3.2.1. Traditional Calculation Method of Internal Force

The traditional method for calculating the bending moment of the cantilever anti-slide pile was mostly based on the Euler–Bernoulli beam theory, which assumed that the elastic assumption and plane section assumption were satisfied in the whole loading process of the cantilever anti-slide pile [24], and the bending moment *M*(*y*) can be obtained according to the following Equations (7)–(11).

$$
\sigma(\mathbf{x}) = \frac{\mathcal{M}}{I\_{\mathbf{x}}} d = E \varepsilon \tag{7}
$$

$$M(y) = \frac{E\Delta\varepsilon}{d}I\_{\text{x}}\tag{8}$$

$$d = h - \left(a\_s + a'\_s\right) \tag{9}$$

$$I\_x = \frac{bh^3}{12} \tag{10}$$

$$
\Delta \varepsilon = \varepsilon\_s - \varepsilon'\_s \tag{11}
$$

where *σ*(*x*) is the section stress of the pile; *Ix* is the inertia moment of pile section; *b* and *h* are the pile sectional width and sectional height, respectively; *d* is the distance between the tensile steel bars and structural steel bars (i.e., N1 and N2 reinforcement); *as* and *a s* are the thickness of concrete cover of reinforcement N1 and N2, respectively; *E* is the overall elastic modulus of the pile; *ε*<sup>1</sup> and *ε*<sup>2</sup> are the strains of reinforcements N1 and N2, respectively.

It can be found that the traditional method always regards the cantilever anti-slide pile as an elastic body in the calculation of the bending moment. However, in the indoor experiment of Section 3.1, the concrete or steel bar exhibited plastic deformation characteristics when the pile reached Stage II and Stage III, and the pile also presented plastic failure characteristics when destroyed. Therefore, the traditional method is not reasonable when the pile reaches Stage II and Stage III. It is imperative to analyze the stress and deformation characteristics of each stage of the pile separately and then establish a reasonable calculation method for the bending moment.

#### 3.2.2. Optimized Calculation Method of Internal Force

#### (1) Stage I

When the trapezoidal thrust load behind the cantilever anti-slide pile is small, the concrete and steel bars in the tension area bear the tensile force together, and both are in an elastic state. The concrete strain at the edge of the tensile area of the pile is less than the concrete's ultimate tensile strain, and there are no cracks on the pile. Based on the assumption of elastic body, the cross-sectional stress and strain state in the tensile area of the pile at this stage meets Equation (12), and Figure 15 shows the stress and strain distributions of the pile section.

$$\begin{array}{l} \mathfrak{e}\_{\mathfrak{s}} = \mathfrak{e}\_{\mathfrak{t}} \preceq\_{\mathfrak{t}u} \mathfrak{e}\_{tu} \\ \sigma\_{\mathfrak{s}} \le \mathfrak{e}\_{tu} E\_{\mathfrak{s}} \end{array} \tag{12}$$

where *εtu* is 0.0001; and *σ<sup>s</sup>* and *Es* are the stress and elastic modulus of the reinforcement N1, respectively.

**Figure 15.** Stress and strain diagrams of the pile normal section in stage I. (**a**) Pile section; (**b**) transformed section; (**c**) distribution of average strain; (**d**) distribution of normal stress.

The height of the compression area (*xc*) is determined using the plane section assumption, and it is given as follows:

$$\frac{\mathbf{x}\_{\varepsilon} - a'\_{s}}{\varepsilon'\_{s}} = \frac{h\_{0} - a'\_{s}}{\varepsilon\_{s} + \varepsilon'\_{s}} \tag{13}$$

$$\mathbf{x}\_{\varepsilon} = \frac{\varepsilon\_{s}^{\prime}}{\varepsilon\_{s} + \varepsilon\_{s}^{\prime}} h\_{0} + \frac{\varepsilon\_{s}}{\varepsilon\_{s} + \varepsilon\_{s}^{\prime}} a\_{s}^{\prime} \tag{14}$$

where *h*<sup>0</sup> is the effective height of the pile section, *h*<sup>0</sup> = *h–as*. Making *ε <sup>s</sup>*/*ε<sup>s</sup>* = *ζ* and simplifying Equation (14), then *xc* is:

$$\mathbf{x}\_c = \frac{\mathbb{Z}}{1+\mathbb{Z}}h\_0 + \frac{1}{1+\mathbb{Z}}a'\_s \tag{15}$$

In Section 3.2.1, the contribution of reinforcements to the inertia moment of the pile (*Ic*) section was ignored in the traditional calculation method for pile bending moment. In fact, as a composite material, the pile inertia moment (*Ic*) is larger than that of plain rectangular concrete beam. Therefore, based on the conversion section method, the reinforcements N1 and N2 with the cross-sectional areas of *As* and *A s* were converted into concrete with the areas of *nAs* and *nA s* at the same position respectively. Then, *Ic* is:

$$I\_{\varepsilon} = (n-1)A\_s(h\_0 - \mathbf{x}\_{\varepsilon})^2 + (n-1)A\_s' \left(\mathbf{x}\_{\varepsilon} - a\_s'\right)^2 + \frac{bh^3}{12} + bh(\mathbf{x}\_{\varepsilon} - h/2)^2 \tag{16}$$

where *n* = *Es*/*Ec*, and *Ec* is the elastic modulus of pile concrete.

At this time, the tensile stress of the concrete at the same level of the reinforcement N1 (*σt*) is:

$$
\sigma\_t = E\_c \varepsilon\_s = \sigma\_s \frac{E\_c}{E\_s} = \frac{M\_1(h\_0 - \chi\_c)}{I\_c} \tag{17}
$$

The bending moment of the test pile (*M*1) was further obtained as:

$$M\_1(h\_0 - \chi\_c) = \sigma\_s \frac{E\_c I\_c}{E\_s} \tag{18}$$

Similarly, the stress of the concrete at the same level of the reinforcement N2 (*σ s*) should meet Equation (19), namely

$$M\_1 \left(\mathbf{x}\_{\mathbf{c}} - a'\_{\ \mathbf{s}}\right) = \sigma'\_{\ \mathbf{s}} \frac{E\_{\mathbf{c}} I\_{\mathbf{c}}}{E\_{\mathbf{s}}} \tag{19}$$

Combining Equations (18) and (19), and noting *σ<sup>s</sup>* = *Esε<sup>s</sup>* and *σ <sup>s</sup>* = *Esε <sup>s</sup>*, *M*<sup>1</sup> in Stage I can be obtained as:

$$M\_1 = \frac{E\_\varepsilon l\_c}{d} \left(\varepsilon\_s + \varepsilon'\_s\right) \tag{20}$$

It can be seen that the structure of Equation (20) is similar to that of Equation (8), and the difference between the two is the inertia moment of the pile section. Equation (20) is more objective and reasonable to consider the increasing effect of the steel bars on the inertia moment of the pile section.

(2) Stage II

When the concrete stress in the pile tensile area exceeds its tensile strength, cracks will occur and develop. At this moment, the concrete in the tension area basically no longer bears the tensile stress, the steel bar stress increases rapidly, and the concrete stress in the compression area increases statistically, which finally leads to the redistribution of the pile section stress, as shown in Figure 16. At this stage, the concrete in the pile compression area starts to enter the plastic compression stage, and the stress–strain relationship of the concrete changes from linear to nonlinear.

**Figure 16.** Stress and strain diagrams of the anti-slide pile in normal section Stage II. (**a**) Distribution of average strain; (**b**) distribution of normal stress.

At this stage, the concrete in the pile compression area does not conform to the elastic body assumption, and its stress–strain constitutive relationship should be adopted according to Chinese specification (GB 50010-2019) [25], as shown in Figure 17 and Equation (21).

$$\begin{aligned} \sigma\_{\mathfrak{c}} &= \sigma\_0 \left[ 1 - \left( 1 - \mathfrak{c}\_{\mathfrak{c}} / \mathfrak{c}\_0 \right)^2 \right], \; 0 < \mathfrak{c}\_{\mathfrak{c}} \le \mathfrak{c}\_0 = 0.002\\ \sigma\_{\mathfrak{c}} &= \sigma\_0 = \text{const}, 0.002 \le \mathfrak{c}\_{\mathfrak{c}} \le \mathfrak{c}\_{\mathfrak{c}u} = 0.033 \end{aligned} \tag{21}$$

where *σ<sup>c</sup>* and *ε<sup>c</sup>* are the concrete stress and strain in the pile compression area, respectively; *εcu* is the concrete ultimate compressive strain; *σ*<sup>0</sup> is the concrete peak compressive stress, which is taken as the design value of the axial compressive strength.

**Figure 17.** Typical stress–strain constitutive relationships of reinforcement and concrete.

Since the reinforcement N1 is still in an elastic state at this stage, the simplified elastoplastic stress–strain expression can be used for its stress–strain relationship, as shown in Equation (22) and Figure 17.

$$\begin{aligned} \sigma\_s &= E\_s \varepsilon\_s, \ \varepsilon\_s \le \varepsilon\_y \\ \sigma\_s &= \sigma\_y, \ \varepsilon\_s \ge \varepsilon\_y \end{aligned} \tag{22}$$

where *ε<sup>y</sup>* and *σ<sup>y</sup>* are the yield strain and yield strength of steel bars, respectively.

The stress and strain state characteristics of the compressed concrete and tensile steel bars of the pile at this stage are shown in Equation (23). Combining Equations (12) and (23), it can be seen that the strain (*ε<sup>s</sup>* = *εtu*) or stress (*σ<sup>s</sup>* = *εtuEs*) of the tensile steel bars can be used as the boundary sign between Stage I and Stage II.

$$\begin{aligned} \varepsilon\_{\varepsilon} \le \varepsilon\_{0}, \ \varepsilon\_{s} \ge \varepsilon\_{t\mu} \\ \sigma\_{\varepsilon} \le \sigma\_{0}, \ \sigma\_{s} \ge \varepsilon\_{t\mu} E\_{s} \end{aligned} \tag{23}$$

The calculation method of the bending moment of the pile in Stage II is introduced as follows. Firstly, based on the plane section assumption, the concrete strain at *xc*(*εxc*) and the concrete strain with a distance *y* from the neutral axis (*εcy*) are determined as follows:

$$
\varepsilon\_{\rm xc} = \varepsilon'\_s \frac{\mathfrak{X}\_c}{\mathfrak{X}\_c - a'\_s} \tag{24}
$$

$$
\varepsilon\_{\mathcal{E}\mathcal{Y}} = \varepsilon'\_{\,s} \frac{\mathcal{Y}}{\varkappa\_{\,c} - a'\_{\,s}} \tag{25}
$$

Then, the solution expression for the resultant force of the compressive stress of the pile concrete (*Fc*) is established by integrating the section stress of compressive concrete, as shown in Equation (26). Substituting Equations (21) and (25) into Equation (26), *Fc* is obtained as:

$$F\_{\mathfrak{c}} = \int\_{0}^{\chi\_{\mathfrak{c}}} \sigma\_{\mathfrak{c}} b \mathbf{d}y \tag{26}$$

$$F\_c = b\sigma\_0 x\_c \left[ \frac{\varepsilon'\_s}{\varepsilon\_0 (1 - \frac{a'\_s}{x\_c})} - \frac{\varepsilon'\_s}{3\varepsilon\_0^2 \left(1 - \frac{a'\_s}{x\_c}\right)^2} \right] \tag{27}$$

Since *a <sup>s</sup>* << *xc* in stage II, <sup>1</sup> <sup>−</sup> *as xc* <sup>≈</sup> 1 in Equation (27). Making *<sup>η</sup>* <sup>=</sup> *<sup>ε</sup> s <sup>ε</sup>*<sup>0</sup> and simplifying Equation (27) to get *Fc*:

$$F\_c = \sigma\_0 b \chi\_c \left(\eta - \frac{1}{3}\eta^2\right) \tag{28}$$

Secondly, the bending moment of *Fc* to the center of tensile steel bars is obtained as follows:

$$\mathcal{M}\_{\mathfrak{c}} = \int\_{0}^{\mathbf{x}\_{\mathfrak{c}}} \sigma\_{\mathfrak{c}} b (h\_{0} - \mathbf{x}\_{\mathfrak{c}} + \mathfrak{y}) \mathbf{d}y = F\_{\mathfrak{c}} \left( h\_{0} - \frac{1 - 0.25 \eta}{3 - \eta} \mathbf{x}\_{\mathfrak{c}} \right) \tag{29}$$

Thirdly, the bending moment of the compressive steel bars to the center of the tension steel bars is obtained as follows:

$$M\_{\mathfrak{s}} = E\_{\mathfrak{s}} \varepsilon\_{\mathfrak{s}}^{\prime} A\_{\mathfrak{s}}^{\prime} \left( h\_0 - a\_{\mathfrak{s}}^{\prime} \right) \tag{30}$$

Finally, the bending moment of the section of the test pile in Stage II (*M*2) is obtained by combining Equations (29) and (30), as shown in Equation (31).

$$M\_2 = M\_\varepsilon + M\_s = \sigma\_0 b x\_\varepsilon \left(\eta - \frac{1}{3}\eta^2\right) \left(h\_0 - \frac{1 - 0.25\eta}{3 - \eta} x\_\varepsilon\right) + E\_s \varepsilon\_s' A\_s' \left(h\_0 - a\_s'\right) \tag{31}$$

(3) Stage III

When the thrust load behind the cantilever anti-slide pile is large enough, the tensile steel bars gradually yield, and the strains of the steel bars increase rapidly. The concrete in the tension area will no longer bear the tension, and the cracks on the pile gradually rise and expand. Part of the concrete in the compression area enters the stable plastic stage and tends to fail, as shown in Figure 18. When the strain reaches the ultimate compressive strain, the compressive concrete is crushed, and the test pile can no longer bear the external load, which indicates that the test pile has reached the failure state. The stress–strain state characteristics of the steel bars and concrete in this stage are shown in Equation (32), and it can be seen that the strain (*ε<sup>s</sup>* = *εy*) or stress (*σ<sup>s</sup>* = *σy*) of the tensile steel bars can be used as the boundary sign between Stage II and Stage III.

$$\begin{aligned} \varepsilon\_{\mathfrak{c}\mathfrak{c}} & \ge \varepsilon\_{0\prime} \cdot \varepsilon\_{\mathfrak{s}} \ge \varepsilon\_{\mathfrak{y}} \\ \sigma\_{\mathfrak{c}} & \ge \sigma\_{0\prime} \cdot \sigma\_{\mathfrak{s}} \ge \sigma\_{\mathfrak{y}} \end{aligned} \tag{32}$$

In the process of calculating the resultant force of compressive stress of the test pile concrete (*Fc*), the stress distribution curve of compressive concrete should be divided into two sections. The first and second sections are respectively the curve and the straight line (Figure 18), where the straight line section indicates that the stress in the concrete has reached *σ*0. Based on the plane section assumption and referring to Equation (25), the height of the boundary point of the concrete stress distribution curve is *ε*0*xc*/*εxc*. Based on Equations (21), (24) and (25), *σc*, *εxc* and *ε<sup>c</sup>* can be further obtained respectively. Substituting *σc*, *εxc* and *ε<sup>c</sup>* into Equation (26) and integrating in sections, *Fc* at this stage is:

$$F\_c = \int\_0^{x\_c} \sigma\_c b \mathbf{d}y = \int\_0^{\sigma\_0 \frac{\pi}{12}} \sigma\_c b \mathbf{d}y + \int\_{\sigma\_0 \frac{\pi}{12}}^{x\_c} \sigma\_c b \mathbf{d}y = \sigma\_0 b x\_c \left(1 - \frac{1}{3\eta}\right) \tag{33}$$

$$\left[\sum\_{i \in \mathcal{I}} \sum\_{j \in \mathcal{J}} \overline{\sigma\_i}^c\right]\_{i,j} \tag{34}$$

$$M \subseteq \left[\sum\_{i \in \mathcal{I}} \overline{\sigma\_i}^c \sum\_{j \in \mathcal{J}} \overline{\sigma\_i}^c \overline{\sigma\_j}^c, \overline{\sigma\_i}^c\right]\_{i,j} \tag{45}$$

$$\mathbf{(a)} \tag{5}$$

**Figure 18.** Stress and strain diagrams of the anti-slide pile in normal section stage III. (**a**) Distribution of average strain; (**b**) distribution of normal stress.

The bending moment of *Fc* to the center of the tensile steel bars in this stage is obtained as follows:

$$M\_{\varepsilon} = \int\_{0}^{\mathbf{x}\_{\varepsilon}} \sigma\_{\varepsilon} b (\hbar\_{0} - \mathbf{x}\_{\varepsilon} + \mathfrak{y}) \mathrm{d}\mathfrak{y} = F\_{\varepsilon} \left( \hbar\_{0} - \frac{6\eta^{2} - 4\eta + 1}{12\eta^{2} - 4\eta} \mathbf{x}\_{\varepsilon} \right) \tag{34}$$

The calculation of *Ms* in this stage is the same as Equation (30), so the bending moment of the test pile in Stage III is:

$$M\_3 = M\_\varepsilon + M\_s = \sigma\_0 b x\_\varepsilon \left( 1 - \frac{1}{3\eta} \right) \left( h\_0 - \frac{6\eta^2 - 4\eta + 1}{12\eta^2 - 4\eta} x\_\varepsilon \right) + E\_s \varepsilon\_s' A\_s' \left( h\_0 - a\_s' \right) \tag{35}$$

In summary, the specific calculation steps of internal force or bending moment in each stage during the whole process of stress and deformation of the cantilever anti-slide pile are represented in Figure 19.

#### *3.3. Bending Moment Distribution of Test Pile*

According to the optimized calculation method for the bending moment of the cantilever anti-slide pile proposed in this paper, the bending moment distribution of the test pile in the loading process of the trapezoidal load was obtained, as shown in Figure 20. It is obvious that the pile bending moment increased gradually with the trapezoidal load, increasing first and then decreased gradually along the pile length. The maximum bending moment was located at the sliding surface, and the farther away from the sliding surface, the faster the attenuation. At the same time, the evolution process of the pile bending moment mainly included three stages. When the resultant force of the trapezoidal thrust load (*FT*) was less than the pile cracking load, the maximum bending moment gradually increased, and increased to 0.86 kN·m. When *FT* reached 10.2 kN, subsequently, the test pile entered the working stage with cracks, and the pile bending moment increased rapidly. When *FT* increased to 36.6 kN, the maximum bending moment was about 4.16 kN·m. When the steel bars yielded, the external trapezoidal load behind the test pile remained basically unchanged, the test pile tended to be damaged, and the maximum bending moment increased slowly to 4.49 kN·m when the test pile was destroyed.

**Figure 19.** Calculation flow chart of bending moment of anti-slide pile at each stage.

**Figure 20.** Bending moment distributions along test pile.

Equation (8) in Section 3.2.1 was used to calculate the pile bending moment at the sliding surface, and further compared with the optimized calculation method established in Section 3.2.2, and the comparison results are represented in Figure 21. It can be found that the two are basically the same when the test pile worked in the elastic stage (Stage I). The maximum bending moment at the sliding surface obtained by using the optimized calculation method established in Section 3.2.2 (Equation (20)) is slightly larger. The reason was that the increasing effect of tensile steel bars and compressive steel bars on the inertia moment of pile section was considered in Equation (20), resulting in the calculation result of the inertia moment of pile section larger than that of Equation (8). When the test pile worked in the stage of crack emerging and developing (Stage II), the gap between the two grew gradually with the trapezoidal load. The reason was that the anti-slide pile in stage II was regarded as an elastoplastic body in the optimized calculation method built in this paper (Equation (31)), and the elastoplastic characteristics of compressive concrete were considered. The pile compressive concrete was treated as a linear elastomer during the whole loading process in the traditional method (Equation (8)), resulting in a larger calculation value of bending moment. When the values of *FT* were 16.2 kN and 36.6 kN, respectively, the bending moments calculated by the traditional method (Equation

(8)) were 2.87 times and 5.24 times the bending moment obtained by the optimized calculation method established in Section 3.2.2 (Equation (31)). Moreover, the theoretical ultimate flexural bearing capacity of the test pile(*Mu*) was 3.93 kN·m according to the flexural strength formula of reinforced concrete beam, which is not much different from the maximum bending moment obtained by the optimized calculation method established in this paper (Equation (35)). The bending moment at the sliding surface of the test pile calculated by the traditional method was as high as 35.37 kN·m, which is obviously inconsistent with the fact.

**Figure 21.** Comparison of the maximum bending moments of the test pile based on the traditional method and the optimized calculation method.

The above results show that when the cantilever anti-slide piles work in the elastic stage or there are no cracks in the pile during the service life, the errors of the bending moment calculated by the traditional method or the optimized calculation method established in Section 3.2.2 are small, and both methods can be used. However, when the cantilever anti-slide pile cracks in service, the traditional method will produce large errors and mislead the evaluation of the service status of the cantilever anti-slide pile in service and the structural design, resulting in a waste of economy and resources.

## **4. Numerical Analysis and Discussion**
