## 2 The Numerical Model of Shield Tunnel with Stagger-Jointed Assembly
### Material Properties  
In the calculation model, different elastic-plastic characteristics are used to describe the mechanical behavior of segment rebar and joint bolts during loading. For the segment rebar, the elastic modulus and Poisson's ratio are 210 GPa and 0.3, respectively. The yield strength and tensile strength of the rebar are 335 MPa and 455 MPa, respectively. For the joint bolt, the elastic modulus and Poisson's ratio are 200 GPa and 0.3, respectively. The yield strength and tensile strength of the bolt are 400 MPa and 500 MPa, respectively. During operation of the shield tunnel, cracks may appear on the surface of the segment due to the effect of the surrounding environment. In order to analyze the load-bearing performance of the shield tunnel under unloading conditions more accurately, the elastic-plastic damage constitutive model is used to calculate the cracking of the segment concrete [24; 25]. The principle is that the material remains in the elastic stage until the ultimate compressive or tensile stress of the concrete is reached. When the stresses exceed the threshold, the material begins to yield and enter the plastic stage. To describe the damage-softening characteristic of the concrete material, compression damage factor \(d_{c}\) and tensile damage factor \(d_{t}\) are introduced. The greater the damage factor, the more severe the material damage, as shown in Figures 8 and 9. Figure 8: Tensile stress–strain curve of the concrete. Figure 7: Diagram of the size of the tunnel segment rebar. When the segment concrete is damaged, the stiffness of the concrete will decrease. The compressive stress \(\sigma_{c}\) and tensile stress \(\sigma_{t}\) can be describe as follows:  
\[\sigma_{c}=(1-d_{c})E_{0}(\varepsilon_{c}-\varepsilon_{c}^{pl}) \tag{1}\]  
\[\sigma_{t}=(1-d_{t})E_{0}(\varepsilon_{t}-\varepsilon_{t}^{pl}) \tag{2}\]  
\[\varepsilon_{c}^{pl}=\varepsilon_{c}^{in}-\frac{d_{c}}{(1-d_{c})}\frac{\sigma_ {c}}{E_{0}} \tag{3}\]  
\[\varepsilon_{t}^{pl}=\varepsilon_{t}^{ck}-\frac{d_{t}}{(1-d_{t})}\frac{\sigma_ {t}}{E_{0}} \tag{4}\]  
where \(\sigma_{c}\) is the compressive stress; \(\sigma_{t}\) is the tensile stress; \(\varepsilon_{c}^{pl}\) is the equivalent plastic compressive strain; \(\varepsilon_{t}^{pl}\) is the equivalent plastic tensile strain; \(\varepsilon_{c}^{in}\) is the compressive inelastic strain; \(\varepsilon_{t}^{ck}\) is the tensile cracking strain; and \(E_{0}\) is the initial elastic modulus. The definition \(\sigma_{c}\)-\(\varepsilon_{c}^{in}\), \(\sigma_{t}\)-\(\varepsilon_{t}^{ck}\) is adopted to describe the characteristic of the strain softening after the cracking of concrete, and \(d_{c}\)-\(\varepsilon_{c}^{in}\), \(d_{t}\)-\(\varepsilon_{t}^{ck}\) is adopted to describe the stiffness degradation characteristics of concrete caused by damage. The calculation expressions of \(\varepsilon_{c}^{in}\), \(\varepsilon_{t}^{ck}\) are as follows:  
\[\varepsilon_{c}^{in}=\varepsilon_{c}-\varepsilon_{0c}^{pl}=\varepsilon_{c}^{ pl}+\frac{d_{c}}{(1-d_{c})}\frac{\sigma_{c}}{E_{0}} \tag{5}\]  
\[\varepsilon_{t}^{ck}=\varepsilon_{t}-\varepsilon_{0t}^{el}=\varepsilon_{t}^{ pl}+\frac{d_{t}}{(1-d_{t})}\frac{\sigma_{t}}{E_{0}} \tag{6}\]  
\[\varepsilon_{0c}^{el}=\sigma_{c}/E_{0} \tag{7}\]  
\[\varepsilon_{0t}^{el}=\sigma_{t}/E_{0} \tag{8}\]  
where \(\varepsilon_{0c}^{el}\) is the elastic recovery strain under compression, and \(\varepsilon_{0t}^{el}\) is the elastic recovery strain under tension. Figure 7:0 Compressive stress–strain curve of the concrete. The concrete damage plasticity model assumes a non-associated potential plastic flow in order to control the inelastic volume deformation.