## 3 Validation  
In this section, the validities of finite-element models in terms of capturing the capacities of a single bucket foundation and tetrapod bucket foundations in undrained clay were demonstrated and discussed. When \(s=0\), the tetrapod bucket foundations become equivalent to a single bucket foundation. The uniaxial vertical, horizontal and moment capacity factors of a single bucket foundation are defined as \(N_{V,s}=V_{ult,s}/A_{s}s_{u0}\), \(N_{H,s}=H_{ult,s}/A_{s}s_{u0}\) and \(N_{M,s}=M_{ult,s}/A_{s}Ds_{u0}\), respectively. \(V_{ult,s}\), \(H_{ult,s}\) and \(M_{ult,s}\) are the uniaxial vertical, horizontal and moment capacities of the single bucket foundation, respectively. \(s_{u0}\) is taken as the soil undrained shear strength at the skirt tip level in this section. The values of \(N_{V,s}\), \(N_{H,s}\) and \(N_{M,s}\) with rough contact conditions and \(d=0\) obtained from the finite-element results in this study are validated, and they are basically coincident with the existing solutions [17, 42], as listed in Table 1.  
The differences of \(N_{V,s}\) between the results presented in this study and exact solutions reported by [PERSON] and [PERSON] [42] are within 2.4%. For \(N_{H,s}\), the differences between the results presented in this study and exact solutions \(N_{H,s}=1.0\) were within 7.5%. Ideally, for a single circular footing under horizontal loading, a sliding failure occurred in a very thin layer of soil along the soil-foundation interface. When its thickness is assumed 0, \(N_{H,s}\) is equal to 1.0. However, in finite-element models in this study, the thickness of this thin layer was meshed as 0.003 \(D\), which inhibited soil sliding and induced a slightly higher value for \(N_{H,s}\). \(N_{M,s}\) in this study was no more than 10.5% higher compared to that given by the solution based on plastic theory [17]. The slight overprediction of \(N_{M,s}\) was caused by representing a circular-scoop failure mechanism with hexahedron elements [21]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\kappa\)} & \multicolumn{3}{c}{\(N_{V,s}\)} & \multicolumn{2}{c}{\(N_{H,s}\)} & \multicolumn{2}{c}{\(N_{M,s}\)} \\ \cline{2-7}  & **This Study** & \multicolumn{2}{c}{**Exact Solutions**} & \multirow{2}{*}{**This Study**} & \multicolumn{2}{c}{**Exact Solutions**} & \multirow{2}{*}{**This Study**} & \multirow{2}{*}{**Solutions [17]**} \\  & & **Solutions [42]** & & & & (\(N_{H,s}=1.0\)) & \\ \hline
0 & 5.91 & 6.05 & 1.06 & 1.00 & 0.74 & 0.67 \\
1 & 6.81 & 6.95 & 1.06 & 1.00 & 0.86 & 0.78 \\
2 & 7.50 & 7.63 & 1.06 & 1.00 & 0.97 & 0.88 \\
6 & 9.65 & 9.67 & 1.07 & 1.00 & 1.33 & 1.24 \\
10 & 11.42 & 11.33 & 1.07 & 1.00 & 1.67 & 1.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparisons between finite-element results in this study and existing theoretical solutions for uniaxial capacities of single circular footing. Figure 3: Example of 3D finite-element models for tetrapod bucket foundations when (**a**) \(\varphi=0^{\circ}\) (**b**) \(\varphi\
eq 0^{\circ}\). For the single bucket foundation with \(d/D=0.2\), the finite-element results in this study were compared to lower-bound (LB) solutions reported by [PERSON] and [PERSON] [3] and UB solutions presented by [PERSON] et al. [5], as shown in Figure 4a. The comparison showed that the finite-element results are situated between the upper- and lower-bound solutions. [PERSON] [43] used the Dagang drilling rigs based on the tetrapod bucket foundation as prototypes and conducted a 1:40 model test in the mud pool. The main physical and mechanical indexes of the indoor soil model were as follows: effective unit weight \(\gamma^{\prime}=6\ \mathrm{kN/m^{3}}\), undrained shear strength \(s_{\mathrm{u}}=1.2+0.157\ z\ \mathrm{kPa}\) and [PERSON]'s modulus \(E_{\mathrm{u}}=1000\,s_{\mathrm{u}}\). The vertical and horizontal load tests were conducted, respectively.