## 2 Asymmetric Scour Hole Modeling  
Differing from the traditional strain wedge model that assumes a linear variation in lateral displacement of a pile with depth, an improved nonlinear strain wedge model was employed [28]. As shown in Figure 1, under the action of a lateral load, the passive compressive soil behind the pile foundation can be regarded as a three-dimensional wedge. The governing differential equation of a laterally loaded single pile is as follows:  
\[EI\frac{\partial^{4}y}{\partial^{4}z}+k_{i}y_{i}=0 \tag{1}\]  
where \(EI\) is the flexural stiffness of a single pile, \(y_{i}\) is the lateral displacement of the \(i^{\text{th}}\) layer of a single pile, and \(z\) is the depth below the ground surface. The subgrade reaction modulus of the strain wedge at the depth \(z\) of any layer is  
\[k_{i}=\frac{p_{i}}{(y_{i-1}+y_{i})/2} \tag{2}\]  
where \(p_{i}\) represents the soil resistance provided via the unit pile length at the \(i^{\text{th}}\) layer. Drawn from Figure 1a, \(h\) is the maximum depth of the wedge, corresponding to the first zero point of the pile displacement curve. The subgrade reaction modulus in the area below the wedge is calculated with \(k_{i}=\eta_{\text{h}}z_{i}\), where \(\eta_{\text{h}}\) is the constant of subgrade reaction [21]. Under the action of a lateral load, the three-dimensional wedge with a height of \(h\), a base angle of \(\beta_{\text{m}}\), and a fan angle of \(\varphi_{\text{m}}\) is formed in front of the pile foundation in Figure 1a. The wedge changes dynamically with the varying load, for which the base angle \(\beta_{\text{m}}\) and fan angle \(\varphi_{\text{m}}\) follow. \(\beta_{\text{m}}=45^{\circ}+\varphi_{\text{m}}/2\). As shown in Figure 1b, the top view of the \(i^{\text{th}}\) layer wedge satisfies the following equilibrium relationship:  
\[p_{i}=(\Delta\sigma_{\text{h}})_{i}BC_{i}S_{1}+2\tau_{i}dS_{2} \tag{3}\]  
where \(S_{1}\) and \(S_{2}\) are pile shape factors for a circular pile, versus \(S_{1}=0.75\) and \(S_{2}=0.5\) for a square pile, \(S_{1}=S_{2}=1.0\), \(d\) represents the diameter of a circular pile or the width of the square pile shaft, and \((\Delta\sigma_{\text{h}})_{i}\) is the lateral stress increment at the \(i^{\text{th}}\) layer of the wedge. Using Duncan-Chang model, the stress-strain state of the soil is given as  
\[\left(\Delta\sigma_{\text{h}}\right)_{i}=\left(\sigma_{1}-\sigma_{3}\right)_{ i}=\frac{\varepsilon_{i}}{\frac{1}{\text{E}_{\text{ini}}}+\frac{\varepsilon_{i}}{ \left(\sigma_{1}-\sigma_{3}\right)_{\text{ult}}}} \tag{4}\]  
where \(\varepsilon_{i}\) represents the lateral strain at the \(i^{\text{th}}\) layer in the wedge, corresponding to the axial strain in triaxial tests, and \(E_{ini}\) represents the initial tangent modulus during the hyperbolic stage [29]. The asymptotic value of the deviatoric stress \(\left(\sigma_{1}-\sigma_{3}\right)_{\text{ult}}\) is calculated with  
\[\left(\sigma_{1}-\sigma_{3}\right)_{\text{ult}}=\frac{\left(\sigma_{1}-\sigma _{3}\right)_{\text{f}}}{R_{\text{f}}} \tag{5}\]  
where \(R_{\text{f}}\) is the failure ratio, and \(\left(\sigma_{1}-\sigma_{3}\right)_{\text{f}}\) is the deviatoric stress at failure, calculated as  
\[\left(\sigma_{1}-\sigma_{3}\right)_{\text{f}}=\Delta\sigma_{\text{hf}}=\sigma _{v0}\Big{[}\tan^{2}(\frac{\pi}{4}+\frac{\varphi}{2})-1\Big{]} \tag{6}\]  
where \(\sigma_{v0}\) is the vertical effective stress.