# Lateral Loading of a Rock-Socketed Pile Using the Strain Wedge Model Based on Hoek-Brown Criterion
## 3 Basic Concept of Strain Wedge  
The interaction between the pile and the rock is the key to studying the lateral bearing capacity of the rock-socketed piles. The pile foundation embedded in the rock mass undergoes flexural deformation under the action of the lateral load, which causes the rock around the pile to be displaced and produces a reaction force distributed along the pile. The subgrade reaction affects the pile to prevent its deformation from increasing. Based on the Winkler beam on elastic foundation (BEF), the rock/soil around the pile is regarded as a series of discrete, nonlinear springs, and the differential equation for the pile can be obtained as reference [19]:  
\[EI\frac{d^{4}y}{dx^{4}}+p(x,y)=0 \tag{26}\]  
where \(p(x,y)\) is the rock resistance in the direction of the pile displacement and is a function of the depth of rock and the lateral displacement of the pile. When the pile lateral displacement is small, it can be assumed that the relationship between rock resistance and the pile lateral displacement is linear elastic:  
\[p(x,y)=E_{s}(x)y \tag{27}\]  
where \(E_{s}(x)\) is the subgrade reaction modulus, which is a function of depth of rock. Therefore, Equation (26) can be rewritten as the following expression:  
\[EI\frac{d^{4}y}{dx^{4}}+E_{s}(x)y=0 \tag{28}\]  
The strain wedge model (SWM) is used to analyze the response of the pile under lateral load based on the BEF. It is necessary to consider the wedge in front of the pile, the stress-strain relationship of the rock and the pile-rock. [PERSON] et al. [20] first proposed the strain wedge model in homogeneous rock. Under static lateral load, the soil in front of the pile is under passive compression. [PERSON] et al. [21] and [PERSON] et al. [20] simplified it into a three-dimensional wedge, named strain wedge model. As shown in Figure 3, the main control parameters of the passive wedge shape are wedge fan angle \(\varphi_{m}\); base angles \(\Theta_{m}\), and \(\beta_{m}\); wedge depth \(h\). Figure 3: Basic strain wedge in uniform soil (Modified from [PERSON] et al. [21]). The configuration of the wedge at any instant of load and, therefore, mobilized friction angle, \(\varphi_{m}\), and wedge depth, \(h\), is given by the following equation:  
\[\Theta_{m}=45-\frac{\varphi_{m}}{2}\text{ or }\beta_{m}=45+\frac{\varphi_{m}}{2} \tag{29}\]  
The width, \(\overline{BC}\), of the wedge face at any depth is:  
\[\overline{BC}=D+(h-x)2\tan\beta_{m}\tan\varphi_{m} \tag{30}\]  
where \(D\) is the width of the pile cross-section. The main stress parameters: wedge-shaped body lateral stress change \(\Delta\sigma_{h}\) and pile side shear stress \(\tau\). It is assumed that the pile in the rock control depth close to the pile head is simplified to be linear, and the linear deflection angle is \(\delta\), thus as to ensure that the vertical and lateral strains of the rock in the strain wedge are evenly distributed. In the process of loading and pile deflection, the strain changes uniformly with the shape and depth of the strain wedge. Due to the pile installation effects, the lateral rock pressure coefficient \(K_{0}=1.0\). The major principal stress change (\(\Delta\sigma_{h}\)) in the wedge is in the direction of pile movement, and it is equivalent to the deviatoric stress change in the triaxial test. The vertical stress change (\(\Delta\sigma_{v0}\)) equals zero, corresponding to the standard triaxial compression test where deviatoric stress if increased while confining pressure remains constant. As the lateral load on the pile top increases, the shape of the strain wedge in front of the pile and the stress-strain conditions of the rock in the strain wedge change accordingly. At the same time, \(\varphi_{m}\) is the mobilized friction angle of the rock, the friction angle of the rock under a certain stress condition. When the rock is broken, it reaches the maximum value, which is the friction angle \(\varphi\) of the rock.