## 4 Theoretical Verification  
In the example of a certain cast-in-place reinforced concrete pile [32], Table 1 provides the physical and mechanical parameters of a single pile and the surrounding soil. The displacement curve of the pile obtained using the modified strain wedge method (SWM) is compared with the results from references [22; 32] to validate the accuracy of the SWM method. It is evident that the displacement curve obtained using the SWM method exhibits a similar variation trend to the results in references [22; 32], and the displacement values are highly consistent in Figure 5. \begin{table}
\begin{tabular}{l c} \hline \hline \multicolumn{1}{c}{**Physical Meaning**} & \multicolumn{1}{c}{**Numerical Value**} \\ \hline Lateral load at the top of the pile/kN & 200 \\ Length, \(L\)/m & 20 \\ Diameter, \(d\)/m & 0.5 \\ Flexural stiffness, \(E1\)/kN-m\({}^{2}\) & \(1.17\times 10^{6}\) \\ Cohesive force, \(c\)/kPa & 0 \\ Relative density, \(D_{\mathrm{r}}\)/\% & 65 \\ Poisson ratio of the pile, \(\
u\) & 0.2 \\ Poisson ratio of soil, \(\
u_{s}\) & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of a single pile and the surrounding soil. Figure 4: Flow chart of the simplified model. To verify the correctness of the SWM method under the condition of three-dimensional asymmetric local scour, the calculation results of the proposed method are compared with those of the simplified method [7] and the equivalent-strain wedge method [28] using the parameters given in Tables 2 and 3. Figure 6 demonstrates the excellent agreement between the groundline displacement obtained through the calculation method proposed in this paper and the results from the field test [7]. The figure unmistakably demonstrates that, with an increase in the scouring size, there is a remarkable escalation in the groundline displacement. When factoring in the scour effect, it becomes apparent that the calculation results in this paper closely align with those results in the references [7,28] when the groundline displacement of the pile is maintained within the 20 mm range. However, when the groundline displacement exceeds 20 mm, the calculated lateral bearing performance appears to be lower. The primary reason behind this discrepancy could be attributed to the continuous compression and densification of the passive soil in front of the pile during the actual loading process. \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Meaning** & **Numerical Value** \\ \hline Type of pile foundation & Steel pipe pile \\ Length, \(L/\)m & 21.3 \\ Diameter, \(d/\)m & 0.61 \\ Wall thickness, \(t/\)m & 0.0095 \\ Flexural stiffness, \(EI/\)MN\(\cdot\)m\({}^{2}\) & 163.32 \\ Foundation type & Saturated medium-density fine sand \\ Effective gravity, \(\gamma/\)kN/m\({}^{3}\) & 10.4 \\ Internal friction angle, \(\varphi/^{\circ}\) & 39 \\ Relative density, \(D_{\mathrm{r}}/\%\) & 60 \\ Poisson ratio of the pile, \(\
u\) & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical parameters of single pile and soil. Figure 5: Lateral displacement curves of single pile, [PERSON] et al. [22]; [PERSON] et al. [32]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Meaning** & **Numerical Value** \\ \hline Width of upstream bottom of pile foundation, \(S_{\mathrm{t1}}/\)m & 1 \\ Width of scour hole, \(2b/\)m & 7.46 \\ Slope of upstream scour hole of pile foundation, \(\beta_{1}/^{\circ}\) & 39 \\ Slope of downstream scour hole of pile foundation, \(\beta_{2}/^{\circ}\) & 39 \\ Slope on both sides of pile foundation, \(\beta_{3}/^{\circ}\) & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Three-dimensional asymmetric local scour hole parameters. Consequently, the soil stiffness increases, leading to an enhanced bearing performance on the side of the pile. However, this paper does not take into account this particular effect. Figure 3b shows a simplified model of three-dimensional asymmetric local scour around the pile.