Spatial Deformation Calculation and Parameter Analysis of Pile–Anchor Retaining Structure

: Scholars often consider the deformation of a foundation pit retaining structure as a signiﬁcant indicator of its stability. However, the current theoretical prediction formula for pit with pile–anchorretaining structure deformation is not yet perfect. This study utilizes a simpliﬁed spatial deformation model of a pile–anchorretaining structure and the principle of minimum potential energy to derive a prediction formula for the retaining structure’s spatial deformation. Afterwards, a numerical simulation model is developed based on actual engineering practices. On-site monitoring data is compared with the results of theoretical calculation formulas and numerical simulation models to validate their applicability. The research ﬁndings reveal minimal discrepancies between the theoretical calculation results, numerical simulation outcomes


Introduction
As the urban population continues to grow, so does the demand for urban space. However, urban space is limited, prompting scholars to propose the concept of underground urban space. The goal of this concept is to alleviate the scarcity of urban space by developing underground space. Foundation pit engineering is considered the most significant underground space engineering in cities. Foundation pit engineering plays a crucial role in creating initial construction space for a variety of structures such as high-rise building foundations, subway stations, underground shopping malls, and underground parking lots. However, the excavation process of a foundation pit can potentially lead to alterations in the initial state of the surrounding soil, posing a risk of damage to nearby buildings and structures [1][2][3]. As a result, construction risks during foundation pit excavation have been a major topic of interest among scholars. The study of Momeni et al. [4] showcased the practicality of utilizing the random set finite element method (RS-FEM) and event tree analysis (ETA) to effectively evaluate the reliability and potential lethality risk associated with urban excavations; Lin et al. [5] developed a risk assessment model 2 of 17 for excavation systems; Sun et al. [6] proposed a fuzzy hierarchical analysis (AHP)-based model for pit health assessment. In their work, Long and Li [7] integrated the uncertainty and correlation of evaluation indices with the fuzzy nature of expert evaluation. They devised an evaluation model grounded in fuzzy theory.
The stability of a foundation pit can be directly inferred from the deformation characteristics exhibited by its retaining structure during excavation. Various research methods are employed to analyze such deformations, including numerical simulation, theoretical analysis, and indoor testing. In their study, Wu et al. [8] introduced a combined structural model that incorporates an elastic roadbed beam and a continuous beam. They incorporated elastic springs to simulate the interaction between the piles and the basement wall. Finite difference equations and calculation methods were developed to accurately calculate the internal forces and displacements of the combined pile-wall structure. This approach enables precise calculation of excavation-induced internal forces and deformations of retaining piles during the excavation phase. Wang et al. [9] used the PLAXIS 3D numerical simulation software to study the deformation characteristics of foundation pits under asymmetric loading. Li et al. [10] conducted a study on the deformation characteristics of a shaped foundation pit support structure using the deep foundation pit of Malian North Station on Xiamen Metro Line 2. They employed field observation and numerical simulation methods to analyze the deformations. On the other hand, Su et al. [11] designed and executed a large-scale model test of a pile-anchor-supported foundation pit with a geometric similarity ratio of 1:10. Wei et al. [12] used the virtual image technique to calculate the soil displacement caused by the deformation of the support structure, and at the same time, introduced the rotational misalignment collaborative deformation model to analyze the longitudinal deformation of the tunnel and the maximum displacement equivalent field of the tunnel caused by different support structure deformation methods. The theoretical analysis method can provide designers with prediction values quickly using prediction formulas, significantly enhancing the efficiency of predicting construction risks. An improved support system known as cast-in-place pile and anchor cable (pole) support utilizes anchor bolts and anti-slide piles to provide anchoring and sliding resistance forces. This system effectively resists soil sliding behind the support structure. Due to its exceptional support effectiveness and stability, the cast-in-place pile and anchor cable support system is highly favored by designers and researchers. Numerous studies have investigated the disturbance effects of foundation pit excavation using this support system, employing laboratory tests, numerical simulations, and on-site monitoring. However, limited research exists regarding theoretical prediction methods and formulas for the spatial deformation of pile-anchorretaining structures.
In this study, the spatial deformation mode of the pile-anchorretaining structure is thoroughly analyzed, and a spatial deformation prediction formula is derived based on the principle of minimum potential energy. A numerical simulation model is created using FLAC3D, a finite difference software. The primary objective is to present theoretical prediction methods and formulas for predicting the spatial deformation of pile-anchorretaining structures, addressing the research gap in previous studies. The practicality of the theoretical prediction formula is validated through a comparison of its results with numerical simulation outcomes and on-site monitoring data. Furthermore, a comprehensive analysis of the control effect of relevant factors is performed. These findings establish a robust theoretical foundation for predicting deformations in foundation pits utilizing pile-anchorretaining structures.

Spatial Deformation of Retaining Structure
Based on the foundation pit measurements, it was observed that the retaining structure exhibited distinct spatial deformation characteristics. The structure as a whole underwent lateral bending due to the active soil pressure. However, the deformation at the corners and fulcrum positions of the foundation pit was limited due to the constraints, resulting Appl. Sci. 2023, 13, 6637 3 of 17 in significantly smaller deformation of the wall compared to the middle section of the foundation pit. It is evident that the spatial deformation characteristics of the retaining structure are mainly attributable to its lateral bending resistance. Hence, the collective lateral bending resistance of the retaining wall is referred to as the spatial deformation effect of the retaining structure.
The pile-anchorretaining structure consists of a crown beam, a waist beam, piles, and anchor cables. The crown beam is a reinforced concrete continuous beam arranged on top of the retaining structure surrounding the foundation pit. Its purpose is to connect all of the piles together (such as bored piles and rotary excavation piles) to prevent the top edge of the foundation pit from collapsing. The waist beam is a reinforced concrete or steel beam located below the top of the retaining structure. Its function is to transmit the bearing capacity of the support and anchor cable, which changes the support effect from a point to a straight line, thereby improving the stability and integrity of the retaining structure. Therefore, the spatial deformation of the pile-anchorretaining structure is formed by the combined action of the crown beam, waist beam, piles, and anchor cables, as shown in Figure 1.

Spatial Deformation of Retaining Structure
Based on the foundation pit measurements, it was observed that the retaining structure exhibited distinct spatial deformation characteristics. The structure as a whole underwent lateral bending due to the active soil pressure. However, the deformation at the corners and fulcrum positions of the foundation pit was limited due to the constraints, resulting in significantly smaller deformation of the wall compared to the middle section of the foundation pit. It is evident that the spatial deformation characteristics of the retaining structure are mainly attributable to its lateral bending resistance. Hence, the collective lateral bending resistance of the retaining wall is referred to as the spatial deformation effect of the retaining structure.
The pile-anchorretaining structure consists of a crown beam, a waist beam, piles, and anchor cables. The crown beam is a reinforced concrete continuous beam arranged on top of the retaining structure surrounding the foundation pit. Its purpose is to connect all of the piles together (such as bored piles and rotary excavation piles) to prevent the top edge of the foundation pit from collapsing. The waist beam is a reinforced concrete or steel beam located below the top of the retaining structure. Its function is to transmit the bearing capacity of the support and anchor cable, which changes the support effect from a point to a straight line, thereby improving the stability and integrity of the retaining structure. Therefore, the spatial deformation of the pile-anchorretaining structure is formed by the combined action of the crown beam, waist beam, piles, and anchor cables, as shown in Figure 1.

Crown Beam Deformation
Multiple sets of measured data and numerical simulation results have consistently shown that the deformation of the crown beam exhibits substantial spatial effects [13][14][15]. At the specific corner of a rectangular foundation pit, it is notable that the crown beam exhibits minimal to no horizontal displacement. Conversely, the maximum displacement is observed precisely at the middle of the foundation pit. This simplified deformation mode can be visually understood by referring to Figure 2.

Crown Beam Deformation
Multiple sets of measured data and numerical simulation results have consistently shown that the deformation of the crown beam exhibits substantial spatial effects [13][14][15]. At the specific corner of a rectangular foundation pit, it is notable that the crown beam exhibits minimal to no horizontal displacement. Conversely, the maximum displacement is observed precisely at the middle of the foundation pit. This simplified deformation mode can be visually understood by referring to Figure 2.
Based on the foundation pit measurements, it was observed that the retaining s ture exhibited distinct spatial deformation characteristics. The structure as a whole un went lateral bending due to the active soil pressure. However, the deformation at the ners and fulcrum positions of the foundation pit was limited due to the constraints sulting in significantly smaller deformation of the wall compared to the middle sectio the foundation pit. It is evident that the spatial deformation characteristics of the retai structure are mainly attributable to its lateral bending resistance. Hence, the collectiv eral bending resistance of the retaining wall is referred to as the spatial deformation e of the retaining structure.
The pile-anchorretaining structure consists of a crown beam, a waist beam, piles, anchor cables. The crown beam is a reinforced concrete continuous beam arranged on of the retaining structure surrounding the foundation pit. Its purpose is to connect all o piles together (such as bored piles and rotary excavation piles) to prevent the top edg the foundation pit from collapsing. The waist beam is a reinforced concrete or steel b located below the top of the retaining structure. Its function is to transmit the bearing ca ity of the support and anchor cable, which changes the support effect from a point straight line, thereby improving the stability and integrity of the retaining structure. Th fore, the spatial deformation of the pile-anchorretaining structure is formed by the c bined action of the crown beam, waist beam, piles, and anchor cables, as shown in Figu   Figure 1. Pile-anchor retaining structure.

Crown Beam Deformation
Multiple sets of measured data and numerical simulation results have consiste shown that the deformation of the crown beam exhibits substantial spatial effects [13 At the specific corner of a rectangular foundation pit, it is notable that the crown b exhibits minimal to no horizontal displacement. Conversely, the maximum displacem is observed precisely at the middle of the foundation pit. This simplified deforma mode can be visually understood by referring to Figure 2.   After excavation of the foundation pit, the crown beam has a large rotation at the corner of the foundation pit, resulting in bending and torsion. The corner can be assumed to be a fixed support, which is a constrained torsion problem. The crown beam deformation curve can be expressed as Equation (1): In this formula, δ is the displacement of the crown beam at a certain distance x from the corner of the foundation pit, δ 0 is the displacement of the crown beam in the middle of the foundation pit towards the foundation pit, which is also the displacement of the pile top in the middle of the foundation pit, and l is the side length of the foundation pit.

Pile Deformation
According to research, the deformation of piles in pile-anchorretaining structures consists of both horizontal and vertical deformations. However, horizontal deformation is generally considered to be the main deformation mode of piles [16][17][18]. Previous studies have shown that the maximum horizontal deformation of piles usually occurs near the pit bottom [19][20][21]. Therefore, this study assumes that the horizontal deformation curve of piles can be approximated by a quadratic curve fitting. The simplified horizontal deformation mode of the pile is shown in Figure 3. After excavation of the foundation pit, the crown beam has a large rotation at the corner of the foundation pit, resulting in bending and torsion. The corner can be assumed to be a fixed support, which is a constrained torsion problem. The crown beam deformation curve can be expressed as Equation (1): In this formula, δ is the displacement of the crown beam at a certain distance x from the corner of the foundation pit, δ0 is the displacement of the crown beam in the middle of the foundation pit towards the foundation pit, which is also the displacement of the pile top in the middle of the foundation pit, and l is the side length of the foundation pit.

Pile Deformation
According to research, the deformation of piles in pile-anchorretaining structures consists of both horizontal and vertical deformations. However, horizontal deformation is generally considered to be the main deformation mode of piles [16][17][18]. Previous studies have shown that the maximum horizontal deformation of piles usually occurs near the pit bottom [19][20][21]. Therefore, this study assumes that the horizontal deformation curve of piles can be approximated by a quadratic curve fitting. The simplified horizontal deformation mode of the pile is shown in Figure 3. Let us make the excavation depth of the foundation pit h = λH. H is the length of the pile. λ is the ratio of the depth of the foundation pit to the length of pile, abbreviated as the depth coefficient of the foundation pit. The displacement of the pile top is the horizontal deformation of the crown beam δ. Assume that the embedded end of the support pile does not displace and the maximum displacement of the support pile occurs at the bottom of the pit. Therefore, the displacement of the pit bottom u2 = ξδ, and ξ is the deformation curve shape coefficient. The lateral displacement u(x,z) of the pile can be expressed as Equation (2): In this formula, u (x, z) is the horizontal deformation of the pile at a depth z from the corner x of the foundation pit; n1, n2, and n3 are coefficients of the quadratic curve equation, n1 = (1 − λ − ξ)/(λ(1 − λ)H 2 ), n2 = (λ2 + ξ + 1)/(λ(1 − λ)H), and n3 = 1. The coefficients n1 and n2 Let us make the excavation depth of the foundation pit h = λH. H is the length of the pile. λ is the ratio of the depth of the foundation pit to the length of pile, abbreviated as the depth coefficient of the foundation pit. The displacement of the pile top is the horizontal deformation of the crown beam δ. Assume that the embedded end of the support pile does not displace and the maximum displacement of the support pile occurs at the bottom of the pit. Therefore, the displacement of the pit bottom u 2 = ξδ, and ξ is the deformation curve shape coefficient. The lateral displacement u(x,z) of the pile can be expressed as Equation (2): In this formula, u (x,z) is the horizontal deformation of the pile at a depth z from the corner x of the foundation pit; n 1 , n 2 , and n 3 are coefficients of the quadratic curve equation, n 1 = (1 − λ − ξ)/(λ(1 − λ)H 2 ), n 2 = (λ 2 + ξ + 1)/(λ(1 − λ)H), and n 3 = 1. The coefficients n 1 and n 2 of the quadratic curve equation can be obtained through the boundary conditions and the two determined points.

Waist Beam Deformation
Assuming the height of the waist beam above the pile cap is h y = ξH, where ξ is the height coefficient of the waist beam, and that the deformation curve of the waist beam is consistent with that of the crown beam, the deformation calculation formula of the waist beam can be expressed as Equation (3):

Anchor Cable Deformation
When assuming that the anchor cable is connected to the waist beam and remains stationary without displacement, the horizontal deformation of the waist beam can be determined by calculating the horizontal component of the anchor cable deformation. The deformation of the anchor cable, represented as ∆, can be mathematically expressed using Equation (4): In this formula, θ is the angle between the anchor cable and the horizontal plane.

Distribution of Soil Pressure behind the Structure
The soil pressure model has a significant impact on the deformation analysis of foundation pits. Numerous monitoring data indicate that the displacement of retaining piles directly affects the distribution and magnitude of soil pressure. Previous studies suggest that the soil pressure on the back wall of the foundation pit retaining structure can be approximated as Rankine active soil pressure [22,23], as shown in Figure 3. The calculation formulas for active soil pressure P a and passive soil pressure P b are: In the given formula, the variables represent the following parameters: c: Cohesive force of the soil mass (kPa). These parameters play a significant role in determining the horizontal deformation of the waist beam based on the horizontal component of the anchor cable deformation.

Calculation of Strain Energy of Support Structure
In this study, the total potential energy of the support structure system is referred to as the resistance potential energy, while the potential energy generated by the soil on the support structure is referred to as the active potential energy. Specifically, the resistance potential energy comprises the bending strain energy of the crown beam, the bending strain energy of the waist beam, the bending strain energy of the support pile, and the elastic strain energy of the anchor cable. On the other hand, the active potential energy is determined by the effect of active and passive soil pressures on the support structure system. The expression for each strain energy is presented below.
Crown beam bending strain energy, W g : The setting heights of anchor cables and waist beams are h i = ξ i H. If there are M rows of waist beams in total, the strain energy, W yi , of the waist beam at the i-th row is: Assuming that a total of n piles is set in the retaining structure and the pile spacing is s, the bending strain energy, W z , of all piles is expressed as: Generally, the anchor cable is set at the same position as the waist beam. Assuming that the number of anchor cables and piles is the same, there are n anchor cables in a row with the same anchor cable spacing of s, and the stiffness K m = EA/L, where E, A, and L are the elastic modulus, cross-sectional area, and length of the anchor cable, respectively. The elastic strain energy of the anchor cable, W m , is given by: In summary, the total resistance potential energy, W D , of the foundation pit retaining structure is given by Equation (11): The expression of active soil pressure potential energy, W pa , is given by Equation (12): The expression of passive soil pressure potential energy, W pp , is given by Equation (13): The expression of the active potential energy received by the retaining structure is given by Equation (14):

Deformation Prediction Formula
Based on the expressions of the strain energy for the crown beam, waist beam, pile, and anchor cable, as well as the potential energy from the active and passive soil pressures, the total potential energy of the pile-anchorbeam support system can be expressed as Equation (15): The principle of minimum potential energy [24][25][26] states that the critical state of a structure corresponds to the first derivative of the total potential energy being 0. Therefore, the optimal equilibrium state of the structure can be determined by solving for the stationary value of the potential energy. The expression for the minimum potential energy principle is given by Equation (16): The displacement of the crown beam in the middle of the foundation pit, denoted as δ 0 , can be determined by solving Equation (16): In this formula, T pa is the unit active external potential energy of retaining structure. T pp is the unit passive external potential energy of the retaining structure. T gb is the unit deformation strain energy of the crown beam. T wbi is the unit deformation strain energy of the i-th row of waist beams. T sb is the unit deformation strain energy of the pile. T mi is the unit strain energy of the i-th row of anchor cables. After calculating the displacement of the crown beam in the middle of the foundation pit towards the foundation pit, denoted as δ 0 , the deformation curves of the crown beam and the horizontal deformation curves of each pile can be determined. Ultimately, the spatial deformation of the retaining structure can be determined.

Project Profile
The Changsha International Financial Center, located in Hunan Province, China, stands as the tallest building in the region, reaching a maximum height of 452 m. The foundation pit associated with this structure has an excavation depth of 31.80 m and covers a substantial area of 76,700 square meters. With its considerable depth and size, the foundation pit can be classified as deep and large. The foundation pit exhibits a predominantly rectangular shape, with the following approximate lengths for each side: east (168 m), south (488 m), west (136 m), and north (546 m). Please refer to Figure 4 for a visual representation.
The principle of minimum potential energy [24][25][26] states that the critical state of a structure corresponds to the first derivative of the total potential energy being 0. Therefore, the optimal equilibrium state of the structure can be determined by solving for the stationary value of the potential energy. The expression for the minimum potential energy principle is given by Equation (16): The displacement of the crown beam in the middle of the foundation pit, denoted as δ0, can be determined by solving Equation (16): In this formula, Tpa is the unit active external potential energy of retaining structure. Tpp is the unit passive external potential energy of the retaining structure. Tgb is the unit deformation strain energy of the crown beam. Twbi is the unit deformation strain energy of the i-th row of waist beams. Tsb is the unit deformation strain energy of the pile. Tmi is the unit strain energy of the i-th row of anchor cables. After calculating the displacement of the crown beam in the middle of the foundation pit towards the foundation pit, denoted as δ0, the deformation curves of the crown beam and the horizontal deformation curves of each pile can be determined. Ultimately, the spatial deformation of the retaining structure can be determined.

Project Profile
The Changsha International Financial Center, located in Hunan Province, China, stands as the tallest building in the region, reaching a maximum height of 452 m. The foundation pit associated with this structure has an excavation depth of 31.80 m and covers a substantial area of 76,700 square meters. With its considerable depth and size, the foundation pit can be classified as deep and large. The foundation pit exhibits a predominantly rectangular shape, with the following approximate lengths for each side: east (168 m), south (488 m), west (136 m), and north (546 m). Please refer to Figure 4 for a visual representation. The challenging aspect of this project is that the foundation pit is situated within a densely populated urban area, where existing buildings, roads, and underground pipes The challenging aspect of this project is that the foundation pit is situated within a densely populated urban area, where existing buildings, roads, and underground pipes are in close proximity. The strata within the foundation pit are characterized by their unstable nature, and any instability in the retaining structure can potentially result in severe disasters. The specific parameters of the strata materials are outlined in Table 1, which have been compiled based on the field investigation report.  are in close proximity. The strata within the foundation pit are characterized by their unstable nature, and any instability in the retaining structure can potentially result in severe disasters. The specific parameters of the strata materials are outlined in Table 1, which have been compiled based on the field investigation report.

Numerical Simulation Model
According to the engineering background in Section 4.1, the area on the west side of the foundation pit was selected to build a corresponding numerical simulation model. The applicability of the theoretical prediction formula is verified by comparing theoretical prediction results, numerical simulation results, and on-site monitoring data.

Numerical Simulation Model
According to the engineering background in Section 4.1, the area on the west side of the foundation pit was selected to build a corresponding numerical simulation model. The applicability of the theoretical prediction formula is verified by comparing theoretical prediction results, numerical simulation results, and on-site monitoring data.

Model Size and Boundary Conditions
To establish a numerical simulation model, the western area of the foundation pit is chosen as the focus. The length of the retaining structure on the west side of the foundation pit measures 136 m. Consequently, the model dimensions are adjusted accordingly, with a length of 140 m, a longitudinal length of 120 m, and a height of 60 m. The arrangement of soil layers, as well as the positioning of piles and anchor cables, aligns with the depiction shown in Figure 5. This approach ensures consistency between the numerical simulation model and the specified soil layer distribution and support system layout.
For this study, a numerical model is created utilizing the FLAC 3D finite difference software. The model consists of a total of 4,859,822 nodes, 3,689,843 elements, and is divided into seven groups. Figure 6 presents a visual representation of the comprehensive numerical simulation model. To establish a numerical simulation model, the western area of the foundation pit is chosen as the focus. The length of the retaining structure on the west side of the foundation pit measures 136 m. Consequently, the model dimensions are adjusted accordingly, with a length of 140 m, a longitudinal length of 120 m, and a height of 60 m. The arrangement of soil layers, as well as the positioning of piles and anchor cables, aligns with the depiction shown in Figure 5. This approach ensures consistency between the numerical simulation model and the specified soil layer distribution and support system layout.
For this study, a numerical model is created utilizing the FLAC 3D finite difference software. The model consists of a total of 4,859,822 nodes, 3,689,843 elements, and is divided into seven groups. Figure 6 presents a visual representation of the comprehensive numerical simulation model.
(a) overall model (b) retaining structure model

Material Parameters
The model comprises of five types of materials: soil, crown beam, waist beam, pile, and anchor cable. Solid elements are used to simulate the soil, crown beam, and pile. The soil follows the Mohr-Coulomb yield criterion, while the crown beam and pile follow the linear elastic constitutive equation. The waist beam and anchor cable are simulated using structural elements in the software. The waist beam uses beam elements, while the anchor cable uses cable elements, both following the linear elastic constitutive equation [27,28]. The soil layer parameters of this model, as per the design report, are listed in Table 1, while other material parameters are displayed in Table 2.

Material Parameters
The model comprises of five types of materials: soil, crown beam, waist beam, pile, and anchor cable. Solid elements are used to simulate the soil, crown beam, and pile. The soil follows the Mohr-Coulomb yield criterion, while the crown beam and pile follow the linear elastic constitutive equation. The waist beam and anchor cable are simulated using structural elements in the software. The waist beam uses beam elements, while the anchor cable uses cable elements, both following the linear elastic constitutive equation [27,28]. The soil layer parameters of this model, as per the design report, are listed in Table 1, while other material parameters are displayed in Table 2.

Calculation Steps
The model's calculations do not consider the incremental impact of excavation; instead, they focus on the final deformation of the retaining structure once the excavation reaches the design elevation. To achieve this, the calculations are divided into three steps: Step 1: Material parameters corresponding to the soil along the depth are assigned, and a geostatic step is performed to obtain the initial stress field.
Step 2: The displacement field from step 1 is cleared while retaining the initial stress field. Material parameters are assigned to the crown beam, pile, waist beam, and anchor cable. The contact surface between the pile and soil is activated, and calculations are performed to simulate the excavation and construction of the retaining structure.
Step 3: In step 3, the displacement field is reset to zero. The soil within the design elevation is completely removed, and calculations are conducted to simulate the excavation to the bottom of the pit.
By following these three steps, the model captures the final deformation of the retaining structure accurately as the excavation progresses, while considering the specific stages and parameters involved in the process. Figure 7 illustrates the calculation process employed for the theoretical predictions. It provides a visual representation of the steps and procedures involved in generating the theoretical predictions. By referring to this figure, one can easily grasp the sequential flow and methodology utilized in the calculation process for the theoretical predictions.

Calculation Steps
The model's calculations do not consider the incremental impact of excavation; instead, they focus on the final deformation of the retaining structure once the excavation reaches the design elevation. To achieve this, the calculations are divided into three steps: Step 1: Material parameters corresponding to the soil along the depth are assigned, and a geostatic step is performed to obtain the initial stress field.
Step 2: The displacement field from step 1 is cleared while retaining the initial stress field. Material parameters are assigned to the crown beam, pile, waist beam, and anchor cable. The contact surface between the pile and soil is activated, and calculations are performed to simulate the excavation and construction of the retaining structure.
Step 3: In step 3, the displacement field is reset to zero. The soil within the design elevation is completely removed, and calculations are conducted to simulate the excavation to the bottom of the pit.
By following these three steps, the model captures the final deformation of the retaining structure accurately as the excavation progresses, while considering the specific stages and parameters involved in the process. Figure 7 illustrates the calculation process employed for the theoretical predictions. It provides a visual representation of the steps and procedures involved in generating the theoretical predictions. By referring to this figure, one can easily grasp the sequential flow and methodology utilized in the calculation process for the theoretical predictions.

Comparative Analysis of Results
During on-site construction, horizontal monitoring points were arranged at the crown beam and pile. The horizontal monitoring points of the crown beam were measured using a level gauge with a horizontal spacing of 20 m, and the monitoring points were symmetrical about the middle of the foundation pit. The deep horizontal deformation of the retaining pile was monitored using an inclinometer, which consists of an inclinometer tube, a clinometer, and a digital readout. The inclinometer tube is made of 90 mm PVC and is buried and fixed in the pile wall, which is placed into the pile hole together with the reinforcement cage. The clinometer used is a CX-03E type borehole inclinometer with high sensitivity (total system accuracy: ±4 mm/15 m) and strong stability. The spacing of the inclinometers is 8 m, with five horizontal inclinometers within a depth of 40 m. The horizontal layout position is the same as the position of the horizontal monitoring points of the crown beam.

Comparative Analysis of Results
During on-site construction, horizontal monitoring points were arranged at the crown beam and pile. The horizontal monitoring points of the crown beam were measured using a level gauge with a horizontal spacing of 20 m, and the monitoring points were symmetrical about the middle of the foundation pit. The deep horizontal deformation of the retaining pile was monitored using an inclinometer, which consists of an inclinometer tube, a clinometer, and a digital readout. The inclinometer tube is made of 90 mm PVC and is buried and fixed in the pile wall, which is placed into the pile hole together with the reinforcement cage. The clinometer used is a CX-03E type borehole inclinometer with high sensitivity (total system accuracy: ±4 mm/15 m) and strong stability. The spacing of the inclinometers is 8 m, with five horizontal inclinometers within a depth of 40 m. The horizontal layout position is the same as the position of the horizontal monitoring points of the crown beam.
In the theoretical calculations, it is assumed that the strata are uniformly distributed, and the equivalent soil layer method can be used to convert non-uniform strata into uniformly distributed soil layers. Numerous studies have proven the applicability of the equivalent soil layer method [29][30][31]. Other parameters can be calculated based on design data, but currently, there is no research indicating a clear value for the pile curve shape coefficient ξ. In this study, a value of 1.25 was chosen for ξ.

Comparison of Crown Beam Deformation
It is observed from Figure 8 that the deformation of the crown beam is symmetrically distributed, with smaller deformations occurring at the corners of the foundation pit and larger deformations occurring at the middle position of the crown beam. This is because the numerical simulation model imposes a constraint on the deformation in the Y direction at the corner of the foundation pit. With the constraint on the deformation in the Y direction of the two points at the corner, the crown beam is equivalent to a simply supported beam. Under the combined influence of the soil pressure exerted on the retaining wall and the deformation of the pile, the crown beam demonstrates a deformation pattern akin to that of a simply supported beam subjected to a uniform load. This deformation behavior is a result of the forces and displacements acting on the crown beam due to the surrounding soil pressure and the response of the supporting pile system.
In the theoretical calculations, it is assumed that the strata are uniformly distributed, and the equivalent soil layer method can be used to convert non-uniform strata into uniformly distributed soil layers. Numerous studies have proven the applicability of the equivalent soil layer method [29][30][31]. Other parameters can be calculated based on design data, but currently, there is no research indicating a clear value for the pile curve shape coefficient ξ. In this study, a value of 1.25 was chosen for ξ.

Comparison of Crown Beam Deformation
It is observed from Figure 8 that the deformation of the crown beam is symmetrically distributed, with smaller deformations occurring at the corners of the foundation pit and larger deformations occurring at the middle position of the crown beam. This is because the numerical simulation model imposes a constraint on the deformation in the Y direction at the corner of the foundation pit. With the constraint on the deformation in the Y direction of the two points at the corner, the crown beam is equivalent to a simply supported beam. Under the combined influence of the soil pressure exerted on the retaining wall and the deformation of the pile, the crown beam demonstrates a deformation pattern akin to that of a simply supported beam subjected to a uniform load. This deformation behavior is a result of the forces and displacements acting on the crown beam due to the surrounding soil pressure and the response of the supporting pile system.  Figure 9 shows the crown beam deformation curve obtained by the theoretical calculation method, numerical simulation method, and on-site data. It can be concluded that: (1) Theoretical calculations of the crown beam's deformation align with the observed deformation curve derived from on-site monitoring data. The horizontal deformation curve of the crown beam exhibits a distinctive "V"-shaped distribution. Conversely, the numerical simulation method produces a horizontal deformation curve that shows a "U"shaped distribution. These findings highlight the correspondence between the theoretical predictions and the actual behavior of the crown beam, as well as the insights gained from the numerical simulation method. This is because the pile and the crown beam are integral structures in the numerical simulation, resulting in a greater stiffness of the retaining structure. The deformation gap in the middle area of the foundation pit is small, while the deformation gap in the surrounding area is large. During the theoretical calculation and on-site construction, there is no tie contact between the crown beam and the pile, which differs from the situation in the numerical simulation method.
(2) Comparing the results obtained from the theoretical calculation, numerical simulation, and on-site data, it is evident that the maximum horizontal deformations of the crown beam are 13.23 mm, 12.44 mm, and 12.60 mm, respectively. These values demonstrate that the differences between the three methods are relatively small. This finding  Figure 9 shows the crown beam deformation curve obtained by the theoretical calculation method, numerical simulation method, and on-site data. It can be concluded that: (1) Theoretical calculations of the crown beam's deformation align with the observed deformation curve derived from on-site monitoring data. The horizontal deformation curve of the crown beam exhibits a distinctive "V"-shaped distribution. Conversely, the numerical simulation method produces a horizontal deformation curve that shows a "U"shaped distribution. These findings highlight the correspondence between the theoretical predictions and the actual behavior of the crown beam, as well as the insights gained from the numerical simulation method. This is because the pile and the crown beam are integral structures in the numerical simulation, resulting in a greater stiffness of the retaining structure. The deformation gap in the middle area of the foundation pit is small, while the deformation gap in the surrounding area is large. During the theoretical calculation and on-site construction, there is no tie contact between the crown beam and the pile, which differs from the situation in the numerical simulation method.
(2) Comparing the results obtained from the theoretical calculation, numerical simulation, and on-site data, it is evident that the maximum horizontal deformations of the crown beam are 13.23 mm, 12.44 mm, and 12.60 mm, respectively. These values demonstrate that the differences between the three methods are relatively small. This finding supports the practicality of both the theoretical calculation and the numerical simulation model. It is important to note that the theoretical calculation results and the numerical simulation results are presented as smooth curves, as they do not consider the variation in soil layers or construction steps. supports the practicality of both the theoretical calculation and the numerical simulation model. It is important to note that the theoretical calculation results and the numerical simulation results are presented as smooth curves, as they do not consider the variation in soil layers or construction steps.  Figure 10 shows that the distribution of horizontal deformation of a single pile is small at the top and bottom, but large in the middle region, consistent with the assumed pile deformation curve in Figure 3. The top area of the pile undergoes less soil pressure behind the pile, resulting in relatively small deformation at the top of the pile. At the bottom of the pile, the pile mainly experiences passive soil pressure, leading to limited deformation. Due to the high active soil pressure in the middle of the pile and the limited deformation on both sides of the pile, the maximum deformation ultimately occurs in the middle of the pile. Furthermore, it can be inferred that the maximum horizontal deformation of piles increases for piles closer to the center of the foundation pit. This observation aligns with the assumption made in the theoretical calculation. Moreover, the maximum horizontal deformation of the pile is directly related to the maximum horizontal deformation of the crown beam. The point of maximum horizontal deformation for the crown beam corresponds to the centerline of the foundation pit, where the horizontal deformation of the pile also reaches its maximum value. This correlation further supports the relationship between the deformations of the pile and the crown beam in the context of the foundation pit. Figure 11 illustrates the horizontal deformation curve of piles at x = 0 m, x = 35 m, and x = 70 m Several observations can be made from the figure:  Figure 10 shows that the distribution of horizontal deformation of a single pile is small at the top and bottom, but large in the middle region, consistent with the assumed pile deformation curve in Figure 3. The top area of the pile undergoes less soil pressure behind the pile, resulting in relatively small deformation at the top of the pile. At the bottom of the pile, the pile mainly experiences passive soil pressure, leading to limited deformation. Due to the high active soil pressure in the middle of the pile and the limited deformation on both sides of the pile, the maximum deformation ultimately occurs in the middle of the pile. supports the practicality of both the theoretical calculation and the numerical simulation model. It is important to note that the theoretical calculation results and the numerical simulation results are presented as smooth curves, as they do not consider the variation in soil layers or construction steps.  Figure 10 shows that the distribution of horizontal deformation of a single pile is small at the top and bottom, but large in the middle region, consistent with the assumed pile deformation curve in Figure 3. The top area of the pile undergoes less soil pressure behind the pile, resulting in relatively small deformation at the top of the pile. At the bottom of the pile, the pile mainly experiences passive soil pressure, leading to limited deformation. Due to the high active soil pressure in the middle of the pile and the limited deformation on both sides of the pile, the maximum deformation ultimately occurs in the middle of the pile. Furthermore, it can be inferred that the maximum horizontal deformation of piles increases for piles closer to the center of the foundation pit. This observation aligns with the assumption made in the theoretical calculation. Moreover, the maximum horizontal deformation of the pile is directly related to the maximum horizontal deformation of the crown beam. The point of maximum horizontal deformation for the crown beam corresponds to the centerline of the foundation pit, where the horizontal deformation of the pile also reaches its maximum value. This correlation further supports the relationship between the deformations of the pile and the crown beam in the context of the foundation pit. Figure 11 illustrates the horizontal deformation curve of piles at x = 0 m, x = 35 m, and x = 70 m Several observations can be made from the figure: Furthermore, it can be inferred that the maximum horizontal deformation of piles increases for piles closer to the center of the foundation pit. This observation aligns with the assumption made in the theoretical calculation. Moreover, the maximum horizontal deformation of the pile is directly related to the maximum horizontal deformation of the crown beam. The point of maximum horizontal deformation for the crown beam corresponds to the centerline of the foundation pit, where the horizontal deformation of the pile also reaches its maximum value. This correlation further supports the relationship between the deformations of the pile and the crown beam in the context of the foundation pit. Figure 11 illustrates the horizontal deformation curve of piles at x = 0 m, x = 35 m, and x = 70 m Several observations can be made from the figure:

Comparison of Pile Deformation
(1) The horizontal deformation curve of the pile exhibits a distinctive 'V'-shaped distribution, but the position of the maximum deformation differs between the numerical simulation, theoretical calculation, and on-site monitoring data. Specifically, the numerical simulation indicates that the maximum deformation occurs at z = −20 m, whereas the on-site monitoring data show the maximum deformation at z = −26 m. In contrast, the theoretical calculation predicts the maximum horizontal deformation of the pile to occur at z = −33 m. This discrepancy in the results can be attributed to the simplifications made in the model assumptions and the limitations of the calculation method. It is clear that further research is necessary to enhance the accuracy and reliability of the results.
(2) The horizontal deformation curve of a pile positioned at x = 70 m exhibits slight variations compared to a pile situated at x = 35 m. This phenomenon is further analyzed in Section 4.3.1. However, both the theoretical calculation and on-site monitoring data demonstrate that the deformation of a pile at x = 70 m is considerably larger than that of a pile at x = 35 m. In other words, for piles closer to the center line of the foundation pit, the horizontal deformation increases significantly. This observation confirms the trend that the horizontal deformation of the pile becomes more pronounced as it approaches the center of the foundation pit. The close agreement between the theoretical calculation, numerical simulation, and on-site monitoring data demonstrates the practicality of the proposed models. However, it should be noted that the current calculation method has limitations. Simplifications have been made in the analysis process, such as neglecting the influence of pit existence, precipitation, stratification of the foundation soil, and nonlinear effects of the pit and soil. Additionally, the non-uniformity of soil convergence has not been taken into account, which can introduce errors in the theoretical results. Future studies can explore these factors to improve the accuracy of the predictions. Furthermore, the current model assumes that the maximum horizontal deformation of the retaining structure occurs at the bottom of the pit, which may differ from the actual site conditions. Therefore, further improvements can be made to enhance the equation of the vertical deformation curve of the retaining structure.
(1) The horizontal deformation curve of the pile exhibits a distinctive 'V'-shaped dis tribution, but the position of the maximum deformation differs between the numerica simulation, theoretical calculation, and on-site monitoring data. Specifically, the numer cal simulation indicates that the maximum deformation occurs at z = −20 m, whereas th on-site monitoring data show the maximum deformation at z = −26 m. In contrast, th theoretical calculation predicts the maximum horizontal deformation of the pile to occu at z = −33 m. This discrepancy in the results can be attributed to the simplifications mad in the model assumptions and the limitations of the calculation method. It is clear tha further research is necessary to enhance the accuracy and reliability of the results.
(2) The horizontal deformation curve of a pile positioned at x = 70 m exhibits sligh variations compared to a pile situated at x = 35 m. This phenomenon is further analyze in Section 4.3.1. However, both the theoretical calculation and on-site monitoring dat demonstrate that the deformation of a pile at x = 70 m is considerably larger than that of pile at x = 35 m. In other words, for piles closer to the center line of the foundation pit, th horizontal deformation increases significantly. This observation confirms the trend tha the horizontal deformation of the pile becomes more pronounced as it approaches th center of the foundation pit. . The clos agreement between the theoretical calculation, numerical simulation, and on-site monitor ing data demonstrates the practicality of the proposed models. However, it should be note that the current calculation method has limitations. Simplifications have been made in th analysis process, such as neglecting the influence of pit existence, precipitation, stratificatio of the foundation soil, and nonlinear effects of the pit and soil. Additionally, the non-un formity of soil convergence has not been taken into account, which can introduce errors i the theoretical results. Future studies can explore these factors to improve the accuracy o the predictions. Furthermore, the current model assumes that the maximum horizontal de formation of the retaining structure occurs at the bottom of the pit, which may differ from the actual site conditions. Therefore, further improvements can be made to enhance th equation of the vertical deformation curve of the retaining structure. Figure 11. Horizontal deformation curves of piles. Note: Ns in Figure 11 represents the numerical simulation results; Tc represents the theoretical calculation results; Fm represents on-site monitor ing data. Figure 11. Horizontal deformation curves of piles. Note: Ns in Figure 11 represents the numerical simulation results; Tc represents the theoretical calculation results; Fm represents on-site monitoring data.

Parameters Analysis on Deformation
The control effect of the pit enclosure structure is influenced by various factors, including stratigraphic parameters and enclosure structure design parameters. This study specifically focuses on the influence of enclosure design parameters on the control effect, considering the proposal of a new enclosure system. Other factors affecting the control effect of the enclosure structure have been extensively studied in previous research. The subsequent analysis primarily examines the impact of the embedment depth of the pile, the size of the rib beam, and the size of the waist beam on the control effect based on the design parameters of the new enclosure. Building upon the engineering case discussed in Section 4, this study investigates the influence of crown beam size and material, waist beam size and material, and pile characteristics on the horizontal deformation of the piles located in the center of the foundation pit. The objective is to provide valuable insights for parameter design and material selection in similar engineering projects.

Waist Beam Size and Strength
The impacts of the size and material strength of the waist beam on the horizontal deformation of the pile at x = 70 m are presented in Figure 12a and Figure 12b, respectively. It can be observed that as the size and strength of the waist beam increase, the horizontal deformation of the pile gradually decreases. This is attributed to the increase in overall stiffness of the structure. However, since the waist beam has a limited effect on the overall deformation of the structure, the horizontal deformation of the pile does not change significantly with the increase in size and strength. The findings of this analysis can serve as a useful reference for the design of parameters and selection of materials for similar engineering projects.

Parameters Analysis on Deformation
The control effect of the pit enclosure structure is influenced by various factors, including stratigraphic parameters and enclosure structure design parameters. This study specifically focuses on the influence of enclosure design parameters on the control effect, considering the proposal of a new enclosure system. Other factors affecting the control effect of the enclosure structure have been extensively studied in previous research. The subsequent analysis primarily examines the impact of the embedment depth of the pile, the size of the rib beam, and the size of the waist beam on the control effect based on the design parameters of the new enclosure. Building upon the engineering case discussed in Section 4, this study investigates the influence of crown beam size and material, waist beam size and material, and pile characteristics on the horizontal deformation of the piles located in the center of the foundation pit. The objective is to provide valuable insights for parameter design and material selection in similar engineering projects.

Waist Beam Size and Strength
The impacts of the size and material strength of the waist beam on the horizontal deformation of the pile at x = 70 m are presented in Figure 12a and Figure 12b, respectively. It can be observed that as the size and strength of the waist beam increase, the horizontal deformation of the pile gradually decreases. This is attributed to the increase in overall stiffness of the structure. However, since the waist beam has a limited effect on the overall deformation of the structure, the horizontal deformation of the pile does not change significantly with the increase in size and strength. The findings of this analysis can serve as a useful reference for the design of parameters and selection of materials for similar engineering projects.

Pile Size and Strength
In Figure 13a, the horizontal deformation curve of the pile at x = 70 m is depicted for various pile sizes, while Figure 13b displays the same curve for piles with different material strengths. It can be observed that the horizontal deformation of the pile gradually decreases as the pile size and strength increase. This phenomenon is attributed to the fact that larger and stronger piles possess higher bending stiffness, resulting in reduced horizontal deformation. Consequently, modifying the pile size and material strength has the potential to significantly enhance the overall stiffness of the retaining structure, leading to a reduction in the horizontal deformation of the pile.

Pile Size and Strength
In Figure 13a, the horizontal deformation curve of the pile at x = 70 m is depicted for various pile sizes, while Figure 13b displays the same curve for piles with different material strengths. It can be observed that the horizontal deformation of the pile gradually decreases as the pile size and strength increase. This phenomenon is attributed to the fact that larger and stronger piles possess higher bending stiffness, resulting in reduced horizontal deformation. Consequently, modifying the pile size and material strength has the potential to significantly enhance the overall stiffness of the retaining structure, leading to a reduction in the horizontal deformation of the pile.

Crown Beam Size and Strength
In Figure 14, the horizontal deformation curve of the pile at x = 70 m is shown for varying sizes and material strengths of the crown beam. It can be observed that these parameters have minimal impact on the horizontal deformation of the pile. Based on this analysis, it can be concluded that, within the pile-anchorretaining structure system, increasing the size of the pile is the most effective measure for controlling the deformation of the retaining structure.

Conclusions
This study focuses on the analysis of pile-anchorretaining structures in foundation pit engineering. A simplified spatial deformation model of the retaining structure is used, along with the principle of minimum potential energy, to derive a theoretical prediction formula for deformation. The formula is validated through comparisons with on-site monitoring data and numerical simulation results. The study also investigates the influence of various parameters on the deformation of the retaining structure. The key findings are as follows:

Crown Beam Size and Strength
In Figure 14, the horizontal deformation curve of the pile at x = 70 m is shown for varying sizes and material strengths of the crown beam. It can be observed that these parameters have minimal impact on the horizontal deformation of the pile. Based on this analysis, it can be concluded that, within the pile-anchorretaining structure system, increasing the size of the pile is the most effective measure for controlling the deformation of the retaining structure.

Crown Beam Size and Strength
In Figure 14, the horizontal deformation curve of the pile at x = 70 m is shown for varying sizes and material strengths of the crown beam. It can be observed that these parameters have minimal impact on the horizontal deformation of the pile. Based on this analysis, it can be concluded that, within the pile-anchorretaining structure system, increasing the size of the pile is the most effective measure for controlling the deformation of the retaining structure.

Conclusions
This study focuses on the analysis of pile-anchorretaining structures in foundation pit engineering. A simplified spatial deformation model of the retaining structure is used, along with the principle of minimum potential energy, to derive a theoretical prediction formula for deformation. The formula is validated through comparisons with on-site monitoring data and numerical simulation results. The study also investigates the influence of various parameters on the deformation of the retaining structure. The key findings are as follows:

Conclusions
This study focuses on the analysis of pile-anchorretaining structures in foundation pit engineering. A simplified spatial deformation model of the retaining structure is used, along with the principle of minimum potential energy, to derive a theoretical prediction formula for deformation. The formula is validated through comparisons with on-site monitoring data and numerical simulation results. The study also investigates the influence of various parameters on the deformation of the retaining structure. The key findings are as follows: (1) The theoretical prediction formula and numerical simulation model demonstrate practical applicability, with small errors observed when compared to on-site monitoring results.
(2) The crown beam exhibits a 'V'-shaped horizontal deformation curve, and the proximity to the center line of the foundation pit correlates with greater horizontal deformation. The maximum horizontal deformation of the crown beam is measured at 13.23 mm (theoretical), 12.44 mm (numerical simulation), and 12.60 mm (on-site monitoring). The differences in the horizontal distribution arise due to the treatment of the crown beam and pile as a whole in the numerical simulation model. (3) The horizontal deformation curve of the pile also exhibits a 'V'-shaped distribution.
The maximum horizontal deformation of the pile is observed at different depths (z = 33 m, z = 20 m, and z = 26 m) in the theoretical calculation, numerical simulation, and on-site monitoring, respectively. This discrepancy is attributed to the simplifications made in the theoretical calculation and numerical simulation, which do not account for the actual excavation steps. Future research can focus on modifying the spatial deformation curve of the retaining structure based on these findings. (4) The size and material strength of the waist beam and crown beam have negligible influence on the horizontal deformation of the pile. However, increasing the size and material strength of the pile significantly enhances the overall stiffness of the retaining structure, thereby reducing the horizontal deformation of the pile. Therefore, improving the size and material strength of the pile is crucial in minimizing the deformation of the retaining structure.
Overall, this study provides valuable insights into the prediction and control of deformation in pile-anchorretaining structures, offering guidance for parameter design and material selection in similar engineering projects.