## 2 Methodology
### Calculation of \(\Delta z_{k}^{1}\)  
Considering that the group effect has been proven to be elastic [17], it is reasonable to calculate the additional pile response by Poulos elastic analysis. According to Equation (4), \(\Delta z_{k}^{1}\), the additional elastic settlement of pile \(k\) due to the existence of other piles in the group, should be solved by Equation (11) in turn to predict the settlement of the pile group. \[\Delta z_{k}^{1}=z_{kge}^{1}-z_{k}^{1} \tag{11}\]  
where \(z_{kge}^{1}\) and \(z_{k}^{1}\) are the elastic head settlements of the group pile \(k\) and the isolated pile equivalent to pile \(k\), respectively, both calculated by Poulos elastic analysis. In the [PERSON] elastic analysis, the soil is assumed to be an isotropic, elastic semi-space, and pile-soil interaction is analyzed by [PERSON]'s solution of the displacement in the soil due to a vertical load applied below the soil surface. As depicted in Figure 2, an isolated  
Figure 1: Concept of the NFC Model. pile with length \(L\) is discretized into \(n\) uniform segments with each segment length of \(\delta\). The skin friction \(t^{i}\) of element \(i\) along the pile shaft is simplified to be evenly distributed on the element. The pile end bearing stress \(t_{b}\) is also considered to be evenly distributed on an enlarged pile tip with a diameter of \(d_{b}\). As a result of \(n\) skin friction stresses along \(n\) pile segments and an end bearing stress, the PSI function for an isolated pile can be written as Equation (12). \[\{z_{r}\}=\frac{d}{E_{s}}[I_{s}]\{t\} \tag{12}\]  
where \(z_{r}\) is the relative pile-soil displacement, including the midpoint displacements of \(n\) pile segments and a pile tip displacement; \(d\) is the pile diameter; \(E_{s}\) is the soil modulus; \([I_{s}]\) is the pile-soil interaction matrix; and \(t\) is the pile-soil interaction force, including \(n\) skin friction stresses \(t^{i}\) and a tip bearing stress \(t_{b}\). The element \(I_{sij}\) of such a matrix signifies the displacement of the midpoint of element \(i\) induced by a unit load of skin friction or the tip bearing stress and can be obtained by integration of [PERSON]'s solution [4]. When taking tunneling into consideration, as a result of pile-soil displacement compatibility, \(z_{r}\) equals the free-field soil settlement \([z_{bs}]\) minus pile settlement \([z_{bs}]\) for the group pile \(k\), and Equation (12) can be rewritten as Equation (13). \[\{z_{bs}\}=\frac{d}{E_{s}}[I_{s}]\{t\}+\{z_{bs}\} \tag{13}\]  
Given that the pile is applied with a vertical head load of \(P\), the relationship between the displacement and stress along the pile shaft in its differential form is given in Equation (14). By substituting Equation (13) into Equation (14), Equation (15) can be obtained to calculate \(\{t\}\), and in turn, \(z_{bs}^{1}\) of the isolated pile subjected to both tunneling and the vertical pile head load can be easily calculated. \[\{t\}=\frac{d}{4\delta^{2}}E_{p}A_{p}\big{[}I_{p}\big{]}\{z_{bs}\}+\{Y\} \tag{14}\]  
where \(E_{p}\) is the pile modulus, \(A_{p}\) is the pile sectional area, \([I_{p}]\) is the \((n+1)\times(n+1)\) pile action matrix, \(\{Y\}\) is a \((n+1)\) column vector, and \(\{Y\}=\{\frac{nP}{nP^{2}\delta},0,0,\ldots 0\}^{-1}\)([PERSON] and [PERSON], 1980). \[\Bigg{(}[I]-\frac{d^{2}E_{p}A_{p}}{4E_{s}\delta^{2}}\big{[}I_{p}\big{]}[I_{s} ]\Bigg{)}\{t\}=\frac{dE_{p}A_{p}}{4\delta^{2}}[I_{p}]\{z_{bs}\}+\{Y\} \tag{15}\]  
For a pile group consisting of \(m\) group piles, the group effect of group pile \(i\) on the target pile \(k\) can be considered by superposition of the displacements of segments on pile \(k\) induced by the skin friction and end bearing loads on pile \(i\). With tunneling further taken into consideration, the pile-soil interaction Equation (13) can be modified by adding the  
Figure 2: Concept of the elastic analysis method. contribution of \(\{t_{i}\}\) along the group pile \(i\), and the pile-soil interaction equation for group pile \(k\) can be given as Equation (16).