## 2 Materials and Methods
However, the mechanical behavior of segment joints has been studied analytically [2], taking into account the variation of the axial load in the joint. This relation was used in this study, showing that these iterations are not necessary. For each longitudinal joint, the moment-rotation relation proposed in [PERSON] et al. [2] was used to represent its mechanical behavior. This constitutive behavior is presented in Equations (2)-(4). \[M=\left\{ \begin{array}{ll}\frac{\pi b^{2}bE}{32(1-\
u^{2})}\varphi&if\;e \leq\frac{1}{4}h\\ \frac{\pi bE\left(\frac{1}{2}-e\right)}{2(1-\
u^{2})}\varphi e&if\;e>\frac{1}{4 }h\end{array} \right. \tag{2}\]  
where  
\(M=\) flexural moment;  
\(\phi=\) rotation;  
\(h=\) longitudinal joint thickness;  
\(b=\) longitudinal joint width;  
\(\
u=\) [PERSON]'s ratio;  
\(E=\) modulus of elasticity;  
\(e=\) load eccentricity. On the other hand, Equation (3) determines the initial stiffness of the longitudinal joint. \[M_{max}=P_{max}e=\sqrt{\frac{A_{2}}{A_{1}}}f^{\prime}cA_{1}e\sqrt{\frac{A_{2}} {A_{1}}}\leq 2 \tag{3}\]  
where  
\(M_{max}=\) maximum moment;  
\(P_{max}=\) theoretical maximum load;  
Figure 5: Modeling strategy for the proposed model based on beam and spring elements (1D model) considering coupling between rings: (**a**) top view; (**b**) lateral view. \(A_{1}\) = contact area in function of eccentricity;  
\(A_{2}\) = area of the joint;  
\(f\) = nominal compressive strength. The yield rotation is calculated with Equation (4). \[\varphi_{y}=\left\{ \begin{array}{cc}\frac{32\left(1-\
u^{2}\right)}{\pi l^{2} bF}M_{\text{max}}&if\ e\leq\frac{1}{4}h\\ \frac{2\left(1-\
u^{2}\right)}{\pi bE\left(\frac{1}{2}-e\right)}\ M_{\text{max}}&if\ e>\frac{1}{4}h\end{array} \right. \tag{4}\]  
where  
\(\phi_{y}\) = yield rotation. The maximum moment \(M_{\text{max}}\) and the yield rotation \(\varphi_{y}\) were calculated by applying Equations (3) and (4). In Table 2, the geometrical and mechanical properties of the segment joint used in the proposed model are depicted. The circumferential joints were modeled by means of elastic springs to detect slips between rings (Figure 5). In these joints, the spring stiffness used was \(10^{7}\) N/mm according to [PERSON] [33], in whose work the results of the experimental tests of the Botlek Railway Tunnel are presented. Figure 6 shows the simplified numerical model used to represent the experimental model depicted in [PERSON] and Oosterhout [28] and in [PERSON] [33]. The four tangential supports were considered to stabilize the model, as in the experimental model, and the load protocol was assigned in the simplified structural model to validate this modeling strategy. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Material** & \(\
u\) & \( \begin{array}{c}E\\ \textbf{(MPa)}\end{array} \) & \(\begin{array}{c}\textbf{\emph{f}}_{\textbf{c}}  
On the other hand, in the three-dimensional model (3D model), the reinforcing bars of the segments were included by means of a smeared approach (volume ratio), and the nonlinear material model proposed by [PERSON] and [PERSON] [35] was used to simulate the concrete behavior of crushing in compression and cracking in tension. In addition, contact elements were used to represent the mechanical behavior of joints; with these elements, the stiffness of each joint is updated during the loading process, making the structural analysis \"more precise\" than in the simplified models, albeit with more computational costs. The segments were modeled by using an eight-node solid element identified as SOLID65, while the joints were considered by means of the contact elements identified as CONTA173 and TARGE170. In longitudinal joints, the contact between the surfaces was considered as perfectly rough; this corresponds to an infinite friction coefficient. Thus, only the rotation of the segment joints was considered, which matched the behavior obtained by experimental tests [2; 20; 21; 22; 24; 12].