# Analytical Method for Capped Pile–Soil Interaction considering the Load Action of Soil under the Pile Cap

Authors: Shilin Luo; Mingquan Liu; Jianqing Jiang; Ailifeila Aierken; Jin Chang; Xuewen Zhang; Rui Zhang
Journal: Geofluids
 (2022)

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## Abstract

To study the pile–soil interaction mechanism of capped pile, an analytical method considering the load action of soil under the pile cap is proposed. The shearing displacement method is used to derive the lateral friction of pile body under the pile top load, and the Boussinesq solutions is used to derive the lateral friction of pile body considering the load of soil under the cap. The theoretical expressions of axial force and load-settlement curves are also achieved by means of establishing and solving the equilibrium differential equation of the pile body. Comparison of calculation results with the ordinary pile indicates that soil load under the pile cap reduces the lateral friction value; the influenced depth is about four times of the load action radius. The axial force and the load-settlement curve data are verified by a case history data. The results show that the computed data agree well with the measured data. The proposed method can direct the design of capped pile composite foundations.


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## Body

## 1. Introduction

Capped pile is a type of reinforcement body in foundation; it is formed by placing a certain size of pile cap on the top of an ordinary pile body. The pile body is usually a prestressed pipe pile or solid concrete pile. The pile cap is normally a round or a square concrete slab. Capped piles have been widely used in soft soil treatment [1–3]. The original purpose of setting a cap on the pile top is to reduce the pile top penetration into the upper cushion and regulate the load sharing ratio of the pile and soil [4, 5]. With the increasing engineering applications, pile caps are found to raise the bearing capacity of piles and influence the working behavior of composite foundations, which many scholars have further researched [6–10]. The research methods include field test [11], model test [12–16], experiment test [17–19], and numerical simulation approaches [20–29].When the pile settled under a vertical load, there is a displacement difference between the pile body and soil of the ordinary pile, while there is no such displacement difference between pile body and soil of capped pile due to the coordination of the cap. This leads to the pile body–soil interaction difference between capped pile and ordinary pile. Shang et al. analyzed the interaction of a flexible subsoil matrix and a piled box (raft) using Geddes stress factor [30]. The results show that the method is reliable. Cui et al. studied the response of pile groups in layered soil under the assumption that pile–cap–soil interaction is linear elastic [31]. The study used the shearing displacement method, and computed load-settlement curves were consistent but beneath the measured ones. Jin-bo and Cong-xin analyzed pile–soil interaction with a stiffness matrix method based on differential displacement of pile and soil. The pile lateral friction and pile tip load were assumed to be proportional to the respective displacement [32]. Wang et al. assumed that the distribution of skin friction on capped pile is simplified as two force triangles; by combining Mindlin-Geddes and Boussinesq solutions, the equation of the additional side stresses for a single capped pile foundation is derived [33]. Chen et al. used a slider displacement method to consider the overall bearing capacity of capped piles. In their study, the Plandel shear slip failure model is used in the upper foundation soil bearing capacity analysis, and the load transfer method is used to analyze the bearing capacity of the lower pile body. This method is suitable for short, rigid piles with large pile caps under the condition of smooth soil interface with the pile cap [34].To sum up, the pile–soil interaction is often considered in the settlement calculation of composite foundation. Compared with the large number of engineering applications, theoretical researches of capped pile–soil interaction are not extensive.In this paper, we propose an analytical method to further study the pile–soil interaction mechanisms of capped piles with consideration to the load action of soil under the cap. Upper load is transmitted to the pile body and the soil under the cap through pile cap. The lateral frictional resistance of the capped body is superposed by friction caused by the load on the pile body and the load on the soil under the pile cap. By establishing the differential equilibrium equation of the pile body, the theoretical expressions of pile axial force and the load-settlement curve are deduced. The feasibility of this method is verified by comparing computed results with other methods and field test data of a case history. We anticipate that the present research could help to better understand the pile–soil interaction mechanism issues of capped pile, such as the relationship of load-settlement curves and treatment of soft soil foundation, and then provide some practical experience and enlightenment on the studying of the pile engineering.


## 2. Analysis Method of Pile–Soil Interaction

### 2.1. Analysis Model

When a vertical load is acting on the top of a pile cap, the load is transmitted downward by the pile cap. Because the pile cap is directly in contact with the pile body and the soil, the upper load is transferred to the pile body and the soil under the cap simultaneously. Due to the differential resistance stiffness of the pile body and the soil, each of them supports a different amount of load. Existing test results and finite element simulation results had confirmed that the load sharing phenomenon occurs throughout the pile body settlement process. The influence of the load on the soil under the pile cap cannot be ignored when analyzing the pile–soil interaction. To appropriately simplify the analysis, the following assumptions are used: (1) the pile cap is rigid and without bending and shearing deformation, (2) the pile body is rigid and without radial deformation, (3) the load transmitted to the pile top is equivalent to a concentrated load, and the load transmitted to the soil under the cap is equivalent to a uniformly distributed load, (4) the stress in foundation soil is calculated according to the elastic theory, and (5) the pile tip stress is calculated using the Winkler foundation model.The mechanical model of a pile with cap is established, as shown in Figure1, where R and r are the radii of the pile cap and pile body, respectively, q and q1 are uniform loads acting on the pile top and soil under the cap, respectively, and Pt is the equivalent concentrated load acting on the pile top. The relationship among the variables is described:
(1)Pt+q1×πR−r2=q×πR2.Figure 1
Mechanical model of capped pile.
(a)
Prototype of load(b)
Load distribution modelAccording to Figure1 and assumption (3), a pile–soil interaction analysis model (Figure 2) is established on the above basis, where Pb is equivalent concentrated load at the pile tip and qb is uniform load at the bottom of the soil under the cap.Figure 2
Pile–soil interaction analysis model.When a concentrated loadPt is acted on the pile body alone, the lateral friction will appear on the pile side. When a uniform load q1 is acted on the soil under cap individually, there will also be friction on the pile side. When they act together, the above two states can be superimposed. Then, the lateral friction on the pile–soil interface can be calculated by the superposition of τp and τq, which are derived from the concentrated load Pt acting on the top of the pile and the uniform load q1 acting on the top of the soil under the cap, respectively.


### 2.2. Solution for Lateral Friction

First, we study the case of concentrated loadPt acting on the top of a pile. Rondoloph and Worth assumed that when a concentrated load is acting on a single pile, the soil around the pile only produces annular shear displacement, which can be approximately treated as a plane problem [35]. Thus, a rectangular coordinate system is employed to establish the calculation schematic diagram (Figure 3), and the origin of the coordinate axis is located at the pile body center.Figure 3
Calculation schematic diagram ofPt.
(a)
Load schematic(b)
Calculation schematic [35]According to the shearing displacement method, ifl is the pile length, Ap is the area of the cross section of the pile body, Up is the pile perimeter, Ep is the elastic modulus of the pile body, Gs is the shearing deformation modulus of the soil, ν is the Poisson ratio of the soil, ωb is displacement of the pile tip, and rm is the maximum influence radius that can be assigned as 0.51−νl, the expressions for lateral frictional resistance of the pile body τpz, displacement of the pile body ωpz, axial force of the pile body Npz, and equivalent concentrated load at the pile tip Pb are shown, respectively, as follows:
(2)τpz=k1Upωpz,(3)ωpz=ωbcoshλl−z,(4)NPz=Pbk1k2sinhλl−z+coshλl−z,(5)Pb=k2ωb,where λ2 is the quotient of k1 and ApEp and k1 is the shear stiffness between the pile and soil, calculated as follows:
(6)k_1=2πGsln−1r_m/r.The compressive stiffness of soil (k2) beneath the pile tip can be calculated as follows:
(7)k2=4rGs/1−υ.Then, we study the case of uniform loadq1 acting on the soil under the cap. The same coordinate system is employed to establish the calculation schematic diagram (Figure 4). The pile–soil interface is located on the vertical plane with distance from the origin r; q1 is effective in the range between r and R, and the coordinates of an arbitrary point P underground on the pile–soil interface are r,z. The load on the micro unit dx is replaced by linear load q1dxr≤x≤R.Figure 4
Calculation schematic of soil loadq1.
(a)
Load schematic(b)
Calculation schematicAccording to the plane stress solution [36], the additional shear stress τqz induced by linear load q1 at point P at depth z (0≤z≤l) on the pile–soil interface is
(8)τqz=q1πsin2θ,where θ is the angle between the connection line of point P to unit dx and the pile–soil interface (0≤θ≤π/2).Substituting the geometrical relations in Figure4(b) into Equation (8) produces
(9)τqz=q1πR−r2z2+R−r2.Finally, the lateral friction on the pile–soil interfaceτz is the sum of τpz and τqz:
(10)τz=τpz+τqz.Considering the directions ofτpz and τqz produces
(11)τz=k1Upcoshλl−z⋅ωb−q1π⋅R−r2z2+R−r2.


### 2.3. Solutions for Axial Force and Settlement

When the expression of lateral friction is obtained, take the pile body in Figure2 as a calculation isolator (Figure 5).Figure 5
Calculation isolator.According to the force balance of the pile body, the equilibrium equation is established:(12)Nz=Pt−∫0zUpτzdz=Pt−∫0zUpτpz+τqzdz.By solving the integral, the axial force expressionNz is obtained:
(13)Nz=NPz+2rq1R−r−zarctanR−rz,where Npz is the axial force of an ordinary pile body.According to the elastic compression conditions of a pile body, the relationship of displacement and frictional resistance of the pile side is expressed as Equation (14), and the relationship of axial force and displacement of the pile body is expressed as
(14)ωz=1ApEp∫lzUp⋅τzdz,(15)Nz=ApEp⋅dwzdz.By substituting Equations (2), (3), and (9) into Equation (10), the lateral friction of the capped pile is obtained as
(16)τz=Pt⋅k2⋅coshλl−zUp⋅k1⋅sinhλl+k2⋅coshλl−q1⋅R−r2π⋅z2+R−r2.By substituting Equation (11) into Equation (14), the settlement of the capped pile is calculated:
(17)ωz=Ptcoshλl−zk1sinhλl+k2coshλl+2q1πrEpR−rarctanlR−r−arctanzR−r.Then substituting Equation (17) into Equation (15), the axial force of the capped pile is calculated:
(18)Nz=Pt⋅coshλl‐zk1⋅sinhλl−z+k2⋅coshλl−z+2rq1R−r−zarctanR−rz.Whenz=0, the settlement of the pile top st is equal to ω0.
(19)ω0=st=Pt⋅coshλlk1⋅sinhλl+k2⋅coshλl+2q1π⋅r⋅EpR−rarctanlR−r.Equations (18) and (19) are the theoretical expressions of the axial force and the load-settlement curve of a capped pile, respectively. Based on the above analysis, the corresponding computational program is compiled to facilitate the comparison and verification of the calculation results.


## 2.1. Analysis Model

When a vertical load is acting on the top of a pile cap, the load is transmitted downward by the pile cap. Because the pile cap is directly in contact with the pile body and the soil, the upper load is transferred to the pile body and the soil under the cap simultaneously. Due to the differential resistance stiffness of the pile body and the soil, each of them supports a different amount of load. Existing test results and finite element simulation results had confirmed that the load sharing phenomenon occurs throughout the pile body settlement process. The influence of the load on the soil under the pile cap cannot be ignored when analyzing the pile–soil interaction. To appropriately simplify the analysis, the following assumptions are used: (1) the pile cap is rigid and without bending and shearing deformation, (2) the pile body is rigid and without radial deformation, (3) the load transmitted to the pile top is equivalent to a concentrated load, and the load transmitted to the soil under the cap is equivalent to a uniformly distributed load, (4) the stress in foundation soil is calculated according to the elastic theory, and (5) the pile tip stress is calculated using the Winkler foundation model.The mechanical model of a pile with cap is established, as shown in Figure1, where R and r are the radii of the pile cap and pile body, respectively, q and q1 are uniform loads acting on the pile top and soil under the cap, respectively, and Pt is the equivalent concentrated load acting on the pile top. The relationship among the variables is described:
(1)Pt+q1×πR−r2=q×πR2.Figure 1
Mechanical model of capped pile.
(a)
Prototype of load(b)
Load distribution modelAccording to Figure1 and assumption (3), a pile–soil interaction analysis model (Figure 2) is established on the above basis, where Pb is equivalent concentrated load at the pile tip and qb is uniform load at the bottom of the soil under the cap.Figure 2
Pile–soil interaction analysis model.When a concentrated loadPt is acted on the pile body alone, the lateral friction will appear on the pile side. When a uniform load q1 is acted on the soil under cap individually, there will also be friction on the pile side. When they act together, the above two states can be superimposed. Then, the lateral friction on the pile–soil interface can be calculated by the superposition of τp and τq, which are derived from the concentrated load Pt acting on the top of the pile and the uniform load q1 acting on the top of the soil under the cap, respectively.


## 2.2. Solution for Lateral Friction

First, we study the case of concentrated loadPt acting on the top of a pile. Rondoloph and Worth assumed that when a concentrated load is acting on a single pile, the soil around the pile only produces annular shear displacement, which can be approximately treated as a plane problem [35]. Thus, a rectangular coordinate system is employed to establish the calculation schematic diagram (Figure 3), and the origin of the coordinate axis is located at the pile body center.Figure 3
Calculation schematic diagram ofPt.
(a)
Load schematic(b)
Calculation schematic [35]According to the shearing displacement method, ifl is the pile length, Ap is the area of the cross section of the pile body, Up is the pile perimeter, Ep is the elastic modulus of the pile body, Gs is the shearing deformation modulus of the soil, ν is the Poisson ratio of the soil, ωb is displacement of the pile tip, and rm is the maximum influence radius that can be assigned as 0.51−νl, the expressions for lateral frictional resistance of the pile body τpz, displacement of the pile body ωpz, axial force of the pile body Npz, and equivalent concentrated load at the pile tip Pb are shown, respectively, as follows:
(2)τpz=k1Upωpz,(3)ωpz=ωbcoshλl−z,(4)NPz=Pbk1k2sinhλl−z+coshλl−z,(5)Pb=k2ωb,where λ2 is the quotient of k1 and ApEp and k1 is the shear stiffness between the pile and soil, calculated as follows:
(6)k_1=2πGsln−1r_m/r.The compressive stiffness of soil (k2) beneath the pile tip can be calculated as follows:
(7)k2=4rGs/1−υ.Then, we study the case of uniform loadq1 acting on the soil under the cap. The same coordinate system is employed to establish the calculation schematic diagram (Figure 4). The pile–soil interface is located on the vertical plane with distance from the origin r; q1 is effective in the range between r and R, and the coordinates of an arbitrary point P underground on the pile–soil interface are r,z. The load on the micro unit dx is replaced by linear load q1dxr≤x≤R.Figure 4
Calculation schematic of soil loadq1.
(a)
Load schematic(b)
Calculation schematicAccording to the plane stress solution [36], the additional shear stress τqz induced by linear load q1 at point P at depth z (0≤z≤l) on the pile–soil interface is
(8)τqz=q1πsin2θ,where θ is the angle between the connection line of point P to unit dx and the pile–soil interface (0≤θ≤π/2).Substituting the geometrical relations in Figure4(b) into Equation (8) produces
(9)τqz=q1πR−r2z2+R−r2.Finally, the lateral friction on the pile–soil interfaceτz is the sum of τpz and τqz:
(10)τz=τpz+τqz.Considering the directions ofτpz and τqz produces
(11)τz=k1Upcoshλl−z⋅ωb−q1π⋅R−r2z2+R−r2.


## 2.3. Solutions for Axial Force and Settlement

When the expression of lateral friction is obtained, take the pile body in Figure2 as a calculation isolator (Figure 5).Figure 5
Calculation isolator.According to the force balance of the pile body, the equilibrium equation is established:(12)Nz=Pt−∫0zUpτzdz=Pt−∫0zUpτpz+τqzdz.By solving the integral, the axial force expressionNz is obtained:
(13)Nz=NPz+2rq1R−r−zarctanR−rz,where Npz is the axial force of an ordinary pile body.According to the elastic compression conditions of a pile body, the relationship of displacement and frictional resistance of the pile side is expressed as Equation (14), and the relationship of axial force and displacement of the pile body is expressed as
(14)ωz=1ApEp∫lzUp⋅τzdz,(15)Nz=ApEp⋅dwzdz.By substituting Equations (2), (3), and (9) into Equation (10), the lateral friction of the capped pile is obtained as
(16)τz=Pt⋅k2⋅coshλl−zUp⋅k1⋅sinhλl+k2⋅coshλl−q1⋅R−r2π⋅z2+R−r2.By substituting Equation (11) into Equation (14), the settlement of the capped pile is calculated:
(17)ωz=Ptcoshλl−zk1sinhλl+k2coshλl+2q1πrEpR−rarctanlR−r−arctanzR−r.Then substituting Equation (17) into Equation (15), the axial force of the capped pile is calculated:
(18)Nz=Pt⋅coshλl‐zk1⋅sinhλl−z+k2⋅coshλl−z+2rq1R−r−zarctanR−rz.Whenz=0, the settlement of the pile top st is equal to ω0.
(19)ω0=st=Pt⋅coshλlk1⋅sinhλl+k2⋅coshλl+2q1π⋅r⋅EpR−rarctanlR−r.Equations (18) and (19) are the theoretical expressions of the axial force and the load-settlement curve of a capped pile, respectively. Based on the above analysis, the corresponding computational program is compiled to facilitate the comparison and verification of the calculation results.


## 3. Comparison and Validation

A case history is introduced to validate the computed results of this method. The Su-Kun-Tai Expressway is located beside Yangcheng Lake in China, and the majority of the route is constructed on a deep, soft, soil foundation. The distribution and physical parameters of the foundation soil layers are shown in Table1.Table 1
Distribution and physical parameters of soil layers.
Soil layerDepth (m)Water content, (%)Bulk density (kN/m3)Compression modulus (MPa)Poisson ratioSandy silt1.230.919.25.830.3Silty clay1.2-1.733.918.43.470.3Clay1.7-8.938.318.12.620.3Muddy clay8.9-10.648.317.22.80.3Silty clay10.6-12.135.8183.180.3Silty clay12.1-26.42419.86.970.3Silty clay26.4-31.423.619.98.230.3Capped piles were used to reduce settlement under the embankment load. The pile body is a PTC-A400-65 concrete pipe pile with lengths of 25 m (T2) and 29 m (T4) in two different test sections. The caps are prefabricated 1.5 m square concrete slabs with thickness of 40 cm. Full-scale tests of capped piles were performed for lateral friction, axial force, and bearing capacity of T2 and T4 piles. The axial force was measured by prefabricated stress gauge at different depths on steel bars in the pile body, and the soil pressure under the pile cap was measured by embedded soil pressure boxes under the pile cap. The bearing capacity of capped piles was determined by load testing [11]. The measured data of the T2 and T4 piles are shown in Table 2 and are used to compare and validate the proposed method.Table 2
Measured data of T2 and T4.
Total load (kN)Settlement (mm)Load on pile topPt/(kN)Uniform load on soil under the capq1/(kN/m2)T24002.036317.66002.954525.98004.873132.610007.091738.912008.7110047.1140010.4128454.9160014.6144871.5180035.11107326.3200056.91258349.1T45001.746915.07502.669525.910004.392933.612505.5115644.015007.4138155.917509.6160071.0200012.4181089.1225018.01978128.5250049.21840310.8


### 3.1. Comparison

In order to compare the influence of pile cap on the lateral friction, the capped pile and the ordinary pile are taken into account. The influence of load action on the soil under the pile cap is considered in the lateral friction analysis. The shearing displacement method [35] is adopted for the ordinary pile calculating; the proposed method is adopted for capped pile calculating. The lateral friction of two types of piles is calculated under the same condition, and the results are shown in Figure 6.Figure 6
Comparison of lateral friction along pile length of T4.The lateral friction curve of two types of piles in Figure6 shows obvious difference near the pile top. The lateral friction of capped pile is less than that of ordinary pile. The reason for this phenomenon is that the load on the soil under the pile cap (q1) reduced the soil deformation, which lead to the pile lateral friction becomes small. The value difference between them is the largest at the pile top and decreases with the depth. This indicates that the load (q1) affected lateral friction in a certain depth. In this case, the equivalent action radius of q1 is 0.645 m; the max influenced depth is 2.5 m, which is about 4 times of the load action radius. Beyond this depth, the effect on the lateral friction caused by q1 is very little and can be ignored. In addition, the load level has a great influence on the lateral friction difference. The higher the load level, the greater the difference. For example, it is 8.5 kPa, 19.5 kPa, and 32.5 kPa when the load level is 500 kN, 1000 kN, and 1500 kN, respectively.All of the above indicate that when the soil under the pile cap is carrying a load, the lateral friction is affected by the load level and the action radius.


### 3.2. Validation

#### 3.2.1. Axial Force and Settlement

In order to verify the availability of this method, the calculation results of axial force and the settlement are validated with the measured data. The parameters used in calculation are shown in Table3.Table 3
Calculation parameters.
Pile numberPile lengthl/(m)Pile radiusr/(m)Soil deformation modulusEs/(Mpa)Pile deformation modulusEp/(MPa)Soil shear modulusGs/(Mpa)Soil Poisson ratioνT2250.25.21300002.210.3T4290.25.45300002.230.3As seen from Figure7, the variation of axial force calculated by this method is consistent with the measured data. With the load increasing, the axial force of the pile body increases evenly in most stages. But when the load changes from 2000 kN to 2250 kN, the increase of axial force at the pile tip is smaller. Under the two load levels, the pile body begins to settle faster, and the measured settlement data in Table 2 reflect this phenomenon clearly. This indicates that the soil under the pile tip began to yield and produced plastic deformation, so the axial force increase of pile tip is small. At this time, the load increment of pile body decreased, and the load on soil under the pile cap (q1) increased correspondingly to balance the total load. When q1 is increased, the lateral friction near the pile top decreases, leading the axial force to increase. On the whole, the results of axial force computed by this method are consistent with the experimental phenomenon and the theoretical analysis.Figure 7
Axial force curves of T4 along pile length.
(a)
Load 500kN-1250kN(b)
Load 1500kN-2000kNThe calculated load-settlement curves (Figure8) are as follow:Figure 8
Load-settlement curves of T2 and T4.Figure8 demonstrates that the variation of load-settlement curves computed by this method is consistent with that of the measured curves, but the calculated settlements are slightly less than the measured data. In the two cases (T2 and T4), the error of settlement increases when the soil is close to failure. The reason for this phenomenon is that when the capped pile is approaching the ultimate load, the differential displacement between pile and soil is increased gradually, the condition of equal strain cannot be strictly satisfied, which leads to a larger error.


## 3.1. Comparison

In order to compare the influence of pile cap on the lateral friction, the capped pile and the ordinary pile are taken into account. The influence of load action on the soil under the pile cap is considered in the lateral friction analysis. The shearing displacement method [35] is adopted for the ordinary pile calculating; the proposed method is adopted for capped pile calculating. The lateral friction of two types of piles is calculated under the same condition, and the results are shown in Figure 6.Figure 6
Comparison of lateral friction along pile length of T4.The lateral friction curve of two types of piles in Figure6 shows obvious difference near the pile top. The lateral friction of capped pile is less than that of ordinary pile. The reason for this phenomenon is that the load on the soil under the pile cap (q1) reduced the soil deformation, which lead to the pile lateral friction becomes small. The value difference between them is the largest at the pile top and decreases with the depth. This indicates that the load (q1) affected lateral friction in a certain depth. In this case, the equivalent action radius of q1 is 0.645 m; the max influenced depth is 2.5 m, which is about 4 times of the load action radius. Beyond this depth, the effect on the lateral friction caused by q1 is very little and can be ignored. In addition, the load level has a great influence on the lateral friction difference. The higher the load level, the greater the difference. For example, it is 8.5 kPa, 19.5 kPa, and 32.5 kPa when the load level is 500 kN, 1000 kN, and 1500 kN, respectively.All of the above indicate that when the soil under the pile cap is carrying a load, the lateral friction is affected by the load level and the action radius.


## 3.2. Validation

### 3.2.1. Axial Force and Settlement

In order to verify the availability of this method, the calculation results of axial force and the settlement are validated with the measured data. The parameters used in calculation are shown in Table3.Table 3
Calculation parameters.
Pile numberPile lengthl/(m)Pile radiusr/(m)Soil deformation modulusEs/(Mpa)Pile deformation modulusEp/(MPa)Soil shear modulusGs/(Mpa)Soil Poisson ratioνT2250.25.21300002.210.3T4290.25.45300002.230.3As seen from Figure7, the variation of axial force calculated by this method is consistent with the measured data. With the load increasing, the axial force of the pile body increases evenly in most stages. But when the load changes from 2000 kN to 2250 kN, the increase of axial force at the pile tip is smaller. Under the two load levels, the pile body begins to settle faster, and the measured settlement data in Table 2 reflect this phenomenon clearly. This indicates that the soil under the pile tip began to yield and produced plastic deformation, so the axial force increase of pile tip is small. At this time, the load increment of pile body decreased, and the load on soil under the pile cap (q1) increased correspondingly to balance the total load. When q1 is increased, the lateral friction near the pile top decreases, leading the axial force to increase. On the whole, the results of axial force computed by this method are consistent with the experimental phenomenon and the theoretical analysis.Figure 7
Axial force curves of T4 along pile length.
(a)
Load 500kN-1250kN(b)
Load 1500kN-2000kNThe calculated load-settlement curves (Figure8) are as follow:Figure 8
Load-settlement curves of T2 and T4.Figure8 demonstrates that the variation of load-settlement curves computed by this method is consistent with that of the measured curves, but the calculated settlements are slightly less than the measured data. In the two cases (T2 and T4), the error of settlement increases when the soil is close to failure. The reason for this phenomenon is that when the capped pile is approaching the ultimate load, the differential displacement between pile and soil is increased gradually, the condition of equal strain cannot be strictly satisfied, which leads to a larger error.


## 3.2.1. Axial Force and Settlement

In order to verify the availability of this method, the calculation results of axial force and the settlement are validated with the measured data. The parameters used in calculation are shown in Table3.Table 3
Calculation parameters.
Pile numberPile lengthl/(m)Pile radiusr/(m)Soil deformation modulusEs/(Mpa)Pile deformation modulusEp/(MPa)Soil shear modulusGs/(Mpa)Soil Poisson ratioνT2250.25.21300002.210.3T4290.25.45300002.230.3As seen from Figure7, the variation of axial force calculated by this method is consistent with the measured data. With the load increasing, the axial force of the pile body increases evenly in most stages. But when the load changes from 2000 kN to 2250 kN, the increase of axial force at the pile tip is smaller. Under the two load levels, the pile body begins to settle faster, and the measured settlement data in Table 2 reflect this phenomenon clearly. This indicates that the soil under the pile tip began to yield and produced plastic deformation, so the axial force increase of pile tip is small. At this time, the load increment of pile body decreased, and the load on soil under the pile cap (q1) increased correspondingly to balance the total load. When q1 is increased, the lateral friction near the pile top decreases, leading the axial force to increase. On the whole, the results of axial force computed by this method are consistent with the experimental phenomenon and the theoretical analysis.Figure 7
Axial force curves of T4 along pile length.
(a)
Load 500kN-1250kN(b)
Load 1500kN-2000kNThe calculated load-settlement curves (Figure8) are as follow:Figure 8
Load-settlement curves of T2 and T4.Figure8 demonstrates that the variation of load-settlement curves computed by this method is consistent with that of the measured curves, but the calculated settlements are slightly less than the measured data. In the two cases (T2 and T4), the error of settlement increases when the soil is close to failure. The reason for this phenomenon is that when the capped pile is approaching the ultimate load, the differential displacement between pile and soil is increased gradually, the condition of equal strain cannot be strictly satisfied, which leads to a larger error.


## 4. Conclusions

Compared with the ordinary pile, the pile–soil interaction of capped pile is more complicated. In this study, we proposed an analysis method of pile–soil interaction for capped piles considering the load action of the soil under the pile cap. We established an analysis model of pile–soil interaction according to the mechanical characteristics of capped piles and analyzed the lateral friction by combining the shearing displacement method and Boussinesq solutions. Theoretical analytical expressions of lateral friction, axial force, and load-settlement curves were obtained by establishing the equilibrium differential equation of a pile body and solving it. By comparing the variation of lateral friction curve between ordinary pile and capped pile, it is found that the load acting on the soil under the cap reduces the lateral friction value in a depth of about 4 times the load action radius. Beyond this depth, the influence can be ignored. This is unfavorable to the bearing capacity of pile body; for short piles with large pile caps, this phenomenon should be fully considered in design. The axial force and load-settlement curves calculated by this method are validated by a case history of reinforcing subgrade with capped piles. The result shows that the variations of axial forces agree with the experimental date and theoretical analysis and the variation of load-settlement curves are consistent with that of the measured curves. The error of load-settlement increases when the soil is close to failure, which need to be further studied.



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*Source: 1007731-2022-03-24.xml*