the process, i.e., under the construction conditions of the earth pressure for the (E₀ − E_a). The active earth pressure E_a can be obtained by Equation (2) and the static earth pressure can be calculated with the help of Equation (3).

For the inner supports, each inner support will be removed one by one during the construction of the basement wall, and the reinforced concrete floor slabs of each basement level will be established after the removal of the inner supports. Summarizing the above, the “pile-wall” calculation sketch can be obtained when the construction condition of the basement wall is completed, as shown in Figure 5. In the actual project, what is in place between the retaining pile and the basement wall is not just a superposition. Through the planting of reinforcement and other ways to connect the retaining pile and the basement wall, the reinforcement plays a role in the “pile-wall” connection node so that the retaining pile and the basement wall will not produce staggered gaps in the deformation of contact surfaces in the deformation of the pile and the wall, i.e., “pile-wall” common deformation. Therefore, the role of the force on the basement wall and the role of soil pressure on the retaining piles can be dealt with equivalently, i.e., the form of the force on the basement wall is the same as that on the retaining piles, and the size of the piles and walls is distributed according to the stiffness ratio of the “pile-wall” structure [23].

According to the literature [23], the cross-section thickness of the retaining piles can be equated as follows:

$$h=0.838\cdot d\cdot \sqrt{{\displaystyle \frac{d}{d+t}}}$$

(20)

where d is the diameter of the retaining pile, t is the clear distance between piles, and h is the equivalent thickness of the retaining pile section.

During the construction of the basement wall, the external force on the basement wall mainly comes from the soil pressure outside the excavation, transmitted by the retaining piles; the spring support force of the inner support released when the inner support is removed; and the floor slab support force after the floor slab is constructed. Based on the stiffness of the retaining pile and the stiffness of the basement wall derived from Equation (20), shown above, the earth pressure on the retaining pile and the basement wall is distributed according to the principle of stiffness ratio. Assuming that the floor slabs are installed at each location of inner support removal in the excavation, i.e., the number of floor slabs installed is n, and the basement wall is divided into n + 1 units with the help of floor slabs at each level, the deflection equation of the basement wall under the construction condition of basement wall is shown in Equation (21):

$$E{I}_2{\displaystyle \frac{d^4\omega (z)}{d{z}^4}}+(k_{d,i}-k_{T,i})b_{0}z={\displaystyle \frac{I_2}{I_1+I_2}}\left(E_0-E_a\right)$$

(21)

where E is the modulus of elasticity of the basement wall (kPa); I₂ is the sectional moment of inertia of the basement wall (m⁴); k_T,i is the stiffness of the inner support (kN/m/m); k_d,i is the stiffness coefficient of the floor slab; E_a is the active earth pressure outside the excavation, as seen in Equations (2)–(4); and E₀ is the static earth pressure outside the excavation, as seen in Equation (3).

The fourth-order differential transformation in Equation (21) is simplified with the help of Equation (9), and the bending moment and shear force of the basement wall at the surface location are 0, i.e., Equations (10)–(12) are also applicable to the basement wall. Therefore, the displacement stiffness matrix of the basement wall is the same as that of the retaining pile, see Equation (13).

When the inner support is removed, the inner support spring reaction force will be released, the numerical magnitude is the same as the inner support force, and the direction of action is opposite; the total stiffness matrix of the inner support [K_T] is shown in Equation (14).

For the floor support after construction of the floor slab, the floor stiffness coefficient k_d can be determined by obtaining the floor stiffness coefficient k_d from the following equation:

$$k_d={\displaystyle \frac{E{A}_d}{L_d}}$$

(22)

where A_d is the cross-sectional area of the floor slab within the calculation; E is the modulus of elasticity of the floor slab (i.e., the modulus of elasticity of reinforced concrete); and L_d is the calculated length of the floor slab, which can be generally taken as half of the width of the excavation.

Therefore, the total stiffness matrix of the floor slab [K_d] is obtained through Equation (22):

$$\left[K_d\right]=k_d\cdot \left[\begin{array}{l}1 0\\ 1\\ \cdots \cdots \cdots \\ 0 1\end{array}\right]$$

(23)

The final equilibrium equations of forces for the basement wall can be obtained as follows:

$${\displaystyle \frac{I_2}{I_1+I_2}}\cdot \left[E_0-E_a\right]=\left[K\right]\cdot \left[\Delta \right]+\left[K_d\right]\cdot \left[{\Delta }_d\right]-\left[K_T\right]\cdot \left[{\Delta }_T\right]$$

(24)

where [E₀ − E_a] is the earth pressure increment matrix of the earth pressure outside the excavation, from active earth pressure to static earth pressure; [K] is the overall stiffness matrix of the basement wall; [Δ] is the deformation matrix of the basement wall; [K_d] is the overall stiffness matrix of the floor slab; [Δ_d] is the deformation matrix of the floor slab; [K_T] is the total stiffness matrix of the inner support; and [Δ_T] is the deformation matrix of the inner support.

According to the deformation coordination conditions at the corresponding positions of the basement wall, the soil outside the excavation and the floor slab, and the lateral deformation of the three at the same depth is the same, and due to the removal of the inner support after the completion of the construction of the basement wall in the excavation, the deformation of the basement wall will be generated at a time when the floor slab has not been constructed. The floor slab will be built after the completion of the current construction conditions and then the floor slab will be built after the completion of the next excavation conditions; in essence, the floor slab is lagging behind the deformation of the basement wall. Therefore, the deformation of the basement wall when the floor slab is set up should be deducted (i.e., lateral deformation of the basement wall under the previous construction condition), and Equation (25) can be corrected as follows:

$${\displaystyle \frac{I_2}{I_1+I_2}}\cdot \left[E_0-E_a\right]=\left(\left[K\right]+\left[K_d\right]-\left[K_T\right]\right)\cdot \left[\Delta \right]-\left[K_d\right]\cdot \left[{\Delta }^{\prime }\right]$$

(25)

where [Δ′] is the deformation matrix at the previous excavation condition.

Since the deformation of the basement wall is approximately 0 when the basement wall is not constructed and no floor slabs are provided, the initial equilibrium displacement [Δ₀] can be found by the following formula:

$$\left[{\Delta }_0\right]={\displaystyle \frac{I_2}{I_1+I_2}}\cdot \left[E_0-E_a\right]\cdot {\left(\left[K\right]-\left[K_{T0}\right]\right)}^{-1}$$

(26)

where [K_T0] is the inner support stiffness matrix when the first basement wall is constructed and the floor slab has not yet been constructed.

Since the earth pressure and the displacement of the basement wall are coupled with each other, the calculation needs to iteratively solve for the deformation of the basement wall. From the force equilibrium equation of the elastic foundation beam method, the displacement [Δ_n¹] of the basement wall structure when the basement wall is constructed to the “nth” floor is solved as

$$\left[{{\Delta }_n}^1\right]={\left(\left[K\right]+\left[K_d\right]-\left[K_T\right]\right)}^{-1}\cdot \left(\left[{E_{(0-a)n}}^0\right]+\left[K_d\right]\cdot \left[{\Delta }^{\prime }\right]\right)-\left[{\Delta }_0\right]$$

(27)

where [E_(0–a)n⁰] is the earth pressure matrix obtained from the deformation of the basement wall in the previous condition.

After establishing the relevant matrix equations, the computational program edited in the MATLAB language is applied to solve the equations iteratively, and the iterative process is shown in Figure 6, where the solved [Δ_n¹] is substituted into the active earth pressure Equation (11) to calculate the corresponding active earth pressure value [P_an¹] at that displacement, and then into the above displacement solving Equation (12) to obtain the corresponding basement wall displacement [Δ_n²]. Repeat the iterative solution until the deformation of the basement wall [Δ_n^p−1] and [Δ_n^p] of the two previous and two previous solutions satisfy the set allowable error requirements. The lateral displacement curves of the retaining pile and the basement wall solved iteratively by MATLAB (v2024) software are compared with the numerical simulation results established in Section 3 to verify the correctness of the lateral displacement of the “pile-wall” structure derived from the theoretical derivation.

2.4. Analysis of Theoretical Derivation Results of Bending Moment

For the retaining piles, the total external force acting on the pile comprises two parts: the active earth pressure E_a* (as quantified in Equation (2)) during the excavation stage of the excavation, and the earth pressure increment from active to at-rest state E₀ − E_a (as shown in Equations (3) and (4)) during the construction of the basement wall. The variation in internal forces of the retaining piles is caused by the combined action of these two external forces. Therefore, the internal force calculation for the retaining piles must comprehensively consider both the foundation excavation stage and the basement wall construction stage.

In contrast, for the basement wall, the total external force acting on the wall arises solely from the change in earth pressure from active to at-rest state E₀ − E_a during the wall construction stage. Accordingly, the internal force variation in the basement wall is determined by this specific external force, and only the wall construction stage should be considered in its internal force analysis.

Conventionally, previous studies on the internal forces of “pile-wall” structures consider only the external earth pressure acting on the retaining piles, while the basement wall is assumed to share the external pressure in proportion to the stiffness ratio of the pile and the wall. This means that, after the basement wall is constructed, the final external earth pressure on the retaining pile is assumed to be E₀⋅I₁/(I₁ + I₂), and the pressure on the basement wall is E₀⋅I₂/(I₁ + I₂), where I₁ and I₂ are the moments of inertia of the retaining pile and the basement wall, respectively, and E₀ is given in Equation (3).

Such an approach leads to an underestimation of the internal force in the retaining piles and an overestimation in the basement wall. Based on the above analysis of the external earth pressure effects on both the retaining piles and the basement wall, the theoretically derived bending moment diagram of the “pile-wall” system, as shown in Figure 7, and the differences in internal force between this theoretical approach and the conventional stiffness ratio-based method are presented.

Research by ref. [24], based on an excavation project in the Hongqiao Business District of Shanghai, was conducted on-site stress monitoring of an excavation employing a “pile-wall” structure. The measured reinforcement stresses were converted into bending moments of the retaining piles and basement wall, resulting in an in situ bending moment curve of the “pile-wall” system.

Since the engineering conditions of the Hongqiao Business District project differ from those in this study, the numerical discrepancy between the measured results from the literature [24] and the theoretical results derived herein is not considered. Instead, the focus is on validating the overall trend of the “pile-wall” internal force curves proposed in this study. A schematic diagram of the measured bending moment curve from the literature [24] is shown in Figure 8.

Overall, the theoretical calculations presented in this paper show good agreement with the numerical simulation results in terms of the lateral deformations of the retaining piles and basement wall, as well as the corresponding deformation depths. Furthermore, the trend of the calculated bending moment curves for the “pile-wall” system closely matches the measured data. These comparisons indicate that the proposed method can reasonably simulate the deformation characteristics of the “pile-wall” structure during excavation.

3. Overview of the Project and Introduction to Finite Element Modeling

3.1. Engineering Background

The project is located in Leopard Creek Greenland Park at the intersection of Shendun 3rd Road and Guanggu 4th Road in Guanggu Center City, Wuhan City, Hubei Province, and the geomorphological unit of the site is the Yangtze River III terraced stripped Longgang zone; the site is formed by multi-phase landfill which forms a sloped landscape which is high in the northeast and low in the southwest. According to the site conditions-the proposed site soil type is soft soil–medium-hard soil-this project’s excavation depth is deeper, due to the narrow space around the basement perimeter caused the red line of the site being closer, and perimeter piles and internal support joint blocking are used for excavation retaining; the specific form, tightly adhering to the type “pile-wall” excavation engineering structure section, is shown in Figure 9.

3.2. Engineering and Hydrogeologic

According to the drilling data, the stratum of the proposed site area is Quaternary Holocene Artificial Fill (Q4ml), Quaternary Holocene Floodplain (Q4al + pl), Quaternary Upper Pleistocene Floodplain (Q3al + pl), Quaternary Upper Pleistocene Residual Layer (Q3el), and the lithology is mainly heterogeneous fill, pulverized clay, gravelly pulverized clay, gravelly clay, clay, and red clay, and the values of physical parameters of each layer are shown in

Table 1. Values of parameters for each soil layer in numerical simulation.

Materials	H/m	γunsat/(kN·m−3)	E′/MPa	C′/kPa	φ /(°)
Filling	4.8	19.0	8.9	36	13.4
Silty clay	5.0	19.3	8.4	40	13.5
Clay	4.0	19.3	12.1	54	14.7
Red clay	11.2	18.3	19.9	84	16.4

. The values of physical parameters of each soil layer are selected based on the “Geotechnical Engineering Survey Report for Construction Drawing Design of Wuhan Optics Valley Center City Natural Gas Distributed Energy Project Project”, and the values of some parameters are shown in Table 1. The bottom of the excavation is situated on clay, and the characteristic value of the ground bearing capacity meets the requirements, which can be used as the natural foundation of the excavation.

Within the depth range revealed by the exploration holes in the site, the groundwater type of the site is upper layer stagnant water. The upper layer of stagnant water mainly exists in the low-lying fill in the southwestern part of the site, and its water is directly recharged by the vertical infiltration of atmospheric rainfall, with no unified free water surface. The water level varies with the precipitation season and the magnitude is not uniform, which has a small impact on the construction of the excavation.

3.3. Finite Element Simulation Model

3.3.1. Intrinsic Modeling and Parameter

The finite element simulation software Plaxis 3D (v 2024) is used to establish the numerical calculation model and for all the strata in the finite element model, the hardening soil model is used for the calculation [25]; the actual elastic modulus of the soil is greater than the compression modulus, due to its structural nature as well as anisotropy; according to the actual engineering experience, the actual elastic modulus of the soil in the excavation can be taken as 2 times to 5 times of the compression modulus of the soil, which is the compression modulus of the soil, cohesion, angle of internal friction, and other parameters are given by the geotechnical test report table in the “Geotechnical Engineering Survey Report on Construction Drawing Design of Wuhan Optics Valley Center City Natural Gas Distributed Energy Project Project”; and the tensile strength of the soil is taken as 0.

The intrinsic model of the soil body is the hardening soil model (HS model), and its soil parameters are determined according to the parameters given in Table 1. The soil shear expansion angle ψ is taken as 0 [26] because the internal friction angle φ of each layer of the soil body has not exceeded 30°; the unloading Poisson’s ratio of the soil ν is taken as 0.2; the stiffness parameter of the soil is firstly determined as follows the reference tangent compression modulus E_oed^ref, which is determined according to the compression modulus E_s¹⁻² between 100~200 kPa given in the project investigation report [27], i.e., E_oed^ref ≈ E_s¹⁻²; the reference tangent modulus E₅₀^ref is taken to be 1 times the reference tangent compression modulus E_oed^ref [28]; the unloading reloading modulus E_ur^ref is taken to be 4 times the reference tangent modulus E₅₀^ref [29]; the power index m for each stiffness parameter is suggested in

Table 2. Values of parameters for hardening soil model.

	①	②	③	④
Materials	Fillings	Silty Clay	Clay	Red Clay
Thickness/m	4.8	5.0	4.0	11.2
γunsat/kN·m−3	19.0	19.3	19.3	18.3
Cohesion (C)/kPa	36	40	54	84
The angle of internal friction (φ)/°	13.4	13.5	14.7	16.4
Es1−2/kPa	8.9	8.4	12.1	19.9
ψ	0	0	0	0
Eoedref/kPa	8.9	8.4	12.1	19.9
E50ref/kPa	8.9	8.4	12.1	19.9
Eurref/kPa	35.6	33.6	48.4	79.6
m	0.8	0.8	0.8	0.8
Rf	0.9	0.9	0.9	0.9
Rinter	0.65	0.65	0.65	0.65
ν	0.2	0.2	0.2	0.2
Drainage state	drainage	drainage	drainage	drainage

, 0.8 for cohesive soils and 0.5 for sandy soils [26]; and the damage ratios R_f are all taken as 0.9

Since both the retaining piles and the basement wall are simulated by solid units, the solid units are given the concrete material parameters. In the actual project, when the retaining piles are driven into the ground, the cement slurry needs to be injected between the piles to reduce the deformation of the excavation during excavating, at the same time preventing soil leakage between the piles. In Plaxis 3D, the Linear Elastic model is generally selected for concrete and cement slurry, so the parameters of concrete and cement slurry in this numerical simulation model are determined according to

Table 3. Table of concrete and liquid cement parameters in numerical simulation.

Materials	Drainage Type	γunsat/kN·m−3	Elastic Modulus (E)/kPa	ν
Concrete	non-porous	25	31.00 × 106	0.1
liquid cement	non-porous	12	4.0 × 103	0.1

. In addition, we mainly study the force of the retaining piles in order to prevent the elastic modulus of the liquid cement being set too large, which can affect the deformation and the force of the retaining piles; the value of the elastic modulus of the liquid cement is much smaller than the elastic modulus of the concrete.

3.3.2. Model Building

In order to explore the influence of various factors on the force and deformation of the clingy “pile-wall” structure, relying on this project and referring to the numerical simulation of the excavation project in the literature [30], the finite element model structures including the basement wall, the perimeter piles, the foundation base plate, the foundation roof plate, and the connection node between the top and the bottom were established, and the specific model is shown in Figure 10.

The specific excavation model dimensions are shown in Figure 11, where Figure 11a is the front view of the model and Figure 11b is the top view of the model.

In the model, the basement wall and the retaining row piles are simulated by solid units, the excavation floor and excavation roof are simulated by slab units, the connection nodes are simulated by slab units, and the simulation of each component of the finite element model is shown in

Table 4. Parameter information of model components.

Member	Analog Unit	γunsat	H	E
Excavation roof	Plate unit	25	0.4	4 × 108
Excavation floor	Plate unit	25	0.4	4 × 108
Internal support	Plate unit	25	0.2	2.1 × 108
Connection node	Plate unit	25	0.2	2.1 × 108

.

Because the connection node simulated by the point-to-point anchor unit in Plaxis 3D finite element simulation software can only view the axial force, the bending moment and shear force of the connection node cannot be viewed, so the connection node simulated by the point-to-point anchor unit at the bottom and the top of the excavation is inserted into the connection node along the Y direction through the length of the X direction and the connection node of the same length of the plate unit, which can be reflected through the plate unit of the internal force of the bending moment and shear force of the connection node; the specific location of the connection node is shown in

Table 5. Model connection node locations.

	Soil	Elevation (±0 at Ground Level)	γunsat/(kN·m−3)
1	Layer 1 connection node	±0	25.0
2	Layer 2 connection node	−2.5	25.0
3	Layer 3 connection node	−5.0	25.0
4	Layer 4 connection node	−7.8	25.0
5	Layer 5 connection node	−9.8	25.0
6	Layer 6 connection node	−11	25.0

. For the excavation bottom plate and excavation top plate, because the stiffness is substantial in the actual project, the deformation along the X-direction can be almost ignored, which can be regarded as a hinge bearing [31]; therefore, the X-direction fixed constraints are applied to the excavation top plate and the excavation bottom plate in the model, so that the pit bottom and the pit top plate do not produce displacement in the X-direction, and thus the hinge bearing is simulated.

To ensure the numerical stability and physical fidelity of the finite element simulation, the boundary conditions of the Plaxis 3D model are carefully designed based on actual construction constraints and excavation sequence characteristics. The boundary conditions are divided into three categories: global model boundaries, local structural boundaries, and load transfer interfaces, as described below.

(1)

Global model boundary settings: The global boundaries of the finite element domain are set to avoid any artificial reflections or displacement constraints that may influence the stress–deformation behavior within the excavation region. ① Bottom boundary: The entire bottom surface of the model is set as a fully fixed boundary (i.e., zero displacement in X, Y, and Z directions), simulating the reaction from the deep, undeformable subsoil. This prevents any rigid body movement of the entire model and provides realistic vertical support to the excavation structure. ② Lateral boundaries: The lateral boundaries (parallel to the X- and Y-directions) are modeled as normal fixities, where horizontal displacements perpendicular to each boundary face are restricted, while allowing in-plane deformation: specifically, the front and back boundaries (Z direction) are fixed in the Y-direction, free in X and Z, and the left and right boundaries (X direction) are fixed in the X-direction, free in Y and Z. This setting aims to mimic the semi-infinite behavior of the surrounding soil without inducing artificial lateral confinement.

(2)

Local structural boundary conditions: Local boundary conditions were assigned to key structural elements to reflect their functional stiffness and constraint characteristics in real construction. ① Foundation base slab ang top slab: In practical excavation engineering, both the base slab and the top slab of the basement exhibit extremely high in-plane stiffness, making their horizontal (X-direction) deformation negligible. Therefore, X-direction fixed displacement boundaries were applied to the nodes along the base and top slabs in the model to simulate a hinged constraint condition. This effectively limits lateral deformation without introducing full fixity, thus reflecting realistic rotational freedom and vertical settlement allowance. ② Retaining piles: Retaining piles were modeled with full continuity to the surrounding soil, allowing natural interaction without additional artificial constraints. Pile bottoms were embedded in the deep soil layer and tied to the fixed bottom boundary. No lateral fixity was applied along the shaft, allowing natural mobilization of soil-pile interaction along the depth. ③ Basement walls: The basement walls were connected to the foundation and perimeter piles via interface elements and connection nodes (detailed below), without additional boundary constraints, allowing them to freely interact with excavation-induced soil deformation and structural reactions.

(3)

Load transfer interface and connection node constraints: As discussed earlier, the connection node between the perimeter piles and the basement wall plays a critical role in force transmission. Given the modeling limitation of node-to-node anchors in Plaxis 3D (which only transmit axial force), the following composite modeling strategy was adopted. ① Connection nodes: At both the top and bottom interfaces, where piles and basement walls connect, short plate elements were inserted along the Y-direction, with a length equal to the actual connection width in the X-direction. These plate segments were embedded between the pile and wall solid elements. This allows the transfer of axial force, bending moment, and shear force across the interface. The ends of the inserted plate elements were tied via interface elements to the adjacent solid parts, ensuring continuous deformation and accurate load transfer. ② Inner supports and slabs: During staged excavation, internal horizontal supports (e.g., slabs or struts) were sequentially activated and modeled with X-direction fixed constraints at their ends to simulate their bracing function.

The model generated 82,845 cells and 150,619 nodes in the meshing stage, and in the hydraulic stage, the water head decreased to 2 m below the excavation surface in each excavation condition; in other words, the effect of the ground water in the model was not considered. For varying the parameters of different structures in the model and analyzing the effects of the structure under different parameters, different construction conditions are established [32] and the effects of the structure under different parameters are analyzed by controlling a single variable to vary the parameters of different structures in the model: ① balance the initial ground stress; ② zero initial displacement (apply the perimeter piles); ③ excavation roof construction (apply the first horizontal support and at the same time activate the X-direction fixed displacement constraints on the excavation roof so that the excavation roof can be regarded as a hinged bearing with no displacement in the X-direction); ④ step-by-step excavation of the soil (and at the same time activate the constraints on the internal support plates and plates in the excavation stage step); ⑤ construction of the pit footing (and at the same time activate the construction of basement floor slab and activation of X-direction fixed displacement constraint of basement floor slab so that basement floor slab can be regarded as hinge bearing without displacement in X-direction); ⑥ construction of basement outer wall in step-by-step phases (and activation of connection nodes between basement perimeter piles and basement outer wall by phase-by-step steps at the same time).

4. Analysis of Numerical Simulation

In the traditional excavation structure, there is no permanent connection structure between the retaining piles and the basement wall, which means that the retaining piles are no longer involved in the structural stress after the underground structure is built, and the role of the retaining piles is often ignored in the basement design, resulting in the basement wall and the retaining piles being too thick, which is undoubtedly a huge waste of resources and space utilization. The biggest difference between the tight-fitting “pile-wall” support structure and the traditional structure lies in the setting of connection nodes. By implanting several reinforcing bars between the retaining piles and the basement wall and setting up the connection nodes, the retaining piles not only play a role in protecting the excavation during the excavation stage but also play a role in protecting the excavation during the permanent use stage, which can bear the lateral soil and water pressure together with the basement structure, including the basement wall. In the permanent use stage, it also bears the lateral soil and water pressure and the vertical load from the upper building together with the basement structure including the basement wall, which has better economic benefits.

Due to the effects of earth pressure and other external loads, both the retaining piles and the basement walls experience bending moments and shear forces. However, in this finite element simulation, both the retaining piles and the basement walls are modeled using solid elements rather than plate elements, making it impossible to directly obtain bending moment values. Nevertheless, bending moments induce vertical stresses in the support structure, allowing their magnitudes to be inferred from the vertical stress distribution. Therefore, an analysis of the vertical stress in the “pile-wall” structure is conducted. Additionally, under the combined action of external earth pressure, support structure forces, and connection node forces, horizontal shear forces develop within the “pile-wall” structure. These shear forces can be assessed by extracting the horizontal shear stress from the “pile-wall” structure. In the “pile-wall” excavation process, the most critical working condition typically occurs when the basement wall is constructed, and the support structures are removed after excavation reaches the designed final depth. To investigate this scenario, vertical and horizontal stress data of the retaining piles and basement walls were extracted from the simulation results under this specific working condition. To minimize the influence of boundary conditions, stress data were collected from the central region of the structure. Nodal data were taken at 1 m intervals from the top of the structure to the bottom of the structure [33] and curves were generated for these data for comparative analysis.

4.1. Lateral Displacement of “Pile-Wall” Structure

With the step-by-step excavation of the excavation, the effective stress of the soil increases, and the stress state of the soil around the excavation changes, which leads to the complexity of the force between the external soil pressure and the excavation support structure. The numerical simulation model is divided into six working conditions: ① the retaining piles are driven into the soil; ② the excavation is excavated to the elevation of −2.5 m; ③ the excavation is excavated to the elevation of −5.0 m; ④ the excavation is excavated to the elevation of −7.8 m; ⑤ the excavation is excavated to the elevation of −11.0 m, which is the bottom of the excavation; ⑥ the basement wall is constructed to the surface of the ground (elevation of ±0 m). For different construction conditions, the lateral horizontal displacement data of the retaining pile unit in the numerical simulation model varying with depth Z are extracted, and the specific data graph is shown in Figure 12.

Figure 12 presents a comparative analysis of the lateral horizontal displacement of the retaining piles and basement wall, derived from both theoretical calculations and numerical simulations, under two working conditions: (1) after excavating to the final depth of the pit and (2) after completing the basement wall construction.

In the excavation stage, the lateral displacement of the retaining piles increases with depth and reaches its maximum value of approximately 16.7 mm at about 1.0 m below the excavation surface (Z = −11 m). This “spindle-shaped” displacement pattern-characterized by larger displacements in the mid-depth zone and smaller values near the top and bottom boundaries-is clearly evident in both the theoretical derivation curve and the numerical simulation curve.

Following the construction of the basement wall, the earth pressure condition transitions from active earth pressure to at-rest pressure, inducing additional lateral load transfer to the structure. As shown in the curves, the retaining pile continues to displace laterally, with an incremental displacement of about 0.7 mm, while the basement wall, modeled independently in theory, contributes additional stiffness to the system. This process results in a reduced overall lateral deformation compared to retaining piles without connection nodes.

The numerical results confirm that after the connection nodes are activated and the “pile-wall” system acts integrally, the displacement distribution remains spindle-shaped, but with a slightly reduced peak displacement of ~17.4 mm, suggesting the load-sharing capacity of the basement wall helps in restraining further pile movement.

The good agreement between theoretical and numerical results, in terms of both curve shape and peak value location, validates the proposed analytical model’s ability to accurately capture the lateral deformation behavior of the “pile-wall” system across different construction stages.

4.2. Soil Pressure Action During Construction of “Pile-Wall” Structure

The main role of deep excavation supporting structure is to resist the deformation of surrounding soil and bear lateral horizontal load. Earth pressure is one of the main loads, and the theoretical earth pressure is often used as the external load when designing the supporting structure. The theoretical earth pressure is calculated according to the pressure value applied on the retaining structure when the retaining structure reaches the ultimate displacement, while the horizontal displacement of the retaining pile in the field construction is a variable and the earth pressure behind the pile varies with different construction conditions, so the size and distribution curve of the earth pressure are dynamically distributed [19].

Figure 13 presents the distribution curve of external earth pressure around the excavation. From the construction stage where the retaining piles were installed to the excavation stage reaching −11 m, the following observations can be made:

• Peak earth pressure: The maximum external earth pressure is inversely related to the excavation depth. As the excavation depth increases, the peak external earth pressure decreases. At the excavation depth reaching the final excavation level, the peak earth pressure reaches its minimum value of approximately 293 kPa. The reason for this trend is that as excavation progresses, the unloading effect leads to an increasing difference in earth pressure between the inside and outside of the retaining piles. This differential pressure causes horizontal displacement of the piles, resulting in a gradual decrease in external earth pressure.
• Trend of the pressure curve: The external earth pressure exhibits a near-linear increase from the pile top (ground level, ±0 m) to 0.7–0.8 times the length of the retaining pile, where it reaches its peak value. Below this, within 0.2–0.3 times the pile length from the pile bottom (−22 m), the pressure sharply decreases. Overall, the distribution follows a nonlinear downward-convex pattern, which is consistent with the findings of Riqing Xu et al. [34] and Yingying Zhou et al. [35]. Additionally, significant pressure spikes are observed at locations where inner supports are installed. This can be attributed to the fact that inner supports restrict the lateral deformation of the retaining piles, leading to increased active earth pressure at the support locations. In contrast, below the excavation bottom, the embedded segment of the piles experiences reduced horizontal displacement, resulting in a lower active earth pressure in the embedded portion.
• Numerical values of earth pressure: The maximum external earth pressure at different excavation stages is as follows:

  (1)

  Earth pressure of 352 kPa when the retaining piles are installed;

  (2)

  Earth pressure of 326 kPa at an excavation depth of −2.5 m;

  (3)

  Earth pressure of 315 kPa at an excavation depth of −5 m;

  (4)

  Earth pressure of 289 kPa at an excavation depth of −7.8 m;

  (5)

  Earth pressure of 267 kPa at an excavation depth of −11 m.

  These values indicate that the peak earth pressures comply with the allowable earth pressure limits for excavation support structures.
• Earth pressure after basement wall construction: After the basement wall is constructed, the peak external earth pressure increases to approximately 379 kPa, which is higher than the peak pressure at −11 m excavation depth. The curve shape remains downward-convex, but the pressure spikes at inner support locations are slightly lower compared to the excavation process stages. This can be explained by the fact that during the construction of the basement walls, the original inner support structures are removed, eliminating the constraint effect of the inner supports on earth pressure. Additionally, during excavation, the earth pressure transitions from active to at-rest conditions. After the basement wall construction is completed, the active earth pressure fully transforms into at-rest earth pressure, which is greater than the active earth pressure. Consequently, the peak earth pressure after the basement wall construction exceeds that observed at the −11 m excavation depth.

4.3. “Pile-Wall” Structural Forces

4.3.1. Internal Forces in the Retaining Piles

The internal force of the excavation support structure is an important index affecting the safety and stability of the whole excavation project and it is also closely related to the horizontal displacement of the excavation support structure. Under different construction conditions, the force changes in the excavation support structure will affect the development of its horizontal displacement. Comparing the distribution and development of shear stress and bending moment along the depth direction of “pile-wall” support structure under different construction conditions can help excavation designers to have a deeper understanding of the force law of a “pile-wall” support structure from the excavation of the excavation to the completion of the establishment of the basement wall.

Figure 14 shows the variation curve of shear stress with depth of the excavation under different working conditions. Since the internal force of the basement wall during the construction of the basement wall is not considered during the excavation of the excavation, and only the internal force of the “pile-wall” structure is considered after the completion of the excavation and the completion of the construction of the basement wall, the internal force analysis of the “pile-wall” structure is carried out in three construction working conditions: the excavation of the excavation to the bottom of −11 m, the completion of the construction of the basement wall with the connecting nodes, and the completion of the construction of the basement wall without the connecting nodes. Meanwhile, through the theoretical derivation in Section 2, the internal force curves of the “pile-wall” structure derived from the MATLAB software (v 2024) are compared with the numerical simulation results, which can verify the correctness of the theoretical derivation results, and further reveal the law of the internal force of the “pile-wall” structure.

In Figure 14, analysis of peak values in the curve reveals that the shear stress of the retaining piles reaches its maximum at the excavation stage of −11 m. However, after the basement wall is completed, the shear stress in retaining piles with connection nodes is lower than that in piles without connection nodes. The underlying reason is that, after excavation reaches the final depth, the external hydrostatic and earth pressure fully act on the retaining piles, leading to an increase in shear stress, reaching the peak under construction conditions. Once the basement wall is completed, the external pressure is shared between the basement wall and the retaining piles, resulting in a reduction in pile shear stress. Moreover, the connection nodes facilitate the transfer of external earth pressure to the basement wall according to the load-sharing ratio between the piles and the walls. The “pile-wall” structure formed by the connection nodes enhances structural performance, thereby reducing the shear stress in retaining piles with connection nodes compared to those without connection nodes.

From the curve trend, it can be observed that shear stress experiences noticeable abrupt changes at the locations of inner supports, forming distinct peak points. This is due to the inner supports constraining the lateral displacement of the retaining piles, causing a sudden change in shear force at these locations, thereby affecting the smoothness of the shear stress curve.

The numerical simulation results for three different conditions all exhibit a “triangular” distribution, where shear stress initially increases and then decreases. The peak shear stress appears around the excavation bottom (−11 m). Below this point, as soil disturbance diminishes, the shear stress of the retaining pile gradually decreases. The theoretical derivation of the shear stress curve shows a similar trend, where the shear stress first increases, reaching its peak within 2 m–3 m below the excavation bottom, before gradually decreasing, forming a “bulging” shape. This indicates a high degree of consistency between the numerical simulation results and the theoretical derivation in terms of shear stress distribution trends.

Figure 15 presents the variation in bending moment along the depth of the retaining piles under different conditions. Similarly to the shear stress in retaining piles, the analysis does not consider the internal force changes in the retaining piles and basement walls during excavation and basement wall construction. Instead, only two conditions are analyzed: post-excavation completion and post-basement wall construction. Therefore, the structural internal forces of the “pile-wall” system are examined under four construction scenarios: excavation to −11 m, basement wall construction with connection nodes, basement wall construction without connection nodes, and theoretical derivation.

(1)

Peak bending moment analysis: Excluding theoretical results, the maximum bending moment in retaining piles occurs at the excavation depth of −11 m. However, after the basement wall is constructed, the bending moment in retaining piles with connection nodes is lower than in those without connection nodes. The reason is that before the basement wall is constructed, the external hydrostatic and earth pressure act entirely on the retaining piles, increasing the bending moment. Once the basement wall is completed, it shares part of the external pressure, leading to a reduction in the bending moment of the retaining piles. The connection nodes allow the external earth pressure to be distributed between the retaining piles and the basement wall according to their respective load-sharing ratios. The “pile-wall” structure formed by the connection nodes enhances structural integrity, thereby reducing the bending moment in retaining piles with connection nodes compared to those without connection nodes.

(2)

Curve trend analysis: The bending moment in the retaining piles shows distinct abrupt changes at the locations of inner supports, forming sharp peak points in the curve. This occurs because inner supports constrain lateral displacement, leading to sudden changes in bending moment at these locations and affecting the smoothness of the curve. Below the excavation bottom, where no inner supports exist, the bending moment curve is smoother, and its magnitude gradually decreases.

(3)

Numerical simulation results: The bending moment distribution above the excavation bottom exhibits a “zig–zag” pattern with multiple inflection points, where each inner support location corresponds to a turning point. Below the excavation bottom, the bending moment follows a “bulging” shape, first increasing and then decreasing. The peak bending moment appears approximately 5 m below the excavation bottom, beyond which soil disturbance diminishes, and the bending moment gradually decreases.

(4)

Comparison with theoretical results: Compared to the numerical simulation results, the theoretical bending moment curve is smoother. Above the excavation bottom, the bending moment values are smaller, while below the excavation bottom, the values gradually increase, reaching a peak at approximately 5 m below the excavation bottom before decreasing. This trend is consistent with the “bulging” shape observed in numerical simulations, indicating a high degree of agreement between numerical simulations and theoretical derivations.

4.3.2. Internal Force in Basement Wall

In the “pile-wall” structure, the internal force of the basement wall is an important factor to determine the thickness and reinforcement rate of the basement wall, and analyzing the distribution law of the basement wall has a far-reaching impact on the safety and economy of the excavation project; comparing the distribution and development law of the shear stress and bending moment of the basement wall along the depth direction under different construction conditions can help the excavation designers to have a deeper understanding of the stress law of the “pile-wall” supporting structure.

Figure 16 shows the variation curves of shear stress with depth of the basement wall under different working conditions. Since the basement wall has not been constructed at the stage of the excavation, the internal force analysis of the basement wall only considers the internal force of the “pile-wall” structure under the two working conditions after the construction of the basement wall is completed, so the internal force analysis of the “pile-wall” structure is carried out under the two construction conditions of the basement wall completed with connecting nodes and the basement wall completed without connecting nodes. At the same time, through the theoretical derivation in Section 2, the internal force curves of the basement wall derived from MATLAB (v 2024) software are compared with the numerical simulation results, which can verify the correctness of the theoretical derivation results and further reveal the law of the internal force of the “pile-wall” structure.

Figure 14 illustrates the variation in shear stress of the basement exterior wall under different conditions.

(1)

Peak shear stress analysis: The shear stress peak of the basement wall with connection nodes is higher than that of the wall without connection nodes. This occurs because the connection nodes enable the external earth pressure to be distributed between the retaining piles and the basement wall based on a load-sharing ratio. The “pile-wall” structure enhances the ability of the basement wall to absorb the external earth pressure transferred from the retaining piles. Consequently, the shear stress in the basement wall with connection nodes is slightly greater than that in the wall without connection nodes.

(2)

Curve trend analysis: The shear stress curve exhibits distinct peaks at the locations of inner supports, indicating sudden changes in shear force. This phenomenon occurs because the inner supports restrict the lateral displacement of the basement wall, causing a shear force discontinuity at these points, which affects the smoothness of the curve.

(3)

Numerical simulation and theoretical results: In both numerical simulation cases, the shear stress in the basement wall gradually increases with depth, reaching its peak at the excavation bottom (−11 m). However, the theoretical derivation follows a different pattern; the shear stress first decreases, reaching a local extremum at around −4 m, and then gradually increases, forming a convex-shaped distribution. Although the shear stress curve shapes differ between numerical simulations and theoretical derivations, the overall increasing trend in the numerical simulations aligns well with the theoretical results.

Figure 17 shows the bending moment of the basement wall with depth under different working conditions, which is the same as the basement wall shear stress because the basement wall has not been constructed in the excavation stage of the excavation, meaning that the internal force analysis of the basement wall is only considered after the completion of the construction of the basement wall under two working conditions of the internal force of the “pile-wall” structure, i.e., from the completion of the construction of the basement wall with a connecting node and the completion of the basement wall without a connecting node, and only then are the two construction conditions are analyzed. Therefore, the internal force analysis of the pile wall structure is carried out from the two construction conditions of the completed basement wall with connection nodes and completed basement wall without connection nodes.

From the perspective of peak bending moments, the peak bending moment in the basement wall with connection nodes is higher than that in the wall without connection nodes. This occurs because the connection nodes enable the external earth pressure to be transferred to the basement wall through the retaining piles according to the “pile-wall” load-sharing ratio. The “pile-wall” structure formed by the connection nodes allows the basement wall to better share the external earth pressure transferred from the retaining piles. Consequently, the bending moment in the basement wall with connection nodes is slightly greater than that in the wall without connection nodes.

From the curve trend, it can be observed that the bending moment curve of the basement wall with connection nodes exhibits distinct inflection points at the connection node locations. This phenomenon is due to the inner supports constraining the lateral displacement of the basement wall, resulting in abrupt changes in bending moment at the connection node positions, which affects the smoothness of the bending moment curve. In contrast, the bending moment curve of the basement wall without connection nodes gradually increases with depth. The reason is that without connection nodes, the “pile-wall” structure cannot function effectively, meaning that the basement wall only resists the lateral deformation transferred from the retaining piles without sharing the external earth pressure according to the load distribution ratio at the connection nodes. As a result, the bending moment distribution of the basement wall without connection nodes resembles that of a cantilever retaining pile under earth pressure, showing a gradual increase in bending moment with depth.

The theoretical derivation of the basement wall bending moment curve first increases, then decreases, and then increases again, forming a “double convex” shape. In terms of overall trend, the numerical simulation results for the basement wall with connection nodes show good agreement with the theoretical derivation.

5. Conclusions

This study proposed a stage-wise analytical and numerical framework for analyzing the mechanical behavior of closely attached “pile-wall” structural systems in deep excavations. Based on the Wuhan Optics Valley Central City project, a finite difference method and Plaxis 3D numerical simulations were used to examine the load-sharing mechanism, internal force evolution, and deformation characteristics of the “pile-wall” system. The main conclusions are as follows:

(1)

A simplified analytical model coupling elastic foundation beams with spring supports was developed to simulate the staged excavation and basement wall construction processes. The model accounts for the active-to-at-rest earth pressure transition and connection node effects, and its predictions of lateral displacement and internal force distributions showed good agreement with Plaxis 3D numerical simulations, validating its applicability and accuracy.

(2)

The horizontal displacement of the retaining piles under excavation loads exhibited a spindle-shaped distribution, with the maximum displacement occurring approximately 1 m below the excavation surface. After basement wall construction, due to the load-sharing effect of the connection nodes, the overall displacement increased slightly (by 4.2%), but remained within acceptable limits, confirming the structural stability of the integrated “pile-wall” system.

(3)

Earth pressure distributions during excavation followed a nonlinear concave-downward profile, with pressure peaks occurring at support locations due to lateral constraint effects. After wall construction, the earth pressure transitioned to at-rest conditions, resulting in a peak pressure of 379 kPa. These observations align with theoretical predictions and previous research and support the rationality of the staged modeling approach.

(4)

Theoretical and numerical results consistently indicated that the internal force peak in retaining piles occurred during the final excavation stage, while the peak in basement walls occurred post-construction. The agreement in force trends between both methods confirms that the proposed analytical model can reliably capture the force evolution and interaction mechanisms within the “pile-wall” system.

In summary, the study establishes a validated theoretical–numerical method for evaluating the force–deformation behavior of integrated “pile-wall” systems with connection nodes. The findings provide theoretical support and practical guidance for the design and analysis of such systems in deep excavation engineering, particularly where structural synergy and space-saving are critical concerns.