# Seismic Damage Analysis of Box Metro Tunnels Accounting for Aspect Ratio and Shear Failure
## 3 Numerical Modeling  
The pseudo-static procedure presented in [PERSON] et al. [21] was adopted to perform nonlinear analyses of the tunnel structures using two-dimensional (2D) frame elements. The tunnel structures were modeled using SAP2000 (ver. 15), a finite element analysis program [38]. We also modeled the joint-offset of the wall-slab and column-slab connections to simulate their relatively rigid behaviors. Details on the nonlinear concrete and steel models for the tunnels are presented in the study by [PERSON] et al. [21]. Plastic flexural and hinge models were used for all frame elements. The performance of the plastic hinge was dictated by the moment-curvature and shear force-shear deformation relationships that were determined from a nonlinear section analysis. For the shear strength-shear deformation relation of a section, we adopted the definitions and guidelines of ACI-318 [39], FEMA-356 [40] and Park and Paulay [41] to determine the yield strength (\(V_{y}\)), ultimate strength (\(V_{u}\)), and the corresponding yield deformation (\(\Delta_{y}\)) and ultimate deformation (\(\Delta_{u}\)). The idealized shear model is illustrated in Figure 2. A series of normal and shear springs were attached to the nodes of the nonlinear frame elements to simulate soil-structure interaction, as defined according to the Seismic design code for metropolitan subway of Korea [42]. The horizontal and vertical normal spring constants, \(K_{H}\) and \(K_{V}\), were defined as:  
\[K_{H}=k_{h0}\left(\frac{h}{30}\right)^{-3/4}, \tag{1}\]  
\[K_{V}=k_{h0}\left(\frac{b}{30}\right)^{-3/4}, \tag{2}\]  
where \(k_{h0}=\left(\frac{1}{30}\right)\)\(E_{D}\), \(h\) and \(b\) are the height and width of the tunnel, and \(E_{D}\) is the elastic modulus of surrounding soil. The shear springs for vertical and horizontal, \(K_{SS}\) and \(K_{SB}\), frames were defined as:  
\[K_{SS}=\frac{1}{4}K_{H}, \tag{3}\]  
\[K_{SB}=\frac{1}{4}K_{V}. \tag{4}\]  
Figure 1: Sections of cut-and-cover box tunnels modeled: (**a**) Single, (**b**) double, and (**c**) triple boxes. Figure 3 shows the boundary conditions and imposed loads on a tunnel. Five uniform site profiles with shear wave velocities (\(V_{s}\)) of 50, 100, 200, 300 and 400 m/s were used. The Poisson's ratio (\(\
u\)) and unit weight were set to 0.3 and 20 kN/m\({}^{3}\) for all profiles. Since the frame-spring model is an approximation of the continuum domain, there is a need to verify whether the simplified model realistically simulates the tunnel-soil interaction. The calculated values of the racking ratio (\(R\)) of box tunnels were compared with the finite difference analysis computations [43], a closed form solution [44], and finite element simulations [26] in Figure 4. R is defined as follows:  
\[R=\frac{\Delta_{Box}}{\Delta_{Free-field}}, \tag{5}\]  
where \(\Delta_{Box}\) is the relative displacement of top and bottom of the tunnel frame, and \(\Delta_{Free-field}\) is the relative displacement of soil profile at the top and bottom level of the tunnel. The calculated \(R\) is plotted against the flexibility ratio (\(F\)) of a rectangular tunnel, which is defined as [26]:  
\[F=\left(\frac{G_{m}}{K_{s}}\right)\left(\frac{B}{H}\right), \tag{6}\]  
where \(B\) and \(H\) are the width and height of the tunnel, \(G_{m}\) is the shear modulus of the surrounding soil, \(K_{S}\) is the racking stiffness of tunnel, and \(K_{S}=P/\Delta\), in which \(\Delta\) is the lateral displacement due to the applied lateral force \(P\) at the roof of the tunnel. Figure 3: Boundary conditions and applied loads to the box tunnel. \[K_{H}=k_{h0}\left(\frac{h}{30}\right)^{-3/4}, \tag{1}\]0 Shear force–deformation relationship of the cross section. The curve of [PERSON] [43], defined as \(R=2F(1+F)\), is based on pseudo-static analyses using four types of single box tunnels and five different burial depths. The [PERSON] [44] curve, \(R=[4(1-v)F]/[3-4v+F]\), is derived from an analytical procedure.