## 2 Literature Review
### p-Multiplier  
[PERSON] and [PERSON] [12] stated that the behavior of piles in groups that received lateral loads were different from that of a single pile. The lateral load borne by each pile in the group varies according to its position. This effect is commonly referred to as the pile group effect. To account for the loss of soil resistance in the pile design, [PERSON] et al. [13] further proposed a p-multiplier approach. As shown in Figure 2, the p-multiplier or \(\mathrm{p_{m}}\) is used to modify a single pile lateral soil spring to obtain pile springs in a group. Usually, the p-multiplier is constant in the same row.  
The p-multiplier can be calculated from the p-y curve by dividing the single pile soil resistance by the pile group. The p-y curve is the soil reaction to the pile (\(p\)), which is related to the pile deflection (\(y\)) through a non-linear load-deflection curve. The p-y formulation based on API [7] can be calculated as follows:  
\[\frac{p}{p_{u}D}=\tanh a\Big{(}\frac{y}{D}\Big{)}^{b} \tag{7}\]  
where \(p\) = lateral soil resistance per unit length of the pile, \(y\) = pile deflection, and \(p_{u}\) = lateral bearing capacity.  
New development of the p-y method can be used to describe better soil-pile behavior based on API [7], as cited in [PERSON] et al. [6]:  
\[\begin{array}{l}\mathrm{a}=8.62\Big{(}\frac{L_{p}}{D}\Big{)}^{-0.18}\\ \mathrm{b}=0.5\\ \mathrm{L_{p}}=\mathrm{pile\ depth}\end{array} \tag{8}\]  
Figure 2: Comparison of the p-y curves of single and group piles.  
Based on full-scale testing, [PERSON] et al. in FEMA p-751 [14] recommended a p-multiplier, \(\mathrm{p_{m}}\), where D is the diameter or width of the pile and s is the center-to-center distance between the piles in the direction of loading.  
First (leading) row piles:  
\[\mathrm{P_{m}}=0.26\ln\Bigl{(}\frac{\mathrm{s}}{\mathrm{D}}\Bigr{)}+0.5{\leq}1.0 \tag{9}\]  
Second row piles:  
\[\mathrm{P_{m}}=0.52\ln\Bigl{(}\frac{\mathrm{s}}{\mathrm{D}}\Bigr{)}{\leq}1.0 \tag{10}\]  
Third or higher row piles:  
\[\mathrm{P_{m}}=0.60\ln\Bigl{(}\frac{\mathrm{s}}{\mathrm{D}}\Bigr{)}-0.25{\leq}1.0 \tag{11}\]  
[PERSON] and [PERSON] [15] formulated the prediction equation by adjusting the curve to the data, representing efficiency with pile distance. There is no distinction for soil type, pile diameter, or penetration.  
Leading piles:  
\[\mathrm{P_{m}}=0.7\Bigl{(}\frac{\mathrm{s}}{\mathrm{D}}\Bigr{)}^{0.26} \tag{12}\]  
Trailing piles:  
\[\mathrm{P_{m}}=0.38\Bigl{(}\frac{\mathrm{s}}{\mathrm{D}}\Bigr{)}^{0.38} \tag{13}\]  
where \(\mathrm{p_{m}}\) = p-multiplier, s = distance between piles, and D = diameter of the pile.  
The p-multiplier (\(\mathrm{p_{m}}\)) is taken from the center-to-center distance of the pile in the loading direction group, expressed in multiples of the pile diameter (D). The calculation of group effects based on the American Association of State Highway and Transportation Officials [16] is shown in Table 1.  
The p-multiplier depends on the pile spacing and row number according to the direction of loading. In order to assign the \(\mathrm{p_{m}}\) value to the other pile spacing values, interpolation between values must be carried out [16].