Review of the Double-Row Pile Supporting Structure and Its Force and Deformation Characteristics

Abstract

The conventional support forms of foundation pit retaining piles include single-row piles, double-row piles, anchor-row piles, and so on. The double-row pile supporting structure is widely used in the deep foundation pit supporting the engineering of wharves, bridges, subways, tunnels, and high-rise and super-high-rise buildings. This study on double-row pile supporting structures mainly focuses on four aspects: (1) The influence of dimension parameters, such as pile diameter and pile length, and engineering parameters, such as pile spacing and row spacing, on the deformation control of a double-row pile structure and the stability control of foundation pits; (2) Influence of the soil arch effect on the stress and deformation of the double-row pile supporting structure; (3) Study on the deformation characteristics and rules of the components and the whole structure of the double-row pile supporting structure; (4) Study on the calculation model of pile-soil interactions. Based on the above four aspects, this paper summarizes the latest research status of the existing double-row pile supporting structure and its stress and deformation characteristics. The deformation characteristics and calculation model of the pile-soil interaction of double-row piles are reviewed and evaluated. Finally, the problems and deficiencies in the research on double-row pile support are summarized. These results provide a reference for future research on the double-row pile supporting structure of the foundation pit and the numerical analysis and calculation model and lay a solid foundation for further development of the theory.

1. Introduction

Double-row pile supporting systems are mainly used in temporary engineering as a protection measure to ensure the smooth construction of the underground section of the main part of the structures and the safety of the environment around the foundation pit [1,2,3,4]. During the process of urbanization, the density of high-rise, super-high-rise, and underground buildings in cities has increased, and the utilization rate of underground space has been increasing. Influenced by the foundation of adjacent structures and the complex underground environment, the double-row pile supporting structure has attracted wide attention [5,6,7,8]. Based on the existing single-row pile, and by adding a row of piles behind it, the double-row pile supporting structure evolves from this. The double-row pile structure is mainly composed of the front pile, the back pile, the crown beam, and the connecting beam [9,10,11,12,13,14]. As far as the single-row pile-supporting structure is concerned, the new type of pile-row-supporting structure has the good stress and deformation characteristics of the single-row pile-supporting structure and has better stiffness and excellent anti-lateral displacement ability [15,16,17,18,19,20,21]. This makes the horizontal displacement and vertical displacement of the support section better controlled and able to meet the requirements of “solving the large height difference in a small space”; that is, it can have more prominent advantages in the case of a narrow construction space, strict deformation, and a deep excavation depth [22,23,24,25].

In recent years, research on double-row pile supporting structures has achieved significant results. For example: (1) The control effect of the pile diameter, the pile length, the crown beam, the connecting beam, and other basic structural dimensions, as well as pile spacing and row spacing, on the horizontal deformation and vertical settlement of the whole double-row pile structure system and the cost control of foundation pit engineering [26,27,28,29,30]; (2) The influence of the development of the soil arch effect on the deformation of the double-row pile supporting structure during the excavation of the foundation pit [31,32,33]; (3) Based on the numerical simulation analysis, the deformation mechanism and deformation characteristics of the overall double-row pile structure, the front pile, the back pile, the crown beam, the connecting beam, etc. [34,35,36]; (4) Optimization and innovation of the pile-soil interaction calculation model [37,38,39,40]. Most of the above studies have been based on numerical simulation analysis, experimental research, and other methods. Generally, they researched the dimension parameters of double-row pile supporting structures, pile-soil stress and deformation characteristics, and calculation models.

According to the available data, in the same foundation pit wall protection project, the size parameters of the components, such as pile diameter and pile length, and the design parameters, such as pile spacing and row spacing, are often variable. However, there is a node near which the structural parameters can give full play to their supporting role in controlling the deformation of the foundation pit. When it is far away from this node, the supporting effect of the double-row pile supporting structure will not only be greatly reduced but also increase the project cost in terms of cost control [41,42,43,44]. For example, Wang et al. [29] found through numerical modeling that the control effect of a double-row pile supporting structure on the stability of the foundation pit is similar to that of a single-row pile when the spacing of the double-row pile supporting structure is too small. Appropriately increasing the double-row pile gap can have a good control effect on reducing the level shift of the supporting structure. In addition, based on numerical modeling and scale tests, Lin et al. [45] found in their study on the connected beams of a double-row pile supporting structure that properly increasing the stiffness of the connected beams can play a certain control role in reducing the horizontal displacement of the supporting structure. However, when the stiffness increases, there is a node, and when the node is exceeded, the deformation control effect of the connecting beam on the double-row pile structure is no longer obvious, which provides a certain reference for the selection of the design parameters of the double-row pile members.

In the process of foundation pit excavation, it must be accompanied by the release, transfer, and balance of soil stress, which is the direct cause of the deformation of the retaining wall structure [46,47]. Zhao et al. [31] and Hu et al. [36] believed that this transmission of stress will inevitably lead to a relative shift between the earth and the pile and continues to expand outward, which causes the double-row pile supporting structure to bear the direct soil pressure and the indirect soil pressure transmitted by soil between piles. The supporting pile is deformed under the combined action of these two earth pressures. To study the development process and deformation law of double-row pile supporting structures, many scholars have used centrifugal testing machines to carry out centrifugal simulation tests and other methods to carry out deformation coordination-related research, gradually carried out a theoretical analysis of force deformation of double-row pile supporting structures, and optimized the related pile-soil calculation model [48]. In terms of theoretical analysis and calculation, the pile-soil calculation model is mainly based on classical earth pressure theory, earth arch effect theory, and Winkler theory [49,50,51]. Among the three calculation theories, the Winkler model can easily consider the change in foundation reaction, so the elastic foundation beam method based on the Winkler assumption has been the most widely used [52]. In the Winkler model, the double-row pile supporting structure is approximately assumed to be a vertically placed elastic foundation beam embedded in the soil, the earth pressure and other constraints are assumed to be equivalent springs, and the rod system finite element method is used to calculate the deformation and internal force. However, for example, how the soil shear effect is reflected in the equivalent spring stiffness coefficient k and the influence of the spatial effect of the connected beam and crown beam on the double-row pile support system still need to be further studied [53,54]. The above work specifically includes the following research directions: (1) study of double-row pile structure; (2) the influence of the soil arch effect on the double-row pile supporting structure; (3) deformation characteristics of double-row pile supporting structures; (4) the calculation model of pile-soil interaction. This article will be systematically sorted and summarized from the above four directions, with a focus on reviewing and commenting on the current research methods and conclusions on the mechanical deformation characteristics and pile-soil interaction calculation model of double-row pile supporting structures. Finally, the opportunities and challenges facing research on double-row pile supporting structures are described, and the application prospect and development trend of double-row pile supporting structures are forecasted.

2. Study of Double-Row Pile Structures

The design of basic parameters such as the pile diameter, the pile length, the size of the crown beam and connecting beam, the pile spacing, and the row spacing of double-row pile supporting systems is very important. A double-row pile structure model is shown in Figure 1. The change of these basic structural design parameters will often have a great impact on the whole pull pile supporting structure. Using reasonable structural design parameters will play an important role in the deformation control of the whole system. The use of reasonable structural design parameters will play a significant role in the deformation control of the entire system [55,56].

In the design of a double-row pile supporting system, the design of optimal row spacing plays an important role, and the change in row spacing directly affects the horizontal bearing capacity of the front pile. For example, Ilyas et al. [57], Ooi et al. [58], and Qian et al. [59] studied the horizontal displacement of double-row pile supporting structures under different row spacings and found that the appropriate increase in row spacing had a good control effect in reducing the horizontal displacement of the double-row pile structure. However, the internal force of the double-row pile supporting structure will increase with the increase in row distance between the front and back piles, and the retaining effect of the back pile will also decline [60]. The maximum horizontal displacement of piles can be effectively reduced by reducing the row spacing of double-row piles. However, when the row spacing is too small, the supporting role of the back row pile may be greatly reduced, and its anti-lateral movement ability cannot be fully brought into play. At this time, the double-row pile can be regarded as a single-row pile in terms of its supporting effect [38]. Because it is difficult to obtain actual engineering monitoring data, finite element software can be used to conduct numerical simulation research and qualitative analysis of the optimal row and pile spacing. Through numerical modeling analysis, optimal structural design parameters with certain reliability can be obtained, and the structural parameters with reasonable economy can be given in combination with engineering practice [61,62,63]. However, optimal row spacing has regional and project limitations in engineering, and different scholars place different emphases on the value of optimal row spacing based on the qualitative analysis of the numerical simulation model established. For example, Peng et al. [64] believed that when the pile diameter of a double-row pile in a deep foundation pit is 600–1200 mm, the most reasonable row spacing of the double-row pile supporting structure is three to four times the pile diameter. Rong et al. [65] believed that the horizontal displacement is most significantly controlled when the row spacing between the front and back piles is four to five times the pile diameter. Dong et al. [66] comprehensively considered that, from the perspectives of technology and economy, the optimal row spacing between the front and back piles was 1.6–2 times the pile diameter.

For the study of horizontal pile spacing of front and back piles, it is generally believed that increasing pile spacing can reduce the lateral displacement of double-row pile supporting structures, but the greater pile spacing means that greater earth pressure will be borne. Therefore, the pile spacing cannot be increased without restraint to support the control effect [67,68]. Ying et al. [69] found, through finite element numerical model analysis, that the displacement of double-row pile supporting structures could be reduced to a certain extent by appropriately reducing the distance between horizontal piles in the back row. However, reducing the horizontal pile spacing of the back row piles is equivalent to increasing the overall stiffness of the back row piles. At this time, the earth pressure on the back of the row piles will also increase, which will also increase the project cost. In summary, the deformation control function of a double-row pile structure can only be fully brought into play by matching the distance between rows and piles and having a reasonable construction scheme [70].

When the supporting structure is in the optimal row and pile spacing, the control effect of the connecting beam becomes a new research object. When there is no connecting beam, the supporting structure system only acts as the single-row pile supporting structure, and the deformation characteristics of the structure are not significantly different from those of the single-row pile supporting structure. For the pile row support structure with connected beams, connected beams can not only reduce the range of soil stress concentration between two groups of double-row piles, but also increase the overall mechanical performance of the load section and the overall stiffness of the structure [71,72,73]. In addition, increasing the stiffness of the beam can control the horizontal displacement of the supporting structure. However, the increase in the stiffness of connected beams has a node, and when the node is exceeded, the control effect of connected beams on the horizontal displacement and lateral torsion of the double-row pile supporting structure is no longer obvious, which also plays a certain reference role in the cost control [74,75]. In a study on crown beams, Qian et al. [59] found that reducing the stiffness of crown beams greatly increases the lateral displacement of double-row pile supporting structures. However, the effect of increasing the stiffness of the crown beam on reducing the lateral displacement of the double-row pile supporting structure was not obvious. Later, Gong et al. [76] studied the crown beam and connecting beam, considering their integrity based on their rigid connection. The rigid connection between the crown beam and the connecting beam makes the two a whole, which can effectively improve the lateral support stiffness coefficient of the rectangular double-row pile double beam and also plays a good control role in the lateral displacement constraint of the double-row pile supporting structure.

Drawing from a large amount of the literature and field monitoring data, it can be understood that the soil pressure of the front pile and the back pile in the double-row pile support system is different. For this reason, many scholars have carried out a lot of research on the characteristics of double-row pile support systems. Through the establishment of a numerical model to carry out numerical simulation analysis, it was found that the different embedment depths of the rear pile have an effect on the internal force distribution of the front and back piles, which further leads to the conclusion that the combination of long and short piles can give full play to the overall performance of the row pile structure [77,78,79]. However, Ying et al. [80] and Shen et al. [81] successively found, through physical simulation tests, that when the double-row pile structure adopts the front long and back short composite pile structure, the change of the back short pile length does not significantly improve the overall stability of the double-row pile support structure.

The selection of the pile section size is also one of the parameters that cannot be ignored in the control of foundation pit deformation and engineering costs. With the increase in the height-diameter ratio of the supporting pile, the horizontal displacement of the pile top of a double-row pile also increases. The excessive height-diameter ratio will pose a serious threat to the stability of the double-row pile retaining structure itself [82]. If the pile length is too short, resulting in an insufficient embedment depth and not meeting the requirements of the foundation pit anti-uplift, it will lead to a large horizontal displacement of the pile and even to the stability of the whole structural system declining. If the pile length is too long, it can meet the control requirements for pile displacement, but because the structure design is too conservative, the performance of the supporting pile will not only not be fully utilized but will also waste the project’s investment. The same principle applies to the increase in pile diameter, which has little effect on controlling the horizontal displacement of the pile top. The same is true for increasing the pile diameter. Excessive pile diameter has little effect on the horizontal displacement control of the pile top [83,84]. In recent years, due to the limitations of the site, a double-row pile supporting structure with an arched beam has been gradually developed, as shown in Figure 2.

Therefore, Li et al. [85,86] conducted a comparative study on the double-row pile supporting structure with an arched beam and the ordinary double-row pile supporting structure. The spatial supporting structure pile formed by the combination of an arched beam and a double-row pile was more conducive to controlling the deformation of the foundation pit. Thus, the arch beam has been widely used in all kinds of double-row pile layouts. This provides some engineering experience and theoretical reference for reducing cost expenditure and selecting reasonable design schemes and design parameters in restricted sites [87].

3. Influence of the Soil Arch Effect on Double-Row Pile Supporting Structures

With the excavation of the foundation pit, the soil arch effect is gradually generated and expanded, which is bound to have a certain influence on the stability and supporting effect of the double-row pile supporting structure. This section mainly analyses the relative motion state and interaction between the soil and the pile when the soil arching effect occurs and expands. The deformation mechanism of the double-row pile structure caused by the soil arch effect and the influence of the soil arch effect on the stress and deformation characteristics of the double-row pile supporting structure are reviewed.

According to the available data, Karl [88] first named the arch effect phenomenon, which means that the load in the unstable soil will be transmitted to the surrounding rigid boundary. Wang [89] believed that the soil arching effect is a phenomenon of soil shear stress transfer. In the process of foundation pit excavation, the double-row pile supporting structure will produce displacement and flexural deformation in the direction of the pit when it bears the earth pressure generated by the soil mass under the action of dead weight or overload. Then, the relative displacement between the supporting pile and the soil behind the pile will occur, and the soil will also deform due to the shear friction between the soil particles. At this time, the stress of the deformed soil will be released and transferred to the undeformed soil; that is, the earth arch effect occurs, and the collapse area before the arch is formed [90,91,92], as shown in Figure 3. Wang et al. [93] and Hu et al. [36] found that the outward expansion of the soil arch effect would make supporting piles bear additional indirect soil pressure transmitted between adjacent piles or even between pile groups based on direct soil action. The superposition of heavy earth pressure will cause the double-row pile supporting structure to continue to produce displacement and deflection in the direction of the pit.

Different from single-row pile supporting structures, double-row pile supporting structures will produce a soil arch effect around the front and back piles, and the supporting system will be affected by the superimposed soil arch effect of these two parts. Yang et al. [94] found through research that properly increasing the horizontal pile spacing in rows of piles could reduce the area of multiple soil arches as a whole, thus reducing the stress of the multiple soil arch effect as a whole. However, if the soil arch effect zones of the front pile and the back pile are compared and analyzed, the soil arch effect zone around the back pile is reduced, while the soil arch effect zone of the front pile is increased. At this time, the residual load of the front pile will increase, which is unfavorable to the horizontal bearing performance of the front pile [31,32,33]. Chen et al. [95] and Wu et al. [47] considered the double soil arching effect under the influence of the double-row pile support system, and the theoretical calculation value of the earth pressure is closer to the monitoring value of the earth pressure.

In foundation pit wall protection engineering, there is soil pressure distribution transfer and balance at every stage of the development of the soil arch effect, which will always accompany the engineering. To simplify the calculation and facilitate analysis, the soil arching effect on a two-dimensional model is usually studied with emphasis; that is, a vertical section is taken out for analysis. However, the three-dimensional effect of soil cannot be ignored. The three-dimensional soil arching effect has a great influence on the supporting effect of the supporting structure [53,84]. Furthermore, the mechanism of the soil arching effect on the spatial effect and deformation coordination of double-row pile retaining structures is not very clear, and more theoretical analysis and sample data are needed to verify it.

4. Deformation Characteristics of Double-Row Pile Supporting Structures

In the process of foundation pit excavation, the deformation of the double-row pile structure will run through the whole project. Due to the disharmony between pile displacement and soil deformation, the displacement of the pile body is greater than that of the soil, which leads to the phenomenon of pile-soil separation and further causes the bending and horizontal displacement of the pile end [60,92]. This section mainly summarizes and analyses the deformation characteristics of the whole double-row pile structure and each component, which can play a certain reference role in how to effectively control the deformation of the foundation pit.

He et al. [96], Zhao et al. [97], and Han et al. [98] found that double-row pile supporting structures can give better play to their overall rigidity characteristics and have a better deformation control effect than single-row pile supporting structures through comparative analysis. Based on the excavation project of the square in front of Huaian East Station, Zhou et al. [99] selected different excavation sequences for different pit locations and found that the excavation sequence from far to near could reduce the surface settlement and reduce the deformation of the foundation pit supporting and retaining structure. However, it is not sufficient to control the horizontal displacement of the foundation pit only using a reasonable excavation sequence, which should be combined with other influencing factors. The specific application effect needs to be further studied [100,101]. Based on the analysis of practical engineering examples, the deformation and stress of the double-row pile structure in the excavation process can be numerically simulated by using three-dimensional finite element theory to analyze the stress and deformation characteristics of the double-row pile structure [102,103,104]. Zhou et al. [37], Zheng et al. [105], and Hu et al. [106], through numerical simulation tests, concluded that the strain and lateral displacement of piles in the front row were larger than those in the back row. Piles in the front row were mainly subjected to bending deformation, while piles in the back row were subjected to tensile deformation. Zhou et al. [107] and Zheng et al. [108] concluded that the front pile in the middle of the long side of the foundation pit would bear a larger bending moment and have a larger bending deformation, while the back pile would mainly play the role of uplift. Li et al. [44] used the “space-time effect” to reduce the deformation of the supporting structure in the double-row pile supporting project of a deep foundation pit adjacent to the subway and concluded that the deformation rule of the double-row pile in the supporting process is similar to that of a cantilever pile. Under the influence of pipeline excavation, shifting, and backfilling, the top of the supporting pile will appear deformed and show a sudden change. Chen et al. [109] and Zhao et al. [110] further found that when the overall stiffness of the supporting structure is too small, it will cause excessive horizontal deformation as well as pile torsional deformation. The deformation of a double-row pile supporting structure is mainly horizontal, and the deformation of the pile structure accounts for a large proportion. The maximum horizontal displacement usually occurs at the top of the supporting pile, and the horizontal displacement along the deep direction of the pile shows a decreasing trend [35,111].

Xiong et al. [112] studied the influence of different embedment depths of supporting piles on the deformation characteristics of double-row pile support structures and the stability of foundation pits through numerical modeling and other methods. The results showed that the deeper the embedment depth of supporting piles, or the harder the soil or rock in the embedment area of the foundation pit, the higher the stability of the double-row pile support structure and foundation pit. Wei et al. [113] further studied the influence of expansive soil moisture content on the supporting structure of foundation pits through numerical modeling and found that the maximum displacement and bending moment of supporting piles were positively correlated with the moisture content of expansive soil. Moreover, once the moisture content of expansive soil exceeds 24.4%, the maximum displacement and bending moment of supporting piles will significantly increase. In deep foundation pit engineering, deformation control is often the most important factor. Nie et al. [114] also studied the influence of soil properties between the front and back rows of piles on the displacement of the supporting structure through numerical modeling, and concluded that the improvement of soil between piles within 2~4 m below the top of the supporting piles has little effect on reducing the flexural deformation of the piles.

Du et al. [115] and Wang et al. [116] analyzed the support forms of negative angles and positive angles. The deformation characteristics of the negative-angle double-row pile supporting structure are the same as those of the double-row pile supporting structure in the general position. The deformation characteristics of the front and back piles of the double-row pile supporting structure with a positive angle are the same as those of the negative-angle pile. However, the displacement relationship between the lateral piles of the double-row pile supporting structure with a positive angle is opposite to that of the negative-angle type. The total deformation of the pillar pile at the corner is the largest, and the further away from it, the smaller the deformation. It is a pity that the deformation characteristics of the crown beam and connecting beam were not analyzed in the analysis. According to the statistics, the horizontal deformation characteristic curve of the crown beam of the double-row pile supporting structure is a sinusoidal function [117,118,119]. However, with the development of The Times, the excavation area of foundation pit wall protection engineering is constantly expanding, and the size of the crown beam is also constantly increasing, which extends the deformation mode of the new crown beam with curves on both sides and a straight line in the middle. Ruan et al. [120] and Shen et al. [121] analyzed the difference in deformation modes between large and small crown beams, and the research showed that the intensity of the “waveform characteristics” of crown beams was controlled by the depth of the foundation pit and the expression of “waveform characteristics” was affected by its size. The spatially indeterminate structure composed of crown beams, connecting beams, and front and back piles can effectively reduce the horizontal displacement of the pile top. However, the deformation characteristics of crown beams and connecting beams are still in the description stage, and deformation characteristics such as ultra-large, ultra-long crowns, and linked beams are still to be explored [122,123].

5. Calculation Model of Pile-Soil Interaction

At present, the calculation model of pile-soil interaction is mainly based on the classical earth pressure theory, the soil arch effect theory, and the Winkler theory. The process of innovation and optimization of computational models essentially goes through theoretical field investigation, theoretical analysis, numerical simulation verification, and model establishment. The pile-soil effect is taken into account in the calculation model of the earth arch effect and the Winkler theory, so it has been more widely used [124]. This section mainly reviews the calculation model based on three theories and introduces the optimization process and development status of the calculation model based on the Winkler theory.

In the calculation model based on the classical theory of earth pressure, it is usually considered that excessive soil displacement is one of the main reasons for excessive lateral displacement and even the failure of double-row pile supporting structures. To quantitatively study the relationship between soil displacement and pile internal force, Tschebotarioff et al. [125], based on the Rankine earth pressure theory, which is based on soil pressure in the limiting state of a retaining wall, assumed that soil pressure was triangularly distributed along the depth direction of the pile in the pile row support, and then calculated the displacement and internal force of the pile, as shown in Figure 4.

Subsequently, domestic scholar Huang et al. [126] first proposed to assume the soil as a rigid-plastic body for rigid-plastic analysis. On this basis, He et al. [127] proposed the “volume ratio coefficient method” to determine the distribution coefficient according to the ratio of the weight of the sliding soil on both sides of the row pile.

$$\mathit{\alpha }=\frac{2\mathit{L}}{{\mathit{L}}_0}-{\left(\frac{\mathit{L}}{{\mathit{L}}_0}\right)}^2,$$

(1)

In the formula, L is the row distance and L₀ is the width of the sliding crack surface. However, because the distribution coefficient of this model is simple, it often causes an uneven distribution of passive earth pressure between the front and back rows of piles, resulting in an unclear value of the maximum bending moment.

Then, based on the research of Huang and He et al., Cheng et al. [128] assumed the sum of earth pressure between piles as Rankine active earth pressure and the pile row supporting structure as a steel truss statically indeterminate structure embedded in the soil [129], and then calculated the earth pressure.

$${\mathit{e}}_{\mathit{\alpha }\mathit{F}}{=\mathit{e}}_{\mathit{\alpha }}{+\mathit{\alpha }\mathit{e}}_{\mathit{\alpha }}$$

(2)

$${\mathit{e}}_{\mathit{\alpha }\mathit{B}}{=\mathit{e}}_{\mathit{\alpha }}{-\mathit{\alpha }\mathit{e}}_{\mathit{\alpha }}$$

(3)

$$\mathit{\alpha }=\frac{2·\mathit{b}·\mathit{z}·{\mathit{ctg}\mathit{\eta }-\mathit{b}}^2}{{(\mathit{z}·\mathit{ctg}\mathit{\eta }-\mathit{b})}^2}$$

(4)

In the formula, η is the rupture angle; z is the depth; b is the width; α is the distribution coefficient of earth pressure; and e_α is the main earth pressure distribution intensity. In the assumption of the model, it is considered that the steel frame structure only bears the horizontal load, and the method of automatically adjusting the internal force of the double-row pile supporting structure is different in the process of not considering the change of external force. The model considers the beam connected by pile row and pile top as a steel frame structure with only the bottom fixed together, so that the pile row supporting system only shifts under the soil pressure without considering the angle caused by the uneven force, and then ignores that the pile row structure also bears the force couple action, the effect of soil between piles on supporting piles is ignored, and the soil arch effect is also ignored. Because it is based on the solution of force balance or moment balance under simple boundary conditions, the calculation is relatively simple, but the error is large, and the application is limited.

In the exploration of the mechanical model of double-row piles and earth pressure by foreign scholars, Springman [130] and Stewart [131] and other scholars calculated the distribution of earth pressure, the distribution of the bending moment, and the displacement of piles to explore the distribution of the pile-soil interaction force but did not consider the influence of the soil arching effect. With the gradual establishment of the soil arching effect theory, Richard et al. [132], Kingsley et al. [133], and Hu et al. [91] provided the stress calculation formula considering the soil arching effect, in which the vertical average stress σ_av and the pile side soil pressure σ_h are, respectively:

$${\mathit{\sigma }}_{\mathit{av}}=\frac{\mathit{V}}{\mathit{B}}=\frac{\mathit{\sigma }\mathit{B}}{{2\mathit{\mu }\mathit{K}}_{\mathit{w}}}{[1-\mathit{e}}^{(-\frac{{2\mathit{K}}_{\mathit{w}}\mathit{\mu }}{\mathit{B}}\mathit{h})}{]+\mathit{q}}_0{\mathit{e}}^{(-\frac{{2\mathit{K}}_{\mathit{w}}\mathit{\mu }}{\mathit{B}}\mathit{h})}$$

(5)

$${\mathit{\sigma }}_{\mathit{h}}{=\mathit{K}}_{\mathit{w}}\frac{\mathit{V}}{\mathit{B}}=\frac{\mathit{\gamma }\mathit{B}}{2\mathit{\mu }}{[1-\mathit{e}}^{(-\frac{{2\mathit{K}}_{\mathit{w}}\mathit{\mu }}{\mathit{B}}\mathit{h})}{]+\mathit{K}}_{\mathit{w}}{\mathit{q}}_0{\mathit{e}}^{(-\frac{{2\mathit{K}}_{\mathit{w}}\mathit{\mu }}{\mathit{B}}\mathit{h})}$$

(6)

In the formula, h is the soil depth along the vertically downward direction of the surface; V is the total vertical load when h = 0, V = q₀; K_w is the lateral earth pressure coefficient; γ is the weight of soil; B is the influence range of the soil stress arch; μ is the friction coefficient of the wall back, $\mathit{\mu }=\mathit{tan}\mathit{\delta }$, $\mathit{\delta }$ is the friction angle of wall soil. For rough wall backs, $\mathit{\delta }=\mathit{\phi }$ (friction angle) can be taken. The model assumes that the side wall of two adjacent piles is a parallel wall with a certain length, roughness, and no deformation. When the influence range of the soil arching effect in a row pile retaining structure is B, the soil pressure can be calculated by the geotechnical principle. However, the model is only for the analysis of lateral earth pressure when a single foundation pit is supported by a row pile. The deformation and internal force of the double-row pile supporting structure is not fully considered by the connection beam and crown beam.

In analyses of pile-soil interaction in double-row pile-supporting structures, the elastic foundation beam method proposed by Liu based on pile-soil interaction is widely used [134]. Liu et al. [135] considered the front pile, back pile, and connecting beam as three independent parts and, based on first studying the joint action of the connecting beam, lower elastic soil, and pile side soil pressure on the front and back pile, equivalent foundation soil was used as an independent equivalent spring to simulate its deformation properties to calculate the lateral resistance of soil to pile, as shown in Figure 5.

$${\mathit{p}}_{\mathit{x}}{=\mathit{b}}_0·{\mathit{c}}_{\mathit{y}}$$

(7)

In the formula, b₀ is the calculated width of the pile; c_y is calculated according to China’s method; and x is the lateral displacement of the pile. The model assumes the soil is a linear elastic body, which is quite different from the actual situation, so the calculation model does not apply to all cases. Scholars then, based on Liu Zhao’s model, further assumed that the soil around the pile was a discrete linear spring, and the internal force and deformation of the pile body were calculated by numerical modeling according to the finite element method of the elastic foundation beam [136]. At this time, the earth pressure borne by the supporting pile can be regarded as the reaction force caused by the spring deformation caused by the deformation of the pile supporting structure. However, the model assumes that the soil pressure, horizontal displacement, and angle of the pile are all distributed in a parabolic shape, neglecting the cohesive force and friction resistance between the pile and soil.

In addition, Zheng et al. [137] argued that the anti-overturning ability of double-row piles was very high, which could be assumed to be a steel frame embedded in the soil, and the foundation reaction coefficient was used to replace the equivalent spring stiffness for calculation, as shown in Figure 6.

$$\mathit{K}=\frac{{\mathit{E}}_{\mathit{s}}}{\mathit{H}}$$

(8)

In the formula, H is the thickness of soil between two rows of piles, and E_s is the horizontal compression modulus of soil between piles.

Later, Wu [60], based on the beam method of elastic foundation recommended by the code, considered that the pile-soil interaction and the existence of a sliding surface would lead to a triangular distribution of pile and earth pressure P_A above the pit bottom in the front row and a triangular distribution of soil pressure P_B above Z₀ in the active area on the back of the back row pile, as shown in Figure 7.

$${\mathit{Z}}_0=\mathit{H}-\mathit{L}\mathit{cot}(\frac{\mathit{\pi }}{4}-\frac{\mathit{\phi }}{2})$$

(9)

$${\mathit{p}}_{\mathit{F}}{=\mathit{K}}_{\mathit{sp}}{\mathit{\alpha }(\mathit{\gamma }\mathit{HK}}_{\mathit{a}}-2\mathit{c}\sqrt{{\mathit{K}}_{\mathit{a}}})$$

(10)

$${\mathit{p}}_{\mathit{B}}{=\mathit{K}}_{\mathit{sp}}{\left\{\right(1-\mathit{\alpha }\left)\right(\mathit{\gamma }\mathit{HK}}_{\mathit{a}}-2\mathit{c}\sqrt{{\mathit{K}}_{\mathit{a}}}{)+\mathit{K}}_{\mathit{a}}\mathit{q}\}$$

(11)

In the formula, H is the depth of the foundation pit; L is the row spacing; K_a is the main dynamic earth pressure coefficient; q is the additional load on the soil surface outside the rear row piles; K_sp is the spatial effect coefficient of earth pressure. According to the proportion obtained by experience, the formula artificially distributes the earth pressure borne by the rear pile, separately studies the front pile and the rear pile, and divides the pile body into micro sections for equilibrium calculation. In this way, the problem of pile-soil interaction is weakened, and the calculation model cannot truly and effectively reflect the interaction between the whole pile supporting structure and the soil between piles.

Under the action of horizontal load, with the increase in pile displacement, the soil around the pile will also be compressed, and the soil in some large displacement areas enters a plastic state, and the soil resistance will not increase with the increase in displacement. Based on the numerical modeling, the nonlinear expression of the pile displacement and the soil reaction force is obtained by the p-y curve method and the proportional coefficient method, which can well reflect the effect of the double-row pile crown beam [138,139]. On the whole, the calculation model based on the soil arch effect and the Winkler foundation model can obtain more accurate simulation results and is widely used. However, these three classical calculation models do not take the pile row supporting structure as a whole space system to calculate the whole, and the relationship between the deformation coordination effect of the pile, crown beam, and connected beam and earth pressure needs to be further studied.

Based on Wu et al. [60], Deng et al. [140], and other scholars who established a calculation model of pile-soil interaction in pile-soil supporting structures from different angles, scholars initially studied the spatial effect in the excavation process of the foundation pit and the influence of the crown beam on soil pressure and preliminarily analyzed the pile-soil interaction mechanism. After that, Nie et al. [141] and Bai et al. [142] initially established a calculation model considering the overall spatial effect of the row pile supporting structure.

$${\mathit{p}}_{\mathit{F}}{=\mathit{K}}_{\mathit{sp}}{\left\{\right(1-\mathit{\alpha }\left)\right(\mathit{\gamma }\mathit{Z}}_{\mathit{H}}{\mathit{K}}_{\mathit{a}}-2\mathit{c}\sqrt{{\mathit{K}}_{\mathit{a}}}{)-\mathit{K}}_{\mathit{a}}\mathit{q}\}$$

(12)

$${\mathit{p}}_{\mathit{B}}{=\mathit{K}}_{\mathit{sp}}{\left\{\right(1-\mathit{\alpha }\left)\right(\mathit{\gamma }\mathit{Z}}_{\mathit{H}}{\mathit{K}}_{\mathit{a}}-2\mathit{c}\sqrt{{\mathit{K}}_{\mathit{a}}}{)+\mathit{K}}_{\mathit{a}}\mathit{q}\}$$

(13)

$${\mathit{P}}_{\mathit{B}}{=\mathit{B}}_{\mathit{d}}·{\mathit{p}}_{\mathit{F}}$$

(14)

$${\mathit{P}}_{\mathit{B}}{=\mathit{B}}_{\mathit{d}}·{\mathit{p}}_{\mathit{B}}$$

(15)

$${\mathit{B}}_{\mathit{s}}=\mathit{H}\mathit{tan}(45°-\frac{\mathit{\phi }}{2})$$

(16)

In the formula, B_s is the calculated long depth affected by the spatial effect; H is the foundation pit depth; φ is the weighted internal friction angle of soil; P_F and P_B are the maximum earth pressure borne by a single pile. The progress of the calculation model is that the spatial coordination of the crown beam is taken into account, and the internal force and deformation characteristics of the pile body under the influence of the stiffness coefficient K_si of the crown beam are considered [143].

$${\mathit{K}}_{\mathit{si}}=\frac{{\mathit{q}}_{\mathit{i}}}{{\mathit{d}}_{\mathit{i}}}$$

(17)

In the formula, q_i is the resultant horizontal force acting on the crown beam by the top of the pile i; di is the horizontal displacement of the crown beam of pile i. Inheriting the previous equivalent treatment of soil between piles, the model has certain credibility in reflecting the compression performance of soil.

However, the above models ignore the shear effect of soil, so the calculation model cannot truly reflect the influence of the effect on the spatial system of the supporting structure [144,145]. Later, Zhu et al. [146] and other scholars continuously optimized the stiffness coefficient of soil springs based on the beam theory of elastic foundations.

$${\mathit{K}}_{\mathit{sn}}=\frac{{\mathit{I}}_{\mathit{n}}{+\mathit{I}}_{\mathit{n}+1}}{2}{\mathit{bK}}_{\mathit{hn}}$$

(18)

$$\mathit{K}=\frac{{\mathit{bE}}_0}{\left({1-\mathit{\nu }}_{\mathit{s}}^2\right)\mathit{\omega }}$$

(19)

In the formula, K_hn is the horizontal subgrade coefficient of the foundation soil at the unit node; I_n and I_n+1 are the lengths of two adjacent elements at the node; b is the width of calculation; E₀ is the deformation modulus of soil; ν_s is the Poisson ratio of the soil. Then the deflections of the part above the bottom of the pit and the part below the bottom of the pit are calculated, respectively. In the process of verifying the reliability of the theoretical value based on the actual value, it is found that the actual value will be smaller than the theoretical value due to the discounting of the load and the reduction of the related safety factor.

Based on previous theoretical studies, Zhou et al. [37] assumed that the front and back piles of the double-row pile foundation pit supporting the project in Changchun were vertically placed Euler-Bernoulli base beams. It was found that there is a stress concentration phenomenon at the bottom of the foundation pit, and there is a maximum earth pressure. With the excavation of the foundation pit, the maximum earth pressure will gradually decrease, and the earth pressure below the bottom of the foundation pit is the minimum. The results obtained by numerical modeling and model testing are much larger than those required in the specification. Therefore, a revised formula is proposed to correct the parts above and below the rear bottom of the front pile.

$${\mathit{p}}_{\mathit{c}1}{=\mathit{k}}_{\mathit{c}}{\Delta \mathit{v}+\mathit{p}}_{\mathit{co}}$$

(20)

$${\mathit{p}}_{\mathit{c}2}{=\mathit{k}}_{\mathit{c}}{\Delta \mathit{v}+\mathit{p}}_{\mathit{ak}}$$

(21)

$${\mathit{k}}_{\mathit{c}}=\frac{{\mathit{E}}_{\mathit{S}}}{{\mathit{s}}_{\mathit{y}}-\mathit{d}}$$

(22)

$${\mathit{p}}_{\mathit{co}}{=(2\mathit{a}-\mathit{a}}^2{)\mathit{p}}_{\mathit{ak}}$$

(23)

$$\mathit{a}=\frac{{\mathit{s}}_{\mathit{y}}-\mathit{d}}{{\mathit{h}\mathit{tan}(45}^{°}{-\mathit{\phi }}_{\mathit{m}}/2)}$$

(24)

$${\mathit{p}}_{\mathit{ak}}{=\mathit{\sigma }}_{\mathit{ak}}{\mathit{K}}_{\mathit{ai}}{-2\mathit{c}}_{\mathit{i}}\sqrt{{\mathit{K}}_{\mathit{ai}}}$$

(25)

$${\mathit{K}}_{\mathit{ai}}{=\mathit{tan}}^2{(45}^{°}{-\mathit{\phi }}_{\mathit{i}}/2)$$

(26)

In the formula, c_i and φ_i are the distance and internal friction angle of the ith layer of soil, respectively. σ_ai is the standard value of vertical stress in point soil. p_ak is the standard value of passive earth pressure strength inside the supporting structure. K_ai is the active earth pressure coefficient.

Zhang et al. [92] also assumed that the front and back piles of the row pile supporting structure were vertically placed Euler-Bernoulli base beams and calculated the earth pressures of the back piles in the upper and lower parts of the pit bottom, respectively, based on Rankine earth pressure theory, as shown in Figure 8.

$${\mathit{p}}_{\mathit{U}}{=(\mathit{FK}}_{\mathit{a}}{+\mathit{\gamma }\mathit{zK}}_{\mathit{a}}-2\mathit{c}\sqrt{{\mathit{K}}_{\mathit{a}}})\mathit{b}$$

(27)

$${\mathit{p}}_{\mathit{D}}{=(\mathit{qK}}_{\mathit{a}}{+\mathit{\gamma }\mathit{zK}}_{\mathit{a}}-2\mathit{c}\sqrt{{\mathit{K}}_{\mathit{a}}})\mathit{b}$$

(28)

In the formula, p_U and p_D are the active earth pressures acting on the back pile above and below the pit bottom, respectively; b is pile spacing; F is overload; z is soil thickness; K_a is the Rankine active earth pressure coefficient, and q is the interaction force between piles and soil. Finally, the deflection differential equation is established to calculate the deformation value and internal force value of the pile. The calculation method of the above model takes into account the continuous deformation coordination effect of the row pile supporting structure and the pile-soil interaction. However, when the equivalent spring model is used to simulate the soil between piles, the distribution of the soil layer is not considered in the value of the equivalent spring stiffness coefficient k, which is uneven and nonlinear. It also ignores the influence of objective conditions such as soil thickness between piles, soil stress release during foundation pit excavation, unloading direction during foundation pit excavation, and internal and external precipitation.

Based on the equivalent truss theory proposed by Sun [147] and Wu et al. [148], scholars later realized that this equivalent compression spring failed to fully consider the interaction between piles and soil, so they added an equivalent soil column (virtual pile) into the inter-pile soil between the front row of piles, the back row of piles, and between two adjacent piles to consider the flexural stiffness of the soil between piles, as shown in Figure 9.

Furthermore, the earth pressure distribution coefficient is further optimized to calculate the earth pressure distributed by the front and rear piles [149].

$${\mathit{K}}_{\mathit{c}}=\frac{{\mathit{K}}_{\mathit{c}}^{\prime }}{2}$$

(29)

$${\mathit{\alpha }}_{\mathit{r}}=\frac{{2(\mathit{s}}_{\mathit{y}}-\mathit{d})}{\mathit{h}\mathit{tan}\left({45°-\mathit{\phi }}_{\mathit{m}}/2\right)}-{\left[\frac{\left({\mathit{s}}_{\mathit{y}}-\mathit{d}\right)}{\mathit{h}\mathit{tan}\left({45°-\mathit{\phi }}_{\mathit{m}}/2\right)}\right]}^2$$

(30)

$${\mathit{P}}_{\mathit{F}}{=\mathit{\alpha }}_{\mathit{r}}{\mathit{P}}_{\mathit{a}}$$

(31)

$${\mathit{P}}_{\mathit{B}}=(1-{\mathit{\alpha }}_{\mathit{r}}{)\mathit{P}}_{\mathit{a}}$$

(32)

In the formula, K′_c is the spring stiffness of the spring connected between the front and back piles and the virtual piles; d is the pile diameter; s_y is row spacing; φ_m is the internal friction angle; P_a is the earth pressure. Later, Liu et al. [54] established the finite element model of the plane bar system of pile rows based on the Winkler foundation beam theory. The model considered the pile-soil interaction effect and obtained the soil pressure distribution rule based on the sliding proportional coefficient method. However, in the process of establishing the calculation model, the vertical friction effect of pile soil was neglected, and the spatial effect was also not considered.

The change in the soil layer is also one of the factors that affect pile-soil stress. Given the performance change of the soil layer between piles, Wang et al. [150] put forward the influence of the reinforcement effect of soil between piles on the calculation of deformation and the internal force of the pile rows supporting structure. Based on an equivalent truss, the model simulated the soil between piles with simplified rods and further optimized the equivalent spring stiffness coefficient of pile subsoil and soil between piles, as shown in Figure 10.

$${\mathit{k}}_{\mathit{c}}=\frac{{2\mathit{E}}_{\mathit{s}}}{\mathit{L}-\mathit{d}}$$

(33)

$${\mathit{k}}_{\mathit{s}}=\mathit{m}(\mathit{z}-\mathit{h})$$

(34)

$${\mathit{K}}_{\mathit{v}}{=\mathit{k}}_{\mathit{v}}\mathit{bhm},$$

(35)

In the formula, k_c is the horizontal stiffness coefficient of soil between piles; k_s is the front row pile passive zone level reaction coefficient; k_v is the vertical subgrade coefficient of foundation soil. L is row spacing; d is pile diameter; m is the proportional coefficient of the horizontal reaction coefficient of soil; z is the calculation depth; h is the depth of the foundation pit; b is the spacing of the equivalent spring. Therefore, the calculation formula takes into account the shear effect and bending resistance of the soil between piles. Cao et al. [144,151] proposed four improved equivalent truss calculation models so that the soil shear effect and pile side friction could be well reflected. In the later model-building process, many scholars further analyzed the influence of various parameters of soil between piles on the bearing performance of pile row supporting structures [39,40]. However, it is neglected that the equivalent truss model is equivalent to the pile as a bar, so the influence of the pile diameter on the pile-soil interaction is not shown. The value of the soil layer parameter is too simple, ignoring the nonlinear soil itself. By comparison, there is still a gap between the theoretical value and the data in the actual project.

Based on the nonlinear node load function array proposed by Zhao et al. [152], the internal forces of the pile and the soil pressure at the side of the pile are inventively calculated. Zhou et al. [153], in summarizing the advantages and disadvantages of previous calculation models, proposed a nonlinear function between the soil pressure inside and outside the pit and pile displacement and took the nonlinear interaction between pile and soil into account to calculate the strength of the principal earth pressure behind double rows of piles:

$${\mathit{P}}_{\mathit{a}}{=\mathit{k}}_0\mathit{rz}-\frac{{2\mathit{\eta }\left(\mathit{z}\right)(1+2\mathit{k}}_0)\mathit{rz}}{3\mathit{a}\mathit{\nu }\sqrt{{\mathit{k}}_{\mathit{a}}}\left(\mathit{d}-\mathit{z}\right)+6\mathit{b}\mathit{\eta }\left(\mathit{z}\right)}$$

(36)

In the formula, k₀ is the pressure coefficient of static earth; k_a is the main earth pressure coefficient; and $\mathit{\eta }\left(\mathit{z}\right)$ is the horizontal displacement of the pile. Unlike the Winkler foundation beam method, the calculation model considers the linear functional relationship between the pile side’s soil resistance and the pile’s displacement itself, and does not overestimate the soil resistance value. Zhang [92] and Ou et al. [154], based on their previous studies and Euler-Bernoulli double-layer beam theory, also took the layering property of foundation soil and the linear change of the foundation reaction coefficient into account to establish the plane rod system finite element model, as shown in Figure 11.

The soil pressure on the back row piles can be calculated according to the following formula,

$${\mathit{p}}_{\mathit{i}}=[\left(\mathit{F}+\sum _{\mathit{n}=1}^{\mathit{i}-1}{\mathit{\gamma }}_{\mathit{n}}{\mathit{H}}_{\mathit{n}}{+\mathit{\gamma }}_{\mathit{i}}{\mathit{z}}_{\mathit{i}}\right){\mathit{k}}_{\mathit{ai}}{-2\mathit{c}}_{\mathit{i}}\sqrt{{\mathit{k}}_{\mathit{ai}}}]\mathit{b}$$

(37)

In the formula, F is the additional load; γ_i is the natural bulk density of soil layer I; z_i is the thickness of soil layer i; b is the distance between piles; c_i is the cohesion of soil layer i. k_ai is the Rankine active earth pressure coefficient of soil layer i (1 ≤ I ≤ N). The interaction force between piles and soil can be expressed as,

$${\mathit{q}}_{1,\mathit{i}}{=\mathit{k}}_{1,\mathit{i}}·{\mathit{b}}_0{(\mathit{y}}_{1\mathit{i}}{-\mathit{y}}_{2\mathit{i}})$$

(38)

$$\left\{\begin{array}{c}\begin{array}{cc}{\mathit{k}}_{1\mathit{i}}=\frac{{\mathit{E}}_{\mathit{si}}}{\mathit{D}}& ,\mathit{L}>4\mathit{D}\end{array}\\ \begin{array}{cc}{\mathit{k}}_{1\mathit{i}}{=\mathit{k}}_{\mathit{i}}^0{+\mathit{m}}_{\mathit{i}}{\mathit{z}}_{\mathit{i}}& ,\mathit{L}<4\mathit{D}\end{array}\end{array}\right.$$

(39)

In the formula, b₀ is the calculated width of the supporting pile, which can be calculated according to

Table 1. Calculated width of supporting pile.

Pile Type	b* ≤ 1/(m)	b* > 1/(m)
Rectangular pile	b0 = 1.5b + 0.5	b0 = b + 1
Circular pile	b0 = 0.9(1.5b + 0.5)	b0 = 0.9(b + 1)

^* When the supporting pile is a rectangular pile, b is the pile width; when the supporting pile is a circular pile, b is the pile diameter.

; y_1i is the flexural deformation of the pile body of the i section of the rear pile; y_2i is the flexural deformation of section i of the front pile; k_1i is the foundation reaction coefficient of soil layer i. L is the length of the supporting pile; D is the row spacing; m_i is the foundation proportion factor of soil layer i. The passive earth pressure on the front row piles can be expressed as,

$${\mathit{q}}_{2,\mathit{i}}{=\mathit{k}}_{2,\mathit{i}}·{\mathit{b}}_0{\mathit{y}}_{2\mathit{i}}$$

(40)

$$\left\{\begin{array}{c}\begin{array}{cc}{\mathit{k}}_{2\mathit{i}}=0& ,1\le \mathit{i}\le {\mathit{n}}_1\end{array}\\ \begin{array}{cc}{\mathit{k}}_{2\mathit{i}}{=\mathit{k}}_{\mathit{i}}^0{+\mathit{m}}_{\mathit{i}}{\mathit{z}}_{\mathit{i}}& {,\mathit{n}}_1+1\le \mathit{i}\le \mathit{N}\end{array}\end{array}\right.$$

(41)

In the calculation model, the pile-soil interaction is calculated by assuming that the soil between piles is also equivalent to a spring with a vertical nonlinear direction. In the process of considering the soil nonlinearity, the soil pressure of the embedded pile below the base is calculated using linear changes, which ignores the influence of the pile size and the embedded depth on the soil pressure. As a result, the actual monitored value is often much smaller than the calculated value.

In a double-row pile structure, the soil between the front pile and the back pile is usually defined as the limiting soil, and it is assumed that the load analysis of the row pile supporting structure is limiting, which is not consistent with the actual situation [155,156]. The pile structure and soil are usually non-limiting, so the soil pressure under the non-limiting state cannot be calculated according to the Coulomb soil pressure and Rankine soil pressure theories. Later, Jiang [157] considered the change in displacement when deducing the formula of earth pressure and obtained the nonlinear characteristics of the change of earth pressure with displacement under a non-limiting state.

$${\mathit{p}}_{\mathit{a}}{=\mathit{p}}_0{+\mathit{F}}_{\mathit{a}}{(\mathit{p}}_{\mathit{acr}}{-\mathit{p}}_0)$$

(42)

$${\mathit{p}}_{\mathit{p}}{=\mathit{p}}_0{+\mathit{F}}_{\mathit{p}}{(\mathit{p}}_{\mathit{acr}}{-\mathit{p}}_0)$$

(43)

In the formula, p_a and p_p are the active earth pressure and the passive earth pressure, respectively. F_a and F_p are displacement functions introduced to consider the relationship between displacement, relaxation stress, and compression stress. p₀ is the static earth pressure; p_acr is the relaxation stress. However, the formula only reflects the characteristics of the earth pressure undertaken by the pile-ground interaction as this single variable changes in position, and in fact, the ground pressure is also influenced by many factors, such as the Pile-Land-ground-based interaction.

Later, Cornelius et al. [158] considered the characteristic of the Pille-Land interaction in a non-limit state, the establishment of the earth pressure formula can be expressed as,

$${\mathit{p}}^{*}{=\mathit{p}}_0{(\mathit{p}}_{\mathit{cr}}{-\mathit{p}}_0{)\mathit{tan}}^{\frac{1}{3}}\left[\mathit{A}{\left(\frac{\mathit{\delta }}{{\mathit{\delta }}_{\mathit{cr}}}\right)}^2\right]$$

(44)

In the formula, ${\mathit{p}}^{*}$ is the earth pressure at any time; $\mathit{A}={\mathit{tan}}^2\left(45°-\mathit{\phi }/2\right)$; $\mathit{\delta }$ is the displacement, and the positive or negative value depends on the direction of the relative movement of the soil mass, the active value is negative, and the passive value is positive. ${\mathit{\delta }}_{\mathit{cr}}$ is the allowable displacement when the soil reaches the limit state. As A only considers the friction angle of soil mass without considering other factors such as cohesion, compression modulus, and shear effect, there is a large discrepancy compared with the actual situation, and the calculation model needs to be further optimized.

In the relevant specification of the double-row pile supporting structure, it is considered that the pile spacing should be two to five times the pile diameter, so it is assumed that the soil between the front pile and the back pile is not semi-infinite soil but finite soil. Based on the previous research results and the advantages and disadvantages of the calculation model, Xu et al. [159] took the cohesiveness, friction angle, splitting surface angle, and soil arch effect of soil mass under a non-limiting state into consideration, derived the variation law of the included angle of the fracture surface, and then obtained the active earth pressure formula of a finite soil mass. Based on this, Xue et al. [160] established a calculation method for post-pile landslide thrust under non-limiting conditions that can accurately characterize the force characteristics of pile-soil interaction. However, the model is based on the support structure of a row pile without a contact beam, the soil layer is too simple, ignoring the nonlinear stress and deformation characteristics of the soil layer, and the calculation method is complicated and has low universality in engineering.

Based on the Fortan language of deep learning, Isbuga [161] established the calculation model of the finite element difference method, numerically simulated the displacement field of soil around the pile, and further considered the influence of the pile group effect on pile-soil interaction for numerical modeling. Its established Fortan language numerical calculation model based on deep learning can improve the efficiency of computer work to a certain extent and can clearly observe the influence of pile group effect on pile-soil interaction. However, there is also a certain gap between the accuracy and effectiveness of the pile-soil interaction solution and the measured value due to the too-simple model assumption. Moreover, the influence of the lateral squeezing effect and soil performance on pile-soil interaction is not further studied, and the calculation model written in Fortran language needs to be further optimized.

All of the above calculation models can consider the pile-soil interaction mechanism of double-row pile support structures to a first-class degree. The calculation theory and method have accumulated a large amount of engineering experience in years of engineering practice and have a certain feasibility, which also plays an indelible role in the further development of the theory of pile row support. However, the calculation process for these theories is complicated and tedious, and other problems still exist. For example, the influence of pile material type, soil layer property, pile row structure size effect, the size and connection mode of the connecting beam between pile row and crown beam, lateral soil squeezing effect, and pile group effect on the deformation and force of the pile row supporting structure has not been further explored [162,163,164,165]. In the analysis of the mechanism of increasing flexural stiffness of double-row piles, the mechanical mechanism of supporting the pile and soil layer still lacks relevant research.

6. Conclusions and Prospect

The continuous development of underground space has attracted more attention in terms of the design and construction of foundation pit wall protection engineering. Among them, double-row pile supporting structures are widely used in the infrastructure construction process of wharves, bridges, subways, tunnels, and high-rise and super-high-rise buildings due to their advantages of large stiffness, small lateral stiffness, and no need for internal support [166]. However, the mechanical and deformation characteristics and mechanisms of double-row pile supporting structures are extremely complex, and the mechanical analyses of the pile-soil interaction and corresponding calculation models are constantly optimized; the system is also relatively complex. Therefore, this reviewed and summarized the double-row pile structure, the influence of the soil arch effect on double-row pile-supporting structures, the deformation characteristics of double-row pile-supporting structures, and four aspects of the calculation model of pile-soil interaction.

• This paper reviewed the research status of double-row pile structures. The influences of structural parameters such as the pile diameter, the pile length, row spacing, pile spacing, and the crown beam size on the performance of double-row pile supporting structures were introduced. It also conducted a comparative analysis of the double-row pile supporting structure under different conditions, such as the combination of long and short piles, whether there were connected beams or not, and different pile layout forms, and then summarized and evaluated the control effect of the double-row pile supporting structure on deformation under different conditions.
• The mechanism of the soil arching effect was introduced, the relative motion of the soil and pile body and the mechanism of the deformation of double-row pile structures caused by interactions when the soil arching effect occurs was analyzed, and the influence of the soil arching effect and the double soil arching effect on the deformation of row pile structure was reviewed.
• The research status of the deformation characteristics of the front pile, back pile, connecting beam, and crown beam by numerical simulation and field monitoring was further introduced, and their influencing factors were reviewed.
• The paper focused on pile-soil force analysis and the calculation model based on classical earth pressure theory, soil arch effect theory, and Winkler theory, summarized the advantages and disadvantages of various calculation models, and put forward our own opinions. The study can provide a reference for the continuous improvement and optimization of the calculation model.

The double-row pile supporting structure is a complex spatial integration framework [167,168], which presents huge challenges for researchers in exploring pile-soil, pile-pile interaction, and deformation. With the development of engineering technology, different forms of double-row pile supporting structures are also being upgraded, such as the newly emerging prefabricated double-row pile supporting structure and multi-row pile supporting structure in recent years, which necessitates higher requirements for researchers in the study of deformation rules and mechanisms of foundation pit supporting structures and the establishment of force analysis and calculation models. Rooted in environmental protection and sustainable development, the new materials used for foundation pit engineering [169,170,171,172] and the construction method based on artificial intelligence are also constantly being updated and iterated [173,174,175], which will also promote the development of the industry and greater impetus to reduce carbon emissions.