## 5 Capacities of Tetrapod Bucket Foundations
### Moment Capacities  
#### 5.3.1 Effects of the Foundation Separation Distance Ratio on Moment Capacities  
If the uniaxial moment capacity of tetrapod bucket foundations is denoted as \(M_{ult}\), the moment capacity factors can be defined as \(N_{M}=M_{ult}/ADs_{u0}\). Take \(\varphi=0\) as an example: the moment capacities of tetrapod bucket foundations with various foundation separation distance ratios (_s/D_) at \(\kappa=10\) are shown in Figure 15. The results indicate that \(N_{M}\) and the group effect factor of the tetrapod bucket foundations (\(E_{M}=N_{M}/N_{M,s}\)) both increase remarkably with the increase in _s/D_, because of the increase in the moment arm. \(E_{M}\) is greater than 1.0 and its enhancement is up to 27-fold when _s/D_ = 3.5 and _d/D_ = 0.25, manifesting that the moment capacity of the tetrapod bucket foundation greatly enhances at large _s/D_, compared to that of the single bucket foundation.  
Figure 14: Comparisons of horizontal capacity factors for tetrapod bucket foundations between finite-element results and fitted curves when (**a**) \(\kappa=0\) and (**b**) \(\kappa=10\).  
Take \(\varphi=0\) as an example: the vector fields of the displacement of the single bucket foundation and tetrapod bucket foundations under ultimate moment loading at \(\kappa=10\) when \(d/D=0.25\) and \(1.0\) are presented in Figure 16. When \(s/D=0.75\), the tetrapod bucket foundations and soils among the buckets can be regarded as a single foundation and rotate around a point under ultimate moment loading. When \(s/D\) increases to \(3.5\), the failure mode of the tetrapod bucket foundations evolves into a push-pull failure mode, which means that some buckets are pushed downwards while some buckets are pulled upwards under ultimate moment loading as shown in Figure 16.  
The moment capacity of tetrapod bucket foundations in the push-pull failure mode (\(M_{max,pp}\)) can be calculated by the vertical tensile and compressive resistances of individual buckets (\(V_{ult,s}\)) as shown in Figure 17 and Equations (9) and (10).  
\[\begin{array}{l}M_{\max,pp}=2[\sin(45^{\circ}-\varphi)+\cos(45^{\circ}- \varphi)]\,V_{ult,s}s\\ =2\sqrt{2}\sin(90^{\circ}-\varphi)N_{V,s}A_{s}s_{u0}\,\,\,(0^{\circ}\leq\varphi <45^{\circ})\end{array} \tag{9}\]  
Figure 16: The vector fields of the displacement of the single bucket foundation and tetrapod bucket foundations under ultimate moment loading at \(\kappa=10\) when (**a**) \(d/D=0.25\) and (**b**) \(d/D=1.0\).  
Figure 15: Moment capacities of tetrapod bucket foundations at \(\kappa=10\): (**a**) moment capacity factor, (**b**) moment group effect factor and (**c**) normalized moment group effect factor.  
\[\begin{split} M_{\max,pp}&=2(V_{ult,s}s+M_{ult,s})\\ &=2(N_{V,s}s+N_{M,s}D)A_{s}s_{u0}\ \ (\varphi=45^{\circ})\end{split} \tag{10}\]  
In order to produce simplified formulas to evaluate the moment capacity of tetrapod bucket foundations, the normalized moment group effect factor of tetrapod bucket foundations (\(F_{M}=M_{ult}/M_{max,pp}\)) is defined, as shown in Figure 15c. The interactions of soils surrounding each bucket gradually vanish and the failure mode of tetrapod bucket foundations gradually evolves into a push-pull failure mode with the increase in s/D, so \(M_{ult}\) gradually increases to \(M_{max,pp}\) and \(F_{M}\) finally converges into 1.0.