## 5 Conclusions
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Surcharge**} & \multicolumn{8}{c}{**Maximum of Responses**} \\ \cline{2-9}  & \multicolumn{2}{c}{\(y_{\alpha,\max}\)} & \multicolumn{2}{c}{\(M_{\alpha,\max}\) (\(10^{-3}\))} & \multicolumn{2}{c}{\(p_{\alpha,\max}\)} & \multicolumn{2}{c}{\(p_{\beta,\max}\)} \\ \cline{2-9}  & **Free–Tip** & **Fixed–Tip** & **Free–Tip** & **Fixed–Tip** & **Free–Tip** & **Free–Tip** & **Fixed–Tip** \\ \hline \(p\) = 34.375 kPa & 0.004 & 0.003 & 0.519 & 0.055 & 0.014 & 0.012 & 0.007 & 0.007 \\ \(p\) = 84.375 kPa & 0.006 & 0.006 & 1.810 & 0.794 & 0.046 & 0.034 & 0.030 & 0.026 \\ \(p\) = 134.375 kPa & 0.031 & 0.012 & 4.470 & 1.470 & 0.113 & 0.069 & 0.064 & 0.043 \\ \(p\) = 184.375 kPa & 0.070 & 0.018 & 6.490 & 3.340 & 0.168 & 0.103 & 0.096 & 0.068 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Maximum of responses under different constraint conditions at the pile tip. **Author Contributions:** Methodology, D.Z. ; Investigation, C.W. ; Resources, D.Z. ; Data curation, Q.Z. and C.W. ; Writing-original draft, Q.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This research is part of work supported by grants from the National Natural Science Foundation of China (No. 52278329), the Hunan Provincial Natural Science Foundation of China (Nos. 2020)[J4619], and the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20200835). **Data Availability Statement:** Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request (the data in the graph and the code of calculation, etc.). **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**  
\(b\): pile width;  
\(c\): cohesion;  
\(C_{\rm u}\): coefficient of uniformity;  
\(C_{\rm c}\): coefficient of curvature;  
\(d_{50}\): median particle diameter;  
\(D_{\rm r}\): relative density;  
\(e\): void ratio;  
\(e_{\rm max}\): maximum void ratio;  
\(e_{\rm min}\): minimum void ratio;  
\(E_{\rm p}\): elasticity modulus of piles;  
\(E_{\rm p}I_{\rm p}\): flexural rigidity of piles;  
\(G_{\rm s}\): specific gravity;  
\(I_{\rm p}\): inertia moment of piles;  
\(L\): pile length;  
\(m\): proportional coefficient of subgrade reaction coefficient;  
\(M\): bending moment;  
\(M_{\rm z}\): nondimensional bending moment;  
\(n\): loading stage;  
\(p_{0}\): initial uniformly distributed load caused by the self-weight of the lever loading system;  
\(p_{1}\): uniformly distributed load caused by a balance weight of 60 kg;  
\(p\): interchange magnitudes;  
\(p_{\rm a}\): earth pressure;  
\(p_{\rm p}\): subgrade reaction;  
\(p_{\rm a,\kappa}\): nondimensional earth pressure;  
\(p_{\rm p,\kappa}\): nondimensional subgrade reaction;  
\(y\): pile deflection;  
\(y_{\rm a}\): nondimensional pile deflection;  
\(z\): depth below the ground surface;  
\(z_{\rm a}\): nondimensional depth;  
\(\alpha\): deformation coefficient of piles;  
\(\beta\): pile inclinations;  
\(\gamma\): unit weight;  
\(\gamma_{\rm dmax}\): maximum dry unit weight;  
\(\gamma_{\rm dmim}\): minimum dry unit weight;  
\(\epsilon_{\rm t}\): tension strain;  
\(\epsilon_{\rm c}\): compression strain;  
\(\
u\): poisson's ratio;  
\(\varphi\): internal friction angle;  
\(\omega\): moisture content. * [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON] Role of piles in mitigating the movement of pipes in soft grounds during embankment loading. _Mar. Georesour. Geotechnol._**2018**, _37_, 1019-1031. [CrossRef]
* [PERSON] and [PERSON] (2021) [PERSON]; [PERSON] [PERSON] Effect of rock-socketing on laterally loaded piles installed in the proximity of sloping ground. _Int. J. Geomech._**2021**, _21_, 04020252. [CrossRef]
* GB/T 50145--2007; Standard for Classification of Engineering Soils. Nanjing Hydraulic Research Institute: Nanjing, China, 2021. (In Chinese)
* GB/T 50123--2019; Standard for Geotechnical Testing Method. China Planning Press: Beijing, China, 2019.