## 2 Materials and Methods
### Theoretical Investigation  
#### 2.1.1 Uplift Capacity of Helical Pile  
The equations for the axial capacity of helical piles were first developed by [PERSON] and [PERSON] [10] for designing multi-helix anchors in cohesionless soil and the same by [PERSON] et al. [8] and [PERSON] et al. [9] for designing in cohesive soils. For multi-helix helical piles, two types of failure mechanisms were observed, namely the cylindrical failure model and the individual bearing model, as shown in Figure 2a,b, respectively, for uplift loading conditions.  
The transition between the two failure mechanisms depends on the critical spacing ratio (S/D), which is the ratio of the distance between two helices (S) to the helical plate diameter (\(D\)). The equation that [PERSON] et al. [9] and [PERSON] and [PERSON] [24] came up with to  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & & \\ \cline{3-8} \multirow{2}{*}{**Helical Anchor Type**} & \multirow{2}{*}{**Embedment Ratio**} & \multirow{2}{*}{**Spacing Ratio**} & \multirow{2}{*}{**No. of Plates**} & \multicolumn{2}{c}{**Numerical**} & \multirow{2}{*}{**Experimental Tests**} \\  & & & & & & & & \\ \cline{3-8}  & & & & & & & & \\ \cline{3-8}  & & & & & & & & \\ \hline SHP-1-12 & 12 & 0 & 1 & Yes & Yes & - & - \\ SHP-2-4 & 4 & 3.75 & 2 & Yes & - & - & - \\ SHP-2-6 & 6 & 3.75 & 2 & Yes & - & Yes & - \\ SHP-2-8.75 & 8.75 & 3.75 & 2 & Yes & Yes & Yes & Yes \\ SHP-34 & 4 & 1.875 & 3 & Yes & - & - & - \\ SHP-3-6 & 6 & 1.875 & 3 & Yes & - & Yes & - \\ SHP-3-8.75 & 8.75 & 1.875 & 3 & Yes & Yes & Yes & Yes \\ SHP-4-4 & 4 & 1.25 & 4 & Yes & - & - & - \\ SHP-4-6 & 6 & 1.25 & 4 & Yes & - & Yes & - \\ SHP-4-8.75 & 8.75 & 1.25 & 4 & Yes & Yes & Yes & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 1: Overview of testing program.  
Figure 2: Mode of failure mechanism under uplift loading: (**a**) Cylindrical shear failure, and (**b**) Individual bearing failure.  
figure out the uplift bearing capacity when the cylindrical failure mechanism is taken into account is as follows:  
\[Q_{u}=S_{f}(\pi DL_{c})c_{u}+A_{H}\big{(}c_{u}N_{u}+\gamma^{\prime}H\big{)}+\pi dH _{eff}\alpha c_{u} \tag{1}\]  
where \(Q_{u}\) = ultimate uplift/tensile capacity of helical pile; \(D\) = diameter of helical plate (m); \(d\) = diameter of helical pile shaft (m); \(L_{c}\) = spacing between top and bottom helical plates (m); \(c_{u}\) = undrained shear strength of soil (kN/m\({}^{2}\)); \(A_{H}\) = area of helical plate (m\({}^{2}\)); \(\gamma^{\prime}\) = effective unit weight of soil (kN/m\({}^{3}\)); H = embedment depth to top helix (m); \(H_{\mathit{eff}}\) = effective pile length above top helix (m) (\(H_{\mathit{eff}}\) = \(H-D\)); \(\alpha\) = adhesion factor; \(N_{u}\) = uplift bearing capacity factor = 1.2 \(\frac{\Pi}{D}\leq\) 9, [25,26]; and \(S_{f}\) = spacing ratio factor.  
The helical piles undergo individual bearing failure mechanisms when the helical plates are spaced with a spacing ratio of more than 1.5. Thus, the ultimate uplift capacity equation recommended by [PERSON] and [PERSON] [24] and [PERSON] [27] is expressed as  
\[Q_{u} =Q_{shaft}+\ \Sigma Q_{bearing} \tag{2}\]  
where \(Q_{shaft}=\pi\mathrm{d}L_{c}c_{u}\) and \(Q_{bearing}=A_{h}9c_{u}\).