## 2 Summary of \(p\)-Multipliers and Group Effect Parameters  
In this section, a comprehensive summary of the experiment and simulation-based \(p\)-multipliers and group factors of group piles in sands are presented. The experiment-based results are a combination of both field and centrifuge model tests. Table 1 shows the calculated \(p\)-multipliers from experimental tests for free-head pile groups. The tests were conducted on steel pipe piles except for the tests of [PERSON] and [PERSON] [22], where concrete piles were used. In all of these tests, the range of \(S\)/\(D\) was from 3 to 5.65, whereas \(\varphi\) varied from 32\({}^{\circ}\) to 40\({}^{\circ}\). The tests were performed using 3 \(\times\) 3 piles groups, whereas the tests of [PERSON] and [PERSON] [22], [PERSON] [27], and [PERSON] et al. [24] were performed on 4 \(\times\) 4, 3 \(\times\) 5, and 3 \(\times\) 7 pile groups, respectively. The proposed \(p\)-multipliers range from 0.65 to 1.0 for the leading row, whereas the multipliers range from 0.4 to 0.85 for the first trailing row. Table 2 summarizes the \(p\)-multipliers for the fixed-head pile groups. The \(p\)-multipliers measured for the leading and first trailing rows were 0.8 and 0.4, respectively. It is demonstrated that the multipliers are smaller for the fixed-head piles. Table 3 lists the \(p\)-multipliers calculated from numerical analyses. Whereas \(S\)/\(D\) was mostly fixed to 3 in experimental tests, they were varied from 3 to 6 in the numerical simulations. The dependences on \(S\)/\(D\) and pile fixity condition can be observed, which were not evaluated in the field and centrifuge tests. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \multirow{2}{*}{**Reference**} & **Soll** & \multirow{2}{*}{\(\boldsymbol{\varphi}\) (\({}^{\boldsymbol{C}}\))} & **File** & \multirow{2}{*}{**Type**} & **File** & \multirow{2}{*}{**Dayout**} & **File Type** & **D (cm)** & & & \multicolumn{6}{c}{**Proposed \(p\)-Multipliers for Rows**} & & \multirow{2}{*}{**Group Effect**} \\  & **Type** & & & & & **Layout** & & & & & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** & **6 th** & **7 th** & **Parameter** \\ \hline [PERSON] et al. [17] & Sand & 38.5 & Full-Scale & 3 \(\times\) 3 & Steel Pipe & 27.3 & 3 & 0.8 & 0.4 & 0.3 & - & - & - & - & 0.5 \\ \hline [PERSON] and & & & & & & & & & & & & & & & & & \\ [PERSON] [5] & Sand & 38.5 & Full-Scale & 3 \(\times\) 3 & Steel Pipe & 27.3 & 3 & 0.8 & 0.4 & 0.3 & - & - & - & - & 0.5 \\ \hline \multirow{12}{*}{[PERSON] et al. [19]} & Sand & 30 & Centrifuge & 3 \(\times\) 3 & Steel Pipe & 43 & 5 & 1 & 0.85 & 0.7 & - & - & - & - & 0.85 \\  & Sand & 33 & Centrifuge & 3 \(\times\) 3 & Steel Pipe & 43 & 5 & 1 & 0.85 & 0.7 & - & - & - & - & 0.85 \\  & Sand & 30 & Centrifuge & 3 \(\times\) 3 & Steel Pipe & 43 & 3 & 0.65 & 0.45 & 0.35 & - & - & - & - & 0.48 \\  & Sand & 33 & Centrifuge & 3 \(\times\) 3 & Steel Pipe & 43 & 3 & 0.8 & 0.4 & 0.3 & - & - & - & - & 0.5 \\ \[PERSON] and & & & & & & & & & & & & & & & & \\ [PERSON] [22] & Sand & 32 & Full-Scale & 4 \(\times\) 4 & Square Concrete & 76 & 3 & 0.8 & 0.7 & 0.3 & 0.3 & - & - & - & 0.52 \\ \[PERSON] [27] & Sand & 40 & Full-Scale & 3 \(\times\) 5 & Steel pipe & 32.4 & 3.92 & 1 & 0.5 & 0.35 & 0.3 & 0.4 & - & - & 0.51 \\ \hline [PERSON] et al. [25] & Sand & 38 & Full-Scale & 3 \(\times\) 3 & Steel pipe & 32.4 & 3.3 & 0.8 & 0.4 & 0.4 & - & - & - & - & 0.53 \\ \hline [PERSON] [36] & Sand & 38 & Full-Scale & 3 \(\times\) 3 & Steel pipe & 32.4 & 5.65 & 1 & 0.7 & 0.65 & - & - & - & - & 0.78 \\ \hline \end{tabular}
\end{table} 
Table 1: Previous experimental studies conducted on free-head pile groups (modified after [PERSON] [14]).