## 1 Introduction  
Tunnel excavations adjacent to existing piled foundations are frequently encountered in urban areas. Ground loss induced by tunneling can lead to nonnegligible soil settlements around a tunnel and, in turn, additional stressing and deformation on adjacent piled foundations, usually in the form of a pile group for high rise buildings, which are originally designed to withstand the vertical loads transmitted from the upper structure. When a pile group is only subjected to vertical loads on the pile head, the pile-soil interaction (PSI) is in an active mode [1], and a close pile spacing less than 6 times the pile diameter usually requires considering the group effect in the vertical direction due to overlapping stress in engineering practices [2, 3]. Simplified analysis methods to evaluate the response of an active vertically loaded pile group have extensive applications and can usually be classified into three categories: (1) the elastic analysis method, assuming the soil as an infinite elastic space loaded beneath the surface [4, 5], and limited to analysis of pile-soil interaction in the elastic range; (2) the nonlinear foundation column (NFC) model, which needs to be combined with other methods to incorporate the pile-soil-pile interaction [6], for example, the shear displacement method [7, 8]; (3) and numerical simulation [9, 10, 11]. The modified [PERSON] method [12] avoids the disadvantages of the NFC and elastic analysis methods by combining the above two methods and was recommended by the American Petroleum Institute [13] to evaluate the group effect of active pile groups in ocean engineering practices. The pile head settlement of the pile group is assumed to consist of two components: the nonlinear head settlement of an equivalent isolated pile and the additional settlement induced by pile group interactions. The head settlement of the isolated pile is calculated using the NFC model combined with \(t-z/Q-z\) curves, while the additional settlement is determined based on the elastic analysis method suggested by [PERSON] and [PERSON] (2013), which is proportional to the pile head loads. Furthermore, modified \(t-z/Q-z\) data are developed by multiplying the \(Z\) values of the original \(t-z/Q-z\) data by a \(Z\) multiplier larger than unity. The \(Z\) multiplier is determined by trial calculations so that the isolated pile head settlement equals the pile group head settlement predicted above. The modified \(t-z/Q-z\) curves can account for the pile group effect, and the \(Z\) multiplier is considered the parameter that represents the group effect of the active pile group. With tunneling taken into consideration, although a single pile response can be easily considered by either the NFC model or elastic analysis method, analysis of the pile group response will suffer from the difficulties in load redistribution among the group piles and variations in group effect during tunneling, and the addition pile head settlement is no longer proportional to the pile head loads. In terms of computation, the additional pile head settlement cannot be directly determined based on previous research for an active pile group, leading to difficulties in group effect evaluation. Finite element modelling of both the tunneling and the existing pile group loaded at the pile head is also frequently adopted as a workover in present practices because of both the group effect and complex boundary conditions, but it entails a heavy computational burden, not to mention a complicated parameter determination process ([PERSON], 1994; [PERSON], 1994; [PERSON], 1994). Simplified methods have been proposed to include the additional pile head settlement by introducing a shield settlement based on the shear displacement method ([PERSON], 1994), and it was proven that neglecting the working loads can lead to an unconservative prediction of pile response ([PERSON], 1994; [PERSON], 1994).