# Lateral Loading of a Rock-Socketed Pile Using the Strain Wedge Model Based on Hoek-Brown Criterion
## 4 Analysis the Lateral Bearing Capacity of Rock-Socketed Piles Based on Strain Wedge Model  
The Relationship between the Lateral Displacement \(y\) of the Pile and the Lateral Strain \(\varepsilon\) of Rock  
The existing strain wedge model [20,21] iteratively calculates the maximum depth of the wedge based on the global stability and simplifies the lateral displacement of the pile. It is assumed that the displacement of the pile changes linearly within the depth of the wedge, and the deflection angle is defined as \(\delta\). The linear displacement assumption simplifies the calculation, but it will cause some problems. [PERSON] et al. [22] believed that at the maximum depth \(h\) of the wedge, due to the assumption of linear displacement, the actual deflection angle was much smaller than the assumed deflection angle \(\delta\), which makes the calculated subgrade reaction modulus far too large, and singularities appear. To correct this problem, [PERSON] et al. [22] restricted the upper limit of \(k\) directly. It is noted that with the increase of the lateral load on the pile top, the shallow rock is first destroyed and gradually develops to the deep layer, and the strain is unevenly distributed throughout the wedge. For this reason, this article has made the following amendments to the development of pile deformation:  
Assuming that the lateral deformation of the pile at the depth of each layer no longer changes linearly along the depth but is independent of each other, the strain wedge in front of the pile is shown in Figure 5.  
Suppose the lateral deformation of the \(i\)-th layer of pile is \(y_{i}\), the strain of the \(i\)-th layer of soil is \(\varepsilon_{i}\). Moreover, the relationship between the deflection angle \(\delta_{i}\) of the \(i\)-th layer of pile and the displacement of the pile is:  
\[\tan\delta_{i}=\frac{y_{i-1}-y_{i+1}}{2h} \tag{31}\]  
Correspondingly, the strain of each layer of soil in the strain wedge is also independent of each other and is related to the lateral displacement of the pile. The strain of the \(i\)-th layer of soil is:  
\[\varepsilon_{i}=\frac{y_{i}}{\overline{EF_{i}}} \tag{32}\]  
where \(\overline{EF_{i}}\) is the strain wedge length of the \(i\)-th layer of the rock. Figure 5: Nonlinear displacement of strain wedge model. The Relationship between the Lateral Strain \(\varepsilon\) and the Deflection Angle of the Pile \(\delta_{i}\)  
It can be demonstrated from a Mohr's circle of rock strain, as shown in Figure 6, that shear strain \(\gamma\), is defined as:  
\[\frac{\gamma}{2}=\frac{1}{2}(\varepsilon-\varepsilon_{v})\sin 2\Theta_{m}=\frac{1 }{2}\varepsilon(1+v)\sin 2\Theta_{m} \tag{33}\]  
From the relationship between the deflection angle of the pile and the shear strain of the rock, the relationship between the lateral strain of the rock and the mobilized friction angle of the rock can be obtained as:  
\[\delta_{i}=\frac{\gamma_{i}}{2}=\frac{1}{2}\varepsilon_{i}(1+v)\sin 2\Theta_{mi}= \frac{1}{2}\varepsilon_{i}(1+v)\cos\varphi_{mi} \tag{34}\]  
Relationship between Lateral Stress Change \(\Delta\sigma_{h}\) and Lateral Strain \(\varepsilon\)  
[PERSON] et al. [21] proposed the concept of stress level, combined with the triaxial test of sand and soft clay, and established the relationship between stress-strain and the mobilized friction angle. [PERSON] used a power function stress-strain relationship, which reflects the nonlinear change of stress level (\(SL\)) with axial strain (\(\varepsilon\)) under constant confining pressure. In order to be suitable for the entire soil strain stage, the form of stage change is used. However, for intact rock, when the displacement is small, the stress-strain relationship of the rock is mainly linear elastic change.