Research Article  Open Access
Zhijun Zhou, Yaqin Dong, Peijun Jiang, Dandan Han, Tong Liu, "Calculation of Pile Side Friction by Multiparameter Statistical Analysis", Advances in Civil Engineering, vol. 2019, Article ID 2638520, 12 pages, 2019. https://doi.org/10.1155/2019/2638520
Calculation of Pile Side Friction by Multiparameter Statistical Analysis
Abstract
In this paper, a static load test and a multiparameter statistical analysis method are used to study the value of pile side friction in different soil layers in a loess region. Currently, static load testing is the most commonly used method to determine the bearing capacity of pile foundation. During the test, a vertical load is applied at the top of the pile, the data under each load level are recorded, and a QS curve is drawn to obtain the ultimate bearing capacity of a single pile. Reinforcement stress gauges are installed at different sections of the pile body, and then the axial force and the pile side friction of each section are calculated. Few studies have investigated the calculation of pile side friction in different soil layers using the multiparameter statistical analysis method. Obtaining accurate results using this method will provide an important supplement to the calculation of pile side friction and will also be conducive to the development of theoretical calculation of pile side friction. Therefore, taking Wuding Expressway project in loess region as an example, the lateral friction resistance of six test piles is studied through static load testing and multiparameter statistical analysis. The multiparameter statistical analysis method is compared with the static load test results, and the error is controlled within 20%. The results show that the calculation results of multiparameter statistical analysis essentially fulfill engineering requirements.
1. Introduction
Loess sediment covers a large part of the globe, accounting for onetenth of the land area worldwide. Loess is prevalent in China, with complete strata and heavy thickness, covering an area of approximately 630,000 km^{2} [1, 2]. Loess is a yellow silt sediment which was mainly transported by wind during the Quaternary period. It is rich in carbonate, with large voids, obvious vertical joints, and generally low groundwater level [3, 4]. With the continuous development of China’s economy, the traffic in loess areas is developing rapidly, along with increased construction of large highways and bridges [5–10].
At present, pile foundation is the most commonly used foundation form in highway bridge construction, and it is a durable and effective infrastructure [11–15]. In the loess region of Shaanxi Province, bored castinplace piles are widely used due to their welldeveloped construction technology and high bearing capacity [16–21]. Most of the piles are 30–70 m in length and more than 1 m in diameter. Friction piles or endbearing friction piles are also commonly used. For long piles, the frictional resistance of the pile side accounts for more than 80% of the bearing capacity of piles, and for short piles, the resistance generally accounts for more than 60% [22–26]. Therefore, the calculation of lateral resistance in loess areas is of great significance to the construction of highway bridges in such areas of China [27, 28].
At present, the static load test method is one of the most widely used techniques to determine pile side friction [29–31]. A large amount of research into static load testing has been carried out. Static load testing of two 0.45 m thick steel pipe piles to analyze the pile side friction distribution law suggested that the effective stress method could be used to express frictional resistance around piles [32]. Based on the static load test of two concretedriven piles, a formula for calculating the pile side friction of cohesive soil and remolded soil was also proposed [33]. Through static load testing of large diameter and superlong piles in a soft soil area around Dongting Lake, it was found that the piles showed obvious characteristics of friction piles, and a formula for calculating the lateral load transfer model of linear elasticfully plastic piles was presented [34]. Static load testing of large diameter and superlong bored piles in soft soil areas was undertaken to analyze the load transfer law and bearing characteristics of these piles, and the relative displacement of piles and soils when the pile side friction of different soil layers reached the limit value was presented [35]. Through static load testing of an endbearing pile, it was concluded that pile side friction affected the bearing capacity of the endbearing pile to a certain extent, and the bearing capacity exceeded the design bearing capacity of a single pile [36]. The relationship between the total lateral resistance of piles and the settlement at the end of piles under different bearing levels was obtained by static load testing of bored castinplace piles, which indicated that the total lateral resistance of piles could be increased by increasing the strength of rock or soil at the end of piles [37]. Field load tests on superlong castinsitu piles were also carried out, and the axial force curves of test piles under different load levels were obtained, as well as the relationship between unit friction and relative displacement of pile and soil. Through this experiment, it was shown that the unit frictional resistance under compressive load could be calculated by dividing the difference of two continuous axial forces by the area of the pile body between strain gauges [38].
The multiparameter statistical analysis method collects data from many test piles and establishes the relationship between pile side friction, cohesion, and the internal friction angle of the soil layer [39, 40]. However, few studies have been carried out to calculate pile side friction by the multiparameter statistical analysis method. Therefore, taking Wuding Highway in the Loess Plateau as an example, this paper carries out static load tests on six test piles and measures the size and distribution of pile side friction. The pile side friction in different soil layers is then calculated using the multiparameter statistical analysis method. Finally, the two results are compared. Obtaining a reasonable result using this method will provide an important supplement to the calculation of pile side friction, and it will also be conducive to the development of theoretical calculation of pile side friction.
2. Test Site Engineering
Wuding Expressway is located within Yan’an City and Yulin City in Shaanxi Province, China (Figure 1). It starts from the east of Wuqi County, ends at Shijingzi, Southeast of Dingbian County, and is approximately 922.17 km long. The abutments on both sides are located in the Loess Lianghe subarea, and the topography of the abutment area is relatively small. The elevation of the ground level is between 1629.60 m and 1644.59 m, and the relative elevation difference is approximately 14.99 m. The test site shown in Figure 1 is situated at the separated intersection of Sunkelan Village, Yangjing Town, and Dingbian County. The topographic fluctuation of the test site is small, there is no surface water, the groundwater is very deep, and no groundwater is present in the process of drilling. The strata of the test site consist of the following:(1)Loessial soil (): the soil is brownyellow, relatively uniform, contains macropore, wormhole, plant rhizome, and a small amount of gravel and hard plastic.(2)Old loess (): the soil is brown and yellow and relatively noncomplex. A small amount of hyphae is present in the soil, accompanied by wormholes, pinholes, some shellfish, and hard plastic.
3. Test Contents
3.1. Indoor Test
Laboratory testing of soils in the test area was mainly comprised of a moisture content test (Figure 2(a)), a compression test (Figure 2(b)), and a direct shear test (Figure 2(c)). The drying method was used in the soil moisture content test, and the soil void ratio was obtained by the compression test. Through analyzing the data from the moisture content and compression tests, the stratum characteristics and the main physical properties of the soil layer in the test area were obtained, as shown in Table 1.
(a)
(b)
(c)

Cohesion and internal friction angle are important parameters used in this paper. Therefore, 34 groups of samples were tested by the direct shear test, including eight groups of loessial soil samples and 26 groups of old loess samples. In the direct shear test, the upper and lower boxes were aligned, fixed pins were inserted, and the pervious stones and filter paper were placed in the lower boxes. The knife edges of the ring knife with samples were placed upward, the knife back was downward, and the cutting box mouth aligned. The filter paper and the upper pervious stones were then placed, and the samples were pushed into the shear box slowly. Following this, the ring knife was removed, and the force transfer cover plate was added. Sliding steel balls were then installed, along with the shear box and force measuring ring. A preload of 0.01 was applied, the handwheel was rotated, and the dial reading of the force measuring ring was zeroed. After applying the vertical pressure, the fixed pin was pulled out immediately, the stopwatch was commenced, and the handwheel was rotated at a uniform speed of 0.8 mm/min (shear displacement was 0.2 mm per rotation cycle) so that the specimen was sheared and destroyed within 3–5 min. With every turn of the handwheel, the scale reading in the measuring ring was recorded once until soil sample shear failure. The calculated cohesive force and internal friction angle are provided in Table 2.

3.2. Static Load Test
For the static load test, the anchor piles and test piles were arranged in the form of four anchor piles surrounding one test pile. The spacing between the anchor pile and test pile is illustrated in Figure 3. Six test piles with a diameter of 1.5 m and length of 25 m were set up in the test area, along with anchor piles with a diameter of 1.5 m and length of 30 m. The pile body was constructed with C30 concrete, and C40 concrete was used to reinforce the part 1.5 m away from the top of the pile. According to preliminary field investigation data, the groundwater in this area is deeply buried and there is no surface water. The dry rotary drilling method was thus used to drill test piles and anchor piles. After checking the quality of the hole, the reinforcement cage skeleton was lifted and the piles were poured into the pile. The entire testing process consisted of three parts: installation and layout of test elements before the test, construction of the test piles and anchor piles, and test loading and data collection. The specific process for each relevant component is detailed as follows:(1)According to the test requirements, it was necessary to measure the axial force and lateral resistance of the pile under various loads during the test process. Therefore, before the construction of the anchor piles and test piles, a certain number of reinforcement stress gauges were embedded in the pile. Considering the integrity of the test data collection, seven sections were selected along the main reinforcement in the pile for reinforcement stress gauge placement. Because the upper part of the pile was in direct contact with the jack when loading, the deformation was large, so the first layer of the stress gauge meter was placed 0.5 m below the pile top, and the laying depth was 3.5 m, 6.5 m, 11 m, 15.5 m, 20 m, and 24.5 m in turn (Figure 4), with each section connected to three reinforcement stress gauges. The stress gauges at the bottom 24.5 m were located at the end of the test pile and were used to measure the internal force at the bottom of the pile and the resistance at the end of the pile. The reinforcement stress gauges in the middle part measured the internal force of the pile at each soil layer and at the boundary of the soil layer. In the past, reinforcement stress gauges have been welded directly in series to the main reinforcement in the pile. However, high temperatures produced during welding can easily damage the reinforcement stress gauge, thus affecting the test results. Therefore, damage to the steel bars must be avoided when laying out the steel bars, to avoid the stress gauges being affected. In this experiment, the reinforcement connecting the two ends of the stress gauge was processed, and then the highstrength carbon steel cylinder nuts at the two ends of the stress gauge were connected with the reinforcement to protect the reinforcement stress gauge, and it was ensured whether it was able to collect the relevant data easily.(2)With the development of machinery and equipment, rotary drilling bored piles are often used in the construction of pile foundations (friction piles) in loess areas. Compared with manual drilling and percussive drilling, rotary drilling has positive characteristics including high drilling efficiency, with an average drilling rate of 10 m/h. If the groundwater level in the loess area is relatively low, dry drilling can be used to prevent the loess layer around the pile from losing strength or increasing gravity when it encounters water. Rotary drilling construction in loess areas does not require the construction of mud wall protection as the rotary drilling bit will produce mud in the process of drilling which will keep the hole wall stable and perform holeforming wall protection. Compared with impact drilling, rotary drilling has less effect on soil compaction on the hole side. In rotary drilling, the bit moves back and forth on the bottom of the hole and the ground, which makes the hole wall rougher. Higher roughness of soil around the rotary excavation pile can better reflect the interaction between pile and soil. According to Chinese code [41], when bored by rotary drilling in dry operation (Figure 5(a)), the thickness of bottom sediment of friction piles with diameter less than 1.5 mm should be less than 300 mm, and the inclination of pile holes should not be less than 1%; the diameter should not be less than the design value of the pile diameter; and the hole depth should not be less than the design stipulation. Thus, after checking the quality of holeforming meets the requirements, the steel cage skeleton was hoisted (Figure 5(b)) and poured into piles (Figure 5(c)). In the process of drilling holes by rotary drilling, the protecting barrel is used. The protecting barrel is raised 1.5 m above the ground in the concrete filling process of each test pile. After the completion of concrete filling, the protecting barrel of each test pile is not pulled out for later loading so as to prevent the upper concrete from being damaged by compression due to large load in the loading process.(3)The static load test was carried out using an anchor pile reaction device, as illustrated in Figure 6(a). Firstly, eight hydraulic jacks (Figure 6(b)) were evenly arranged on the steel cushion box with sufficient strength and stiffness, and then the main beam and the secondary beam (Figure 6(c)) were hoisted, respectively, with the middle of the main beam located on the hydraulic jack as far as possible. When the secondary beam was hoisted, it was necessary to ensure that the two ends of the secondary beam were in accordance with the position of the anchor pile. After the datum beam was set in place, the displacement dial gauge (Figure 6(d)) was installed on the steel sheet with a magnetic meter frame, and the settlement of the pile top was measured in real time.
Loading was carried out by the slowspeed method. For this experiment, single stage loading was 1,000 kN, the maximum load was 12,000 kN, and the loading stage was 11. According to Chinese code [42], when the variation of settlement in one hour is less than 0.1 mm under the action of various loads and occurs repeatedly, the settlement of the test pile can be considered to be relatively stable. When the pile is in the process of testing, loading can be stopped when one of the following conditions occurs [42]: (1) when the settlement of the pile top under the load is more than five times that under the previous load, the total settlement of pile top is more than 40 mm and (2) when the maximum loading value required by the design has been reached, the pile top settlement has reached a relatively stable standard.
In this study, the unloading load of the test pile was twice that of the graded load when the loading process was completed, and the unloading load lasted for one hour at each stage. At the same time, the settlement at the top of the pile and the bar gauge were measured. After the unloading process was complete, the residual settlement was measured over three hours.
4. Static Load Test Result Analysis
4.1. Settlement Calculation of Pile Top
The bearing capacity of multiple test piles of the same test site engineering and same size was varied, and the average value was taken to carry out static load test result analysis [39, 40]. Four displacement meters were installed to measure the settlement of the pile top under different loads in real time, and then the average settlement of four pile tops was taken as the settlement of the pile top under different loads.
The calculation results are provided in Table 3. The QS curve is formulated by calculating the settlement value of the pile top. The QS curve is an intuitive manifestation of the loading process of pile static load testing, as shown in Figure 7. Analysis of Figure 7 shows that the settlement of the test pile increases suddenly during the loading process. The QS curve shows a sharp drop point, which can illustrate the ultimate bearing capacity of the pile. The ultimate bearing capacity of the test pile is 9,000 kN.

4.2. Axial Force of Pile Body Calculation
When calculating the axial force of the pile body, it is assumed that the pile body is of equal cross section and that the pile body is made of linear elastic material. Under the action of the arbitrary firstorder load, the stress of each section of the pile can be obtained by measuring the frequency value of the stress gauges in the main reinforcement and calculating the stress value [27, 43, 44] by using the relevant formula. Then the strain value of the pile body at each section can be obtained by using the relevant formula. The axial force of the steel bar at each section of the pile body can be determined according to the following formula:where p_{si} is the axial force of the steel bars, K is the calibration coefficient, F_{i} is the vibration frequency of the steel string at section i under the load, F_{0} is the initial vibration frequency of the steel string, and B is the calculated correction value, which is 0 in this paper. The strain values of the corresponding sections are given by using the following formula:where ε_{si} is the strain of the steel bar and E_{s} is the elastic modulus of steel bar, which is 200 GPa in this test. Additionally, A_{s} is the section area of the steel bar, which is 0.0004909 m^{2}. In the process of calculation, if the deformation of concrete and steel bars is assumed to be in harmony with each other, the axial force of the pile at the section can be obtained as follows:where Q_{i} is the axial force of the pile body at section i and E_{c} is the elastic modulus of concrete. As the strength grade of pile concrete is C30, according to Chinese code [45], the value of E_{c} in this test is 30 GPa and A_{c} is the section area of concrete. Using the above formulas (1)–(3), the axial force curve of the pile body is formulated and is presented in Figure 8. Observing Figure 8, it can be seen that during the transfer process of the pile top load, the pile tip resistance is very small and increases slowly under the first five stage loads, indicating that the pile top vertical load is mainly shared by the soil around the pile, so the pile side resistance begins to play a role before pile tip resistance. As the load continues to increase, the pile tip resistance increases significantly. If the load continues to increase, the curve change of the upper part of the pile is almost parallel, indicating that the pile side friction has fully been exerted. As shown in Figure 8, when the test pile is loaded to 9,000 kN, the pile tip resistance is 1,708 kN, and the pile tip resistance ratio is 18.98%. Therefore, the pile belongs to the endbearing friction pile [46].
4.3. Pile Side Friction Calculation
In the course of the test, the lateral friction resistance between two adjacent sections can be considered approximately equal to the variation of the axial force of the pile body between the sections [27, 47–50]. Therefore, the formula for calculating the side friction resistance of the pile is as follows:where U is the perimeter of the pile body, Q_{i−1} is the axial force value at section i − 1, Q_{i} is the axial force value at section i, and l_{i} is the height between the upper and lower sections. The pile side friction curve is formulated and is presented in Figure 9. As shown in Figure 9, the pile side friction increases gradually in the range of 0 m∼11 m, reaches peak value at 11 m, and then decreases gradually. This is because during the load transfer process, as the depth increases, the pile side friction resistance gradually works and reaches the limit value at 11 m. And then the pile top load begins to be mainly borne by the pile tip resistance, and the pile side frictional resistance decreases gradually.
5. Multiparameter Statistical Analysis
Statistical analysis methods include two kinds. The first is the trial algorithm (interpolation method), in which the maximum and minimum values given by the original code are used for trial calculation and the value of pile side friction is adjusted according to the results of the trial calculation. The existing code in China [51] adopts this trial algorithm for analysis [40, 52]. The second method used is least squares statistical analysis, in which the number of classified soil layers with similar geological characteristics (age, stratum, and genesis) is taken as the number of unknown parameters. As the total lateral friction of each test pile can be expressed by the lateral friction of each layered soil, each test pile can be listed as an equation. When the number of test piles is equal to the number of layers, the equation system can be solved. When the number of test piles is larger than the number of layers, the least squares method can be used to simplify the equation system so that the number of equations is the same as the number of layers and the unknown value can be obtained and then substituted. The lateral friction of piles can be calculated using formula (6) [40].
5.1. Basic Equations
According to the distribution of soil layers and the total resistance of each pile which is equal to the sum of the lateral resistance of each layered soil, the lateral resistance equation of each test pile can be determined [39, 40]:where Q_{f} is the total frictional resistance of the pile side, U is the circumference of the pile, q_{si} is the unit surface friction resistance in the soil layer, l_{i} is the pile length of each soil layer, and m is the soil layer number.
According to the relationship between cohesive force, internal friction angle of shear strength index, and frictional resistance, the following equation can be formulated [40]:where a and b are the empirical coefficients, based on existing results [39, 40], a and b should be between 0 and 1. σ_{i} is the average effective weight of each layer of soil, and F_{i} is the empirical coefficient of pile side friction in different soil layers.
Substitute equation (6) into (5), and make
Equation (7) can be simplified as follows [39, 40]:
Supposing that there are n test piles in the project, the soil layer is divided into m layers. If n > m, the equation can be solved. The following formula can thus be obtained from formula (9) [39, 40]:
In this paper, the principle of the least squares method is applied to the calculation of lateral friction of piles. By using the principle of the least squares method, the equations in (10) can be optimized into m standard equations (13) [39, 40]. The specific optimization process is as follows:(1)Construct the error function(2)To minimize the error value, make(3)The optimized standard equations are as follows:
By solving the formulas in (13), the empirical coefficients of different soil layers F_{i} can be obtained. However, two unknown empirical coefficients a and b remain in formula (13). According to the existing results [39, 40], a and b are between 0 and 1. Therefore, it is necessary to assume there are different combinations of a and b to obtain different combinations of F_{i}. Among the F_{i} values of different combinations, a set of values should be chosen as the optimal solution for formula (13), so the standard deviation σ of the formula should be calculated according to the following formula [39, 40]. When the standard deviation σ is the minimum, a and b are the most appropriate values to get the optimal solution F_{i} [39, 40]:
6. Result Analysis of Multiparameter Statistical Analysis
According to the basic principle of multiparameter statistics and the pile test data collected above, there are six test piles and two layers of soil on the side of the pile. Six conditional equations (15) can then be listed from formula (10):
There are six equations and two unknown parameters in this system. The number of equations is more than the unknown number, so it can be solved by the least squares method. By using the principle of least squares, the equations in (15) can be optimized to two standard equations as follows:
The equations in (16) refer to the equations of F_{1} and F_{2}. F_{1} and F_{2} are the empirical coefficients of pile side friction in different soil layers. In the process of solving, it is necessary to assume different combinations of a and b to get different F_{i}. Their standard deviations can then be calculated according to formula (14), and the optimal solution F_{i} can be determined by taking the values of a and b when the standard deviation σ is the smallest. By changing combinations of a and b, the above calculation methods for standard deviation is compiled into a MATLAB program, and the standard deviation σ under different combinations of a and b is obtained, as shown in Table 4.

The results show that the minimum standard deviation is σ = 95.76 when a = 0.1 and b = 0.6. Then, F_{1} = 31.4 and F_{2} = 38.2 can be obtained by solving the equations in (16). The calculated parameters are replaced by formula (6), in which the internal friction angle and cohesion of each layer of soil is averaged [40], as shown Table 2. For example, the calculated values of pile side friction at 6.5 m pile depth and at 24.5 m pile depth are as follows:(1)Loessial soil ():(2)Old loess ():
Because the size of the six test piles are the same and they are located in the same project, the weighted average value of pile side friction of different soil layers under the maximum loading value 12,000 kN in static load test is taken as the measured value, and the calculation process is as follows:(1)Loessial soil ():(2)Old loess ():
Similarly, the calculation of the pile side friction resistance of each soil layer is also a weighted average. The calculated values are compared with the measured values of different soil layers in the static load test, and the error is provided in Table 5. According to Table 5, it is concluded that the error between the two methods is within 20%. If the parameters are reasonable, the calculation results of the multiparameter statistical analysis method can largely meet the engineering requirements.

When calculating pile side friction by the multiparameter statistical analysis method, without considering the change of shear strength index along the pile depth in the same soil, the average value of them is used to calculate pile side friction [40]. The calculation results are shown in Figure 10. It can be seen from Figure 10 that the pile side friction of the same soil layer varies little along the pile depth, while the pile side friction of different soil layers varies obviously along the pile depth. Therefore, in the multiparameter statistical analysis method, without considering the change of the shear strength index of the same soil layer along the pile depth, the pile side friction resistance of the same soil layer changes very little, while the shear strength index of different soil layers is different, and the pile side friction resistance of different soil layers changes obviously along the pile depth.
7. Conclusions
In this paper, static load testing was carried out on the six test piles, and the size and distribution of pile side friction was measured. Pile side friction in different soil layers was then calculated using the multiparameter statistical analysis method. The main findings are summarized as follows:(1)The static load test results show that the pile side resistance and pile tip resistance are not entirely synchronized to the maximum. In the process of pile top load transfer, the resistance of pile side occurs prior to the resistance of the pile tip. As the loading continues to increase, the pile side resistance is fully exerted, the pile end resistance increases significantly, and the pile side frictional resistance first increases and then decreases from top to bottom.(2)The multiparameter statistical analysis method based on the shear strength index can calculate the pile side friction of different soil layers in loess areas. If the parameters are reasonable, the error between the calculated value and the measured value of the static load test method can be controlled within 20%.(3)In the existing Chinese code [51], the value of pile side friction is determined by the type of pile and soil parameter index (void ratio and liquid index). This paper calculated pile side friction by the multiparameter statistical analysis method. It was found that pile side friction is related not only to the type of pile and soil parameter but also to the shear strength index.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was funded by the National Key R&D Program of China (no. 2018YFC0808606) and the Project on Social Development of Shaanxi Provincial Science (nos. 2018SF378 and 2018SF382).
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