## 3 Verification  
[PERSON] et al. [16] conducted a series of centrifuge model tests to investigate the response of a vertically loaded \(2\times 2\) fixed-head pile group subjected to tunnel excavation at different depths. Although the test series were not designed to investigate the variation of group effect due to tunneling, the measured pile group response can serve to verify the validity of the proposed method. Figure 3: Flow chart of the calculation procedure. (”No” has been positioned in the right place). The case with 18 m excavation depth was analyzed (Figure 4). The soil sample for the test was measured to vary in shear strength linearly from 50 kPa at the surface to 75 kPa at the pile tip, and the bulk modulus was 16.5 kN/m\({}^{3}\). A 2 \(\times\) 2 pile group consisting of 4 18-m long circular concrete piles with a diameter of 0.8 m and an elastic modulus of 30 GPa in the prototype was replicated. The pile spacing was 3.125 times the pile diameter (represented by \(d\)), and the pile head was designed to be connected by a rigid pile cap, applied with a vertical load of 4.46 MN before tunneling. The tunnel with a diameter of 6 m was excavated with a centerline located 18 m below the surface, and the center-to-center distance between the tunnel and the front pile row was 5.5 m. The ground loss ratio \(\varepsilon_{0}\) was designed as 1%. Tunneling-induced free-field soil settlement has been thoroughly investigated by researchers, and various empirical prediction formulas have been proposed [18; 19]. Considering the nonlinear effects of soil deformation, [19] proposed a closed-form analytical solution of subsurface soil settlement due to tunnel excavation, see Equation (19). The free-field soil settlement profiles \(z_{sk}(y)\) at the group pile locations adopted in the following analysis are predicted by the formula. \[ \begin{split}& z_{sk}(y)=\varepsilon_{0}R^{2}\Bigg{(}-\frac{y-H}{x^{ 2}+(y-H)^{2}}+(3-4\mu)\frac{y+H}{x^{2}+(y+H)^{2}}-\frac{2y\left[x^{2}-(y+H)^{2 }\right]}{\left[x^{2}+(y+H)^{2}\right]^{2}}\Bigg{)}\exp\\ &\left[-\frac{1.38x^{2}}{(H+R)^{2}}-\frac{0.69y^{2}}{H^{2}} \right]\end{split}  \tag{19}\]  
where \(R\) is the tunnel radius, \(H\) is the excavation depth of the tunnel center, \(y\) is the depth below the surface, and \(x\) is the distance from the tunnel centerline. In NFC analyzes, considering that the hyperbola relationship is simple and extensively used in pile response analysis [1; 20], the \(t-z\) relationship is assumed to follow a hyperbola in the following NFC analyzes, see Equation (20). \[t(y)=\frac{z(y)}{\frac{1}{k_{z,\mathrm{inf}}(y)}+\frac{t_{y}(y)}{z(y)}} \tag{20}\]  
Figure 4: Test of tunneling adjacent to a fixed head pile group (unit: m) [16]. where \(k_{s,ini}(y)\) and \(t_{u}(y)\) are the initial stiffness and ultimate skin friction of the soil spring in the vertical direction at a depth of \(y\) below the surface and can be determined based on Equations (21) and (22), respectively [7,21]. \[k_{s,ini}=\frac{E_{i}}{2(1+\mu)r_{0}\ln(r_{m}/r_{0})} \tag{21}\]  
\[t_{u}=K_{0}\gamma z\tan\varphi+c \tag{22}\]  
where \(r_{0}\) and \(r_{m}\) represent the pile radius and an empirical distance, respectively, and \(r_{m}\) is typically taken as \(2.5r_{0}\). \(E_{i}\) is the initial modulus of the soil, and \(E_{i}=2E_{50}\) for a hyperbola relationship, where \(E_{50}\) is the soil modulus at 50% of the ultimate shear strength measured in an unconfined compression test. Based on the relationship between the shear strength and \(E_{50}\) recommended by Code JTS167-4-2012 [2], \(E_{i}\) can be easily obtained as 18 MPa. The \(Q-z\) relationship is also described by a hyperbola [7]; see Equation (23). \[Q_{b}=\frac{z_{b}}{\frac{1}{k_{bs,ini}}+\frac{Q_{u}}{z_{b}}} \tag{23}\]  
where \(b\) denotes the pile tip and \(k_{bs,ini}\) and \(Q_{u}\) are the initial stiffness and end bearing capacity at the pile tip, respectively, as determined by Equation (24), proposed by [PERSON] and [PERSON] [7], and Equation (25), the Vesic bearing capacity equation [22].