## 6 Conclusions  
The simplified calculation method for three-dimensional asymmetric local scour based on the improved strain wedge model combined with the Boussinesq solution was used to reflect the influence of scouring on the lateral bearing performance of a pile foundation at an asymmetric local scour site. In the design of the pile foundation, the three-dimensional scour hole size parameters can be considered to optimize the pile foundation, and the decision-making process can be guided when studying the risk related to the scour and its potential impact on the lateral bearing performance. After the accuracy of the proposed method was verified, the influence of the three-dimensional asymmetric local scour size on the lateral bearing performance of a single pile was investigated. The key conclusions of the study are as follows:  
1. The approach offers a comprehensive means to unveil the ramifications of three-dimensional asymmetric local scour on the lateral bearing performance of a single pile;
2. Through an analysis of the scouring dimensions, it becomes evident that alterations in the scour depth and scour bottom area exert a substantial influence on the lateral displacement and bending moment of a single pile;
3. With the expansion of the scour dimensions, the lateral displacement of a single pile undergoes an increase within the impacted depth range, concurrently leading to a reduction in its lateral bearing performance;
4. With an increase in the upstream bottom width, \(S_{\text{wb1}}\), the lateral displacement of a single pile within the impacted depth range increases. At a burial depth of 6 m, the corresponding lateral displacement and bending moment of a single pile increase by 0.41%, 1.65%, and 2.06% and 0.15%, 0.38%, and 0.43%, respectively, when \(S_{\text{wb1}}\) increases by two times, four times, and eight times.  
At present, the theoretical analysis of a single pile's response under cyclic loading conditions with three-dimensional asymmetric local scour remains unresolved. Under the action of a cyclic load, the stiffness of the soil around the pile is weakened, and the pile foundation may produce cumulative deformation. Therefore, it is necessary to introduce the soil stiffness attenuation model under the cyclic load. Combined with the method in this paper, the lateral bearing performance of a single pile placed at an asymmetric scour hole site under a cyclic load will be studied. Laboratory experiments will be carried out to simulate single piles placed in a three-dimensional asymmetric scour hole site under cyclic loading. These experiments can help verify the method in this paper and provide parameters for a numerical simulation to ensure the applicability of the research results.  
Conceptualization, J.M.; methodology, J.M. and S.W.; software, S.W.; validation, S.W., C.W. and F.L.; formal analysis, J.M. and C.W.; investigation, J.M. and S.W.; resources, J.M.; data curation, S.W.; writing--original draft preparation, J.M. and S.W.; writing--review and editing, J.M., S.W. and C.W.; visualization, J.M. and D.L.; supervision, S.W.; project administration, J.M. and F.L.; funding acquisition, J.M., F.L. and D.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (grant number 11502072), the Key R&D and Promotion Special Project in Henan Province-Science and Technology Research Project (grant number 222102320462), and the Key R&D and Promotion Special Project in Henan Province-Science and Technology Research (joint fund) Project (grant number 232103810082).  
\begin{table}
\begin{tabular}{c} **Additional Stress** \\ **(kN/m2)** \\ \end{tabular} \\ \hline
0 & 0.114 & 4 & 0.089 \\
1 & 0.031 & 6 & 0.059 \\
2 & 0.022 & 9 & 0.042 \\
3 & 0.020 & 12 & 0.034 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Additional soil stress around a pile with varying \(S_{\text{TI}}\) and \(2b\) values at the same buried depth.