## 3 Results and Discussion
### Theoretical Calculation of the Ultimate Bearing Capacity  
The theoretical calculations were verified by comparison with the simulation results in this study to provide the corresponding calculation formulae for the design method of HSCM piles. As variations in the strength of the soil-cement affect the failure mode of HSCM piles, the formulae for calculating the uplift capacity under different conditions  
Figure 9: Normalized curves of load and displacement.  
should be considered. The uplift bearing capacity of the HSCM pile was mainly generated by the pile and soil interaction when \(C_{ref}/s_{u}\ \in\ [20,80]\), similar to the SDCM pile, as shown in Figure 10a. The failure mode of the HSCM pile gradually changed from pile and soil interface failure to steel pipe and soil-cement interface failure when \(C_{ref}/s_{u}\ \in\ [5,20)\). The calculation of the uplift bearing capacity was essentially the problem of the interaction between the helical pile, the soil-cement column, and the soil around the pile, as shown in Figure 10b.  
The uplift bearing capacity of the HSCM pile was mainly composed of \(f_{s}\) and the self-weight of the pile \(W_{pile}\) when \(C_{ref}/s_{u}\ \in\ [20,80]\). The uplift bearing capacity of the HSCM pile consisted of the end-bearing force \(Q_{bearing}\) of the helix plate, the skin resistance \(Q_{shaft}\) of the steel pipe, and the skin resistance \(f_{s}\) of the local soil-cement column when \(C_{ref}/s_{u}\ \in\ [5,20)\). The calculation of the uplift bearing capacity for different failure modes is shown in Equations (1)-(4). The adhesion coefficient \(\alpha\) of the interface could be taken as 1.05 when calculating the skin resistance between the soil-cement and the clay, but the adhesion coefficient between each interface in different failure modes should also be considered. Combined with the segmentation function of the HSCM-1-1 normalization curve, the adhesion coefficient could be extrapolated as shown in Equation (5). When \(C_{ref}/s_{u}\ \in\ [5,20)\), a correction value related to the soil-cement strength should be added to the calculation model in Figure 10b, as shown in Equation (6), as the failure of the contact surface between the steel pipe and the soil-cement was gradual.  
\[Q_{u}=\left\{\begin{aligned} & f_{s}+W_{pile},\ \frac{C_{ref}}{s_{u}}\in[20,80]\\ & Q_{bearing}+Q_{shaft}+f_{s}+\delta,\ \frac{C_{ref}}{s_{u}}\in[5,20) \end{aligned}\right. \tag{1}\]  
\[f_{s}=\pi\alpha D_{sc}L_{sc}s_{u} \tag{2}\]  
Figure 10: Diagram of the theoretical calculation model.  
\[Q_{bwing}=N_{u}A_{H}s_{u} \tag{3}\] \[Q_{shaft}=\pi\alpha dH_{eff}s_{u}\] (4) \[\alpha=\left\{\begin{aligned} & 1.05,\ \frac{C_{ref}}{s_{u}}\in[20,80]\\ & 0.036\times\left(\frac{C_{ref}}{s_{u}}\right)+0.03,\ \frac{C_{ref}}{s_{u}}\in[5,20)\end{aligned}\right.\] (5) \[\delta=13.68\times\left(\frac{C_{ref}}{s_{u}}\right)-65,\ \frac{C_{ref}}{s_{u}}\in[5,20) \tag{6}\]  
where \(Q_{u}\) is the calculated value of the uplift ultimate bearing capacity, kN; \(\alpha\) is the adhesion coefficient; \(L_{sc}\) is the soil-cement column length, m; \(D_{sc}\) is the soil-cement column diameter, m; \(N_{u}\) is the uplift bearing capacity coefficient, taken as 9 in this study; \(D_{H}\) is the helix plate diameter, m; \(H_{eff}\) is the effective length, m; \(\delta\) is the correction value related to the soil-cement strength.  
The comparison of the theoretical bearing capacity of the HSCM pile obtained by the formula and the simulation results is shown in Figure 11. The deviations between the calculated values of the uplift capacity of HSCM piles and the simulation results were in the range of 2% to 4%, providing a basis for the design method of HSCM piles.