Research on Bearing Theory of Squeezed Branch Pile

Abstract

Based on the shear displacement method, theoretical research on the load transfer relation of squeezed branch pile is established. In combination with the nonlinear relation of the interaction between piles and soil, the hyperbolic function is used to describe the pile tip, pile skin, and branch. The theoretical method of the load transfer is improved, and the expression of the load transfer coefficient of these three parts is given. Finally, the applicability of the theoretical method is verified by the model test of homogeneous soil and the field test of stratified soil. The verification results show that the theoretical calculation method of the squeezed branch single-pile bearing capacity based on the shear displacement method can better calculate the ultimate load in practical engineering applications.

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1. Introduction

The squeezed branch pile is a kind of friction-end bearing pile, which is formed by squeezing into an expanded body at a specific position based on the cast-in-place pile. Thanks to the function of the expanded body, the load-bearing system of branch resistance-shaft resistance-tip resistance is helpful to realize the high bearing capacity and low settlement. At present, related applied research and engineering practices have been carried out in many engineering fields, such as large-scale factory projects and infrastructure construction in clay soil areas (Yin et al. [1] and Xiao et al. [2]).

With the development of engineering technology, the research on bearing characteristics of squeezed branch piles has gradually deepened. Yang et al. [3] obtained the bearing characteristics of high bearing capacity and low settlement through comparative tests of large diameter piles on the same site. Li et al. [4] analyzed the working mechanism of squeezed branch piles in actual projects and found that the expanded body has a high contribution to the bearing capacity of the pile foundation. Zhang and Wang [5] verified the aforementioned bearing characteristics by field comparison tests. It is also proposed that the reasonable setting of the expanded body and the selection of the diameter of the branch have a significant impact on the bearing capacity, Cui and Wang [6], Zhang et al. [7], and Liu et al. [8] verified the excellent application prospect of the squeezed branch pile under different site conditions through the field static load test. The aforementioned studies can prove the good site adaptability of squeezed branch piles. It is of great significance to carry out corresponding theoretical research and practice based on the bearing characteristics of the squeezed branch pile.

Therefore, many scholars have carried out studies mainly on the bearing mechanism and the load-settlement relationship of squeezed branch piles. Liang et al. [9] analyzed the influence of the pile-soil interaction state on the bearing capacity of the pile and proposed a variable section calculation method of pile load-settlement characteristics. Wu et al. [10] proposed a settlement calculation method for squeezed multibranch piles based on the layered summation method. Ju et al. [11] analyzed the pile-soil interaction under load and proposed a revised reference for the design formula of bearing capacity in the code. Gao and Zhu [12] analyzed the characteristics of the load-settlement curve of the squeezed branch piles in actual projects, predicted the ultimate bearing capacity of the squeezed branch piles through the hyperbolic model, and obtained theory results closer to the measured values. Jiang et al. [13] improved the accuracy of bearing capacity calculation by using the power function-hyperbolic combination model. Other scholars have carried out theoretical research on the bearing characteristics of squeezed branch piles. Liu et al. [14] established the relationship between the bearing capacity and settlement of multistage enlarged piles based on the load transfer law of the pile foundation from top to bottom. The calculation formula is obtained, and the calculation result is more consistent with the measured values. Li et al. [15] combined the load-bearing system of branch resistance-shaft resistance-tip resistance and carried out a nonlinear simplified analysis of the load-settlement characteristics of squeezed branch piles through a segmented displacement coordinated iteration algorithm.

The expanded branch can bear more loads, and the damage of the squeeze branch pile is mainly due to the shear failure of the soil under the branch or the failure of the soil at the pile end, which is different from the conventional bearing capacity design method. The side resistance of the squeezed branch pile is limited by the branch resistance, and the pile foundation will completely fail after the expanded body is damaged. Therefore, this paper establishes the load-settlement calculation theory of squeezed branch pile based on the shear displacement method. Combined with the model test of homogeneous soil and the field test of stratified soil, the calculation theory was verified, respectively. According to the verification results, we can analyze the stress mechanism of the squeezed branch pile.

2. Theoretical Analysis

2.1. Bearing Theory of Pile Foundation by the Shear Displacement Method

As shown in Figure 1, the bearing theory of the shear displacement method proposes that the soil around the pile undergoes shear deformation with the settlement of the pile, and the shear deformation is transferred from the radius r₀ of the pile to the influence radius r_m, and equation (1) represents the transfer relationship of shear stress τ(z) in the radial direction, and equation (2) represents the elastic relationship between shear stress τ(z) and shear strain in the soil at the side of the pile:$$\begin{array}{}\text{(1)}& \tau z={\tau }_{0}z\frac{r_0}{r_m},\text{(2)}& \tau z=G_s\frac{\text{d}wr}{\text{d}r}.\end{array}$$

[figure omitted; refer to PDF]

As shown in equations (1) and (2), Randolph and Wroth [16] put forward the influence radius r_m = 2.5Hρ_m (1 − ν); H is the depth of the calculated section. ρ_m is the correction coefficient of finite-depth soil; ρ_m is taken as 1 in the homogeneous soil layer; ν is Poisson’s ratio of the soil at the side of the pile; τ₀(z) is the shear force of the pile-soil contact surface at depth z; and G_s is the shear modulus of the soil.

Combining equations (1) and (2), the displacement of the soil at the side of the pile is obtained by integral:$$\begin{array}{}\text{(3)}& \begin{array}{cc}wr=\frac{{\tau }_{0}z{r}_0}{G_s}\ln \frac{r_m}{r},& r_0\le r\le r_m,\\ wr=0,& r\ge r_m.\end{array}\end{array}$$

Under the load state, equation (4) takes an element of a single pile for force balance analysis, and the relationship between the pile side displacement $w{r}_0$ and axial force is shown in equation (5):$$\begin{array}{}\text{(4)}& \frac{\text{d}Fz}{\text{d}z}=2\pi r_0{\tau }_{0}z,\text{(5)}& \frac{\partial w{r}_0}{\partial z}=\frac{Fz}{\pi r_0^2{E}_p}.\end{array}$$

As shown in equations (4) and (5), F(z) is the axial force of the pile section at depth z; E_p is the elastic modulus of the pile.

At the same depth z, the displacement of the soil at the side of the pile is equal to the pile side displacement. By combining equations (3)–(5), the displacement balance equation can be established as follows:$$\begin{array}{}\text{(6)}& \frac{{\partial }^{2}w{r}_0}{\partial z^2}-\frac{2G_s}{r_0^2{E}_p\ln r_m/r_0}w{r}_0=0.\end{array}$$

The general solution of equation (6) is$$\begin{array}{}\text{(7)}& w=m_1{e}^{\alpha z}+m_2{e}^{-\alpha z}.\end{array}$$

As shown in equation (7), m₁ and m₂ are undetermined constants; the calculation method for parameter α is shown as follows:$$\begin{array}{}\text{(8)}& \alpha ={2G_s\ln {r_m/r_0}^{1/2}{r}_0^2{E}_p}^{1/2}.\end{array}$$

2.2. Bearing Theoretical Calculation Model of Squeezed Branch Pile

Seed and Reese [17] proposed the load-settlement hyperbolic relationship at the end of the pile, and the relationship between the pile tip resistance P_b and the settlement s_b is shown as follows:$$\begin{array}{}\text{(9)}& P_b=A\frac{s_b}{a_s+b_s{s}_b}.\end{array}$$

In equation (9), the cross-sectional area of the straight hole section A = πd²/4, d is the pile diameter and a_s and b_s are the transferred parameters of the pile tip resistance.

As shown in Figure 2, the length of the pile shaft under the nth expanded body is l_n. Assuming that the pile tip resistance is P_b and pile tip settlement is s_b, and combining the bearing theory of the shear displacement method, the displacement of the section under the nth expanded body can be obtained as$$\begin{array}{}\text{(10)}& s_n=\cosh \alpha l_n{s}_b+\frac{\sinh \alpha l_n}{\alpha A{E}_p}{P}_b.\end{array}$$

[figure omitted; refer to PDF]

In equation (10), L is the pile length and E_p is the elastic modulus of the pile shaft.

Assuming that the shaft resistance f_n of the pile shaft varies linearly along the pile body, f_n can be shown as$$\begin{array}{}\text{(11)}& f_n=\pi d{l}_n\frac{{\tau }_n+{\tau }_b}{2}.\end{array}$$

In equation (11), τ_n is the shear stress at the section under the nth expanded body and τ_b is the shear stress at the pile end.

The shear stress and settlement satisfy the hyperbolic function relationship:$$\begin{array}{}\text{(12)}& {\tau }_n=\frac{s_n}{a_{fn}+b_{fn}{s}_n},\text{(13)}& {\tau }_b=\frac{s_b}{a_{fb}+b_{fb}{s}_b}.\end{array}$$

In the formula, a_fn and b_fn are the transfer parameters of the shear force under the nth expanded body; a_fb and b_fb are the transfer parameters of the shear force at the pile end.

For nonrigid piles, the pile would be compressed when the pile is loaded. Assuming that the load dF acts on the microelement dz of the pile, the static balance relationship of the microelement segment is satisfied as follows:$$\begin{array}{}\text{(14)}& \text{d}F=\pi d\frac{s}{a_{fb}+b_{fb}s}\text{d}z,\text{(15)}& \text{d}s=\frac{F}{E_{p}A}\text{d}z,\end{array}$$where s is the settlement of the microelement section relative to the reference section and F is the axial force difference of upper and lower sections of the microelement section.

From equations (14) and (15), the differential equation for the axial force transfer of the pile shaft is$$\begin{array}{}\text{(16)}& F\text{d}F=\pi \text{d}{E}_{P}A\frac{s}{a_{fb}+b_{fb}s}\text{d}s.\end{array}$$

Combining the initial conditions F = 0 and s = 0 to integrate equation (16), we get$$\begin{array}{}\text{(17)}& F=\frac{\sqrt{2\pi \text{d}{E}_{P}A}}{b_{fb}}\sqrt{b_{fb}s-a_{fb}\ln 1+\frac{b_{fb}}{a_{fb}}s}.\end{array}$$

From the results of the shear displacement of the pile shaft l_n, it can be seen that the compression of the pile shaft l_n is$$\begin{array}{}\text{(18)}& s_c^n=s_n-s_b.\end{array}$$

Therefore, the additional axial force required to provide compression of the pile shaft l_n is$$\begin{array}{}\text{(19)}& F_n=\frac{\sqrt{2\pi \text{d}{E}_{P}A}}{b_{fb}}\sqrt{b_{fb}{s}_c^n-a_{fb}\ln 1+\frac{b_{fb}}{a_{fb}}{s}_c^n}.\end{array}$$

Since the rigidity of the expanded body is much greater than the pile shaft, it is assumed that the settlement of each section of the expanded body is equal under the same load. The load-bearing method of the expanded body can be simplified, as the branch resistance acts vertically on the expanded ring area. Same as pile end, the relationship between the branch resistance P_an and the settlement s_n of the nth expanded body is shown as follows:$$\begin{array}{}\text{(20)}& P_{an}=A_{pn}\frac{s_n}{a_{pn}+b_{pn}{s}_{an}}.\end{array}$$

In equation (20), P_a represents the branch resistance, A_p represents the expanded ring area, and s_n represents the settlement of the nth expanded body. The annular cross-sectional area of the expanded body A_pn = π(D_n²-d²)/4, D_n is the diameter of the nth expanded body, and a_pn and b_pn are the transferred parameters of the branch resistance of the nth expanded body.

Regarding the nth expanded body and the pile shaft under the body as an expanded unit, the axial force transmitted to the top of the unit is R_n:$$\begin{array}{}\text{(21)}& R_n=P_{an}+P_b+f_n+F_n.\end{array}$$

For the squeezed branch pile with n expanded units, the calculation process of the squeezed branch pile is shown in Figure 3.

[figure omitted; refer to PDF]

When any set of the pile tip settlement s_b is given, the corresponding pile top load and pile top settlement can be obtained by the above calculation. Taking any pile top load Q and the settlement s, Q_p and Q_q can be determined, where Q_p = 10/12Q, Q_q = 11/12Q, s_p is the corresponding settlement of the pile top Q_p, and s_q is the corresponding settlement of the pile top Q_q, where Δs₁ = s_q − s_p, Δs₂ = s − s_q, any Q is selected. When Δs₂ > 2Δs₁, the ultimate bearing capacity of the pile can be determined to be 11/12Q.

2.3. Calculation Parameters

2.3.1. a_s and b_s

Randolph and Wroth [18] studied the nonlinear relationship between the pile tip resistance and pile tip settlement:$$\begin{array}{}\text{(22)}& s_b=\frac{P_{b}1-v_b}{2G_{b}d}.\end{array}$$

In the formula, G_b is the shear modulus of the pile-end soil and ν_b is Poisson’s ratio of the pile-end soil.

Therefore,$$\begin{array}{}\text{(23)}& a_s=\frac{1-v_b\pi d}{8G_b}.\end{array}$$

Janbu [19] gives the ultimate tip resistance P_bu of straight hole piles:$$\begin{array}{}\text{(24)}& P_{bu}=c_b{N}_c+q_b{N}_{q}A,\text{(25)}& b_s=\frac{1}{c_b{N}_c+q_b{N}_q}.\end{array}$$

In the formula, c_b is the cohesive force of the pile tip soil, q_b is the average effective compressive stress on the side of the pile tip plane, and N_c and N_q are the bearing capacity coefficients of the pile. The calculation is shown in the following formula, where ϕ is the angle of the failure surface of the pile tip under the ultimate load and the range from soft clay to compact sand is 60° to 105°, where γ is the soil gravity on the side of the pile and φ is the effective internal friction angle of the soil.$$\begin{array}{c}\text{(26)}& N_c=N_q-1\cot \phi ,N_q={\tan \phi +\sqrt{1+{\tan }^2\phi }}^2{e}^{2\varphi \tan \phi }.\end{array}$$

2.3.2. a_f and b_f

The relationship between the shear displacement of the pile side soil and the shear stiffness is given by equation (3). Let r = r₀, then the shaft resistance transfer parameter a_f can be shown as$$\begin{array}{}\text{(27)}& a_f=\frac{d}{2G_s}\ln \frac{2r_m}{d}.\end{array}$$

The shear strength of the pile-soil interface can be shown as$$\begin{array}{}\text{(28)}& {\tau }_f=k{\sigma }_v\tan \delta .\end{array}$$

Therefore,$$\begin{array}{}\text{(29)}& b_f=\frac{R_f}{k{\sigma }_v\tan \delta }.\end{array}$$

In the formula, R_f is the failure ratio, which can be taken as 0.80∼0.95 and k is the horizontal earth pressure coefficient, which can be taken according to Table 1. In Table 1, k₀ is the static earth pressure coefficient, k₀ = 1 − sinφ, ${\sigma }_v$ is the vertical stress of the soil at the calculated height and δ is the friction angle of the pile-soil interface.

Table 1

Suggested value of the lateral earth pressure coefficient k.

Data provider	Working conditions	k
Kulhawy [20]	Steel pipe piles, concrete piles or H-shaped steel piles with a smooth surface, and a small settlement	(0.7∼1.2) k0
Yang [21]	Driven steel piles in alluvial deposit or fully weathered granite	(1.2∼1.5) k0

3. Determination of Disk Resistance Function Based on Model Test

3.1. Test Sand and Model Pile

3.1.1. Preparation of Model Piles

As shown in Figure 4, the model pile is made of aluminum alloy, and the pile end and pile head are sealed with the same material. The pile diameter d = 20 mm, the depth of the pile body into the soil L = 500 mm in the test, the exposed pile head is 50 mm, and the length-to-diameter ratio is 25: 1. The inclination angle of the branch surface α = 45° and a φ20 circular hole is left inside. The dimensions are shown in Figure 1(b). According to the different diameters of the expanded body, the piles were divided into 5 groups. The parameters of each group are shown in Table 2.

[figures omitted; refer to PDF]

Table 2

The parameters of the model pile.

Group	D (mm)	h (mm)	l1 (mm)	l2 (mm)
S1	40	25	137.5	337.5
S2	45	30	135.0	335.0
S3	50	35	132.5	332.5
S4	55	40	130.0	330.0
S5	60	45	127.5	327.5

3.1.2. Preparation of Test Sand

Before the test, the natural sand was cleaned and dried first, and then the test sand was screened with a 2.45 mm aperture fine sieve to remove larger particles and impurities. After fixing the position of the model pile with the self-made device, the test sand was filled in six layers, and the filling mass of each layer was controlled to 80 kg. A small vibratory trowel with a vibration frequency of 2840 times/min was used for tamping for 1 min. The compaction state measured by the light penetrometer was of medium density and was kept still for 12 hours after the landfill is completed.

3.2. Test Loading

3.2.1. Test Loading Device

The loading device is shown in Figure 5.

[figures omitted; refer to PDF]

The clearance size of the model box is 0.8 mm × 0.6 m × 0.58 m (L × W × H), which meets the test error control requirements. The sidewall of the model box is made of a combination of 12 mm thick plexiglass plate and using an 8 mm thick steel bar hoop for reinforcement, and the bottom is sealed with a 12 mm thick plexiglass plate. During the test, the maximum deformation of the sidewall of the model box was measured to be less than 0.1 mm, and it indicates that the stiffness of the model box did not affect the test results.

The test loading device adopts lever-type loading, and a balance system is designed in the loading system to eliminate the weight of the steel beam and the loading plate. To ensure that the center of the pile is under load, a solid alloy steel ball is used to transmit the force at the position of the pile head.

3.2.2. Loading Steps and Limit State Determination

The load of each stage of the test is 0.2 kN. After each stage of loading is completed, the settlement of the model pile is observed every 5 minutes. If the difference between the two adjacent observations is less than 0.01 mm, it means that the stable state is reached and the next level of loading begins. If the corresponding settlement of a certain level of the load exceeds 2 times of the previous level, it means that the load limit state has been reached. We determine the ultimate state of the load from two aspects. One is that the settlement of this stage is more than twice that of the previous stage. The other is that settlement dose not reach a stable state within 1.5 hours. It is determined that the total load to the upper level is the ultimate bearing capacity of the pile.

3.3. The Load Transfer Coefficients of the Branch Resistance

The deflections χ(β) of the branch end load were determined from six groups of indoor model experiments on a small scale. The resistance strain gauges on both sides of the pile body were measured, and the vertical displacement of the pile top was monitored by using a dial indicator.

The layout plan of the axial measuring point is shown in Figure 6.

[figure omitted; refer to PDF]

Determination of resistance load transfer coefficients a_p and b_p at the branch:

The load-bearing method at the expanded body is same as the pile end, but it is only affected by the diameter of ball bearing and the interaction between ball bearing and the soil. Therefore, assuming that the branch resistance is equivalently simplified to the tip resistance form, which is shown as follows:$$\begin{array}{}\text{(30)}& P_a=A{\beta }^2-1\frac{s_a}{a_p+b_p{s}_a}=\frac{R_b}{\chi \beta },\text{(31)}& R_b=A\frac{s_a}{a_{sp}+b_{sp}{s}_a}.\end{array}$$

In the formula, R_b is the equivalent tip resistance, a_sp and b_sp are the transfer parameters of ball bearing, the calculation refers to a_s and b_s, χ(β) is the correct function of the branch resistance, and β is the ratio of the branch bearing diameter to the pile diameter.

From equations (31) and (32), the transfer coefficients a_p and b_p of the expanded body can be shown as follows:$$\begin{array}{}\text{(32)}& a_p={\beta }^2-1\chi \beta a_{sp},\text{(33)}& b_p={\beta }^2-1\chi \beta b_{sp}.\end{array}$$

Equation (31) can be transformed into$$\begin{array}{}\text{(34)}& \frac{s_a}{P_a}=\frac{a_p}{A{\beta }^2-1}+\frac{b_p}{A{\beta }^2-1}{s}_a.\end{array}$$

Let y = s_a/P_a and x = s_a, then y = kx + c$$\begin{array}{c}\text{(35)}& k=\frac{b_p}{A{\beta }^2-1},c=\frac{a_p}{A{\beta }^2-1}.\end{array}$$

According to the principle of least squares, it can be obtained$$\begin{array}{c}\text{(36)}& k=\frac{{\displaystyle {\sum }_{i=1}^n{x}_i{y}_i-n\overline{x}\overline{y}}}{{\displaystyle {\sum }_{i=1}^n{x}_i^2-n{\overline{x}}^2}}=\frac{l_{xy}}{l_{xx}},c=\overline{y}-k\overline{x},\\ \gamma =\frac{l_{xy}}{\sqrt{l_{xx}}\sqrt{l_{yy}}},\end{array}$$where,$$\begin{array}{c}\text{(37)}& \overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i},\overline{y}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{y}_i},\\ l_{yy}={\displaystyle \sum _{i=1}^n{y}_i^2}-n{\overline{y}}^2.\end{array}$$

Combining equations (34) and (38), the function χ(β) is simplified as$$\begin{array}{}\text{(38)}& \chi \beta =\frac{kA}{b_{sp}}.\end{array}$$

The function χ(β) is determined according to the axial force data of the pile of 6 groups of model pile tests, and the correction function value of the branch resistance is shown in Table 3.

Table 3

Correction function value of branch resistance.

No.	S1		S2		S3		S4		S5
	Pa (N)	sa (mm)	Pa (N)	sa (mm)	Pa (N)	sa (mm)	Pa (N)	sa (mm)	Pa (N)	sa (mm)
	478	1.80	714	2.61	550	1.50	585	1.38	631	1.20
	630	3.15	895	4.32	746	2.42	797	2.13	862	1.82
	774	5.77	985	5.63	946	3.82	1017	3.22	1101	2.63
	884	8.20	1075	7.54	1047	4.83	1129	3.93	1224	3.13
			1165	10.64	1148	6.15	1243	4.83	1349	3.73

According to the static load test results, multiple groups of P_a and s_a in the elastic-plastic state are determined. The parameters k and c are obtained according to the least square method. The correction value χ(β) of each diameter is obtained by data fitting and shown in Table 3. The obtained empirical formula of the branch resistance correction for each diameter is$$\begin{array}{}\text{(39)}& \chi \beta =0.36{\beta }^2-2.36\beta +\mathrm{4.20.}\end{array}$$

According to the above calculation and derivation, the transfer parameters of the pile tip resistance, shaft resistance, and branch resistance can be determined.

4. Test Verification and Analysis

4.1. Theoretical Verification and Comparative Analysis of On-Site Large-Scale Test

According to Tang [22], field large-scale test data, the length of the No. 1 pile is 2 m, the pile diameter d = 150 mm, the diameter of the bell bearing D = 300 mm, and the bell bearing is set at 1 m below the top of the pile. The bell bearing is set in dense clay, and the pile tip is in a sandy silt layer. Poisson’s ratio of the clay around the pile is 0.25, the sandy silt of the pile tip is 0.3, and the failure angle of the pile tip is 95°, and the physical parameters of the remaining soil layers are shown in Table 4. According to the measured data, and combining the theoretical method of this paper, the comparison of load-settlement between the theoretical curve and the measured curve of the No. 1 pile is shown in Figure 7, and the theoretical and measured values are shown in Table 5.

Table 4

Basic physical and mechanical parameters of the ground at the test site.

No.	Soil layer	H(m)	h(m)	Es(MPa)	ρ (g/cm3)	e	c(kPa)	φ(°)
1	Clay	1.7	0∼1.6	9.1	1.81	0.64	13.9	23
2	Sandy silt	1.0	1.6∼1.8	39	2.1	—	0	25

[figure omitted; refer to PDF]

Table 5

Comparison of the theoretical and measured value.

s0 (mm)	0.48	0.92	1.42	1.70	2.10	3.04	3.68	7.27
Measured value	20.00	30.00	40.00	50.00	60.00	70.00	80.00	90.00
Theoretical value	27.91	42.24	53.13	57.72	63.03	71.79	75.92	87.91
Relative error (%)	39.54	40.81	32.84	15.45	5.05	2.56	−5.09	−2.32

It can be seen from Figure 6 that the theoretical curve obtained by the calculation method in this paper has a relatively close change trend with the field measured curve. Since the soil under the expanded body has reached an elastoplastic state after rolling at the initial stage of loading, the pile top load increases approximately linearly with the pile top settlement. At the initial stage of loading, the theoretical calculation takes the influence of the elastic state of the pile and soil into account and obtained a slow-change theoretical curve that is more consistent with the actual projects. It can be seen from Table 5 that with the increase of the pile top loading, the relative error between the theoretical value and measured value gradually decreases. In the middle and late loading period, although the theoretical curve and measured value do not completely overlap, they can maintain parallel growth. The relative error can be controlled within 5%. Under the loading limit state, the ultimate calculated bearing capacity of the pile by the theory is 82.45 kN, which is in good agreement with the measured value.

4.2. Theoretical Verification and Comparative Analysis of Indoor Test

4.2.1. Load-Settlement Relationship

The same pile tip settlement is given as the small-scale test. According to the theoretical calculation results, the five groups of pile top load-settlement curves with different diameters are shown in Figure 8. Under the same pile top settlement s₀, the error analysis of the theoretical value and measured value of the pile top load is shown in Figure 9.

[figure omitted; refer to PDF]

As shown in Figure 8, at the initial stage of loading, the theory load of the pile top is lower than the measured value. When the load on the pile top is small, the pile foundation and the soil at the side of the pile are in an elastic state, and the bearing characteristics of the squeezed branch piles are affected by the pile-soil interaction. The theoretical value and measured value are easy to deviate. With the increase of the load on the pile top, the soil around the pile gradually reaches an elastic-plastic state. By the middle and late loading period, the measured curve and theoretical curve have approximately overlapped. As shown in Figure 9, the influence of the pile-soil interaction state on the bearing capacity of the squeezed branch pile is more important at the beginning. As s₀ increases, the relative error between the theoretical value and measured value gradually decreases. When s₀ exceeds 2 mm, the error of the pile top load under each diameter is less than 5%, and in the middle and late loading period, the theory curves and measured curves maintain a relatively consistent trend, and the errors of the two measured values are mostly close to 0.

The error analysis of the ultimate bearing capacity of each group obtained by theoretical calculation is shown in Table 6. As shown in Table 6, the theoretical ultimate bearing capacity under the load limit state is in good agreement with the actual measured value. The relative error of the ultimate bearing capacity under different diameters is less than 2%, which can be controlled within a reasonable range. Therefore, the ultimate bearing capacity of squeezed branch piles can be predicted based on the actual working conditions and the method described in this article.

Table 6

Error analysis of the ultimate bearing capacity.

Group	Measured value (kN)	Theory value (kN)	Relative error (%)
S1	2.20	2.24	2.00
S2	2.60	2.57	−1.15
S3	2.80	2.80	0.00
S4	3.20	3.22	0.63
S5	3.60	3.61	0.28

4.2.2. Bearing Strength Analysis

Taking the S1 (D = 40 mm), S3 (D = 50 mm), and S5 (D = 60 mm) groups of squeezed branch piles as examples, the contribution of each resistance under the same s₀ is shown in Figure 10.

[figures omitted; refer to PDF]

As shown in Figure 10, the theoretical resistance contribution rate is closer to the measured contribution rate under the same settlement. As the branch diameter increases, the coincidence between the measured resistance contribution rate and theoretical contribution rate gradually increases. At the initial stage of loading, the measured resistance contribution rate is in low agreement with the theory result. The contribution rate of each resistance value obtained from the theoretical calculation maintains regular increase or decrease without sudden change. Under the test conditions, the soil at the pile end had been tamped and reached an elastoplastic state, and the contribution rate of the measured resistance value is easily biased due to the influence of the test error. With the increase of top load, when reaching the middle and late loading period, the contribution rate of the measured resistance value and theoretical resistance value shows a coincidence or parallel increase. Based on the relationship between the resistance contribution rate of each group, the bearing strength of each part of the squeezed branch pile can be reasonably predicted by the theory method in actual projects.

5. Conclusions

(1) The calculation theory of pile bearing capacity based on the shear displacement method is established. According to the hyperbolic function relation, the load-settlement relation of pile tip resistance, pile skin resistance, and branch resistance is deduced, reflecting the nonlinear relation between pile and soil, and the theoretical model of the bearing capacity calculation of the single pile of the squeezed branch pile is established.

(2) Theoretical calculation methods of the load transfer function at the pile tip, pile skin, and branch were given. Theoretical calculation and comparative analysis were combined with the results of the laboratory model test, and the rationality of theoretical calculation was verified.

(3) The theory of pile bearing capacity has been verified in the homogenized soil model test. The results show that the squeezed branch single pile based on the shear displacement method can reliably predict the ultimate load of the actual project and provide good theoretical support for the design of bearing capacity.

Acknowledgments

The authors sincerely thank the School of Civil Engineering and Architecture, State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mines, Anhui University of Science and Technology for providing the experimental conditions. This work was supported by the National Natural Science Foundation of China (51408006) and Guangxi Key Laboratory of Geomechanics and Geotechnical Engineering.

References

[1] L. Z. Yin, X. G. Fan, S. J. Wang, "A study on application of squeezed branch pile in clay soil foundation," Earth and Environmental Science, vol. 61 no. 1, 2017.

[2] Q. Xiao, D. H. Wang, J. Xu, "Experimental study on mechanical characteristics of squeezed branch piles for power transmission lines," Journal of ExperimentalMechanics, vol. 30 no. 1, pp. 124-130, 2015.

[3] G. L. Yang, S. H. Chen, Y. L. Lin, "Contrastive experimental study between expanded diameter piles and equal diameter piles," Journal of Railway Science and Engineering, vol. 6 no. 2, pp. 34-42, 2009.

[4] F. Li, H. B. Song, Y. D. Zhou, "Bearing characteristics of squeezed branch pile," Journal of Hohai University, vol. 38 no. 2, pp. 202-205, 2010.

[5] L. P. Zhang, X. M. Wang, "Comparisononload-bearing character is ticsamong equal diameter pile, branched pile and branch pile," Journal of Chang’an University, vol. 36 no. 5, pp. 37-44, 2016.

[6] W. Cui, Z. J. Wang, "Field experiment study on the bearing characteristics of squeezed branch pile in loess area," Applied Mechanics and Materials, vol. 166–169, pp. 1329-1332, DOI: 10.4028/www.scientific.net/amm.166-169.1329, 2012.

[7] Y. Zhang, P. Chen, Z. X. Zhao, "Experimental study on squeezed branch pile foundation in soft soil ground," Chinese Journal of Geotechnical Engineering, vol. 35 no. S2, pp. 994-997, 2013.

[8] X. B. Liu, L. H. Chen, Z. Y. Yu, J. X. Hu, W. Peng, "Experimental study on DX pile performance in frozen soils under lateral loading," Sciences in Cold and Arid Regions, vol. 5 no. 5, pp. 608-613, 2013.

[9] Y. Liang, S. F. Ren, D. X. Geng, X. ShangGuan, "Research on vertical bearing capacity and settlement algorithm of large diameter and variable cross section Piles," Journal of East China Jiaotong University, vol. 26 no. 4, pp. 34-38, 2009.

[10] Y. H. Wu, G. Zheng, S. W. Yan, "A settlement prediction method for multi-under-reamed cast-in-situ pile foundation," Chinese Journal of Geotechnical Engineering, vol. 5, pp. 528-531, 2000.

[11] Y. W. Ju, R. W. Liang, X. H. Bai, S. Y. Zhang, "Experimental study of bearing and deformation behavior of squeezed branch pile and its bearing capacity calculation," Engineering Mechanics, vol. 6, pp. 34-38, 2003.

[12] X. J. Gao, X. R. Zhu, "Forecasting ultimate bearing capacity of single squeezed branch pile by hyperbola method," Rock and Soil Mechanics, vol. 9, pp. 1596-1600, 2006.

[13] J. P. Jiang, G. Y. Gao, B. H. Gu, "Bearing behaviors and description of Q-s curves of piles with expanded diameter based on power-hyperbola combined model," Engineering Mechanics, vol. 27 no. 11, pp. 120-126, 2010.

[14] G. B. Liu, G. J. Hu, Z. Han, Y. D. Zheng, R. Y. Zheng, "Model tests on bearing behaviors and calculation of multi-stage expanded piles," Chinese Journal of Geotechnical Engineering, vol. 41 no. S2, pp. 149-152, 2019.

[15] L. X. Li, X. J. Li, B. Liu, "A simplified nonlinear settlement analysis method for axial loaded squeezed and branch piles," Chinese Journal of Underground Space and Engineering, vol. 14 no. 4, pp. 904-911, 2018.

[16] M. F. Randolph, C. P. Wroth, "Analysis of deformation of vertically loaded piles," Journal of the Geotechnical Engineering Division, vol. 104 no. 12, pp. 1465-1488, DOI: 10.1016/0148-9062(79)90663-6, 1978.

[17] H. B. Seed, L. C. Reese, "The action of soft clay along friction piles," Proceedings of the American Society of Civil Engineers, pp. 73l-754, .

[18] M. F. Randolph, C. P. Wroth, "An analysis of the vertical deformation of pile groups," Géotechnique, vol. 29 no. 4, pp. 423-439, DOI: 10.1680/geot.1979.29.4.423, 1979.

[19] N. Janbu, "Static bearing capacity of friction piles," Proceedings of the European Conference on Soil Mechanics and Foundation Engineering, pp. 479-488, .

[20] F. H. Kulhawy, "Limiting tip and side resistance: fact or fallacy?," Proceedings of the Symposium on Analysis and Design of Pile Foundations,DOI: 10.1016/0148-9062(86)91224-6, .

[21] J. Yang, L. G. Tham, P. K. K. Lee, S. T. Chan, F. Yu, "Behaviour of jacked and driven piles in sandy soil," Géotechnique, vol. 56 no. 4, pp. 245-259, DOI: 10.1680/geot.2006.56.4.245, 2006.

[22] S. T. Tang, "Research on bearing mechanism and settlement of single DX pile and DX pile group," , . Ph.D. thesis, Beijing Jiaotong University, Beijing, China, 2012, in Chinese