# 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
### Strain Wedge Model Solution Method and Flow Chart  
Because the obtained subgrade reaction modulus is too complicated to solve by analytical method, the difference method is used to solve Equation (28) in this paper.  
Use Taylor series to expand the expression \(y=f(x)\) of pile horizontal displacement function forward and backward:  
\[f(x+h)=f(x)+hf^{\prime}(x)+\frac{h^{2}}{2!}f^{\prime\prime}(x)+\frac{h^{3}}{3! }f^{\prime\prime\prime}(x)+\frac{h^{4}}{4!}f^{(4)}(x)+O\!\left(h^{5}\right) \tag{57}\]  
\[f(x-h)=f(x)-hf^{\prime}(x)+\frac{h^{2}}{2!}f^{\prime\prime}(x)-\frac{h^{3}}{3! }f^{\prime\prime\prime}(x)+\frac{h^{4}}{4!}f^{(4)}(x)+O\!\left(h^{5}\right) \tag{58}\]  
\[f(x+2h)=f(x)+2 hf^{\prime}(x)+\frac{(2h)^{2}}{2!}f^{\prime\prime}(x)+\frac{(2 h)^{3}}{3!}f^{\prime\prime}(x)+\frac{(2h)^{4}}{4!}f^{(4)}(x)+O\!\left((2h)^{5}\right) \tag{59}\]  
\[f(x-2h)=f(x)-2 hf^{\prime}(x)+\frac{(2h)^{2}}{2!}f^{\prime\prime}(x)-\frac{(2 h)^{3}}{3!}f^{\prime\prime\prime}(x)+\frac{(2h)^{4}}{4!}f^{(4)}(x)+O\!\left((2h)^{5}\right) \tag{60}\]  
Combining Equations (57) and (58) can be obtained:  
\[f^{\prime}(x)=\frac{f(x+h)-f(x-h)}{2h}+O\!\left(h^{3}\right) \tag{61}\]  
Omit the higher-order infinitesimals to obtain:  
\[f^{\prime}(x)=\frac{f(x+h)-f(x-h)}{2h} \tag{62}\]  
In the same way, combing Equations (57)-(60) can be obtained:  
\[2 hf^{\prime}(x)=-f(x-h)+f(x+h) \tag{63}\]\[h^{2}f^{n}(x)=f(x-h)-2f(x)+f(x+h) \tag{64}\]  
\[2h^{3}f^{m}(x)=-f(x-2h)+2f(x-h)-2f(x+h)+f(x+2h) \tag{65}\]  
\[h^{4}f^{(4)}(x)=f(x-2h)-4f(x-h)+6f(x)-4f(x+h)+f(x+2h) \tag{66}\]  
Divide the total length \(L\) of the pile into \(N\) sections, each section having a length of \(h=L/N\). In order to use the finite difference method to solve the problem, two virtual points were added above the pile top and below the pile tip, numbered 1, 2, and \((N+4)\), \((N+5)\), respectively. There was a total of \((N+5)\) nodes along the pile length. The number of pile top is 3, and the number of pile tip was \((N+3)\).  
Substitute Equation (66) into Equation (28)  
\[EI\frac{y_{i-2}-4y_{i-1}+6y_{i}-4y_{i+1}+y_{i+2}}{h^{4}}+E_{s}(x)y_{i}=0 \tag{67}\]  
where \(y_{i-2},\ y_{i-1},\ y_{i},\ y_{i+1},\ y_{i+2}\) correspond to \(f(x-2h),\ f(x-h),\ f(x),\ f(x+h)\), \(f(x+2h)\) respectively.  
The finite difference form of Equation (28) can be obtained:  
\[y_{i-2}-4y_{i-1}+\left(6+\frac{h^{4}E_{s}(x)}{EI}\right)y_{i}-4y_{i+1}+y_{i+2}=0 \tag{68}\]  
Equation (68) has \((N+1)\) equations in total.  
For long piles, the pile tip bending moment and shear are zero. From the reference [19]:  
\[\frac{dy}{dx}=\theta \tag{69}\]  
\[EI\frac{d^{2}y}{dx^{2}}=-M \tag{70}\]  
\[EI\frac{d^{3}y}{dx^{4}}=-V \tag{71}\]  
When the pile tip bending moment and shear force are zero, the following condition can be obtained:  
\[EI\frac{y_{N+2}-2y_{N+3}+y_{N+4}}{h^{2}}=0 \tag{72}\]  
\[EI\frac{-y_{N+1}+2y_{N+2}-2y_{N+4}+y_{N+5}}{2h^{3}}=0 \tag{73}\]  
Pile head conditions:  
When the bending moment \(M_{0}\) and lateral force \(P_{0}\) of the pile head are known:  
\[EI\frac{y_{2}-2y_{3}+y_{4}}{h^{2}}=M_{0} \tag{74}\]  
\[EI\frac{-y_{1}+2y_{2}-2y_{4}+y_{5}}{2h^{3}}=P_{0} \tag{75}\]  
When the bending moment \(y_{0}\) and rotation angle \(\theta_{0}\) of the pile head are known:  
\[y_{3}=y_{0} \tag{76}\]  
\[\frac{-y_{2}+y_{4}}{2h}=-\theta_{0} \tag{77}\]  
There are total \((N+5)\) equations include the differential Equation (68) pile tip conditions Equations (72) and (73) and pile head conditions Equations (74) and (75) or (76) and (77):  
\[K_{(N+5)\times(N+5)}Y_{(N+5)}=P_{(N+5)} \tag{78}\]where \(K_{(N+5)\times(N+5)}\) is the coefficient matrix of the equations; \(Y_{(N+5)}\) is the lateral displacement vector of the pile including the virtual point; \(P_{(N+5)}\) is the vector formed on the right side of the equations.  
It can be solved by iterative method. The solution process is shown in Figure 11.