## 1 Introduction  
Pile foundations are commonly used to withstand both vertical and lateral loads. The lateral response of a pile-soil system is an important design consideration for pile foundations. Well-known methods for prediction of lateral response of a single pile includes the elastic solution proposed by [PERSON] and [PERSON] [1], strain wedge method proposed by [PERSON] et al. [2], beam-on-nonlinear-Winkler-foundation (BNWF) model framework, and continuum analysis [3; 4; 5; 6; 7; 8]. The most common method to analyze the lateral response of piles is the BNWF approach. In this approach, the interaction of the pile-soil system is represented by the \(p\)-\(y\) curve, where \(p\) is the soil resistance and \(y\) and is the lateral displacement. Various functional forms of \(p\)-\(y\) curves are used for the piles embedded in sands such as [PERSON] et al. [9] and API [10]. American Petroleum Institute API [10] provides some simple guidelines to develop nonlinear \(p\)-\(y\) curves that are most often used in practice. However, a number of researchers [11; 12; 13; 14; 15] reported that the use of these generic curves may produce a high level of error in the prediction of lateral response of pile foundations. Despite their documented shortcomings, practitioners most often use the API curves owing to their ease of use. The BNWF model is also widely used to analyze the pile group response subjected to lateral loading. When piles act in a group, the soil resistance decreases due to \"shadowing effect\" and \"edge effect\", as reported by [PERSON] [16]. This reduction in resistance is accounted in the BNWF model by introducing a reduction factor, termed the \(p\)-multiplier, first proposed by [PERSON] et al. [17]. To account for the shadowing effect, a higher value of \(p\)-multiplier is typically applied to the leading row relative to the trailing rows. Various studies recommended that \(p\)-multipliers for squared vertical groups are a function of center-to-center spacing and soil type [17; 18; 19; 20; 21; 22]. Extensive studies have been performed to derive the \(p\)-multipliers from field and model tests [17; 19; 20; 22; 23; 24; 25; 26; 27]. [PERSON] et al. [28] also reported that it is quite acceptable to use an average \(p\)-multiplier for all the piles in the group, rather than applying \(p\)-multipliers for each row. This averaged \(p\)-multiplier is referred to as the group effect parameter. The group effect parameter is widely used in a dynamic analysis, where the direction of the loading changes, converting \"leading\" rows of pile immediately into \"trailing rows\" [28]. The \(p\)-multiplier is most often determined from an iterative process involving comparisons of the BNWF model result with a reference field load test output. After selection of the \(p\)-\(y\) curves, BNWF analyses are performed with a range of \(p\)-multipliers. The \(p\)-multiplier that produces the most favorable fit with the reference set of data is selected. Because experimental data are required, there is a limitation on the cases that can be considered. Full-scale tests have been mostly performed using 3 \(\times\) 3 free-head group pile with spacing to diameter ratio (_S/D_) of 3. The effect of number of piles, _S/D_, and soil shear strength cannot be evaluated. Additionally, it was reported that the \(p\)-multiplier is sensitive to the \(p\)-\(y\) curves [28]. Considering that the design \(p\)-\(y\) curves do not provide realistic representation of the pile-soil interaction, the \(p\)-multiplier derived from this procedure may not be reliable. Additionally, because only the load-displacement outputs are compared, the BNWF model may not provide an agreeable fit with the bending moment profile. The \(p\)-multiplier can also be directly calculated from the ratio of \(p\) calculated for group and single piles [17; 22; 29; 30]. For this direct extraction, full 3D numerical simulations need to be performed. Because the ratio changes with the depth, an averaged value calculated up to the depth of influence should be extracted.