Research on the Analytical Conversion Method of Q-s Curves for Self-Balanced Test Piles in Layered Soils

Abstract

An analytical conversion method was developed for the self-balanced test results of monopile bearing capacity in layered soils to realize the better applicability of the self-balanced test theory for the bearing capacity test of foundation piles. To the additional settlement of the lower pile bottom brought on by the negative friction of the upper pile soil during the loading process in layered soils, the interaction effect between the upper piles and lower piles is first taken into account. To accurately convert the results of the self-balanced test pile into the traditional static load test curve form and solve the ultimate bearing capacity, the displacements and internal forces at micro-segment piles in each layer of soil were obtained using the finite difference method. Then, for verification, conventional static test piles and indoor model tests were conducted in a multi-layered ground foundation. The outcome demonstrates that the simplified conversion method’s bearing capacity of the test pile is greater than that of the traditional static pressure test, the analytical conversion method’s Q-s curve is relatively similar to the results of the conventional static load test, and the accuracy of the analytical conversion results is increased by about 9.3 percent. At the same time, the analytical conversion method was applied to the self-balanced test project of bored cast-in-place piles in Wutong Garden, Laibin, Guangxi, China, and the accurate bearing capacity and internal force deformation characteristics were obtained. The accuracy of the calculation result is improved by 12% compared with that of the simple conversion calculation result. Therefore, it can be widely promoted and applied in self-balanced pile bearing capacity test projects.

1. Introduction

In order to ensure the safety and reliability of pile foundations, the bearing capacity test of the pile foundation is essential [1,2,3]. The traditional methods for static load testing of foundation piles mainly include the vertical compressive pile load method and anchor pile method [4,5,6,7], the high-strain method for dynamic testing of pile bearing capacity [8,9], the probability analysis model, which predicts and evaluates the accuracy of the single-pile bearing capacity [10,11,12], and the finite element approach of numerical simulation of the pile foundation [13,14,15]. However, a novel technique for evaluating the bearing capability of foundation piles is the self-balanced test pile method. There is no requirement for a pressure platform or anchor pile reaction device, in contrast to the conventional stacking method and the anchor pile method. It is frequently used in practical engineering to establish the final bearing of a single pile, since it can save testing time, labor costs, and manpower requirements [16,17]. Osterberg first tested the monopile bearing capacity using self-balanced test piles in 1989, which has since then been widely used in several parts of the world [18,19,20]. Furthermore, several specifications have been developed for it [21,22,23,24].

A major difficulty in the promotion and use of this technology is how to translate the test findings of the self-balanced approach into results that are comparable to standard static load tests. Based on Mindlin’s displacement solution, Seol et al. proposed a method for the calculation of loads in piles, a technique that takes into account the resistance of the connecting shaft [25,26]. Kim et al. studied the load action law of self-balanced piles, considering, among others, the residual load distribution, and evaluated the displacement variables of prestressed, high-strength concrete piles in deeper foundations [27]. The numerical method was utilized by Seol et al. to determine the conversion coefficients, variation laws, and soil elastic modulus for various length-to-diameter ratios [28]. The properties of the self-balanced pile load transfer function are the main focus of many scholars’ study findings, but the conversion method has not been examined [29,30,31]. Currently, in engineering, the test results of self-balanced test piles are converted to conventional static load test Q-s curves by the exact conversion method and the simplified conversion method. With the simple conversion method, it is difficult to determine the exact value of the conversion coefficient in different soil layers [23,24,32], and it lacks of some theoretical support [33,34,35]. The precise conversion method also increases the burden on the test pile, since each micro-segment‘s displacement and internal force need to be measured with many specific displacement transducers [36]. Therefore, there is still room for improvement on the self-balanced theory within the current study. In order to calculate the ultimate bearing capacity of a single pile more accurately, it is essential to research the load transfer mechanism in the test pile of the self-balanced method and find a suitable theoretical analytical conversion method.

According to the self-balanced loading model, the displacements generated by the upper and lower piles interact with each other under the action of the load box. The skin friction resistance of the pile produces additional stress on the pile end, which causes additional settlement at the bottom of the lower pile. The additional settlement affects the relative displacement of the pile soil of the upper pile [37]. As a result, Mindlin’s solution was employed in this study to determine the additional pile bottom settling brought on by the side friction resistance of the top pile [28]. In addition, it is considered that the load action function follows the Kraft model when taking into account the impact of the top pile’s weight on the ability to resist loading. According to the finite difference method, the differential equation between the loading displacement of the pile element and the additional settlement is established. By using MATLAB to calculate the displacement and lateral friction resistance of each micropile section of the upper and lower piles in a multilayered foundation to draw a self-balanced Q-s curve, and then converting the self-balanced test Q-s curve into a conventional static load test, the Q-s curve can calculate the ultimate load capacity of a single pile [38,39]. To confirm the correctness and application of the theoretical analysis approach, the conversion method from this work was compared with the conventional static pressure test in layered soil, respectively. Finally, the conversion method was applied to an actual project of self-balanced bored grouting test piles in Wutong Garden, Laibin, Guangxi, China.

2. Analytical Conversion’s Theoretical Basis of Self-Balanced Test Pile Q-s Curve

2.1. Calculation of Additional Settlement of Lower Pile Toe

The geometric distribution of the self-balanced upper and lower pile elements is shown in Figure 1. The pile element’s vertical extra displacement caused by the shear act of other unit loads can be described as Equation (1).

$$\left\{w_s\right\}=\frac{D}{E_s}\left[I_b\right]\left\{\tau \right\}$$

(1)

where w_s is the pile element’s extra displacement; D is the test pile’s diameter; E_s is the pile’s elastic modulus; τ is the shear stress of the pile; and I_b is the influence factor of additional stress on the load in the semi-infinite plane.

For the k micro-units of the upper section pile, the lower pile’s extra displacement w_bs end caused by the load of the pile body can be described as Equation (2).

$$\left\{w_{bs}\right\}=\frac{D}{E_s}{\displaystyle \sum _{j=1}^k\left(I_{bj}\overrightarrow{{\tau }_j}\right)}$$

(2)

where $\overrightarrow{{\tau }_j}$ are the shear stress vector functions of any unit j in the upper section pile, and the upper pile is negative and the lower pile is positive. I_bj is the additional settlement influencing factor of the lower pile toe caused by the shear stress of unit j.

$$I_{bj}=\pi {\displaystyle \underset{\left(j-1\right)\Delta L}{\overset{j\Delta L}{\int }}{I}_p}dh$$

(3)

According to Mindlin’s solution for Equation (3), I_p is the effect factor of vertical displacement, ΔL is the length of unit j, ΔL = L/n, and h is the burial depth of unit j.

$$I_p=\frac{\left(1+v_s\right)}{8\pi \left(1-v_s\right)}\left\{\frac{z_1{}^2}{R_1{}^3}+\frac{\left(3-4v_s\right)}{R_1}+\frac{\left(5-12v_s+8v_s{}^2\right)}{R_2}+\frac{\left(3-4v_s\right)z^2-2hz+2h^2}{R_2{}^3}+\frac{6h{z}^2\left(z-h\right)}{R_2{}^5}\right\}$$

(4)

where v_s is the Poisson’s ratio of each unit corresponding to the soil layer; z and z₁ are the calculated lengths; and R₁ and R₂ are the calculation coefficients, R₁² = z₁² + D²/4, R₂² = z² + D²/4.

Substituting Equation (4) into (3) and (2).

$$I_{bj}=\frac{\left(1+v_s\right)}{8\pi \left(1+v_s\right)}\left\{\frac{z_1}{R_1}-4\left(1+v_s\right)\ln \left(z_1+R_1\right)+8\left(1-2v_s+v_s{}^2\right)\ln \left(z+R_2\right)+\frac{2L^{2}z/r^2-4L-\left(3-4v_s\right)z}{R_2}+2\frac{\left(L{r}^2-L^2{z}^3/r^2\right)}{R_2{}^3}\right\}$$

(5)

where r is the test pile’s radius. Equation (5) is substituted for Equation (2). The settlement w_bs at the lower pile toe caused by the side shear stress of n units in the pile body can be calculated as Equation (6).

$$w_b{}_s=\frac{{\displaystyle \sum _{j=0}^k{I}_{bj}{K}_z{}_j{w}_j}}{E_{s}U\Delta L/D+{\displaystyle \sum _{j=0}^k{I}_{bj}{K}_z{}_{,j}}}$$

(6)

where U is the test pile’s perimeter, K_zj is connected to the shear stress vector functions, which is the pile–shear soil’s stiffness vector corresponding to the unit ${\tau }_j$.

Hou et al. researched the negative friction resistance of the side caused by water immersion in expansive soil and considered that the pile–soil interface model proposed by Kraft [40] (as demonstrated in Figure 2) can adequately reflect the negative frictional resistance characteristics [41]. Huang et al. considered that the nonlinear behavior of a single friction pile is described here by a nonlinear model which was proposed by Kraft [42]. Although the pile side resistance to friction is negative in the self-balanced upper pile, the pile–soil interaction is the same as in the positive friction resistance model. The Kraft model can conveniently take into account the stratification of soil and the nonlinear deformation of soil around the pile, and also consider the continuity of soil around the pile. In this study, it is presumed that the relationship between the upper and lower piles’ side friction resistance follows Kraft’s form, and the pile’s side friction is written as Equation (7).

$${\tau }_{\nu ,i}=\frac{w_i{G}_s}{r_0\ln \left(\frac{r_m/r_0-\psi }{1-\psi }\right)}$$

(7)

where r₀ is the pile’s radius, r_m is the zone of influence beyond which the shear stress becomes negligible, r_m = 2.5l(1 − v), l is the pile’s length, and v is the soil’s Poisson’s ratio. R_f is the shear strain curve-fitting constant determined from laboratory tests, and it is unquestionably in the range of 0.9–1.0. The pile’s initial stiffness can be expressed as Equation (9).

$$K_{z,i}=\frac{2\pi G_s}{\ln \left(\frac{r_m/r_0-\psi }{1-\psi }\right)}$$

(8)

2.2. Self-Balanced Upper and Lower Piles’ Calculation Process

Figure 3a depicts the pile–soil interaction model (a). The friction between each unit and the soil in the test pile is represented by nonlinear springs, where w represents the displacement of the pile units. The test pile is made up of n micro-elements.

Figure 3b displays the test pile units’ force model. The stress equilibrium condition of the differential element can be described as Equation (7).

$$Q_{i+1}-Q_i+G_i+R_i-K_{z,i}\left(w_i-w_{bs}\right)=0$$

(9)

where G_i is a unit’s self-weight, R_i is an external force, w_bs is additional pile end settlement, w_i is the unit’s vertical displacement, and s_i = w_i − w_b is the relative displacement of the pile and soil.

2.2.1. Internal Force Calculation of Self-Balanced Upper Pile

According to the finite difference relationship between the force and deformation of each unit:

$$T_i=\frac{{\left(E_p{A}_p\right)}_i}{\Delta L}\left(w_i-w_{i-1}\right)$$

(10)

$$T_{i+1}=\frac{{\left(E_p{A}_p\right)}_{i+1}}{\Delta L}\left(w_{i+1}-w_i\right)$$

(11)

where E_p and A_p stand for the pile unit’s elastic modulus and cross-sectional area, respectively. By substituting Equations (6), (8), (10) and (11) into Equation (9), the control equation of the node i can be described as Equation (12):

$$\frac{{\left(E_p{A}_p\right)}_i}{\Delta L}{w}_{i-1}-\left[\frac{{\left(E_p{A}_p\right)}_i+{\left(E_p{A}_p\right)}_{i+1}}{\Delta L}+K_{z,i}\right]w_i+\frac{{\left(E_p{A}_p\right)}_{i+1}}{\Delta L}{w}_{i+1}+K_{z,i}\frac{{\displaystyle \sum _{j=0}^n{I}_{bj}{K}_z{}_{,j}{w}_j}}{E_p{U}_p\Delta L/D+{\displaystyle \sum _{j=0}^n{I}_{bj}{K}_{z,j}}}=-G_i-R_i$$

(12)

where

$$h_i=\frac{{\left(E_p{A}_p\right)}_i}{\Delta L}$$

(13)

$$g_i=-\left[\frac{{\left(E_p{A}_p\right)}_i+{\left(E_p{A}_p\right)}_{i+1}}{\Delta L}+K_{z,i}\right]$$

(14)

$$k_i=\frac{{\left(E_p{A}_p\right)}_{i+1}}{\Delta L}$$

(15)

$$n_{i,j}=K_{z,i}\frac{{\displaystyle \sum _{j=0}^n{I}_{bj}{K}_{z,j}}}{E_p{U}_p\Delta L/D+{\displaystyle \sum _{j=0}^n{I}_{bj}{K}_{z,j}}}{w}_j$$

(16)

$$m_i=-G_i-R_i$$

(17)

As demonstrated in Equation (18):

$$h_{i-1}{w}_{i-1}+g_i{w}_i+k_{i+1}{w}_{i+1}+{\displaystyle \sum _{j=0}^n\left(n_{i,j}{w}_j\right)}=m_i$$

(18)

Assuming that the load box is installed at i = k (k ≤ n), when k = n, the load box is installed at the bottom of the pile, and for the upper pile boundary condition, there is R₁ = 0 at the top of the upper pile, as demonstrated in Equation (19):

$$\frac{{\left(E_p{A}_p\right)}_1}{\Delta L}\left(-w_0+w_1\right)=0$$

(19)

At the top load box i = k of the upper pile, R_k = −F, as demonstrated in Equation (20):

$$\frac{{\left(E_p{A}_p\right)}_k}{\Delta L}\left(-w_{k-1}+w_k\right)=-F_n$$

(20)

where F_n is the load size value of the load box.

The governing equation of the upper pile is written in matrix form as demonstrated in Equation (21):

$$\left[K_z+K_c\right]\left\{w\right\}=\left\{m_z\right\}\begin{array}{cc}& \end{array}(0\le i\le k)$$

(21)

where [K_z] is the upper pile body’s stiffness matrix, [K_c] is the additional stiffness matrix for coupling effects, and {w} is the axial displacement column vector of the lower pile units’ €node, which can be represented as Equation (22):

$$\left\{w\right\}={\left\{\begin{array}{cccc}{w}_0& w_1& \cdots & \begin{array}{cc}{w}_{k-1}& w_k\end{array}\end{array}\right\}}^T,$$

(22)

where {m_z} is the external load column vector of the upper pile, which can be expressed as Equation (23):

$$\left\{m_z\right\}=\left\{G_i\right\}+\left\{Q_i\right\}={\left\{\begin{array}{cccc}\gamma U\Delta L_1& \gamma U\Delta L_2& \cdots & \begin{array}{cc}\gamma U\Delta L_{k-1}& -F_n+\gamma U\Delta L_k\end{array}\end{array}\right\}}^T$$

(23)

Therefore, the upper pile governing equation can be expressed as Equation (24):

$$\left\{\left[\begin{array}{cccc}\begin{array}{cc}{g}_0& k_0\\ h_1& g_1\end{array}& \begin{array}{cc}& \\ k_1& \end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{cc}& \\ & \end{array}\\ \begin{array}{cc}& \cdots \\ & \end{array}& \begin{array}{cc}\cdots & \cdots \\ h_i& g_i\end{array}& \begin{array}{cc}& \\ k_i& \end{array}& \begin{array}{cc}& \\ & \end{array}\\ \begin{array}{cc}& \end{array}& \begin{array}{cc}& \cdots \end{array}& \begin{array}{cc}\cdots & \cdots \end{array}& \begin{array}{cc}& \end{array}\\ \begin{array}{cc}& \\ & \end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{cc}{h}_{k-1}& g_{k-1}\\ & h_k\end{array}& \begin{array}{cc}{k}_{k-1}& \\ g_k& k_k\end{array}\end{array}\right]+\left[\begin{array}{c}\begin{array}{ccc}& & \end{array}\\ \begin{array}{ccc}& n_{i,j}& \end{array}\\ \begin{array}{ccc}& & \end{array}\end{array}\right]\right\}\left\{\begin{array}{c}{w}_0\\ \begin{array}{c}{w}_1\\ \begin{array}{c}⋮\\ w_i\\ ⋮\end{array}\\ w_{k-1}\end{array}\\ w_k\end{array}\right\}=\left\{\begin{array}{c}{m}_0\\ \begin{array}{c}{m}_1\\ \begin{array}{c}⋮\\ m_i\\ ⋮\end{array}\\ m_{k-1}\end{array}\\ m_k\end{array}\right\}$$

(24)

2.2.2. Self-Balanced Lower Pile Internal Force Calculation

In the uppermost load box of the lower pile, i = k, R_k = F. The bottom pile’s toe can be seen as a rigid briquettes force, with the formula R(n) = K_bW_n [43], where K_b is determined using the Boussinesq method and written as Equation (25).

$$K_{bz}=\frac{P_b}{w_b}=\frac{D{E}_b}{1-v_b^2}(1+0.65\frac{D}{h_b})$$

(25)

where h_b is the distance from the pile toe to the bedrock, and E_b is the soil’s elastic modulus at the bottom of the lower pile. v_b is the soil’s Poisson’s ratio at the bottom of the lower pile.

The matrix form of the governing equation for the lower pile is depicted in Equation (26):

$$\left[K_z\right]\left\{w\right\}=\left\{m_z\right\}\begin{array}{cc}& \end{array}(k\le i\le n)$$

(26)

In Equation (27), [K_z] stands for the lower pile body’s stiffness matrix, [K_c] for coupling effects, and {w} stands for the lower pile node’s axial displacement column vector.

$$\left\{w\right\}={\left\{\begin{array}{cccc}{w}_k& w_{k+1}& \cdots & \begin{array}{cc}{w}_{n-1}& w_n\end{array}\end{array}\right\}}^T,$$

(27)

where {m_z} is the external load column vector of the lower pile, which can be expressed as Equation (28):

$$\left\{p_z\right\}=\left\{G_i\right\}+\left\{Q_i\right\}={\left\{\begin{array}{cccc}{F}_n& 0& \cdots & \begin{array}{cc}0& \frac{d{E}_b}{1-v_b^2}(1+0.65\frac{d}{h_b})w_n\end{array}\end{array}\right\}}^T$$

(28)

Therefore, the lower pile’s governing equation can be expressed as Equation (29):

$$\left\{\left[\begin{array}{cccc}\begin{array}{cc}{h}_k& g_k\\ h_{k+1}& g_{k+1}\end{array}& \begin{array}{cc}{k}_k& \\ k_{k+1}& \end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{cc}& \\ & \end{array}\\ \begin{array}{cc}& \cdots \\ & \end{array}& \begin{array}{cc}\cdots & \cdots \\ h_i& g_i\end{array}& \begin{array}{cc}& \\ k_i& \end{array}& \begin{array}{cc}& \\ & \end{array}\\ \begin{array}{cc}& \end{array}& \begin{array}{cc}& \cdots \end{array}& \begin{array}{cc}\cdots & \cdots \end{array}& \begin{array}{cc}& \end{array}\\ \begin{array}{cc}& \\ & \end{array}& \begin{array}{cc}& \\ & \end{array}& \begin{array}{cc}{h}_{n-1}& g_{n-1}\\ & h_n\end{array}& \begin{array}{cc}{k}_{n-1}& \\ g_n& k_{}\end{array}\end{array}\right]\right\}\left\{\begin{array}{c}{w}_k\\ \begin{array}{c}{w}_{k+1}\\ \begin{array}{c}⋮\\ w_i\\ ⋮\end{array}\\ w_{n-1}\end{array}\\ w_n\end{array}\right\}=\left\{\begin{array}{c}{m}_k\\ \begin{array}{c}{m}_{k+1}\\ \begin{array}{c}⋮\\ m_i\\ ⋮\end{array}\\ m_{n-1}\end{array}\\ m_n\end{array}\right\}$$

(29)

By programming to solve the above analysis theory, the axial force and displacement at each point of the upper and lower section of the pile can be solved.

3. Conversion Method

3.1. The Principle of Equivalent Conversion Equation

Figure 4 outlines the foundation of the precise conversion technique. When the top pile is divided into n units, Equations (30) and (31) can calculate the displacement and internal force of the equivalent transformation at any point of the test pile.

$$Q(i)=Q(0)+{\displaystyle \sum _{m=1}^{i}q\left(m\right)}\left\{U\left(m\right)+U\left(m+1\right)\right\}q\left(m\right)/2$$

(30)

$$s(i)=s(0)+{\displaystyle \sum _{m=1}^i\frac{Q(m)+Q(m+1)}{E_p(m)A_p(m)+E_p(m+1)A_p(m+1)}h}(m)$$

(31)

where Q(0) represents the pile’s axial force at i = 0, which is equal to the load value of the load box, and s(0) represents the pile’s displacement at i = 0, which is equal to the load box’s upward displacement. q(m) is the friction resistance on the pile side at i = m.

According to Equations (32) and (33), the equivalent displacement and load of the upper pile top i = n can be obtained.

$$Q(n)=Q(0)+{\displaystyle \sum _{m=1}^{n}q\left(m\right)}\left\{U\left(m\right)+U\left(m+1\right)\right\}q\left(m\right)/2$$

(32)

$$s(n)=s(0)+\frac{h(n)\left\{2Q(0)+{\displaystyle \sum _{m=1}^{n-1}q\left(m\right)}\left[U\left(m\right)+U\left(m+1\right)\right]h\left(m\right)+q\left(n\right)\left[U\left(n\right)+U\left(n-1\right)\right]h\left(n\right)/2\right\}}{E_p(n)A_p(n)+E_p(n-1)A_p(n-1)}$$

(33)

3.2. Conversion to Traditional Static Pressure Test Pile

MATLAB programming was used to solve the axial force, lateral friction resistance and displacement of the vertical points of the upper and lower test piles. Then, according to the principle of the self-balance accurate conversion method and the relationship between the side friction resistance and the variable displacement, the self-balanced test results are converted into an equivalent pile top load-displacement (Q-s) curve. Using the equivalent load method to convert the load loading value of the upper and lower piles into the Q-s curve of the traditional static pressure pile, the ultimate bearing capacity of the single pile can be calculated, which can be expressed as Equation (34)

$$Q_u=Q_u{{}^t}_{up}+Q_u{{}^l}_{down}$$

(34)

where $Q_u{{}^t}_{up}$ is the upper pile’s maximum bearing capacity; where $Q_u{{}^l}_{down}$ is the lower pile’s maximum bearing capacity.

4. Model Test Verification

In this study, the self-balanced pile and the conventional static pressure pile were both tested using an indoor model. The test pile’s maximum bearing capacity was determined by gradually overloading until it collapsed, the internal force and displacement of the pile were determined by mounting strain gauges on the pile body [44,45,46]. This was verified through a comparison of experimental results with those from theoretical calculations.

4.1. Test Scheme

4.1.1. Test Device

In layered soils, self-balanced test pile and conventional static pressure test pile bearing characteristics were carried out using the self-developed test pile loading apparatus. As demonstrated in Figure 5, the test device consists of a model box, loading device, and data acquisition system. The steel drum used in the model box has a wall thickness of 3 mm, a diameter of 580 mm, and a height of 880 mm. The model box’s bottom was welded with a 3 mm round steel plate to prevent the bottom from deforming due to rammed earth. The loading device was suspended by steel wire. The initial load is the weight of the 10 kg basket. Afterward, 5 and 10 kg weights were added for each level to load step by step. The pile’s displacement was measured by a percentage meter. The TST3822EN static resistance strain data collector is a static signal-processing software based on the VC++ development platform. It can be directly connected to the computer through the WiFi analysis system or data network cable to collect, process, and save the data at the test time. It can be used to calculate the strain data of the test pile (as demonstrated in Figure 6).

4.1.2. Model Pile

The test pile model was constructed from a hollow aluminum alloy tube that had a wall thickness of 2 mm, a diameter of 25 mm, a length of 700 mm, and an elastic modulus of 69.5 GPa. The model piles were then divided in half, and strain gauges were inserted uniformly inside at various depths by running along the length of the pile so that the axial stresses could be back-calculated using the measured strains (as demonstrated in Figure 7). After that, epoxy resin was used to attach the combined pile model, and the pile heads were pasted at both ends of the pile to seal the bottom. By adhering a layer of fine sand, the pile’s surface is mimicked.

The pile’s length above the soil is 30 mm, and its embedded depth is 670 mm.

Table 1. Model test pile parameters.

Model	Pile Length/mm	Pile Diameter/mm2	Mass/g	Elastic Modulus/GPa
The self-balanced upper segment pile Z1	470	25	149.7	69.5
The self-balanced lower pile Z2	200	25	121.4	69.5
The traditional static pressure pile C1	670	25	271.1	69.5

displays the test model’s elastic modulus, mass, and pile length, diameter, and mass. Figure 8 depicts the test model piles’ plan configuration, and A_n, B_n, and C_n represent the numbers of strain gauges pasted inside the conventional static pile, the self-balanced lower pile, and the self-balanced upper pile, respectively.

4.1.3. Test Soil

The disturbed soil collected from the site is dried, crushed, and dried. Then the water and soil ratio is calculated according to the water content of each soil layer in the actual project, and the filling soil samples of each layer of soil is configured. The cohesion, internal friction angle, elastic modulus and Poisson’s ratio of each soil layer are determined according to relevant geotechnical tests and the relevant engineering geological data. Then, according to the original density of the soil in the actual project, the quality of the soil sample filled in a certain volume of the box is calculated, and it is compacted and filled layer by layer from bottom to top every 50 mm. The test soil was manually tamped every 50 mm to a dry density of 1.6 g/cm³. Clay, silty clay, clayey silt, and silty sand are selected as the soil for the layered foundation model test. The physical parameters of various soil samples are demonstrated in

Table 2. Basic physical mechanical properties of layered soils samples.

Soil Layer	Type of Soil Layer	Soil Thickness/mm	Water Content	Force of Cohesion/kPa	Internal Friction Angle/°	Elastic /MPa	Poisson Ratio
①	Mealy sand	150	18	5.5	36.4	14	0.35
②	Clayey silt	150	18	7.2	25	23	0.35
③	Silty clay	200	26	25	19.0	19.0	0.42
④	Clay	450	26	59.6	18.0	28.0	0.40

.

4.2. Test Contents

Conventional hydrostatic and self-balanced test piles were loaded in layered soils to failure step by step. The slow loading method was used in the loading process, and a load of each stage lasted for 1 h. The loading was terminated when the load or displacement reached the specified requirements according to the code of practice [23,24,31].

4.3. Experiment Results and Analyses

4.3.1. Experiment Results

(1)

Loading and displacement of test piles

Taking upward displacement as positive and downward as negative, the load displacement curves of the two soil model tests of traditional static pressure piles and self-balanced test piles can be seen in Figure 9. When the traditional static pressure pile was loaded to 1800 N, the pile top displacement reached 9.80 mm, and the increase of its displacement was significantly larger than the previous level. When the test load reaches 1680 N, the test pile fails, and this value can be determined as the maximum bearing capacity of the single pile. When loaded to 2 × 750 N, the lower pile’s downward displacement reached 6.51 mm, and it was damaged. At this time, the upper pile continues to be loaded to 800 N to reach the ultimate load and take the upper and lower piles’ ultimate bearing capacity of 778 N and 725 N.

(2)

The axial analysis of pile

To explore the bearing characteristics of the measured pile, the internal force of the self-balanced measure pile in the layered soils was analyzed.

Equation (35) illustrates the connection between the strain calculation and the axial force of test value.

$$N_{ij}={\epsilon }_{ij}EA$$

(35)

where N_ij is the axial force of the j-th unit pile under the i-th load; ${\epsilon }_{ij}$ is the strain value of the j-th unit pile under the i-th load. A is the test pile’s cross-sectional area, E is the pile’s elastic modulus.

Under the above-mentioned self-balanced loading conditions, the theoretical analysis of the loading values at each level of the load box was used to determine the axial force at each position of the corresponding pile. Taking 200 N, 400 N, 600 N, 725 N (lower pile bearing capacity), and 758 N (upper pile bearing capacity), respectively. The comparison between the measured value and the theoretical value of the pile axial force under different loads is shown in Figure 10. The self-balanced theoretical axial loading calculation in each of the pile body’s points is relatively coincident with the measured data, and the maximum error of the bottom position of the pile does not exceed 5%. The load transfer law is gradually transferred from the loaded end to the far end. For the upper pile, the axial force curve bulges to the left. The load is transferred from bottom to top, and the closer the loading point is, the larger the axial force is, and it decays to the top of the pile. It can be seen from the linear density that the magnitude of the axial force gradually attenuates with the sparseness of the curve. The upper pile has a sparser curve from the loading point of 0 m to 0.15 m, and the attenuation is 46.67% to 60.85%, indicating that this part of the soil layer is subjected to higher loads. In addition, the upper and lower pile axial force curves at the same location far from the loading position overlap, indicating that the effect of the axial force has nothing to do with the loading direction.

(3)

Distribution of side friction resistance

$$q_{si}=\frac{{\epsilon }_i-{\epsilon }_{i-1}}{\pi d{L}_i}EA$$

(36)

where q_si is the side lateral friction resistance of the i-th section; d is the pile diameter; L_i is the length of the i-th pile body. After determining the average side friction of the i-th section, this value is used to determine the side friction at the section’s midpoint. Each midpoint is connected in turn to obtain a smooth curve as the pile body lateral friction distribution.

Based on the axial force test data, the lateral friction resistance of the pile at each point is calculated by Equation (36) and compared with the theoretically calculated lateral friction resistance curves of the pile at different load levels, as demonstrated in Figure 11. It can be seen that the measured results under different horizontal loads are in good agreement with the theoretical results. The distribution of side friction of the upper pile is “large at the bottom and small at the top”, and side friction increases with load. Proximity to the loading position, the lateral frictional resistance starts to act gradually, first increasing and then decreasing. As the load increases, the frictional resistance of the soil below the pile gradually increases. The upper section pile curve is approximately hyperbolic, and the upper pile lateral frictional force has a maximum value at the bottom of the pile (0.025 m) and maintains a stable value. The increase of pile lateral frictional resistance at the bottom of the pile becomes gradually slower.

4.3.2. Equivalent Conversion Results

Figure 12 shows the Q-s curves of the self-balanced test piles calculated by the analytical theory conversion method and the simple conversion method compared with the results of the conventional static load test.

It can be seen that the analytical conversion method’s Q-s curve is relatively consistent with the traditional static test pile, and the Q-s for comparable conversion curve using the simplified conversion method is in poor agreement with the traditional static pile. The capacity for the bearing of the test pile is 1748 N, using the weighted average method with the conversion coefficient of 0.76, and 1619 N, using the method of conversion used in this work. It is known that the calculation accuracy of the exact conversion method is 9.3% higher than that of the simple conversion method, and the calculation results of the conversion method used in this study are closer to the traditional static load test results.

In summary, due to the complexity of the soil layers in multiple-layered foundations, the lateral friction around the pile and the pile’s displacement have different contributions to each layer of soil. If the simplified method was adopted, it lacks a particular theoretical foundation. The simplified conversion method can achieve certain results in the homogeneous soil layer, but the result of bearing capacity measured by the simplified method is too large in layers of soil. Therefore, the method of conversion used in this study is more accurate, and the internal force and bearing characteristics of the test pile can be reflected.

5. Engineering Applications

5.1. Engineering Situation

The theoretical method of this study was used in a self-equilibrium ultimate bearing capacity test project of cast-in-place piles in Yuda Wutong Garden, Guangxi, China. The diameter of the test pile is 1600 mm, the concrete strength grade of the pile body is C35, the pile’s characteristic bearing capacity value is 14,000 kN, and the self-balanced test pile soil layer distribution is shown in Figure 13.

5.2. Test Result

According to the estimated ultimate bearing capacity, the test is carried out at 10 levels of loading from small to large. When the load reaches the 10th level of 7500 kN, the load box reached a largely steady condition after 1320 min observations. The uplift of the upper plate was 8.83 mm, and the total settlement of the lower plate is 7.63 mm. The settlement logarithm of the time (s-lgt) curve and the load settlement (Q-s) curve are shown in Figure 14.

5.3. Internal Force Analysis

The test pile loading values are 1600 kN, 2300 kN, 3050 kN, 3800 kN, 4550 kN, 5300 kN, 6800 kN, 7550 kN, and the results of the theoretical analysis of the test pile by this study are shown in Figure 15. The distribution curve for side friction resistance of the pile body as demonstrated in Figure 16.

The three soil layers near the loading end of the upper pile, −17.620~−7.000 m underground, accounting for 60.27% of the upper pile length, have a sparse axial force curve, and the axial force curve decays by 48.67%~56.85% in this range, indicating that this part of the soil layer bears about 75% of the load transferred from the load box loading. When loaded to the ninth-level load of 7650 kN, compared with the first six levels, the relative displacement suddenly increases and becomes abruptly deformed into plastic softening, and it appears extremely unstable. The curve of the soil layer near the loading end is sparse, and the distance from the loading end is 2.35 m, increasing rapidly to a peak value of 146.3 kPa. In the 9.01 m soil layer above the peak point, the pile’s side friction rapidly attenuated from the peak by about 77.67%.

5.4. Equivalent Conversion Results

The results of the analytic conversion Q-s curve and comparison with the simplified conversion method are shown in Figure 17. It is clear that the Q-s curve belongs to the category of slowly changing types. In the case of smaller loads, the equivalent conversion displacement also increases taught slowly. As the load increases, the displacement growth amplitude gradually increases until the final phase of the ultimate load displacement changes suddenly and the load is terminated.

In this study, the single pile’s maximum bearing capacity count by the conversion method is 14,510.05 kN, and the single pile’s maximum bearing capacity count by the conversion factor of 0.85 is 15,686 kN. Due to the large differences in the properties and shear strength of various soil layers in actual projects, when using the simplified conversion method for super-long and large-diameter piles, the conversion coefficient method is adopted for the soil layer at the pile side according to the homogeneous soil, and the difference in the soil layer’s bearing properties is ignored. The bearing capacity of the pile can be guaranteed in the pile foundation design, but the analysis of the working performance and the test pile’s maximum bearing capacity is not accurate enough. The calculation accuracy of the analytical conversion method can be improved by about 12%. In addition, the difference in the nature of the soil layers on the side of the pile and the interaction between the upper and lower piles are considered. It has systematic theoretical analysis and can better reflect the deformation characteristics and internal force distribution of the self-balanced test pile. Therefore, the conversion method in this study has certain engineering guidance practices for self-balanced test piles.

6. Conclusions

In this study, an analytical conversion method suitable to self-balanced test piles in layered foundations is derived and applied to practical projects by combining the theoretical research of self-balanced test piles with the method of indoor model tests. The main conclusions are as follows.

(1) The analytical conversion method is verified through indoor model tests. Compared with the traditional static pressure method and simplified conversion method, the Q-s curve results of the analytical conversion method is in good agreement with the traditional static pressure method. At the same time, the bearing capacity obtained by the simplified method is too large, and the accuracy of the bearing capacity obtained by the analytical transformation method is improved about 9.2%, which confirms the accuracy and applicability of the analytical conversion method in multi-layered soil.

(2) The analytical conversion method was used for actual self-balanced pile test engineering. The Q-s curve and s-lgt curve of the test pile was plotted, and the distribution law of axial force and pile side friction at each point of the pile body under different levels of loading was analyzed. The ultimate bearing capacity of the test pile was calculated to be 14,510.05 kN, which is about 12% higher than that of the simplified method. Therefore, the conversion method in this paper has certain applicability in the field of self-balanced test pile engineering.

(3) The analytical conversion method in this work has a complete theoretical basis, and it can better reflect the load transfer characteristics of the upper and lower sections of the self-balanced test pile, as well as the distribution regulation of pile side friction and pile shaft axial force. It can be continuously applied in actual self-balanced test piles to determine the bearing capacity of pile foundations. The method can be extended to the special soil layer and variable section pile self-balanced tests, considering the properties of special soil and pile to find solutions. More actual engineering cases are needed to analyze and verify the bearing characteristics of self-balanced test piles as the next step to be studied in depth.