## 4 Results and Discussions  
### Effect of \(\varphi\), S/D, Pile Head Fixity and Number of Piles on \(p\)-Multipliers  
Friction angle \(\varphi\) is observed to have a primary influence on \(p\)-multipliers. Figure 4 plots the variation of \(p\)-multipliers for 5 \(\times\) 5 free- and fixed-head pile groups with \(\varphi\) and _S/D_. The \(p\)-multiplier is shown to decrease with an increase in \(\varphi\). It was reported in [PERSON] et al. [2] that increasing \(\varphi\) causes a wedge-shaped influence zone in front of each pile to increase, which in turn produces larger a shear zone and ultimately higher level of interaction between piles. Calculated \(p\)-multipliers for free-head pile groups range from a minimum value of 0.36 for \(\varphi\) = 40\({}^{\circ}\) and _S/D_ = 3 to a maximum value of 1.0 for \(\varphi\) = 30\({}^{\circ}\) and _S/D_ = 6. The \(p\)-multipliers are positively correlated to _S/D_ because the shadowing and edge effects reduce with an increase in _S/D_. Figure 4: Numerically calculated \(p\)-multipliers for 5 \(\times\) 5 **(a)** free-head and **(b)** fixed-head pile groups. Figure 3: \(V_{s}\) vs. depth profiles used in the FE model. The \(p\)-multipliers are highly dependent on the pile head condition. The fixed-head pile groups yield lower \(p\)-multipliers compared with those for the free-head pile groups, because a higher level of group interaction is produced. For \(\varphi\) = 30\({}^{\circ}\), the \(p\)-multipliers for 3 \(\times\) 3 fixed-head pile groups are 0.75, 0.42, and 0.3 for the leading, first, and second trailing rows, respectively. On the contrary, for the free-head pile group, the \(p\)-multipliers increase to 0.81, 0.55, and 0.51 for leading, first, and second trailing rows, respectively. A pronounced difference is found in the \(p\)-multipliers of the leading and first trailing rows for the fixed-head condition, especially at \(S\)/\(D\) = 5 and 6, as compared with the free-head condition. For the widely used spacings of \(S\)/\(D\) = 3 and 4, the \(p\)-multipliers converge at the third row. For larger spacings, the \(p\)-multipliers are shown to continuously decrease for third and fourth trailing rows. However, for conservative design, it can be assumed to converge after the second trailing row. The number of piles is shown to have negligible influence on the calculated \(p\)-multipliers. Therefore, the \(p\)-multipliers for 3 \(\times\) 3 can also be applied to 4 \(\times\) 4 and 5 \(\times\) 5 pile groups. It is demonstrated in detail that the \(p\)-multipliers are sensitive on \(\varphi\), pile head fixity, and \(S\)/\(D\). In order to capture the effects of these parameters, the following functional form is proposed to calculate \(p\)-multipliers:  
\[p-\text{multiplier}\ =a\left(\frac{S}{D}\right)^{b} \tag{3}\]  
where \(a\) and \(b\) are curve fitting coefficients, summarized in Table 5. These coefficients are dependent on the fixity condition and \(\varphi\). It should also be noted that only \(p\)-multipliers up to second trailing rows are presented because of marginal variation in the calculated values for the following rows. Figures 5 and 6 show the calculated \(p\)-multipliers plotted against \(S\)/\(D\), along with the proposed functional form. The \(p\)-multipliers obtained using the equation developed matches well with the calculated results. Therefore, this equation can be used to predict the \(p\)-multipliers for both free- and fixed-head pile groups subjected to lateral load. Figure 5: Regression between calculated \(p\)-multipliers vs. \(S/D\) for various rows for free-head condition: (a) \(\varphi=30^{\circ}\), (b) \(\varphi=35^{\circ}\), and (c) \(\varphi=40^{\circ}\). Figure 7: Comparison between calculated \(p\)-multipliers and previous experimental work for free-head pile groups in sand with 3 \(\times\) 3 pile configuration: (**a**) S/D = 3, 3.3 and (**b**) S/D = 5, 5.65.