## 3 Results of the Parametric Study
### Number of Piles in a Pile Group
The observations generally remain the same under different load levels, except  
Figure 13: Mobilisation degrees of average shaft shear stress distribution with depth at 30% of the ultimate load. (**a**) Single pile; (**b**) 2 \(\times\) 2 pile group; (**c**) 3 \(\times\) 3 pile group; (**d**) 5 \(\times\) 5 pile group; (**e**) 9 \(\times\) 9 pile group; (**f**) 15 \(\times\) 15 pile group. that the overall importance of the base capacities becomes significant at higher load levels. Figure 15 shows the settlement ratio of the average settlement of the pile base, \(S_{b}\), and the settlement of the pile cap, \(S_{t}\), with the size of the pile group at a pile slenderness ratio of \(L\)/\(D\) = 20. The settlement ratio increases rapidly up to the \(5\times 5\) group, after which the rate of increase becomes very gradual for the larger groups. Two distinct zones of behaviour can be delineated as indicated by the change in the settlement ratio. The results shown in Figure 15 reinforce the previously described pile group interaction behaviours. Figure 16a-f summarises the pile group behaviours in terms of the normalised pile stiffness \(\gamma p_{avg}/G_{so}Dw\), where \(p_{avg}\) is the average load applied on the pile group, \(D\) is the diameter of the pile, \(G_{so}\) is the shear modulus of the soil at the ground surface, and \(w\) is the pile cap settlement at the corresponding applied load on the cap) with different pile slenderness ratios (\(L\)/\(D\)), for various pile group sizes (single pile, \(2\times 2\), \(3\times 3\), \(5\times 5\), \(9\times 9\),  
Figure 14: Degrees of mobilisation of pile base load with different pile slenderness ratios. (**a**) 30% ultimate load; (**b**) 50% ultimate load; (**c**) 80% ultimate load. Figure 15: Displacement ratios between the base and head settlement with sizes of pile groups. and 15 \(\times\) 15 pile groups) and under different load levels (30%, 50% and 80% of the ultimate load of the pile group). In a general sense, the normalised pile stiffness reflects the overall efficiency of the pile group system, with a higher normalised pile stiffness indicating that the system can support higher loads under a given pile cap settlement or that it deforms to a lesser degree under a given load. The results in Figure 16a-f indicate that the normalised pile stiffness generally decreases with the size of the pile group and loading level. For the single pile and the small pile group cases (up to a size of 5 \(\times\) 5), the normalised pile stiffness initially increases with _L/D_ to a critical pile slenderness ratio, after which the normalised pile stiffness decreases with a further increase in _L/D_. However, for larger sizes of pile groups (9 \(\times\) 9 and 15 \(\times\) 15), such a critical slenderness ratio cannot be observed and the normalised stiffness continuously increases with L/D but at a substantially reduced rate. The critical pile slenderness ratios are 20, 25, 30, and 45 for the single pile, 2 \(\times\) 2, 3 \(\times\) 3, and 5 \(\times\) 5 pile groups, respectively. For the single pile, as mentioned previously (see Figure 13a), the mobilisations of the shaft shear stress profiles change from being almost uniformly distributed for the short pile with _L/D_ = 10 to highly concentrated at the upper portion (with _L/D_ from 0 to approximately 0.6) for the long single pile with _L/D_ = 50. Since most of the pile settlements are developed at the concentrated shear zones, this will cause the resulting normalised pile stiffness to change according to the _L/D_ ratio. Similar behaviours are observed for the smaller pile group cases. When the number of piles in the group increases, the interaction effects among the pile shafts become more substantial. The overall efficiencies of the large pile groups are thus much smaller than those of the small pile groups. For large pile groups with short pile lengths, the axial loads induced on the individual piles are highly nonlinear with a much higher magnitude of loads developed at the corner piles than at the centre piles.