The Calculation Method for the Horizontal Bearing Capacity of Squeezed Branch Piles Considering the Plate–Soil Nonlinear Interaction

Abstract

The m-method is a commonly used method to calculate the internal force and deformation of pile foundations under lateral loads. However, for squeezed branch piles, the increase in the load-bearing plate leads to changes in the pile section and the generation of a resistance bending moment under loading, which means the load–displacement relationship at the load-bearing plate will no longer satisfy the linear relationship. In this paper, a hyperbolic load transfer model is established to describe the nonlinear relationship between the soil resistance and lateral displacement at the branch of the pile, and the m-method is used for the straight section of the pile. Laboratory model tests are used to verify the correlation between theory and experimentation. The results show that the theory is consistent with the measured curve. On the basis of the theoretical calculation, the influence of the bearing plate and pile body parameters on the force of the squeezed branch pile is analyzed. The research shows that that the bearing capacity of the squeezed branch pile is improved by increasing the plate’s diameter, placing the plate closer to the ground, and ensuring that the pile top is embedded. The theoretical calculation method established in this paper can correctly and accurately reflect the bearing capacity characteristics of squeezed branch piles under horizontal loads, and it is more safe than performing measurements. Additionally, it can be applied to squeezed branch piles with different plate diameters, plate positions, plate section forms, and plate quantities as well as for piles with different boundary conditions and soil conditions. Moreover, it can also be applied to other pile shapes. This method is of significance for the analysis of the bearing characteristics of piles with variable sections under lateral loads.

1. Introduction

As important foundation forms, piles are widely used in engineering to bear vertical loads. However, many studies have been limited to the analysis of the behavior of dissimilar piles under vertical loads. Research studies on the behavior of dissimilar piles under horizontal loads are very few in the literature [1]. Indeed, in addition to vertical loads, piles are often subjected to significant horizontal loads applied at the pile head or pile shaft caused by wind, earth pressure, tides, currents, ship impacts, mooring ropes, earthquakes [2], deep excavation [3], and tunnel construction [4]. Therefore, it is of great significance to study the bearing characteristics of piles and the pile–soil interaction mechanism under horizontal loads [5].

Compared with constant section piles, the bearing mechanism of piles with multiple branches has changed greatly. The branch increases the action range of the pile–soil interaction and improves the pile-side friction. The branch enhances the effect of the pile–soil interaction by making full use of the mechanical properties of each layer of soil between the branches. Hence, the bearing capacity of the single square concrete in the pile is significantly improved so as to save on costs and shorten the construction period [6]. Squeezed branch piles have been applied in domestic civil engineering, municipal, shipping, aviation, and highway settings, bringing about enormous social and economic benefits [7]. Many scholars have carried out research on the bearing characteristics of squeezed branch piles under vertical loads. Liu [8] and Ma [9] based their studies on the load transfer method, providing a theoretical expression of the load transfer coefficients of squeezed branch piles, which shows that this method is applicable for predicting the ultimate load in engineering applications. Zhang and Xu [10] explored the effects of the branch position, spacing, number, and diameter on the bearing capacity of piles and investigated the soil failure patterns around the branches through field load testing, suggesting that the branch shares a significant proportion of the transferred load, which greatly improves the bearing capacity of squeezed branch piles. Zhang [11] performed a static load model test on squeezed branch piles, studied the stress changes and surface potential response characteristics, and established a loading system and a potential data acquisition system for squeezed branch piles under laboratory conditions, providing an important experimental basis for assessing concrete damage. Wang [12] discussed the bearing performance of squeezed branch piles through a finite element model and analyzed the influence on the bearing performance of both the branch quantity and the ratio of the horizontal load to vertical load, proposing that within a certain range, increased branch quantities lead to increases in ultimate bearing capacity and decreases in vertical pile displacement. The above research proves the excellent bearing capacity and significant social and economic value of squeezed branch piles. In addition, due to the needs of various projects, the research on the lateral load of squeezed branch piles is increasing gradually. Qian [13,14,15] used the testing method for small-scale half-section model piles with undisturbed soil to observe the whole process from soil loading to destruction and the influence of the bearing disc position, disc spacing, and disc number on the horizontal bearing capacity and soil failure state around the pile, providing a reference for the research and development of pile foundations in oceanographic engineering. Yang [16] designed different slope angles for a model pile through computer simulations and used ANSYS 2020 software to analyze the effect of the bearing capacity of NT-CEP piles and the failure behavior. The results have enriched and improved the design theory for NT-CEP piles and complemented the practical engineering applications, playing a positive role in promoting the research and application of NT-CEP piles. However, at present, the theoretical research on the horizontal bearing characteristics of squeezed branch piles is not sufficient, so it is necessary to establish a calculation method for the bearing capacity of piles under horizontal loads.

The theoretical calculation methods for the bearing characteristics of straight piles under horizontal loads include the elastic–plastic analysis method, ultimate foundation reaction method, and elastic foundation reaction method. In the elastoplastic analysis method, the range of the piles and soil is divided into the elastic zone and plastic zone. In the elastic zone, the elastic foundation reaction method is adopted, while in the plastic zone, the ultimate foundation reaction method is adopted. According to the continuous condition of the boundary of the elastoplastic zone, the lateral resistance of the pile is obtained using the elastic foundation reaction method. The common methods include Broms’ method [17,18] and the p-y curve method [19,20]. The ultimate foundation reaction method assumes that the pile is rigid and the lateral resistance of the pile is calculated according to the static equilibrium under the limit state without considering the foundation deformation. The commonly used methods include the Raes [21] method and the Snitko [22] method. In the elastic foundation reaction method, by using the Winkler foundation model [23], the soil around the piles is simulated as a group of independent springs, and then the displacement and internal force of the piles are solved according to the deflection differential equation of the beam on the elastic foundation [24]. The foundation reaction methods mainly include the constant method, k-method, m-method, and c-method, and the current industry norms [25,26] clearly stipulate that the m-method is to be used to calculate pile foundations under horizontal loads. Based on the m-method, Wang and Su [27] discussed the dynamic and static conversion relationship of the m value during testing and proposed a new lateral dynamic load testing method for offshore piles, which was used to analyze the pile–soil interaction and evaluate the horizontal bearing capacity of the piles. Using mathematical statistics, Wu [28] established the correlation formula between the soil’s m value and the pile’s ground surface displacement under different soil conditions, which was used to determine the mechanical characteristics of the pile under horizontal loads. The m-method is an effective tool to solve the horizontal load pile because it is clear in concept, conforms to the actual situation and the calculation result is more safe.

Because of the existence of the bearing plate, the diameter of the pile body becomes larger, which leads to an increase in flexural rigidity and calculated width. Under the action of horizontal loads, the bearing plate will rotate together with the pile body, resulting in the generation of resisting moments, which makes the loading situation of the pile at the bearing plate complicated. Gao [29] gave the analytical solution of the squeezed branch pile according to Zhang’s method [30], which is based on the analytical solution of straight piles under horizontal loads. Furthermore, she also combined it with the deformation coordination conditions at the node, initial conditions, and boundary conditions to obtain the numerical solution. Based on the Winkler foundation model method, Peng [31] used the m-method to represent the linear relationship between plate–soil interaction at the bearing plate, pointing out that the resistance bending moment generated by the rotation of the bearing plate is proportional to the foundation coefficient m₀, and the results show that the pile top displacement and pile body angle of the squeezed branch pile are significantly reduced compared with ordinary piles. All the above methods are based on the elastic properties of foundation soil, but the actual pile–soil relationship is inelastic, so the Winkler foundation model based on a linear model cannot reflect the nonlinear pile–soil load transfer relationship. At the same time, the plastic failure of the foundation soil and the influence of the plate’s angle on the resistance should be considered under the condition of the displacement of the bearing plate.

Based on the Winkler foundation model and m-method, the nonlinear load transfer relationship between the bearing plate and foundation soil is established in the paper. Based on the derived formula, the internal force and deformation program of the squeezed branch pile is formulated, and the feasibility and accuracy of the theoretical calculation are verified by laboratory model tests. On this basis, the influence of the bearing plate’s diameter, placement position, and pile top constraint on the horizontal bearing capacity of the pile is analyzed. The theoretical calculation method proposed in this paper is applicable to the squeezed branch piles with different plate diameters, plate positions, plate section forms, and plate quantities. It can also be applied to the piles with different boundary conditions and soil conditions and can be extended to other variable section-shaped piles.

2. Theoretical Analysis of Horizontal Bearing Capacity of Squeezed Branch Pile

2.1. Analysis Model

The analysis model of the squeezed branch pile is established as shown in Figure 1. The pile length is L, the pile diameter is d, and the diameter of the bearing plate is D; the pile length on the plate is L₁, the pile length under the plate is L₂, and the height of the bearing plate is L′. The Cartesian coordinate system x0y is established with the top of the pile as the coordinate origin, and the horizontal to the right is the X-axis forward, and the vertical to the downward is the Y-axis forward. When there is a horizontal right concentrated force Q₀ and a clockwise couple moment J₀ at the top of the pile, the pile will bend and deform, which causes the horizontal reaction force in the soil of the straight pile section and the reaction couple moment in the soil of the compacted and bearing plate.

It is stipulated that the deflection of the pile to the right is positive, the rotation of the corner displacement is positive, the rotation of the bending moment is positive, the horizontal force Q₀ of the pile top is positive to the right in the direction of the X-axis, and the shear force of the pile body is positive to the right.

2.2. Analysis of Bearing Plate Resisting Moment Considering Plate–Soil Nonlinear Interaction

If the bearing plate with limited length is regarded as a rigid body, the angular displacements at each point of the plate remain the same, and the loading diagram of the bearing plate of the pile on the foundation soil reaction is shown in Figure 2. Under the action of the concentrated force Q₀ at the top of the pile and the couple moment J₀, assuming that the counterclockwise rotation φ occurs on the compacted plate, the foundation soil will form a trapezoidal normal resistance p on the earth-facing surface of the compacted plate, and the symmetrical normal resistance will eventually affect the internal force distribution of the pile in the form of resisting moments.

In the analysis of the pile–soil nonlinear interaction, the pile–soil load transfer relationship is often expressed in the form of hyperbolic functions [32]:

$$p=\frac{s}{a+bs}$$

(1)

where s is the normal displacement of the pile surface, a and b are load transfer coefficients, where a is the reciprocal of the initial stiffness of the foundation soil under pressure, and b is the reciprocal of the ultimate bearing capacity of the foundation soil.

When the angle displacement φ occurs on the bearing plate, the distance from the point of pile axis r to the neutral axis of pile on the plate is rsin θ, and then the displacement of the plate can be expressed as rφsin θ. By substituting the plate displacement into Equation (1), the plate surface resistance p at the distance r from the pile axis is as follows:

$$p=\frac{r\phi \sin \theta }{a+br\phi \sin \theta }$$

(2)

In the initial pressure state, the foundation soil at the surface of the bearing plate is in an elastic state, so its initial stiffness can be determined by the m-method, and its reciprocal a can be expressed as:

$$a=\frac{1}{mh}$$

(3)

where m is the proportional coefficient of the resistance factor, and h is the depth of the plate surface.

Referring to Janbu’s [33] method of determining the ultimate end resistance of straight piles, Li [34] established the expression of the ultimate bearing capacity of the soil around the plate of the bearing plate, whose reciprocal b can be expressed as:

$$b=\frac{1+\frac{D}{d}}{1+\tan \theta \tan \delta }·\frac{1}{c{N}_{\mathrm{c}}+q{N}_{\mathrm{q}}}$$

(4)

In the formula, δ is the friction angle of the pile–soil interface, and the value is taken according to

Table 1. Reference value of pile–soil interface friction angle.

Literature Sources	Pile-Soil Working Conditions	Reference Value
Neil et al. [35]	Pipe piles, dense sand	δ = 29.4°
Liu et al. [36]	Concrete piles, silty soil and clay	δ = 21.3°–31.6°
Jardine et al. [37]	Drive piles, sand soil	δ = 28°–30°

; c is the cohesive force of the soil around the plate; q is the average effective compressive stress on the side of the plate; Nc and Nq are the bearing capacity coefficients, which can be expressed as [33]:

$$\begin{gathered} N_{\mathrm{q}}={\left(\tan \delta +\sqrt{1+{\tan }^2\delta }\right)}^2{\mathrm{e}}^{2\psi \tan \delta } \\ {N}_{\mathrm{c}}=\frac{N_{\mathrm{q}}-1}{\tan \delta } \end{gathered}$$

where ψ is the angle of the soil failure surface under the ultimate load, and the soft clay to dense sand varies from 60° to 105°.

Thus, the load transfer coefficients a and b in Equation (2) are determined.

Substituting Equation (3) into Equation (2), and letting A = bmh, obtains

$$p=\frac{mhr\phi \sin \theta }{1+Ar\phi \sin \theta }$$

(5)

Take the plate surface element; the radial length is dr, the circumferential length is rdθ, and the distance from the element to the neutral axis of the pile is rsinθ; then, the resistance torque caused by the resistance on the element is

$$\mathrm{d}M=p·\mathrm{d}r·r\mathrm{d}\theta ·r\sin \theta =p{r}^2\sin \theta \mathrm{d}\theta \mathrm{d}r$$

(6)

By using the symmetry of the model, the subsoil resistance moment under the rotational displacement of the bearing plate can be obtained by integrating the subsoil resistance on the plate surface:

$$\begin{array}{cc}M\left(\phi \right)& =4{\displaystyle {\int }_{d/2}^{D/2}{\displaystyle {\int }_0^{\mathsf{\pi }/2}p{r}^2\sin \theta \mathrm{d}\theta \mathrm{d}r}}\hfill \\ & =\frac{mh\left(D^3-d^3\right)}{6A}-\frac{mh\mathsf{\pi }\left(D^2-d^2\right)}{4A^2\phi }-\frac{mh\mathsf{\pi }}{A^4{\phi }^3}\left(\sqrt{4-A^2{\phi }^2{D}^2}-\sqrt{4-A^2{\phi }^2{d}^2}\right)\hfill \\ & -\frac{2mh}{A^4{\phi }^3}\left[A\phi \left(D-d\right)+\sqrt{4-A^2{\phi }^2{d}^2}\arcsin \frac{A\phi d}{2}-\sqrt{4-A^2{\phi }^2{D}^2}\arcsin \frac{A\phi D}{2}\right]\end{array}$$

(7)

2.3. Calculation Method of Horizontal Bearing Capacity of Squeezed Branch Pile

The m-method based on the Winkler foundation beam model is a typical method to calculate the horizontal bearing capacity of straight piles. It is considered that the horizontal resistance of foundation soil to the pile is proportional to the horizontal displacement at this point of the pile body. The interaction between the pile and soil in the straight section of the squeezed branch pile body is exactly the same as that in the straight pile section of the pile body, so the m-method can be used to analyze the force in the straight section. The pile–soil interaction at the bearing plate can be described by the calculation equation for the bearing plate resisting moment considering the nonlinear pile-soil interaction obtained in Section 2.2.

Based on the m-method of the Winkler foundation beam model, Wang [38] established the deflection curve equation of the straight pile:

$$E_{\mathrm{i}}{I}_i\frac{d^{4}x}{d^{4}y}=-b_{i}p=-m_i\left(y_{0i}+y\right)b_{i}x$$

(8)

where E_i is the elastic modulus of the pile body material in section i, I_i is the moment of inertia of the pile section, b_i is the calculated width of the pile body of this section, p is the foundation soil reaction force, m_i is the proportional coefficient of the horizontal foundation coefficient changing with depth, and y_0i is the soil resistance parameter at the top of the pile in section i.

The calculation formula for the horizontal displacement x_i−1, rotational displacement φ_i−1, bending moment M_i−1, and shear force V_i−1 at the top of any pile segment to calculate the horizontal displacement x_i, rotational displacement φ_i, bending moment M_i, and shear force V_i at the bottom of the pile segment is as follows:

$$\left\{\begin{array}{c}{x}_i\\ {\phi }_i\\ M_i\\ V_i\end{array}\right\}=K_i\left\{\begin{array}{c}{x}_{i-1}\\ {\phi }_{i-1}\\ M_{i-1}\\ V_{i-1}\end{array}\right\}$$

(9)

where K_i is the coefficient matrix of the pile in section i, and its expression is

$$K_i=\left[\begin{array}{cccc}{A}_1& \frac{1}{{\alpha }_i}{B}_1& \frac{1}{{\alpha }_i{}^2{E}_i{I}_i}{C}_1& \frac{1}{{\alpha }_i{}^3{E}_i{I}_i}{D}_1\\ {\alpha }_i{A}_2& B_2& \frac{1}{{\alpha }_i{E}_i{I}_i}{C}_2& \frac{1}{{\alpha }_i{}^2{E}_i{I}_i}{D}_2\\ {\alpha }_i{}^2{E}_i{I}_i{A}_3& {\alpha }_i{E}_i{I}_i{B}_3& C_3& \frac{1}{{\alpha }_i}{D}_3\\ {\alpha }_i{}^3{E}_i{I}_i{A}_4& {\alpha }_i{}^2{E}_i{I}_i{B}_4& {\alpha }_i{C}_4& D_4\end{array}\right]$$

(10)

where A₁A₂…D₃D₄—dimensionless coefficient determined by depth, whose value is obtained from the table in the Specifications for Design of Foundation of Highway Bridges and Culverts [39]; α_i is the horizontal deformation coefficient of pile segment:

$${\alpha }_i=\sqrt[5]{\frac{m_i{b}_{0i}}{E_i{I}_i}}$$

(11)

b_0i is the calculated width of the pile body of section i.

For the squeezed single-branched pile, the pile body can be divided into the upper part of the plate numbered 1 and the lower part of the plate numbered 2 along the length direction. And the influence of the resisting moment of the bearing plate existing at the joint of the pile segment is introduced into Equation (8). Thus, the calculation formula for the displacement and internal force of the pile segment applicable to the squeezed branch pile is as follows:

$$\left\{\begin{array}{c}{x}_i\\ {\phi }_i\\ M_i\\ V_i\end{array}\right\}=K_i\left\{\begin{array}{c}{x}_{i-1}\\ {\phi }_{i-1}\\ M_{i-1}-M\left({\phi }_{i-1}\right)\\ V_{i-1}\end{array}\right\}$$

(12)

Since there is no bearing plate and bearing plate resisting moment at the top of pile segment 1 above the squeezed branch plate, letting D = d can make M(φ₀) calculated by Equation (7) = 0, so that Equation (10) is also applicable to pile segment 1.

Obviously, for a squeezed branch single pile with two pile segments, eight equations can be determined by using Equation (10), but there are six unknown displacements and six unknown internal forces. As long as four unknowns are determined by boundary conditions, the equation can be solved.

2.4. Boundary Conditions

2.4.1. Pile Top Boundary Conditions

In practical engineering, the pile top of the squeezed branch pile may be in an unconstrained free state or a constrained fixed state.

(1)

The pile top is unconstrained

When the top of the pile can freely experience horizontal displacement and rotational displacement, the boundary conditions of the top of the pile can be written as:

$$\left\{\begin{array}{l}{M}_0=J\\ V_0=Q\end{array}\right.$$

(13)

where M is the concentrated couple acting on the top of the pile; Q is the horizontal load acting on the top of the pile.

(2)

The top of the pile is constrained

When the pile top is constrained by the cap and other structures and cannot undergo rotational displacement, the boundary conditions of the pile top can be written as:

$$\left\{\begin{array}{l}{\phi }_0=0\\ V_0=Q\end{array}\right.$$

(14)

2.4.2. Pile End boundary Conditions

According to the engineering geological conditions and the bearing capacity requirements of piles, the pile bottom of the squeezed branch pile may be in an unconstrained free state or a constrained fixed state.

(1)

The pile bottom is free

When the pile bottom is in the soft soil layer, the horizontal displacement constraint and rotational displacement constraint provided by the foundation soil on the pile bottom can be ignored. In this case, the reverse torque and horizontal reaction of the pile bottom are 0, and the boundary condition of the pile bottom can be written as:

$$\left\{\begin{array}{l}{M}_2=0\\ V_2=0\end{array}\right.$$

(15)

(2)

The pile bottom is embedded

When the pile bottom is embedded in solid rock or dense soil at a large depth and the rock and soil body restrict the horizontal displacement and rotational displacement of the pile bottom, then the boundary conditions of the pile end can be written as:

$$\left\{\begin{array}{l}{x}_2=0\\ {\phi }_2=0\end{array}\right.$$

(16)

(3)

Pile bottom hinged

When the depth of the pile end embedded in the solid rock or dense soil is not large and the rock and soil body only restrict the horizontal displacement of the pile end, then the boundary conditions of the pile end can be written as:

$$\left\{\begin{array}{l}{x}_2=0\\ M_2=0\end{array}\right.$$

(17)

Obviously, for any squeezed branch pile, the four unknowns in Equation (10) can be determined by using the boundary conditions of the pile top and the boundary conditions of the pile bottom. Thus, for a squeezed branch single pile with two pile segments, the eight equations determined by Equation (10) have exactly eight unknowns remaining to be determined, and the equations satisfy the solution conditions.

3. Model Test Verification

3.1. Introduction to the Test

In order to verify the correctness of the theoretical analysis of the horizontal bearing capacity of squeezed branch piles, a set of small-scale laboratory model tests were carried out. The internal size of the model box used in the test is 800 mm (length) × 600 mm (width) × 580 mm (height). The model box is made of two 15 mm thick steel plates and two 12 mm thick plexiglass plates. The model box and loading device are shown in Figure 3, and the schematic diagram of the model pile is shown in Figure 4.

In order to facilitate the repeated development of the test, river sand is selected as the foundation soil. The direct shear test was carried out according to the Standard for Soil Test Method. The soil sample is placed in the fixed upper box and the movable lower box of the direct shear instrument, and the vertical pressure is applied first and then the horizontal thrust is applied to make the soil sample damaged by the shear. And the cohesion force c is 14.3 kPa, and the internal friction angle of the foundation soil φ is 30.5°. The soil density is obtained by dividing the measured soil volume in the filling process by the fixed volume, and the soil weight γ is 16.242 kN/m³.

The model pile is made of PVC pipe with a diameter of 20 mm and a wall thickness of 2 mm. The elastic modulus of the pile is 3.05 GPa after the bending test by the three-point loading method. The bearing plate is milled from aluminum alloy with an elastic modulus of 63.1 GPa. The parameters of the model pile and bearing plate manufactured by processing are shown in

Table 2. Parameters of model pile and bearing plate.

No.	Pile Diameter d (mm)	Plate Diameter D (mm)	Plate Height L′ (mm)	Plate Diameter Ratio (D/d)	Pile Length on Plate L1 (mm)	Pile Length under Plate L2 (mm)
ZK	20	/	/	/	/	/
ZPJ-35	20	35	20	1.75	90	390
ZPW-25	20	55	40	2.75	230	230

. Among them, the ZPJ-35 pile indicates that the bearing plate diameter is 35 mm, and the ZPW-25 pile indicates that the bearing plate is 25 cm away from the ground.

In order to measure the strain of the pile body during the test loading process, 12 strain gauges with a resistance value of 120 Ω were symmetrically installed in six sections of the model pile buried at different depths. The strain data under loading was collected by the DH3820 strain gauge.

As the foundation soil is filled into the model box layer by layer, the model piles installed with strain gauges are gradually buried in the foundation soil. In order to facilitate the installation of the horizontal load application device and the horizontal displacement measurement device on the pile top, the top of the model pile is exposed to a soil surface of 0.1 m, and the actual length of the buried pile is 0.5 m.

In the test, the horizontal load was applied by the slow maintenance load method, and the graded load was set at 6.24 N. During the test, when the displacement of the top of the pile under a certain level of load does not change more than 0.01 mm within 10 min, the deformation of the pile is considered to be stable, and the next level of load is applied [40]. When the displacement of the top of the pile reaches 6 mm, the loading ends [25]. The horizontal displacement of the pile top was recorded every 5 min after each stage of load was applied, and the strain of the pile body was collected every 1 min.

3.2. Analysis of Test Results

(1)

Displacement analysis of pile top

The relationship between the horizontal load of the pile top and the horizontal displacement of the ZK pile, ZPJ-35 pile, and ZPW-25 pile is shown in Figure 5.

As can be seen from Figure 5, under different pile top horizontal loads, the horizontal displacements of ZPJ-35 and ZPW-25 piles measured by tests are always smaller than ZK piles. Under the eighth level load of 50 N, the measured pile top displacements of the ZPJ-35 pile, ZPW-25 pile, and ZK pile are 5.32 mm, 5.42 mm, and 6.09 mm, respectively. The pile top displacements of the squeezed branch pile are reduced by 14% and 12% compared with those of the straight pile, indicating that the existence of the squeezed branch pile effectively improves the horizontal bearing stiffness of the pile. Through the fitting analysis of the test data, the load–displacement relationship of the ZK pile and ZPW-25 pile conforms to the quadratic parabola relationship, while the curve of the ZPJ-35 pile has good linearity.

Figure 5 shows that the load–displacement data points of the ZPJ-35 pile obtained by the test are scattered around the load–displacement curve of the ZPJ-35 pile obtained by using the calculation theory in this paper. And the two have a good coincidence. Under the eighth-order load, the theoretical calculation result of the pile top displacement of the ZPJ-35 pile is 5.03 mm, which has a relative error of 5.45% compared with the measured result of 5.32 mm. Under the same load, the theoretical calculation result of the pile top displacement of the ZPW-25 pile is 4.45 mm, which has a 17.90% relative error from the measured result of 5.42 mm. The error between the calculation results and the measured results of the pile top horizontal displacement of the two squeezed branch piles is less than 20%, which is ideal in the model test with a small scale. And the theoretical calculation of the pile top horizontal displacement can be considered credible.

(2)

Analysis of pile top rotation angle

The relationship between the pile top horizontal load and pile top angle of the ZK pile, ZPJ-35 pile, and ZPW-25 pile is shown in Figure 6.

As shown in Figure 6, the existence of a bearing plate makes the pile top rotation angle of ZPJ-35 and ZPW-25 piles always smaller than that of ZK piles under different pile top horizontal loads. Under the eighth level load with a size of 50 N, the measured pile top rotation angle of ZPJ-35 pile, ZPW-25 pile, and ZK pile is 3.750°, 3.434°, and 4.542°, respectively. The pile top rotation angle of the squeezed branch pile is reduced by 21% and 32% compared with that of the straight pile. Obviously, the existence of the bearing plate effectively limits the pile top angle displacement. The fitting of the test data shows that the load-top angle of the pile conforms to the quadratic parabola relationship between the straight pile and the squeezed branch pile.

As can be seen from Figure 6, the load-top rotation angle data points of the ZPJ-35 pile obtained by the test are scattered around the load-top angle curve of the ZPJ-35 pile obtained by using the calculation theory in the paper, and the two are in good agreement. Under the eighth-order load, the theoretical calculation result of the pile top rotation angle of ZPJ-35 is 3.314°, which has a relative error of 11.63% compared with the measured result of 3.750°. Under the same load, the theoretical calculation result of the pile top rotation angle of the ZPW-25 pile is 3.047°, which has a 11.27% relative error from the measured result of 3.434°. The errors between the calculation results and the measured results are less than 15%, so the pile top rotation angle calculated by theory can be considered reliable.

(3)

Analysis of pile bending moment

Figure 7 shows the measured and theoretical results of the pile bending moment of the ZK pile, the ZPJ-35 pile, and the ZPW-25 pile under the eighth load (50 N).

According to Figure 7, the maximum bending moment points of the three test piles all appear in the observed section with a buried depth of 0.05 m, and the setting of the bearing plate significantly reduces the maximum bending moment values of the piles. According to the measured results, it can be seen that the maximum bending moment of the pile body of the ZK pile is 8.032 N·m, while that of the ZPJ-35 pile and the ZPW-25 pile is only 5.609 N·m and 5.247 N·m, which are reduced by 43% and 53% compared with the straight pile, respectively. At the same time, the reduction in pile bending moment at the section on the plate further leads to the reduction in bending moment at each point of the pile body under the plate.

It can be seen from Figure 7 that the maximum pile bending moment obtained by the actual measurement and theoretical calculation both occurs at the position where the pile body buried depth is close to 0.05 m, in which the calculated maximum bending moment of the ZPJ-35 pile is 5.504 N·m, which is 2% smaller than the measured value of 5.609 N·m. The calculated maximum bending moment of the ZPW-25 pile is 5.247 N·m, which is 3% larger than the measured value of 5.085 N·m.

On the whole, the horizontal displacement of the pile top, the rotation angle of the pile top, and the bending moment of the pile body calculated by using the horizontal bearing capacity theory established in the paper are in good agreement with the test results, which proves the rationality and accuracy of the calculation method in the paper.

4. Analysis of the Influencing Factors of the Horizontal Bearing Capacity of the Squeezed Branch Pile

In order to explore the bearing mechanism of squeezed branch piles under horizontal load, based on the above theoretical derivation, the changes in pile displacement and pile bending moment under horizontal load with the depth of the model pile are discussed. The influencing factors include the diameter of the bearing plate, the position of the bearing plate, the constraint conditions of the pile top, etc. The theoretical calculation parameters are consistent with the model test parameters.

4.1. Influence of Bearing Disk Diameter

When the horizontal load at the top of the pile is 50 N, the diameter of the bearing plate gradually decreases. Decreases of 55 mm, 45 mm, and 35 mm of the bearing plates are taken, respectively, and then the comparison of pile displacement and bending moment is shown in Figure 8 and Figure 9.

The horizontal displacement at the ground increases with the decrease in plate diameter. When the bearing plate diameter is 55 mm, 45 mm, and 35 mm, the horizontal displacement is 4.62 mm, 4.89 mm, and 5.03 mm, respectively. Compared with the ZPJ-35 pile with the smallest plate diameter, the deformation resistance of the ZPJ-45 pile and the ZPJ-55 pile increased by 8% and 3%, respectively. With the increase in pile depth, the horizontal displacement of the pile body decreases first and then increases and remains unchanged gradually. For piles with different plate diameters and the same plate position, the horizontal displacement is concentrated on the bearing plate, and the displacement of the pile body under the bearing plate is small and stable with depth. It can be inferred that the existence of the bearing disc plays a certain role in “embedment,” resulting in a sharp reduction in pile displacement and pile rotation angle. As shown in Figure 9, the diameter of the bearing plate decreases from 55 to 35 mm, and the maximum bending moment values of the pile body are 4.813 N·m, 5.271 N·m, and 5.504 N·m, respectively. Compared with the ZPJ-35 pile, the resistance capacity increased by 13% and 4%, respectively. It can be seen that increasing the bearing plate diameter can balance the load and effectively reduce the maximum bending moment of the pile body. Therefore, in the actual project, the bearing plate diameter should be appropriately increased to achieve good bearing capacity. The bearing plate is set at 0.1 m from the ground. According to the figures, the bending moment of the pile body changes sharply near 0.1 m. At the same time, increasing the diameter of the bearing plate greatly reduces the maximum bending moment of the pile body and the bending moment under the plate.

4.2. Influence of Bearing Plate Position

When the horizontal load at the top of the pile is 50 N, the diameter of the bearing plate is fixed (55 mm), and the position of the bearing plate from the ground is changed to 25 cm, 20 cm, and 15 cm; then, the comparison of pile displacement and bending moment is shown in Figure 10 and Figure 11.

When the bearing plate is 0.15 m, 0.20 m, and 0.25 m away from the ground, the horizontal displacement at the ground of the squeezed branch pile gradually increases, and the horizontal displacement values at the ground are 4.33 mm, 4.41 mm, and 4.45 mm, respectively. Compared with the ZPW-25 pile with the largest buried depth of the bearing plate, its deformation resistance is increased by 3% and 1%, respectively. It can be seen that the closer the bearing plate is to the ground, the more it can play a role, the smaller the deformation at the ground, and the higher the corresponding horizontal bearing capacity. However, with the deepening of the position, the role of the bearing disk is gradually less obvious. That is, the lower the bearing plate is set, the smaller the lateral resistance to the soil. By observing Figure 11, it can be seen that the bending moments of piles with different branches vary significantly at the setting of the plate, and the changes are intensified before and after the curve trend near the bearing plate. The ZPW-15 curve in Figure 11 shows that the bending moment of the pile body changes slowly at 0.15 m, that is, at the bearing plate, but the curve changes abruptly above and below 0.15 m. In the ZPW-25 curve, due to the lower position of the bearing plate, the lateral movement and rotation angle of the pile body are small, and the resistance of the bearing plate to the bending moment cannot be fully exerted. Therefore, in order to make the bearing plate play the maximum role in the actual project, the plate should be placed as close to the ground as possible.

4.3. Influence of Pile Top Constraint Conditions

For horizontal bearing piles, when the top of the pile is embedded (the top of the pile is completely embedded without rotation), the boundary conditions at the ground are that shear force V₀ equals horizontal force Q₀ and the rotation angle φ₀ = 0. When the pile top is free and fixed, the pile top horizontal load is 50 N, and other conditions are the same. The pile body displacement and bending moment of the ZPJ-55 pile are calculated, respectively, and the calculation results are shown in Figure 12 and Figure 13.

When the pile top is free and embedded, the displacement at the ground is 4.62 mm and 1.29 mm, respectively; the deformation resistance is increased by 2.6 times; the displacement zero depth also decreases; and the horizontal bearing capacity of the pile is significantly increased. Moreover, the displacement curve of the pile body under the state of the pile top is slower than that under the free state, and the value fluctuation is small. The constraint degree of the pile top directly affects its failure. As shown in Figure 13, when the pile top is completely embedded, the node state is approximately rigid, and there is a large negative bending moment at the pile top. Although the bearing capacity of the pile body can be improved in the embedded state, the possibility of pile top failure also increases. According to the figure above, the changes in pile body displacement and bending moment of the two piles are consistent, and the curves above and below the bearing plate vary greatly, indicating that the bearing plate shares a greater soil resistance.

5. Analysis and Discussion

In the paper, PVC pipe is selected as the model pile material, and the pile with a conversion depth ≥ 4.0 is a flexible pile according to the horizontal reaction coefficient of the foundation. Compared with rigid piles with high flexural rigidity and bearing capacity, flexible piles have good flexibility, can be bent and deformed, and are used to resist horizontal forces and shear forces. The above tests and theoretical calculations also prove that flexible piles have good deformation capacity. Therefore, in practical engineering, flexible piles with rigid foundations or rigid piles with flexible foundations can be used to improve the reinforcement effect of pile foundations and achieve the purpose of being suitable for various complex geological environments.

The m value obtained by using the standard [41] formula is 4.055 MN/m⁴, which is small, and the deviation is large compared with the given range. It can be seen from the formula that the value of m is related to the horizontal ultimate bearing capacity and its corresponding critical load. But for a flexible pile, its displacement is greatly affected by the load, so the load corresponding to the ground displacement of 6 mm is small, and the load-bearing capacity of the model pile may not have reached the limit at this time, so the calculated m value is small. Therefore, whether the horizontal bearing capacity can take the load corresponding to the ground displacement of 6 mm still needs to be studied. In addition, the relevant dimensions should be reduced during the theoretical verification: if the horizontal load is a long-term or frequent load, we multiply the m value by a reduction factor of 0.4 [42]. For an indoor model test, considering its scaling effect, for the circular pile with a diameter less than 1 m, the calculated width b0 of the pile body is 0.9 × (1.5 d + 0.05) [43]. The “Technical Code for Building Pile Foundations” only distinguishes the precast pile, steel pile, and cast-in-place pile and does not consider the influence of pile diameter on the m value. When the horizontal bearing capacity of pile foundations are calculated according to the “Technical Code for Building Pile Foundations”, whether the value of m is related to the pile diameter is worth studying [42]. The paper [43] points out that the value of m has a linear relationship with soil force strength c, and the proportional coefficient of horizontal resistance m increases with the increase in the value of c. In the test, due to errors such as stratified vibration and manual operation, the soil layer compaction may be inconsistent, resulting in a change in the value of c, resulting in a large difference between the calculated value of m and the standard value.

According to the experiment and calculation results, the nonlinear method can better reflect the relationship between the pile and soil for the squeezed branch pile. Moreover, the displacement and bending moment curves at the bearing plate change obviously; that is, the soil is compacted at the bearing plate, so that the stress around the pile is concentrated and the soil can bear the pile body pressure in a wider range, thus reducing the deformation and sharing the resistance. However, no matter if it is straight piles or squeezed branch piles, the curve will no longer change after reaching a certain depth. The soil is an elastic–plastic body; when the load increases gradually, the surface soil changes from elastic to plastic. With the increase in load, the plastic zone gradually increases, and the horizontal load is transferred to the deeper soil. The greater the depth, the less affected by the load and the less obvious the change. As can be seen from the figure, the loading range of the pile body is about 10 d, which is also consistent with the conclusion drawn in the paper [44].

Changing the constraint conditions of the pile top can improve the bearing capacity of the pile, but for the pile top damage caused by the negative bending moment of the fixed pile top, the embedded depth of the pile top can be changed from full embedded to partial embedded in the actual project. This change can improve the bearing capacity, reduce the deformation of the pile body, and extend the service life of the pile foundation. The paper [45] gives the test results of pile–cap–soil composite force under different embedded depths. The research shows that for an ordinary connection of the pile top, it is appropriate to take the embedded depth of the pile top as 0.5 D (D is the pile diameter).

6. Conclusions

Based on the m-method, a theoretical calculation method of pile deformation and internal force under horizontal load is established in the paper. Through analyzing the parameters of the pile body, such as diameter, position of bearing plate, and constraint conditions of the pile top, the following conclusions are obtained:

(1)

Compared with the ordinary straight pile, the pile with the same diameter and pile length but with an added bearing plate has a higher horizontal bearing capacity and better resistance to deformation.

(2)

The nonlinear calculation method m-method adopted in the paper can correctly reflect the changes in displacement, rotation angle, and bending moment of the pile and is consistent with the experimental results with a high fitting degree, reflecting the nonlinear transfer relationship between the load and the pile soil.

(3)

The horizontal bearing capacity and displacement of the pile can be improved by increasing the diameter of the bearing plate and decreasing the distance between the plate and the ground. However, when the plate diameter increases to a certain extent, the improvement is not obvious. The load range of the pile body is about 10 d (d is the pile diameter); beyond the range, the bearing plate cannot fully play a role, and the force under the plate is similar to that of a straight pile. Changing the constraint conditions of the pile top and reasonably setting the embedded depth of the pile top can better resist the horizontal load and reduce the zero displacement of the pile body.

(4)

According to the static load test results of “Technical Code for Building Pile Foundation”, the horizontal displacement at the ground is 10 mm (for horizontal displacement sensitive buildings, it is 6 mm) and the corresponding load is the characteristic value of horizontal bearing capacity of a single pile. When calculating the proportional coefficient m of horizontal resistance coefficient of pile foundation soil, consideration should be given to the large deformation of flexible piles and the small corresponding load. Such conditions have not yet reached the ultimate load.