Research on the influence of pile diameter on the pile-anchor support system for deep foundation pit adjacent to the existing structure

2.1. Model testing apparatus

The test device comprises a model box, a retaining wall, a pile-anchor system, a displacement measurement system, and a data acquisition and analysis system. The model box has dimensions of 1200 mm × 400 mm × 700 mm, with 16 mm transparent tempered glass on the front and an opening at the top; the remaining panels are constructed from 16 mm steel plate. Model testing equipment is shown in Fig. 1; the left side of model box serves as the movable retaining wall, which is hinged to the upper and lower transmission shafts on its outer side. Inside the box, slots are set at 10 cm intervals, which can be used to place a fixed retaining wall on the right side to achieve different filling widths. A reaction frame is installed above the model box to facilitate the installation of anchor hollow bolts.

Fig. 1Model testing equipment. Photo taken by Sanxian Liu, geotechnical laboratory of Hunan Institute of Science and Technology, in February 2024

a) Front view

b) Top view

2.2. The design and construction of pile-anchor support structure model

To simulate the pile-anchor support system, PVC pipes were used to simulate concrete piles and PVC plate were utilized to simulate crown beams. Eight PVC pipes were sequentially marked as 1 #~8 #. The pipes have an outer diameter of 40 mm, a wall thickness of 2 mm, and a length of 475 mm. The size of the PVC plate is 25 mm × 400 mm × 40 mm, and a hole is drilled every 50mm along the length direction for placing PVC pipes. Grooves are made between holes 2 #, 3 #, 4 #, 5 #, and 6 #, 7 # for fixing model anchor cables. The arrangement of the model piles, strain gauges, and displacement meters is illustrated in Fig. 2.

Fig. 2The arrangement of the model piles, strain gauges, and displacement meters

As shown in Fig. 2, the strain gauges are symmetrically positioned on the excavation and non-excavation sides of piles No. 3 and No. 6. Horizontal displacement measurement points at the pile tops are located on the excavation side between boreholes No. 1 and No. 2, as well as between boreholes No. 7 and No. 8. PVC pipes No. 1 to No. 8 are inserted into the corresponding holes of the PVC plate, ensuring that the bottoms of the PVC pipes are aligned and that 450 mm is exposed above the surface. Subsequently, the PVC pipes and the PVC plate are securely bonded using adhesive. Then, the strong double-sided tape is used to stick fine mesh gauze on the excavation side of the support model to prevent the non-excavation side soil from leaking between the PVC pipes to the excavation side. The Pile-anchor support structure model with strain gauge is shown in Fig. 3; the anchor cable model is fastened at the steel wire groove of the PVC plate, maintaining a remaining length of 1.5 meters. The strain data for the pile body was acquired using strain gauges and static strain measurement devices.

Fig. 3Pile-anchor support structure model with strain gauge. Photo taken by Sanxian Liu, geotechnical laboratory of Hunan Institute of Science and Technology, in February 2024

2.3. Model test scheme

To investigate the deformation characteristics of pile-anchor support system in deep foundation pit, two comparative model tests were designed for the soil behind the crown beam: “no slope” and “with a slope”. The design elevation view of no slope and a slope condition behind the crown beam is presented in Fig. 4 and Fig. 5, respectively. Compared to the model test without slope, the model test with slope has identical pile row length and excavation depth, but the top of the crown beam has slope to simulate the loading condition.

Fig. 4Design elevation view of no slope condition behind the crown beam

The anchor cable passes through the pulley to connect the crown beam with the tension sensor. The tension sensor is securely fastened to the anchor cable via anchor bolts. A displacement meter is installed at the center of the crown beam to measure the horizontal displacement at the pile’s top. Strain gauges are arranged on the pile body to measure the bending moment of the pile. The total designed excavation depth $H=$275 mm, which is divided into five excavations. The depth of the first excavation is 75 mm, and each of the remaining excavations is 50 mm. In the single-group tests, the amount of sand used without slope and with a slope was 0.207 m³ and 0.217 m³, respectively.

Fig. 5Design elevation view of a slope condition behind the crown beam

2.4. Filling material parameters

The filling material used in this model test is sandy soil, which is taken from Dongting Lake in Yueyang, China. The material was processed by sieving pebbles and impurities, then washing with water and sun-drying to prepare the test filler. The soil physical and mechanical parameters are presented in Table 1. The particle size distribution of the test filler was determined using the sieve analysis method. The particle sizes of the fillers predominantly ranged between 0.5 mm and 2 mm, characterized as poorly graded coarse sandy soil with a non-uniformity coefficient $C_u$= 4.89 and a curvature coefficient $C_c$= 1.71. Through the density test and direct shear test, the natural density of the test filler under loose conditions is $\rho$ = 1.46 g/cm³, and the internal friction angle $\phi$ = 35.4°.

2.5. Testing procedure

Taking the slope on the crown beam top test as an example, the operation and steps of the model test are introduced:

1) Install three sets of pulleys on the fixed retaining wall at the anchor cable position shown in Fig. 5. Seal the excess gaps in the fixed retaining wall using tape.

2) Secure the fixed retaining wall into the slot of the model box, ensuring that the movable retaining wall is perfectly vertical and positioned precisely 900 mm away from the fixed retaining wall; and then install and adjust the reaction frame and anchor bolt status and position in sequence.

3) Weigh 52.56 kg of fill soil, pour it into the model box, and level it to a height of 100 mm to serve as the base filling material.

4) Apply Vaseline to both ends of the pile-anchor support structure model, then install it precisely 350 mm away from the movable retaining wall.

5) Thread the anchor cable through the fixed retaining wall, pulleys, and anchor bolts on the reaction frame, securing the anchor head with the anchor bolts.

6) Place a level on the top surface of the PVC plate to adjust the orientation of pile-anchor support structure model, ensuring its precise vertical alignment.

7) Fill in the sand again, ensuring that the pile anchor support structure model is vertical during the filling process, until the shape distribution of sand matches the configuration illustrated in Fig. 5.

8) Tighten the model anchor cable using the pulley system, cut the excess cable beneath the reaction frame, and install the tensile force sensor.

9) Adjust the tension of the model anchor cable by rotating the anchor bolts on the reaction frame until the tension sensor indicates a reading of 5 N.

(10) Install two sets of displacement dial indicators at the center of the crown beam as shown in Fig. 2, ensuring that the dial indicator probe just touch the crown beam and adjust the initial reading to 0.00 mm.

11) Connect the strain gauge cable and the tensile force sensor data cable to the corresponding data acquisition instruments.

12) After a stabilization period of 24 hours, sequentially remove the sandy soil layers with depths of 75 mm, 50 mm, 50 mm, 50 mm, and 50 mm on the excavation side, and record the readings from the strain gauge, tensile force sensor, and displacement dial indicators.

2.6. An overview of numerical simulation of model test

Numerical simulations were carried out using the ABAQUS software, with the slope condition and no slope condition behind the crown beam test serving as an illustrative example. Boundary constraint conditions of model with the slope condition and no slope condition is shown in Fig. 6, Grid division of model with the slope condition and no slope condition is shown in Fig. 7.

Fig. 6Boundary constraint conditions of model with the slope condition and no slope condition

a) With the slope condition behind the crown beam

b) No slope condition behind the crown beam

A 1:1 modeling analysis was conducted under the following conditions: the soil was modeled using the Mohr-Coulomb constitutive model; the anchor cables and row piles were both modeled as linear elastic materials. To account for the frictional interaction between the piles and soil, a frictional contact with relative slip was implemented, with a friction coefficient of 0.32. The relative deformation between the steel wire and the soil, as well as between the steel wire and the crown beam of the row pile, was constrained using an embedded model.

According to the model test conditions, $X$-directional displacement constraints are set on the $X=$0 mm and $X=$900 mm planes, $Y$-directional displacement constraints are set on the $Y=$0 mm and $Y=$400 mm planes, and $X$-directional, $Y$-directional, and $Z$-directional displacement constraints are set on the $Z=$0 plane. For the anchor cable, displacement constraints in the $X$, $Y$, and $Z$ directions are set on the plane at $X=$900 mm. The pile row and soil models utilize the C3D8 element; the anchor cable model employs the T3D2 element. The model with the slope condition and no slope condition is discretized into a total of 33966 elements and 35950 elements, respectively.

Fig. 7Grid division of model with the slope condition and no slope condition

a) With the slope condition behind the crown beam

b) No slope condition behind the crown beam

3.1. Results and analysis on horizontal displacement of pile top in model test

To study the variation law of pile top horizontal displacement with burial depth, the test results of pile top horizontal displacement when the foundation pit is excavated to the bottom ($H=$275 mm) are summarized in Table 2. The horizontal displacement of the pile top towards the outside of the foundation pit is positive, and towards the inside of the foundation pit is negative.

It can be observed from Table 2 that when the excavation depth $H\le$175 mm, the displacement at the pile top in the model test without slope and with a slope is 0 mm. When the excavation depth $H=$225 mm, the average displacement at the pile top in the no slope model test is –0.015 mm, while in the model test with a slope, the average displacement at the pile top is –0.035 mm, there is an increase of 57.1 % compared to the no slope model. Upon excavation reaching the pit bottom ($H=$275 mm), the average displacement at the pile top in the model test without slope is –0.080 mm. In contrast, in the model test with a slope, the average displacement at the pile top is –0.125 mm, indicating a 36.0 % greater displacement in the model with a slope.

3.2. Results and analysis on bending moment in the pile body in model test

The strain gauge readings on the excavation side and the non-excavation side of the same cross-section of pile body are denoted as ${\epsilon }_a$ and ${\epsilon }_b$, respectively. The bending moment $M$ of the pile body can be calculated based on ${\epsilon }_a$ and ${\epsilon }_b$ using Eq. (1):

where $E$ is elasticity modulus of pile body material, $I$ is pile section moment of inertia, $D$ is the pile diameter.

Taking the bottom elevation of the model test as 0, the top elevation of the pile is 575 mm. The calculated results for the bending moment of the pile body are presented in Table 3. The bending moment is positive when directed towards the inner side of the foundation pit and negative when directed towards the outer side.

The bending moment test results for piles No. 3 and No. 6 of different excavation depths were plotted in Fig. 8 and Fig. 9, respectively. It can be observed from Fig. 8 and Fig. 9 that the variation laws of the bending moments for pile No. 3 and pile No. 6 are the same, both exhibiting an “S” shaped distribution. When the excavation depth $H\le$ 130 mm, the bending moment curve of the pile body can be divided into three segments: the middle segment has a positive bending moment, while the upper and lower segments have negative bending moments. The bending moment increases with the increase of excavation depth. This phenomenon occurs because, at relatively shallow excavation depths, the influence of the backfill on the pile body is minimal, and the stress in the anchor cable primarily governs the bending moment. The anchor cable, fixed to the crown beam, exerts a certain constraint on the horizontal displacement of the pile body. When the excavation depth $H\ge$ 180 mm, the bending moment curve of the pile body can be divided into two segments, the upper segment exhibits a negative bending moment, and the lower segment exhibits a positive bending moment; The bending moment and curve curvature of the pile body increase as the excavation depth increases. At this point, the effect of the filling material on the pile body gradually increases, and the bending moment of the pile body is gradually controlled by the effect of the filling material.

Piles No. 3 and No. 6 are located at the center of the test section, and the deformation results are considered representative. The average bending moments of the pile bodies of No. 3 and No. 6 piles in the model tests without slope and with slope when burial depth $H=$275 mm were taken as the representative values of the pile body bending moments. The representative values of the pile body bending moments are detailed in Table 4.

It can be known from Table 4 that at burial depths of $H=$ 90 mm, $H=$330 mm, and $H=$410 mm, the bending moment of the pile body of the model with slope is greater than that of the model without slope. Conversely, at burial depths of $H=$170 mm and $H=$250 mm, the bending moment of the pile body of the model with slope is smaller than that of the model without slope. Additionally, as the burial depth increases, the difference in the bending moment of the pile body between the two models becomes more pronounced.

Fig. 8Bending moment curve for pile No. 3 of different excavation depths

Fig. 9Bending moment curve for pile No. 6 of different excavation depths

3.3. Comparative analysis of horizontal displacement at the top of pile in model test

The horizontal displacement of the pile top under various excavation conditions was systematically summarized and compared based on the model test and the numerical simulation. The statistical analysis of horizontal displacement at the top of pile is presented in Table 5.

It can be observed from Table 5 that when the excavation depth $H\le$ 175 mm, the horizontal displacement of the pile top in the numerical simulation is positive. When the excavation depth $H>$ 175 mm, the numerical simulation indicates a negative horizontal displacement at the pile top. Given that the dial indicator cannot measure forward displacement, the measured horizontal displacement at the pile top during the model test is 0 mm when the excavation depth $H\le$ 175 mm. As the excavation depth increases, the pile top gradually shifts toward the excavation side of the foundation pit. When the excavation depth $H=$225 mm and $H=$275 mm, the relative difference and root mean square error between the numerical simulation and model test results of the horizontal displacement at the pile top is less than 15 % and 0.33 %, respectively. The variation trend of the horizontal displacement at the pile top remains consistent, demonstrating that the numerical simulation results are in agreement with the model test results.

3.4. Comparative analysis of pile body bending moment in model test

Piles No. 3 and No. 6 are located at the center of the test section, and the deformation results are representative. When the excavation depth $H=$275 mm, the average bending moments of the pile bodies for piles No. 3 and No. 6 obtained from the numerical simulation were compared and validated against those derived from the model test. The test results and calculated values of pile body bending moment are presented in Table 6.

According to Table 6, except for the section at a burial depth of 250 mm, the difference in bending moment between the numerical simulation and the model test results for all other sections is less than 18 %. The bending moment of the cross-section at a burial depth of 250 mm exhibited a deviation of 72 %. The deviation of 72 % was attributed to the separation between the fine-hole gauze and the pile during testing, which allowed sand to infiltrate the gap between the two components. Consequently, during the excavation process, the pile body was subjected to the lateral earth pressure from the non-excavated side and the sand's additional load between the fine-pore gauze and the pile body.

The comparison of the bending moment of pile bodies without slope and with a slope behind the crown beam is presented in Fig. 10 and Fig. 11, respectively.

It can be observed from Fig. 10 and Fig. 11 that both the model test and the numerical simulation of the bending moment distribution along the pile body exhibit an “S” shaped distribution. In the model test, the maximum negative bending moment occurs at $H=$ 140 mm, while in the numerical simulation, it is located at $H=$ 150 mm. Similarly, the maximum positive bending moment is observed at $H=$ 350 mm in the model test and at $H=$ 360 mm in the numerical simulation. Overall, the numerical simulation results are in good agreement with those of the model test.

Fig. 10Comparison of bending moments of pile body without slope behind the crown beam

Fig. 11Comparison of the bending moments of pile body with a slope behind the crown beam

By comparing and analyzing the horizontal displacement at the top of the pile and the bending moment of the pile body, the numerical simulation results exhibit high consistency with the model test results. This indicates that the proposed numerical model is valid and can provide a reliable basis for designing foundation pit support schemes.

4.1. Project summary

A deep foundation pit was formed during the excavation of a specific third-line ship lock on the Xiangjiang River. This foundation pit is close to existing second-line ship locks, ship lock control buildings, and other structures. The location map of the project area is illustrated in Fig. 12. The surrounding environment of the foundation pit is complex; it is essential to have strict control of the deformation of the foundation pit. The preliminary design proposes a support scheme combining row piles with anchor cables. The layout of pile-anchor support section for the third-line ship lock is presented in Fig. 13.

Fig. 12The location map of the project area. Photo taken by Yuchen Liu at the Caijiazhou project site in Wangcheng District, Changsha City, Hunan Province on June 2023

Fig. 13The layout of pile-anchor support section for the third-line ship lock. Photo taken by Yuchen Liu at the Caijiazhou project site in Wangcheng District, Changsha City, Hunan Province on January 2024

The typical section is located adjacent to the existing second-line ship lock and the ship lock control building. The support height is 19.10 m. The foundation pit support scheme is a single-row pile combined with five anchor cables. The lengths of anchor cables are 23 m, 26 m, 23 m, 14 m, and 14 m, respectively. The cross-sectional dimensions of the crown beam are 1.20 m × 0.80 m (width × height), while those of the waist beam are 1.00 m × 0.80 m (width × height). The preliminary design scheme for pile-anchor support in the typical section is shown in Fig. 14.

Fig. 14Preliminary design scheme for pile-anchor support in typical section

4.2. Engineering geological conditions

According to the geological investigation, the rock and soil strata at the site consist of Quaternary residual fill, Holocene Quaternary Orange Isle Formation alluvial layers, and Yanshanian intrusive rocks, forming a total of seven distinct layers. The physical and mechanical properties of each layer rock and soil materials are presented in Table 7.

4.3. An overview of numerical simulation of engineering cases

4.3.3. Boundary conditions and mesh division of numerical model

The numerical simulation is divided into five components: soil, row pile, crown beam, waist beam, and anchor cable. The soil is categorized into the excavated side and the non-excavated side according to the cross-section where the pile axis is located. The slope and the top of the slope are the areas with additional construction loads. Excavation conditions are segmented into six steps in accordance with the construction sequence, with corresponding excavation depths of 2.0 m, 3.4 m, 6.4 m, 10.4 m, 14.4 m, and 19.1 m, respectively.

Fig. 15Boundary constraint condition of engineering case

Fig. 16Grid division of engineering case

The boundary constraint condition of engineering case is shown in Fig. 15. The bottom of the numerical model is subject to displacement constraints in the $X$, $Y$, and $Z$ directions. The $Z$-direction displacement constraint is applied to the boundary perpendicular of the pile row, and the $X$-direction displacement constraint is applied to the boundary parallel of the pile row.

The grid division of engineering case is illustrated in Fig. 16. The soil, row pile, and crown beam adopt three-dimensional solid elements, and the element type is C3D8. The anchor cable adopts truss elements, and the element type is T3D2. The waist beam adopts beam elements, and the element type is B31.

4.4. Numerical simulation results and comprehensive analysis

Through numerical simulation, a systematic analysis was conducted to investigate the influence laws of pile diameter on the horizontal displacement, bending moment, surface settlement, and bottom uplift of the foundation pit.

4.4.1. Results and analysis of horizontal displacement of pile body

To investigate the variation pattern of the pile body’s horizontal displacement, the results of the horizontal displacement were systematically analyzed when the foundation pit was excavated to the bottom ($H=$ 19.1 m). The horizontal displacement curve of pile body under different pile diameters is shown in Fig. 17.

It can be observed from Fig. 17 that the horizontal displacement curves of the pile body exhibit a “bulging belly” shape, characterized by smaller displacements at the top and bottom and a larger displacement in the middle. Additionally, the curvature of displacement curves decreases with the increase of pile diameter; when the diameter of the pile changes from 1 m to 2 m, the extreme value of the horizontal displacement of the pile body reduced from 39.79 mm to 10.63 mm, the burial depth of the displacement extreme point decreased from 9.03 m to 4.01 m, which is located between 0.47$H$ and 0.21$H$ of the foundation pit.

Fig. 17The horizontal displacement curve of pile body under different pile diameters

The “Technical standard for monitoring of building excavation engineering” (GB50497-2019) [28] and the “Technical specification for retaining and protection of building foundation excavations” (JGJ120-2012) [29] primarily provide limitations on the horizontal displacement of the pile top for some support structures and do not explicitly indicate the displacement control criteria for pile-anchor support system. The “Code for design of building foundation” (GB 50007-2011) [30] specifies the maximum lateral deformation of supporting structures based on the clear distance between adjacent buildings and excavation boundaries of foundation pits. In this study, by referring to the aforementioned standards and considering the specific characteristics of this foundation pit, the extreme value of horizontal displacement at the pile top is set to 20 mm, and the extreme value of horizontal displacement within the pile body is determined as 57.3 mm, serving as the displacement control indicators for the pile-anchor support system. Based on the above calculation results, the pile diameter $d=$ 1.5 m is identified as the optimal choice.

4.4.2. Results and analysis of pile body bending moment

To investigate the variation pattern of the pile body bending moment, the results of pile body bending moment were summarized when the foundation pit was excavated to the bottom ($H=$ 19.1 m). The bending moment curve of pile body under different pile diameters is shown in Fig. 18.

It can be known from Fig. 18 that the pile diameter increases from 1.0 m to 2.0 m, the extreme value of the negative bending moment of the pile body increases from 1590 kN·m to 3130 kN·m, and the burial depth of the extreme value point of the negative bending moment decreases from 11.63 m to 10.66 m, which is located between 0.61$H$ and 0.56$H$ of foundation pit. The extreme value of the positive bending moment of the pile body increased from 2730 kN·m to 10180 kN·m, and the burial depth of the extreme point of the positive bending moment increased from 19.33 m to 19.81 m, which was located at the bottom of the foundation pit (–0.01$H$ to –0.03$H$). A larger pile diameter corresponds to a greater stiffness of the pile body, enabling it to withstand higher bending moments and bear greater loads, which results in an increased bending moment of the pile body. The positions of the extreme points of positive and negative bending moments change relatively little. When the pile diameter is 1.5 m $\le d\le$2.0 m, the lateral negative bending moment curves on the pile body almost coincide. To fully leverage the role of the anchor cable in the pile-anchor support structure, the pile diameter $d=$1.5 m is the optimal.

Fig. 18The bending moment curve of pile body under different pile diameters

4.4.3. Results and analysis of surface settlement

To investigate the variation pattern of surface settlement, the surface settlement data obtained upon excavating the foundation pit to the bottom ($H=$ 19.1 m) were systematically analyzed. Fig. 19 shows the maximum ground settlement at the top of the slope under different pile diameters. Notably, vertical displacement is defined as positive for upward movement and negative for downward movement.

It can be observed from Fig. 19 that as the pile diameter increases from 1.0 m to 2.0 m, the surface settlement value decreases from 12.62 mm to 4.02 mm, and the tangent inclination angle of the surface settlement relationship curve gradually decreases. A larger pile diameter results in an increased overall stiffness of the supporting structure, thereby reducing the surface settlement.

Fig. 19The maximum ground settlement at the top of the slope under different pile diameters

The adjacent building is the ship lock control building, with a clear distance of 26.57 meters away from the excavation boundary of foundation pit. By the “Code for Design of Building Foundation” (GB 50007-2011), the surface settlement limit is determined to be 0.25 % of the excavation depth $H$ of foundation pit, that is, 47.75 mm. For various pile diameters, the maximum surface settlement values after the completion of foundation pit excavation are all less than 47.75 mm. Increasing the pile diameter effectively reduces surface settlement at the top of the slope. However, the extent of surface settlement reduction decreases with the pile diameter increase. Excessive pile diameters result in underutilization of support performance and waste of engineering materials. Considering both safety and economic efficiency comprehensively, a reasonable pile diameter is within the range of 1.2 m $\le d\le$ 1.8 m, with $d=$ 1.5 m being the optimal choice.

5.1. Results and analysis of inclination control mechanisms in ship locks

To investigate the variation law of the influence of pile diameter on the top inclination of the ship lock control building, The calculation results of the top inclination of the ship lock control building excavated to the bottom of the pit ($H=$19.1 m) were systematically summarized, The inclination curve of the ship lock control building under different pile diameters is presented in Fig. 21. The formula for calculating the inclination of the ship lock is provided in Eq. (2). In this study, vertical displacement is defined as positive for upward movement and negative for downward movement:

where $\theta$ is the inclination of the ship lock, $S$ and $L$ is lateral displacement and vertical displacement of the measuring point, respectively; $H$ is height difference between the measuring point and the ground surface.

Fig. 21The inclination curve of the ship lock control building under different pile diameters

The overall displacement of the ship lock control building tilts towards the foundation pit. As illustrated in Fig. 21, the pile diameter increases from 1.0 m to 2.0 m, the inclination of the ship lock control building roof gradually decreases from 0.0027 to 0.0009. Both values are significantly smaller than the allowable inclination limit of 0.004 specified in the “Code for Design of Building Foundation” (GB 50007-2011).

5.2. Vertical settlement of the gate chamber foundation

To study the variation law of the influence of pile diameter on the vertical displacement of the gate chamber foundation, the results of the vertical displacement of the gate chamber foundation when the foundation pit was excavated to the bottom of the pit ($H=$19.1 m) were systematically summarized. The vertical displacement of gate chamber foundation under different pile diameters is shown in Fig. 22.

Fig. 22The vertical displacement of gate chamber foundation under different pile diameters

It can be observed from Fig. 22 that as the pile diameter increases from 1.0 m to 2.0 m, the average vertical displacement of the gate chamber foundation correspondingly increases from 0.19 mm to 0.59 mm. Both values remain significantly smaller than the allowable average settlement limit of 200 mm specified in the “Code for Design of Building Foundation” (GB 50007-2011).

5.3. Results and analysis of active deformation in finite soil masses

Due to the limitations of the foundation of the second-line ship lock adjacent to the existing building, the soil outside the foundation pit of the third line ship lock does not meet the semi-infinite soil condition, and there is finite soil failure and deformation [31-32]. To study the variation law of pile diameter on the active deformation of finite soil, the $X$-direction soil displacement nephogram of excavation $H=$19.1 m in the foundation pit was summarized; The nephogram of finite soil displacement in the $X$ direction under different pile diameters is shown in Figs. 23-27. For the convenience of comparative analysis, a unified scale is used for all displacement nephogram.

Fig. 23Cloud map representing soil displacement in the X direction (d= 1.0 m)

Fig. 24Cloud map representing soil displacement in the X direction (d= 1.2 m)

Fig. 25Cloud map representing soil displacement in the X direction (d= 1.5 m)

Fig. 26Cloud map representing soil displacement in the X direction (d= 1.8 m)

Fig. 27Cloud map representing soil displacement in the X direction (d= 2.0 m)

The displacement boundary line of 1.01×10^-2 m is obvious, and is therefore selected as the research object. From Fig. 23 and Fig. 24, it can be seen that the typical sliding surface morphology of the soil after the pile with a diameter of $d=$1.0 and $d=$1.2 m is a curve that starts from the heel of the wall and ends at the surface of the fill soil, satisfying the semi-infinite soil failure. From Fig. 25, it can be seen that the typical sliding surface morphology of active deformation of the soil behind the pile with a diameter of $d=$1.5 m starts from the heel of the wall, turns back to the adjacent building foundation, and ends at the surface of the fill soil, the turning point is located at the foundation pit height of 0.74$H$, which belongs to finite soil failure. From Fig. 26 and Fig. 27, it can be seen that the typical sliding surface morphology of active deformation of soil behind piles with a diameter of $d=$1.8 m and $d=$2.0 m is a curve or broken line that starts from 0.2$H$ and 0.6$H$ on the wall and ends at the surface of the fill soil, which belongs to finite soil failure. The $d=$1.0 m, $d=$1.2 m sliding surface start is located at the wall heel, the $d=$1.5 m sliding surface start point is located at the wall heel, but the slope inclination of sliding surfaces increases sharply, the $d=$1.8 m sliding surface start point is 0.2$H$ higher than the wall heel, and the $d=$2.0 m sliding surface start point is 0.6$H$ than the wall heel. As the pile diameter increases, the active failure mode changes from semi-infinite soil to finite soil, and $d=$1.5 m is the critical pile diameter value for finite soil.

Due to the deformation limitations of adjacent buildings outside the foundation pit, and taking into account internal factors such as horizontal displacement of the pile body, bending moment of the pile body, surface settlement, and uplift of the pit bottom, the optimal choice for the engineering case in this article is a pile diameter of $d=$1.5 m.