Multi-Method Sensitivity Analysis of Influencing Factors on the Lateral Displacement of Retaining Piles in Asymmetric Excavations in Soft Soil Areas

Abstract

Asymmetric structures are widespread in deep excavation engineering and place heightened demands on the deformation control and safety of retaining systems. This study focuses on an asymmetric deep foundation pit project in a soft soil area, using PLAXIS 3D to model the entire excavation process, with model accuracy confirmed by measured values. The study systematically explores the impact of multiple factors—including surcharge loading, external groundwater level, soil internal friction angle and cohesion, and the elastic modulus and embedment ratio of the retaining structure—on the lateral displacement of retaining piles. Orthogonal experimental design is utilized to calculate lateral displacements for various factor combinations, with sensitivity analyzed using the range method and verified by grey relational analysis. The results demonstrate that all factors influence the maximum lateral displacement of retaining piles to varying degrees. Both the orthogonal tests and range analysis consistently identify the influence ranking as soil internal friction angle > soil cohesion > retaining structure elastic modulus > embedment ratio > external groundwater level > surcharge loading. The grey relational analysis yields identical rankings. These results offer theoretical support and practical guidance for the design and monitoring of retaining structures in asymmetric deep excavations within soft soil environments.

1. Introduction

The excavation process of deep foundation pits is unpredictable and risky, and construction safety is affected by various factors such as geological conditions, soil mechanical properties, groundwater level, and the surrounding environment [1]. With the increasing complexity of the environment around foundation pits, the impact of construction on nearby existing buildings, municipal roads, and underground pipelines has become more significant. Therefore, while ensuring the safety of foundation pit engineering, it is especially important to design retaining structures economically and reasonably, and to effectively control the displacement of retaining structures during excavation.

During deep excavation construction, the influence of various factors on retaining structure deformation differs considerably. In recent years, numerous researchers have utilized numerical simulations and field monitoring to investigate the displacement and deformation characteristics of different retaining structures [2,3,4,5,6,7]. Zhang et al. [8] applied the finite element method to forecast structural deformations in foundation pits, revealing the direct impact of structural stiffness and support measures on the deflection and settlement of foundation pit retaining walls. These findings offer a theoretical foundation for the design of deep foundation pit support structures. Qin et al. [9] employed finite element techniques to systematically analyze the influence of reinforcement range within the pit, pile insertion ratio, and stiffness on the deformation behavior of foundation pits in thick soft clay strata, and summarized the deformation control measures and control effects of this deep foundation pit. Wang et al. [10] combined machine learning and finite element simulations to predict deformation during excavation, investigating the primary factors influencing the deformation of foundation pits in soft, deep soil layers, considering key elements such as soil parameters, pile stiffness, and insertion ratio. Their findings offer significant guidance for the design and analysis of deep foundation pits in coastal areas with specialized geotechnical conditions. Liu et al. [11] simulated the excavation process using ABAQUS, analyzing how the diaphragm wall lateral displacement varies with depth. Results showed that wall thickness, excavation depth, and excavation width (in that order) most strongly influenced wall displacement in narrow pits. Sun et al. [12] developed a three-dimensional numerical model using FLAC3D to explore the influence of foundation pit-to-building distance, building loading, and excavation depth on the maximum lateral displacement ratio between opposite sides of the metro station’s eccentrically loaded retaining structures. The distance to buildings proved to be the most sensitive factor affecting deformation. Additionally, Yin et al. [13] Yang et al. [14] and Fan et al. [15] examined support design depth and influence radius in foundation excavation, establishing rational parameter values. Wu et al. [16] used Lizheng software, combined with orthogonal testing, to assess the sensitivity of key factors influencing retaining pile displacement, providing a practical reference for coastal excavation design.

The above domestic and international studies indicate that numerical simulation has been widely applied in analyzing the influencing factors of foundation pit excavation in soft soil areas. Significantly, most research has focused on symmetric structures or ideal loading conditions, with a lack of systematic, multi-method cross-analysis of the sensitivity of lateral displacement in retaining piles of asymmetric foundation pits. Nevertheless, due to the complex load and constraint conditions, asymmetric structures impose higher requirements on the safety and deformation control of retaining structures. Therefore, it is urgently necessary to combine numerical simulations with field measurements and employ multiple analysis methods to systematically evaluate the mechanisms and sensitivities of various factors influencing the lateral displacement of retaining piles in asymmetric foundation pits. To address this gap, this study investigates a deep excavation project with an asymmetric structure in a soft soil region, using PLAXIS 3D to simulate the excavation process, with model validity confirmed by comparison with measured values. The influence of factors such as surcharge loading around the foundation pit, external groundwater level, soil internal friction angle, and cohesion, as well as the elastic modulus and embedment ratio of the retaining structure, on the lateral displacement of retaining piles is quantitatively assessed. An orthogonal experimental design, coupled with range analysis and grey relational analysis, was implemented to quantify factor sensitivities and validate the results. This work enhances the sensitivity analysis methodology for asymmetric foundation pit retaining structures and, through combining multiple methods for cross-validation, offers valuable practical guidance for the rational design, deformation monitoring, and risk management of complex deep excavations in soft soil environments.

2. Project Overview

2.1. Foundation Pit Description

The proposed project is a high-rise building with a height of 94.5 m. The foundation pit features an irregular geometry and encompasses a total area of approximately 8000 m², with excavation depths ranging from 10.95 to 11.90 m. The stratigraphic overview of the excavation area is detailed in

Table 1. Overview of the soil layer in the excavation area.

Soil Layer	Color	Thickness (m)	Description
1-1 Plain fill	Variegated	1.50~6.60	Primarily composed of gravel and brick fragments with minor cohesive soil; exhibits uneven compressibility and poor engineering properties.
1-2 Plain fill	Gray-yellow to gray-brown	0.90~3.00	Mainly cohesive soil interspersed with some gravel and bricks; compressibility is high and uneven; poor engineering behavior.
2 Clay	Dark green to gray-yellow	0.40~2.90	Contains iron manganese concretions and bluish-gray bands; mainly medium plasticity; favorable engineering properties.
3 Silty clay	Bluish-gray to gray-yellow	1.50~3.40	Characterized by iron manganese oxide spots and bluish-gray streaks, locally with thin silt layers; moderate compressibility and engineering behavior.
4-1 Silt mixed with silt sand	Gray	3.70~5.10	Well-developed thin bedding, interbedded with silty sand and containing mica debris; moderate compressibility and engineering performance.
4-2 Silt	Gray	5.30~7.90	Composed mainly of feldspar and quartz, with secondary mica; medium to low compressibility; engineering properties are good.

. Based on results from laboratory geotechnical testing, the principal physical and mechanical properties of the soils within the excavation zone are provided in

Table 2. Basic parameters of pit soil layer [17].

Soil Layer	Weight Density (KN/m3)	Friction Angle (°)	Cohesion (kPa)	Compression Modulus (MPa)
1-1	18.3	10.0	12.0	5.34
1-2	18.4	8.0	15.0	6.38
2	19.5	14.3	52.7	7.43
3	18.7	14.3	27.6	5.94
4-1	18.6	24.5	8.0	10.20
4-2	18.8	30.0	2.2	11.57

.

2.2. Layout of Retaining Piles

In comprehensive consideration of the engineering geology, hydrogeological conditions, and the surrounding environment of the excavation area, the foundation pit adopts a support system of bored cast-in-place piles or interlocking piles combined with two levels of reinforced concrete internal bracing. The space between piles is protected by sprayed concrete with mesh reinforcement, supplemented by a fully enclosed Cutter Soil Mixing cement soil cut-off waterproof curtain for waterproofing. The detailed support scheme is as follows: The inner alignment of the bored piles is set 0.88 m from the outer wall of the main structure and 0.3 m from the edge of the base slab (pile cap), with pile lengths ranging from 19.55 to 20.55 m. The bored piles are constructed flush with the inner side of the foundation pit, with a net spacing of 0.20 m between adjacent piles, and the net distance between the cement soil cut-off wall and the bored piles is also 0.20 m. The interlocking piles are constructed with full-length hydraulic steel casing, with a diameter of 1 m, an overlap length of 0.35 m, and a total pile length of 28.05 m. A guide wall is installed at the pile top, with a height of 0.40 m, as shown in Figure 1.

Based on the pit depth, two levels of internal concrete support are designed, with both the support and top beams cast in place. The top beams, purlins, and support beams all use C30 concrete. The vertical columns adopt 4L140/160 steel lattice columns, with Q345B angle steel. Column piles are bored cast-in-place piles with a diameter of Φ800 mm; some use the structural piles of the main structure, with a concrete grade of underwater C30. The lattice columns are embedded no less than 3 m into the column piles, ensuring straightness and secure connections. The cross-sectional arrangement of the retaining piles is illustrated in Figure 2.

2.3. Calculation of the Equivalent Stiffness of the Retaining Pile

While the bored pile retaining wall consists of individual piles, its load-bearing mechanism is similar to that of an equivalent underground continuous wall. The incorporation of a ring beam at the top of the piles further enhances the structural integrity of the pile system. As a result, for the purpose of internal force analysis, the pile wall can be treated equivalently as an underground continuous wall with a specific thickness, based on the principle of equal bending stiffness.

2.3.1. Bored Cast-in-Place Piles

The equivalent diaphragm wall of bored cast-in-place piles is calculated using Equation (1) through stiffness equivalence. The resulting thickness of the plate element is 650 mm.

$${\displaystyle \frac{1}{12}}(\mathrm{D}+\mathrm{t})h^3={\displaystyle \frac{1}{64}}\pi D^4,$$

(1)

In this equation, $t$ represents the pile spacing, mm; $D$ represents the pile diameter, mm; and $h$ represents the equivalent thickness, mm.

2.3.2. Interlocking Piles

The mechanical behavior of interlocking piles is analogous to that of diaphragm walls. In practical calculations, the pile wall can be equivalently treated as an underground continuous wall of a certain thickness, based on the principle of equal bending stiffness, for the analysis of internal forces and displacements. The calculation diagram is shown in Figure 3, and the equivalent thickness is calculated using Equation (2).

$$E_1{I}_1+E_2{I}_2={\displaystyle \frac{2E_3(d-a)b^3}{12}},$$

(2)

In this equation, $E_1$, $E_2$ and $E_3$, respectively, represent the reinforced concrete pile, plain concrete pile, and the equivalent elastic modulus of diaphragm wall; $b$ represents equivalent wall thickness (mm); $d$ represents the diameter of the interlocking pile, mm; $a$ represents overlapping amount, mm; and $I_1$, $I_2$, $I_3$, $I_4$, respectively, represent the moment of inertia of the reinforced concrete cast-in-place pile section, the moment of inertia of the plain concrete cast-in-place pile section, the moment of inertia of the circular section of the plain concrete cast-in-place pile, and the moment of inertia of the 1/4 overlapping part section.

The calculation diagram for the moments of inertia is illustrated in Figure 4.

$$I_2=I_3-4I_4,$$

(3)

$$I_1=I_3={\displaystyle \frac{\pi d^4}{64}},$$

(4)

$$I_4=2{\displaystyle {\int }_0^{y_1}{y}^2\sqrt{R^2-y^2}}dy-(2R-a){\displaystyle {\int }_0^{y_1}{y}^{2}dy},$$

(5)

Using the above calculation method and considering the actual conditions of this project, the equivalent diaphragm wall thickness of the interlocking piles is calculated to be 928 mm.

2.4. Construction Condition

The excavation of the foundation pit adheres to the principles of pre-support followed by excavation, basin-type excavation, partitioned excavation, stratified excavation, strip extraction, and symmetrical excavation. During earthwork excavation, the retaining piles are required to reach at least 80% of their design strength. After each zone is excavated to the base elevation, the concrete blinding and foundation slab are poured on time to reduce the duration of large-scale pit exposure and to control the rebound uplift of the pit. This study primarily details the process from retaining structure construction to excavation down to the slab elevation. The project encounters shallow phreatic water and slightly confined groundwater issues, with the slightly confined groundwater aquifer distributed in the 4-1 silty sand and 4-2 silty sand layers. The bottom of the cut-off curtain extends into the aquiclude, thereby isolating hydraulic connectivity between the slightly confined groundwater aquifers inside and outside the pit. During construction, a tube well is used to lower the slightly confined groundwater table, ensuring that groundwater is reduced below the bottom elevation of the pit base before excavation. The construction design in this study follows the “Technical specification for retaining and protection of building foundation excavations JGJ120-2012,” with the specific construction working conditions outlined in

Table 3. Construction conditions.

Working Condition Type	Description
Condition 1	CSM cement soil mixing wall, retaining pile and column construction and dewatering well construction and dewatering
Condition 2	Excavation to the bottom elevation of the top beam, casting of top beam and the first reinforced concrete support
Condition 3	Excavation to the bottom elevation of the second support, construction of purling and the second concrete support
Condition 4	Layered earth excavation reaches the bottom elevation of the base slab and in-pit construction works, including the elevator shaft and sump pit.

and the working condition diagram presented in Figure 5.

3. Numerical Modeling and Validation

3.1. Model Development

When creating soil layers by drilling, the finite element model dimensions should be selected according to code requirements, insights from similar projects, and considering the effects of hydrogeological conditions in conjunction with project-specific information. In this research, the chosen model size is 340 m × 280 m × 100 m. The equivalent diaphragm wall thicknesses for bored cast-in-place and interlocking piles are derived as per Section 2.3, and the retaining pile model is constructed based on the actual pile depth, as illustrated in Figure 6.

PLAXIS 3D offers several constitutive models, including the Linear Elastic (LE), Mohr–Coulomb (MC), Hardening Soil (HS), Hardening Soil with Small-Strain Stiffness (HSS), and Modified Cam-Clay (MCC) models. Foundation pit excavation is a typical unloading process, during which the stress state and path of the foundation pit excavation will change with the excavation. Therefore, the choice of constitutive model should reflect the stress–strain behavior of the soil during excavation. The Mohr–Coulomb model incorporates only a single elastic modulus and does not account for the unloading effects during excavation, and cannot accurately represent soil strength. It keeps linear elastic behavior before failure, with stiffness unaffected by stress or strain path. While it is commonly applied for initial analysis and evaluation, its precision and suitability are limited when applied to more complex excavation processes. The Modified Cam-Clay and Hardening Soil models, whose stiffness depends on stress state and path, can more accurately simulate the excavation and support process. Numerous studies [17,18,19] confirm that the Hardening Soil Model with Small-Strain Stiffness (HSS) yields results that better match measured values in projects with stringent deformation control. Accordingly, based on analogous engineering experience, this study adopts the HSS model for numerical simulation of the excavation process. Specific support structure parameters are provided in

Table 4. Supporting structure parameter value.

Support Structure	Equivalent Element	Section Size (mm)	Elastic Modulus (GPa)	Poisson’s Ratio	Density (KN/m3)
Retaining Pile	Plate element	φ950	30	0.3	25
Steel Lattice Column	Embedded pile	φ800	30	0.3	25
Top Beam	Beam element	1200 × 900	30	0.2	25
Purling	Beam element	1200 × 800	30	0.2	25
First Internal Support	Beam element	800 × 800	30	0.2	25
Second Internal Support	Beam element	1000 × 800	30	0.2	25

.

3.2. Model Validation

To validate the reliability of the numerical analysis, PLAXIS 3D was utilized to simulate the excavation process, and a lateral displacement monitoring system (T1–T5) was established for the retaining piles. The foundation pit is excavated to the base, and the numerically simulated maximum lateral displacements of the piles at T1–T5 were compared with in situ monitoring results, as illustrated in Figure 7.

According to Figure 7, for monitoring points T1 to T5, the maximum lateral displacement of the pile body was 18.56 mm, with the absolute error between the field-measured and numerical simulation values ranging from 0.45 mm to 0.71 mm. Based on the “Technical specification for retaining and protection of building foundation excavations JGJ120-2012”, the lateral displacement of the retaining structure in this foundation pit is within the acceptable safety range, and the error is minimal, indicating that the design of the retaining structure is appropriate. The numerical simulation accurately reflects the trend of the maximum lateral displacement of the pile body, showing good agreement with the measured values, with the error having a limited impact on the simulation results. The measured values are generally slightly greater than the simulated results, primarily due to the omission of dynamic loads around the excavation in the numerical model and the advanced simulated excavation sequence relative to the actual construction progress. Consequently, some discrepancies arise, but the errors remain within a controllable range. Therefore, it is feasible to employ numerical simulation to analyze the factors influencing retaining pile displacement.

4. Analysis of Influencing Factors on Lateral Displacement

The displacement of retaining piles results from the combined action of multiple factors. Due to the proximity of surrounding buildings to the foundation pit, interlocking piles are utilized as vertical supports for the lateral bracing system to improve the stability and deformation control of the supporting system. This paper focuses on the variation characteristics of the maximum lateral displacement of interlocking piles. To elucidate the influence of various factors on the lateral displacement of retaining piles, a single-factor analysis approach is adopted: one factor—such as surcharge loading around the foundation pit, external groundwater level, soil internal friction angle, soil cohesion, elastic modulus of the retaining structure, or embedment ratio—is varied individually while all other factors are held constant. The degree of influence of each factor on the lateral displacement of the retaining structure is then assessed via numerical simulation.

4.1. Surcharge Loading Around Foundation Pit

Surface surcharge is an unavoidable factor during construction, and excessive surcharge in the vicinity of the foundation pit poses significant risks to its safety and stability. According to established practice, ground surcharge should generally not exceed 30 kPa [15]. To investigate the influence of ground surcharge on the lateral displacement of the retaining structure, numerical simulations were performed for surcharge values of 0 kPa, 10 kPa, 15 kPa, 20 kPa, and 30 kPa. A surface load is applied to the plane of the existing building when applying a load. Under Working Condition 4, the curves depicting the variation in lateral displacement of the retaining structure under different surcharge levels are shown in Figure 8.

As illustrated in Figure 8, the maximum lateral displacement of the retaining structure increases with rising surface surcharge, with values of 16.56 mm, 16.93 mm, 17.26 mm, 17.53 mm, and 18.15 mm for surcharges of 0 kPa, 10 kPa, 15 kPa, 20 kPa, and 30 kPa, respectively. Relative to the initial external surcharge, the increase in lateral displacement at the top of the retaining pile is pronounced, reaching up to a 36% increase at 30 kPa, while changes at the pile base are comparatively minor. Therefore, it is crucial to strictly control the magnitude of surface surcharge around the foundation pit during construction, and to implement appropriate measures when the pile top displacement approaches the warning value.

4.2. External Groundwater Table

This region is characterized by high rainfall, resulting in the groundwater level varying with rainfall. To evaluate the influence of groundwater level on the lateral displacement of the retaining structure, this study investigates external groundwater table heights of 1 m, 2 m (initial value), 3 m, 5 m, and 7 m. Under Working Condition 4, the variation curves of lateral displacement for the retaining structure at different groundwater levels are presented in Figure 9.

Figure 9 demonstrates that when the external groundwater table is set at 1 m, 2 m, 3 m, 5 m, and 7 m, the corresponding maximum lateral displacements of the retaining structure are 17.23 mm, 16.56 mm, 16.14 mm, 15.79 mm, and 15.35 mm, with respective rates of change of 4.1%, 2.5%, 4.6%, and 7.3%. The results indicate that as the external groundwater level decreases, the maximum lateral displacement of the retaining structure also declines. Therefore, from a cost perspective, it is advisable to lower the external groundwater table appropriately during construction to enhance pit stability. During periods of heavy rainfall, it is important to closely monitor changes in lateral displacement of the retaining structure to prevent overall pit instability.

4.3. Soil Internal Friction Angle and Cohesion

4.3.1. Internal Friction Angle

The internal friction angle is a key parameter reflecting the frictional resistance between soil particles, directly influencing the soil’s shear strength and stability. To explore the influence of internal friction angle on the lateral displacement of the retaining structure, this study considers the internal friction angle at 0.6, 0.8, 1.0, 1.2, and 1.4 times its initial value. The impact of varying friction angles on the lateral displacement of the retaining structure is illustrated in Figure 10.

Figure 10 shows that when the soil internal friction angle around the foundation pit is set to 0.6, 0.8, 1.0, 1.2, and 1.4 times the initial value, the calculated maximum lateral displacements of the retaining structure are 18.73 mm, 17.14 mm, 16.56 mm, 15.12 mm, and 14.59 mm, respectively. Compared with the initial internal friction angle, increasing the friction angle leads to a reduction in the maximum lateral displacement, while decreasing it results in an increase. The corresponding variation rates are 13.1%, 3.5%, 8.7%, and 11.9%, indicating a slightly larger sensitivity to changes in internal friction angle.

4.3.2. Cohesion

Soil cohesion reflects the ability of soil particles to adhere to each other, which significantly influences soil strength, stability, and deformation characteristics. To study the impact of soil cohesion on the lateral displacement of the retaining structure, this paper adjusts the cohesion to 0.6, 0.8, 1.0, 1.2, and 1.4 times its initial value. The effect of varying cohesion values on the lateral displacement of the retaining structure is illustrated in Figure 11.

Figure 11 shows that when the cohesion of the soil around the foundation pit is 0.6, 0.8, 1.0, 1.2, and 1.4 times the initial value, the maximum lateral displacements of the retaining structure are 17.96 mm, 16.90 mm, 16.56 mm, 15.81 mm, and 15.10 mm, respectively. Compared to the initial value, an increase in soil cohesion leads to a reduction in the maximum lateral displacement, while a decrease in cohesion results in an increase. The respective rates of change are 8.4%, 2.1%, 4.5%, and 8.8%. These results indicate that greater soil cohesion enhances the cementation between soil particles, thereby reducing the earth pressure on the retaining structure and limiting its displacement.

4.4. Elastic Modulus

The elastic modulus of the retaining structure is a key parameter that affects its stiffness and resistance to deformation. To investigate the effect of the elastic modulus on lateral displacement, this study varies the elastic modulus of the original concrete material to 0.6, 0.8, 1.0, 1.2, and 1.4 times its initial value, while keeping other material parameters unchanged. The influence of different elastic moduli on the lateral displacement of the retaining structure is shown in Figure 12.

Figure 12 shows that when the elastic modulus of the retaining structure is 0.6, 0.8, 1.0, 1.2, and 1.4 times its original value, the maximum lateral displacements are 17.85 mm, 16.90 mm, 16.56 mm, 15.81 mm, and 15.05 mm, respectively. Compared with the initial elastic modulus, increasing the elastic modulus reduces the maximum lateral displacement, while decreasing the elastic modulus increases the displacement; the respective variation rates are 7.8%, 2.1%, 4.5%, and 9.1%. During construction, attention should be paid to the quality of concrete casting to avoid excessive displacement during excavation.

4.5. Embedment Ratio

The embedment ratio of the retaining structure is defined as the depth of the retaining structure below the excavation surface to the depth of the retaining structure above the excavation surface. Research indicates that an excessively high embedment ratio (i.e., overly long piles) offers limited improvement in deformation control [20,21], while significantly increasing project costs. On the other hand, a low embedment ratio could compromise the overall stability of the foundation pit. In this study, the secant bored pile depth is 28.05 m, and the excavation depth is 11.25 m, resulting in an embedment ratio of 1.49. Five models were established with embedment ratios of 0.89, 1.19, 1.49, 1.79, and 2.01, corresponding to pile lengths of 21.31 m, 24.63 m, 28.05 m, 31.39 m, and 33.86 m, respectively. Figure 13 presents the impact of different embedment ratios on the lateral displacement of the retaining structure.

Figure 13 demonstrates that as the pile length increases from 21.3 m to 33.86 m, the lateral displacement curves display a concave profile. The maximum lateral displacements of the pile are 17.71 mm, 17.02 mm, 16.56 mm, 16.26 mm, and 15.79 mm, respectively. Compared to the original pile design, increasing the pile length marginally reduces the maximum lateral displacement, with the reduction being relatively insignificant. Conversely, reducing the pile length increases the lateral displacement, with the maximum displacement shifting downward; the displacement at the pile top decreases, while the displacement at the pile base increases. This behavior results from insufficient embedment, causing the pile to move inward from top to bottom. Therefore, in pile length design, the pile length may be appropriately reduced to optimize construction costs while ensuring the overall stability of the foundation pit.

According to the numerical simulation results, the maximum lateral displacement of the retaining structure typically occurs at a position 0.29 to 0.34 times the pile length, exhibiting a trend of first increasing and then decreasing from top to bottom along the pile. There is a positive correlation between the surface surcharge and the maximum lateral displacement of the pile. In contrast, the external groundwater level, soil internal friction angle and cohesion, elastic modulus of the retaining structure, and embedment ratio all show negative correlations with the maximum lateral displacement of the pile.

5. Sensitivity Analysis

This study investigates the impact of several factors—including surface surcharge (Factor I), external groundwater level (Factor II), soil internal friction angle (Factor III), soil cohesion (Factor IV), elastic modulus of the retaining structure (Factor V), and pile embedment ratio (Factor VI)—on the maximum lateral displacement of retaining piles in foundation pits. Given the difficulty in directly quantifying the relative impact of each factor on the maximum displacement, a sensitivity analysis is necessary. To achieve a comprehensive and balanced evaluation while minimizing the number of required simulations, an orthogonal experimental design is adopted in this section to systematically investigate the sensitivity of each factor affecting the maximum lateral displacement of the retaining piles.

5.1. Orthogonal Experiment and Results

In Section 4, the maximum lateral displacement of the retaining piles was calculated for five different levels of each factor. When calculating surface surcharge, the surcharge was primarily determined based on relevant engineering standards, experience, and the site’s geological conditions. Considering the surrounding buildings, the surcharge around the foundation pit generally does not exceed 30 kPa. Therefore, 0 kPa, 10 kPa, and 20 kPa were selected for analysis. Based on the distribution of the groundwater table in the actual engineering area, it was found that situations with an outside groundwater table at 1 m were rare, so representative heights of 2 m, 5 m, and 7 m were chosen. The factor levels for the soil internal friction angle, cohesion, and the elastic modulus of the retaining structure were based on the actual variation range of the mechanical properties of the soil and the retaining structure, ensuring both the representativeness and feasibility of the experimental conditions. Consequently, the factor levels were set at 0.6, 1.0, and 1.4 times the initial values. When the embedment ratio is 0.89 (pile length 21.31 m), the lateral displacement of the retaining structure is large, which is not recommended in practical engineering. When the embedment ratio increases, the reduction in maximum lateral displacement becomes insignificant, and the economic benefits are relatively low. Therefore, considering both cost and excavation stability, embedment ratios of 1.19, 1.49, and 1.79 were selected for the optimization analysis, corresponding to pile lengths of 24.63 m, 28.05 m, and 31.39 m, respectively. The factor level divisions for the orthogonal experiment are shown in

Table 5. Classification of factor levels.

Level	Surface Surcharge (Factor I)	External Groundwater Level (Factor II)	Soil Internal Friction Angle Ratio (Factor III)	Soil Cohesion Ratio (Factor IV)	Elastic Modulus Ratio of Retaining Structure (Factor V)	Embedment Ratio of Retaining Structure (Factor VI)
1	0	2.00	1.00	1.00	1.00	1.49
2	10.00	5.00	0.60	0.60	0.60	1.19
3	20.00	7.00	1.40	1.40	1.40	1.79

.

Based on the above factor level divisions, an L₁₈(3⁷) orthogonal array for six factors and three levels was established using SPSS, resulting in 18 experimental groups. The maximum lateral displacement of the retaining structure was calculated for each group. The orthogonal experiment design and results are shown in

Table 6. Orthogonal experiment and results.

Test Number	Factors						Maximum Lateral Displacement (mm)
	I(kPa)	II(m)	III	IV	V	VI
1	10.00	5	1.00	0.60	0.60	1.79	19.93
2	0.00	7	1.40	0.60	0.60	1.19	19.28
3	20.00	2	1.00	0.60	1.40	1.79	16.22
4	20.00	2	1.40	0.60	1.00	1.19	20.36
5	10.00	2	0.60	1.40	1.40	1.19	19.37
6	0.00	5	1.00	1.40	1.40	1.19	12.23
7	0.00	2	0.60	1.00	0.60	1.79	20.58
8	10.00	2	1.40	1.40	0.60	1.49	14.89
9	0.00	5	1.40	1.40	1.00	1.79	10.17
10	20.00	7	0.60	1.40	1.00	1.79	15.88
11	10.00	7	1.00	1.00	1.00	1.19	18.07
12	10.00	5	0.60	0.60	1.00	1.49	21.91
13	20.00	5	1.40	1.00	1.40	1.49	11.11
14	20.00	7	1.00	1.40	0.60	1.49	16.43
15	0.00	2	1.00	1.00	1.00	1.49	16.56
16	20.00	5	0.60	1.00	0.60	1.19	23.61
17	0.00	7	0.60	0.60	1.40	1.49	18.18
18	10.00	7	1.40	1.00	1.40	1.79	10.59

.

5.2. Variance Analysis

According to the orthogonal experiments and results summarized in Table 6, an analysis of variance (ANOVA) was performed using SPSS 28.0 software. The various influencing factors were set as fixed factors, and the maximum lateral displacement of the retaining piles was treated as the dependent variable. The resulting test of between-subjects effects is presented in

Table 7. Results of variance analysis.

Source of Variance	Sum of Squares	df	Mean Square	F	Significance	Significance Level
Factor I	6.00	2	3.00	16.67	0.00	Highly significant
Factor II	9.46	2	4.73	26.23	0.00	Highly significant
Factor III	93.14	2	46.57	258.72	0.00	Highly significant
Factor IV	61.15	2	30.58	169.89	0.00	Highly significant
Factor V	61.07	2	30.54	169.67	0.00	Highly significant
Factor VI	33.79	2	16.90	93.89	0.00	Highly significant
Error	1.95	11	0.18

Coefficient of Determination R² = 0.975.

. In ANOVA, the F-value serves as an indicator of the degree of influence each factor has on the maximum lateral displacement. The larger the F-value, the greater the factor’s impact on the outcome; a significance value closer to 0 indicates a higher level of statistical significance for that factor.

Table 7 demonstrates that all six factors exert a highly significant influence on the maximum lateral displacement of the retaining piles, though their degrees of impact differ. Among these, variations in the soil internal friction angle have the most pronounced effect, while the influences of soil cohesion and the elastic modulus of the retaining structure are essentially equivalent. In contrast, the surcharge outside the pit has the least impact on the maximum lateral displacement. Based on the comparison of F-values, the descending order of influence for each factor is as follows: Factor III > Factor IV > Factor V > Factor VI > Factor II > Factor I.

5.3. Grey Relational Analysis

5.3.1. Grey Relational Degree Analysis Method

Grey relational analysis is an approach used for uncertainty analysis; it evaluates the degree of association between influencing factors and outcomes by calculating their relational grades, thus clarifying the correspondence between each factor and the result [22,23]. Accordingly, this study utilizes grey relational analysis to determine the sensitivity of different influencing factors on the maximum lateral displacement of the retaining structure. The fundamental steps are as follows:

Step 1: Determination of the sequence matrix

We define the sub-sequence matrix $X$ (representing the influencing factors), which comprises the surface surcharge, external groundwater level, soil internal friction angle, soil cohesion, elastic modulus of the retaining structure, and embedment ratio; that is,

$$X=\left[\begin{array}{l}{X}_1\\ X_2\\ ⋮\\ X_i\end{array}\right]=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1j}\\ x_{21}& x_{22}& \dots & x_{2j}\\ ⋮& ⋮& & ⋮\\ x_{i1}& x_{i2}& \dots & x_{ij}\end{array}\right],$$

(6)

The parent sequence matrix $Y$ (representing the maximum lateral displacement of the pile) is formed by taking the maximum lateral displacement values corresponding to each set of influencing factors as sequence $Y$; that is,

$$Y=\left[\begin{array}{l}{Y}_1\\ Y_2\\ ⋮\\ Y_i\end{array}\right]=\left[\begin{array}{cccc}{y}_{11}& y_{12}& \dots & y_{1j}\\ y_{21}& y_{22}& \dots & y_{2j}\\ ⋮& ⋮& & ⋮\\ y_{i1}& y_{i2}& \dots & y_{ij}\end{array}\right],$$

(7)

In the above equations, $i$ denotes the number of selected influencing factors, and $j$ represents the target value corresponding to the variation in the $i$-th influencing factor.

Step 2: Data Normalization

Data normalization is performed using Equations (8) and (9).

$$x_{ij}^{\prime }={\displaystyle \frac{x_{ij}-\min x_{ij}}{\max x_{ij}-\min x_{ij}}},$$

(8)

$$y_{ij}^{\prime }={\displaystyle \frac{y_{ij}-\min y_{ij}}{\max y_{ij}-\min y_{ij}}},$$

(9)

By applying Equations (8) and (9) to process the two sequence matrices, a new difference sequence matrix $\Delta$ is obtained, from which the maximum and minimum values are determined.

$$\Delta ij=\left|x_{ij}^{\prime }-y_{ij}^{\prime }\right|,$$

(10)

$$\begin{array}{l}{\Delta }_{\max }=\max {\Delta }_{ij}\\ {\Delta }_{\min }=\min {\Delta }_{ij}\end{array},$$

(11)

Step 3: Correlation Degree Calculation

Based on the above difference matrix, the grey relational matrix $L$ is calculated according to the following formula:

$$L_{ij}={\displaystyle \frac{{\Delta }_{\min }+\eta {\Delta }_{\max }}{{\Delta }_{ij}+\eta {\Delta }_{\max }}},$$

(12)

In the equations, $\eta$ is the distinguishing coefficient, typically chosen from the interval (0, 1).

The mean value of the grey relational coefficients between $X$ and $Y$ is computed to reflect the degree of association between the two. The relational grade is calculated as follows:

$$Q_i={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n{L}_{ij}},$$

(13)

In the equations, $Q$ is the sensitivity index, taking values in [0, 1].

5.3.2. Case Study

In the grey relational analysis, the increments of each influencing factor from Section 4 (with four levels for each factor) were selected as sub-sequences. Notably, the external groundwater level, soil internal friction angle and cohesion, elastic modulus of the retaining structure, and embedment ratio exhibit a negative correlation with the maximum lateral displacement of the piles. To address this, the reciprocals of these factors were utilized when constructing the sub-sequence matrix, thereby establishing a positive correlation. The variations in the maximum lateral displacement of the piles were designated as the parent sequence. During the excavation process, grey relational analysis was conducted to quantitatively evaluate the influence of each factor on the maximum lateral displacement of the retaining piles. When the value of $\eta$ is smaller, the differences among the correlation coefficients become more pronounced, resulting in greater distinguishing ability. Generally, $\eta$ = 0.5 is adopted. The sensitivity index $Q$ approaches 0 for parameters with low sensitivity, whereas values closer to 1 indicate higher sensitivity.

The sensitivity indices calculated according to Equations (6)–(13) are as follows:

$$Q={\left[\begin{array}{cccccc}0.70& 0.69& 0.94& 0.85& 0.81& 0.74\end{array}\right]}^T,$$

(14)

As all influencing factors are rendered dimensionless through normalization in the grey relational analysis, the resulting sensitivity index matrix exclusively represents the sensitivity of the sub-sequences to the parent sequence—that is, the degree to which each factor influences the maximum lateral displacement of the piles. According to Equation (14), the sensitivity ranking of the influencing factors is Factor III > Factor IV > Factor V > Factor VI > Factor II > Factor I, which is consistent with the results of the orthogonal analysis of variance.

Based on the sensitivity analysis results mentioned above, compared with existing research [16], the soil’s internal friction angle has the most significant impact on the deformation of the retaining structure, followed by soil cohesion and the elastic modulus of the retaining structure. This conclusion is consistent with the findings of existing research, further validating the accuracy of this study’s results. In practical engineering applications, controlling the deformation and stress of the retaining structure requires particular attention to the soil’s internal friction angle. Therefore, before excavation, it is necessary to increase the borehole density, especially in areas where geological variability may exist near the pit. Additional in situ testing and laboratory experiments should be conducted to reduce uncertainties in soil parameters and ensure the reliability of the design. Moreover, strict control over groundwater levels and timely drainage measures should be implemented to prevent water from entering the soil, as this could increase soil weight and drastically reduce the internal friction angle, leading to instability in the retaining structure.

6. Conclusions

By conducting finite element numerical simulations of asymmetric foundation pit excavation, bored cast-in-place piles and secant piles were equivalently modeled as diaphragm walls. The influence of various factors on the maximum lateral displacement of retaining piles was evaluated, and sensitivity analysis was performed using orthogonal experiments and range analysis. The main conclusions are as follows:

(1) The lateral displacement of retaining piles is affected to varying degrees by factors such as surcharge loading around the foundation pit, external groundwater level, soil internal friction angle and cohesion, elastic modulus of the retaining structure, and embedment ratio. Of these, the surrounding surcharge has a greater effect on the pile top displacement, while the internal friction angle has a more significant influence on the maximum lateral displacement of the retaining piles.

(2) The results from orthogonal experiments and variance analysis indicate that the order of influence of each factor on the maximum lateral displacement of the retaining structure is Factor III > Factor IV > Factor V > Factor VI > Factor II > Factor I. Specifically, when the ground surcharge is 0 kPa, the external groundwater level is 5 m, the soil cohesion and internal friction angle are 1.4 times their original values, the elastic modulus of the retaining structure is 1.0 times the original value, and the embedment ratio of the retaining structure is 1.79; the maximum lateral displacement of the retaining pile is therefore minimized.

(3) Grey relational analysis yields a sensitivity ranking of Factor III > Factor IV > Factor V > Factor VI > Factor II > Factor I, which is consistent with the results of the orthogonal experiment and variance analysis. Therefore, it is crucial to focus on the impact of the soil’s internal friction angle on the deformation of the retaining structure during foundation pit excavation. It is recommended to perform detailed soil testing before excavation to minimize the uncertainty of soil parameters and ensure the reliability of the design.

Although this study provides a reference for the sensitivity analysis of lateral displacement in asymmetric foundation pit retaining piles in soft soil areas, several research directions remain worth exploring:

(1) Future studies could incorporate real-time monitoring data to dynamically adjust support schemes, thereby optimizing risk management during the foundation pit excavation process. This approach would be particularly effective in complex geological conditions, as real-time feedback can significantly improve construction safety and efficiency.

(2) With the development of artificial intelligence technologies, machine learning and other advanced techniques could be employed in future research. By analyzing large datasets of historical data, these techniques could be leveraged to optimize the design parameters of asymmetric foundation pit retaining structures, enhancing both the efficiency and safety of the design process.