## 2 Asymmetric Scour Hole Modeling
The connection between the sandy mobilized friction angle, \(\varphi_{\text{mi}}\), and the lateral stress change, \((\Delta\sigma_{\text{h}})_{i}\), is established based on the stress level, \(SL_{i}\)[30]:  
\[SL_{i}=\frac{\left(\Delta\sigma_{\text{h}}\right)_{i}}{\Delta\sigma_{\text{hf }}}=\frac{\left[\tan^{2}(\frac{\pi}{4}+\frac{\varphi_{\text{mi}}}{2})-1\right] }{\left[\tan^{2}(\frac{\pi}{4}+\frac{\varphi}{2})-1\right]} \tag{7}\]  
In Equation (3), \(BC_{i}\) is the bottom width of the wedge at the \(i^{\text{th}}\) layer, which is  
\[BC_{i}=d+2 EF_{i}\tan\varphi_{\text{mi}} \tag{8}\]  
where \(EF_{i}\) is the vertical distance from the \(i^{\text{th}}\) layered pile to the failure surface in front of the wedge in Figure 2, \(\varphi_{\text{mi}}\) is the sandy mobilized friction angle at the \(i^{\text{th}}\) layer, and \(\tau_{i}\) is the lateral shear stress on the pile side at the \(i^{\text{th}}\) layer in sand [9]. Figure 1: Strain wedge model: (**a**) three-dimensional view, (**b**) top view, and (**c**) side view. The stain wedge and pile are equally divided into layers with a height of \(h_{i}\), as shown in Figure 2; \(\varepsilon_{i}\) is the strain of the \(i^{\text{th}}\) layer soil, \(y_{i-1}\) and \(y_{i}\) is the displacement of the single pile at the upper and lower interfaces of the \(i^{\text{th}}\) layer, respectively. Then, the connection between the pile deflection angle, \(\delta_{i}\), and the pile displacement \(y_{i}\) of the \(i^{\text{th}}\) layer is  
\[\tan\delta_{i}=\frac{y_{i-1}-y_{i}}{h} \tag{9}\]  
Corresponding to Figures 1c and 2, the soil strain, \(\varepsilon_{i}\), in the depth range of the wedge changes nonlinearly with the depth, which is  
\[k_{i}=\frac{p_{i}}{(y_{i-1}+y_{i})/2} \tag{2}\]0  
The connection between the pile deflection angle, \(\delta_{i}\), of the pile and the lateral soil strain, \(\varepsilon_{i}\), and the sandy mobilized friction angle, \(\varphi_{\text{mi}}\), is [31]  
\[k_{i}=\frac{p_{i}}{(y_{i-1}+y_{i})/2} \tag{2}\]1  
where \(\
u\) is the Poisson ratio of the soil. Figure 3 shows the simplified model of the three-dimensional asymmetric local scour hole, where \(O\) is the center point of the pile. The geometric parameters include the following: the scour hole width, \(2b\); the upstream scour depth, \(S_{\text{d1}}\); the length of the upstream top of the scour hole, \(S_{\text{wb1}}\); the width of the upstream bottom of the scour hole, \(S_{\text{t1}}\); the length of the upstream bottom of the scour hole, \(S_{\text{wt1}}\); the downstream scour depth, \(S_{\text{d2}}\); the length of the downstream top of the scour hole, \(S_{\text{wb2}}\); the width of the downstream bottom of the scour hole, \(S_{\text{t2}}\); the length of the downstream bottom of the scour hole, \(S_{\text{wb2}}\); the upstream scour hole slope of the pile foundation, \(\beta_{1}\); the downstream scour hole slope of the pile foundation, \(\beta_{2}\), and the slope of both sides of the pile foundation, \(\beta_{3}\). The simplified model is symmetrical about the \(xoz\) plane in Figure 3a; therefore, only half of the geometric dimensions of the model are marked in Figure 3b and used for calculation [31]. \[k_{i}=\frac{p_{i}}{(y_{i-1}+y_{i})/2} \tag{2}\]2 Nonlinear displacement in wedge. The self-weight of the soil above the _xoy_ plane is equivalent to the external load, and the vertical additional stress term, \(\Delta\sigma_{z}\), is generated at any depth, \(z\), on the _xoy_ plane. Based on the Boussinesq solution, the vertical additional stress of the soil around the pile is expressed as  
\[k_{i}=\frac{p_{i}}{(y_{i-1}+y_{i})/2} \tag{2}\]3  
In the equation,  
\[k_{i}=\frac{p_{i}}{(y_{i-1}+y_{i})/2} \tag{2}\]4  
Under three-dimensional asymmetric local scour conditions, the calculation of vertical effective stress in the soil around the pile is divided into two parts: first, the vertical effective stress generated via the soil below the bottom of the scour hole, and second, the additional stress generated via the soil above the bottom of the scour hole.