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.
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 Figure 1, where and are the radii of the pile cap and pile body, respectively, and are uniform loads acting on the pile top and soil under the cap, respectively, and is the equivalent concentrated load acting on the pile top. The relationship among the variables is described:
(a) Prototype of load
(b) Load distribution model
According to Figure 1 and assumption (3), a pile–soil interaction analysis model (Figure 2) is established on the above basis, where is equivalent concentrated load at the pile tip and is uniform load at the bottom of the soil under the cap.
When a concentrated load is acted on the pile body alone, the lateral friction will appear on the pile side. When a uniform load 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 and , which are derived from the concentrated load acting on the top of the pile and the uniform load acting on the top of the soil under the cap, respectively.
2.2. Solution for Lateral Friction
First, we study the case of concentrated load 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.
(a) Load schematic
![(b) Calculation schematic [35]](https://static.hindawi.com/articles/geofluids/volume-2022/1007731/figures/1007731.fig.003b.jpg)
(b) Calculation schematic [35]
According to the shearing displacement method, if is the pile length, is the area of the cross section of the pile body, is the pile perimeter, is the elastic modulus of the pile body, is the shearing deformation modulus of the soil, is the Poisson ratio of the soil, is displacement of the pile tip, and is the maximum influence radius that can be assigned as , the expressions for lateral frictional resistance of the pile body , displacement of the pile body , axial force of the pile body , and equivalent concentrated load at the pile tip are shown, respectively, as follows: where is the quotient of and and k1 is the shear stiffness between the pile and soil, calculated as follows:
The compressive stiffness of soil () beneath the pile tip can be calculated as follows:
Then, we study the case of uniform load 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 ; is effective in the range between and , and the coordinates of an arbitrary point underground on the pile–soil interface are . The load on the micro unit is replaced by linear load .
(a) Load schematic
(b) Calculation schematic
According to the plane stress solution [36], the additional shear stress induced by linear load at point at depth () on the pile–soil interface is where is the angle between the connection line of point to unit and the pile–soil interface ().
Substituting the geometrical relations in Figure 4(b) into Equation (8) produces
Finally, the lateral friction on the pile–soil interface is the sum of and :
Considering the directions of and produces
2.3. Solutions for Axial Force and Settlement
When the expression of lateral friction is obtained, take the pile body in Figure 2 as a calculation isolator (Figure 5).
According to the force balance of the pile body, the equilibrium equation is established:
By solving the integral, the axial force expression is obtained: where 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
By substituting Equations (2), (3), and (9) into Equation (10), the lateral friction of the capped pile is obtained as
By substituting Equation (11) into Equation (14), the settlement of the capped pile is calculated:
Then substituting Equation (17) into Equation (15), the axial force of the capped pile is calculated:
When , the settlement of the pile top is equal to .
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 Table 1.
Capped 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.
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.
The lateral friction curve of two types of piles in Figure 6 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 () 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 () affected lateral friction in a certain depth. In this case, the equivalent action radius of 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 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 Table 3.
As seen from Figure 7, 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 () increased correspondingly to balance the total load. When 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.
(a) Load 500kN-1250kN
(b) Load 1500kN-2000kN
The calculated load-settlement curves (Figure 8) are as follow:
Figure 8 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.
Data Availability
Yes, we checked it carefully and comfirmed that all data, models and code generated at the manuscript could be obtained from the corresponding author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Shilin Luo conceptualized the study and was responsible for the methodology and revision. Mingquan Liu was responsible for the algorithms and analysis and wrote the original draft. Jianqing Jiang and Ailifeila Aierken were responsible for the data curation and language editing. Jin Chang was responsible for the revision. Xuewen Zhang and Rui Zhang were responsible for the software and validation.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (42107166), Hunan Provincial Natural Science Foundation (Nos 22021JJ40632 and 2021JJ30758), and Scientific Research Project from the Education Department of Hunan Province (Nos 21C0753 and 21C0740). We also thank Dr. Yan-lin Zhao for his help in the improvement of the text’s language.