## 2 Methodology  
The modified Poulos method proposed by [PERSON] et al. (2012) is a combination of the NFC model and the Poulos elastic analysis, with both methods modified to use their features applicable to active pile groups considering both nonlinear pile-soil interaction and pile group effect.  
For a group of \(m\) piles only subjected to the pile head load, [PERSON] et al. (2012) rationally predicted the settlement of a pile group using Equation (1), known as the modified Poulos method. It is assumed that the settlement \(z_{kG}^{1}\) of group pile \(k\) consists of settlement \(z_{k}^{1}\) of an equivalent isolated pile considering the nonlinear pile-soil interaction and the sum of elastic additional settlements \(\bar{z}_{ke}^{-1}\sum\limits_{i=1,j\
eq k}^{m}\alpha_{ki}P_{i}\) induced by the other piles in the pile group. The superscripts hereafter represent the node number along the pile shaft, and 1 means the first node at the pile head.  
\[z_{kG}^{1}=z_{k}^{1}+\bar{z}_{ke}^{-1}\sum\limits_{i=1,j\
eq k}^{m}\alpha_{ ki}P_{i} \tag{1}\]where \(z_{k\epsilon}^{-1}\) is the elastic pile settlement for an isolated pile applied by a unit pile head load and \(\alpha_{ki}\) is the interaction factor, obtained by the additional response of pile \(k\) due to the existence of group pile \(i\) divided by the elastic settlement of an equivalent isolated pile \(k\).  
The NFC method used to calculate \(z_{k}^{1}\) assumes that the soil be discretized into a series of soil springs along the pile shaft and the pile interacts with the soil by discontinuous nonlinear springs. The Poulos elastic analysis considers the soil as continuous semi-space, and can thus be used to predict both isolated single piles and pile groups, and is adopted to calculate \(z_{k\epsilon}^{-1}\) and \(\alpha_{ki}\) in Equation (1). It can be indicated that the elastic additional settlement is proportional to the pile head load \(P_{i}\) on pile \(i\) in this equation.  
If \(P_{i}\) is known for each group pile, the group pile settlement \(z_{k\epsilon}^{1}\) can be easily calculated by Equation (1). For a pile group connected by a cap, and the distributional pile head load \(P_{i}\) is unknown, \(z_{k\epsilon}^{1}\) can be obtained by incorporating Equations (2) and (3).  
\[P=\sum_{i=1}^{n}P_{i} \tag{2}\]  
\[z_{k\epsilon}^{1}=z_{i\epsilon}^{1}\quad i=1,2\ldots n,i\
eq k \tag{3}\]  
where \(P\) is the total pile head load for the pile group and Equation (3) means that the settlement for every group pile is equal at the pile head.  
When the vertically-loaded pile group is simultaneously subjected to tunneling, however, the elastic additional settlement is not proportional to the pile head load, and the modified Poulos method Equation (1) should be rewritten as Equation (4) to account for the effects of tunneling-induced soil settlements along the pile shaft.  
\[z_{k\epsilon}^{1}=z_{k}^{1}+\Delta z_{k\epsilon}^{1} \tag{4}\]  
where \(\Delta z_{k\epsilon}^{1}\) is the additional elastic settlement of group pile \(k\) due to the existence of other group piles, when the pile group is subjected to a combination of the pile head load and tunnel excavation, as determined by Poulos elastic analysis.  
Therefore, the prediction of \(z_{k\epsilon}^{1}\) consists of two parts: calculation of the nonlinear isolated pile head settlement \(z_{k}^{1}\) by the NFC method and evaluation of the elastic additional pile head settlement \(\Delta z_{k\epsilon}^{1}\) through Poulos elastic analysis.