## 2 Analytical Solution of LEBS with Residual Jacking Force
### Development of an Analytical Solution  
The analytical solution of the LEBS is derived below by considering the action of both an external bending moment, \(M\), and the residual jacking force, \(N\), as shown in Figure 2. Half the segmental ring width and the joint between the rings are taken for stress and deformation analysis. Figure 1: Schematic diagram of the uniformly distributed stiffness of bolts. Figure 2: Stress and deformation of tunnel segment lining (**a**) stress of joint between the rings, (**b**) deformation of the joint between the rings, and (**c**) tunnel profile. The equilibrium Equations regarding the axial force (1) and bending moment (2), respectively, at the joint between the segmental rings are as follows:  
\[2\int_{\varphi}^{\frac{\pi}{2}}E_{c}t\frac{\varepsilon_{c}}{R_{0}-R_{0}\sin \varphi}(R_{0}\sin\alpha-R_{0}\sin\varphi)R_{0}d\alpha-2\int_{-\frac{\pi}{2}}^{ \varphi}E_{c}t\frac{\varepsilon_{t}}{R_{0}+R_{0}\sin\varphi}(R_{0}\sin\varphi -R_{0}\sin\alpha)R_{0}d\alpha=N \tag{1}\]  
\[2\int_{\varphi}^{\frac{\pi}{2}}E_{c}t\frac{\varepsilon_{c}}{R_{0}-R_{0}\sin \varphi}(R_{0}\sin\alpha-R_{0}\sin\varphi)^{2}R_{0}d\alpha+2\int_{-\frac{\varphi }{2}}^{\varphi}E_{c}t\frac{\varepsilon_{t}}{R_{0}+R_{0}\sin\varphi}(R_{0}\sin \varphi-R_{0}\sin\alpha)^{2}R_{0}d\alpha \tag{2}\]  
\(=M-NR_{0}\sin\varphi\)  
where \(\varepsilon_{t}\) and \(\varepsilon_{c}\) denote the outer fibre tensile strain and compressive strain at the ring joint, respectively; \(\theta\) denotes the cross-section rotating angle; \(l_{s}\) denotes the segmental ring width; \(t\) denotes the tunnel lining thickness; \(R_{0}\) denotes the radius of the tunnel lining centre-line; \(E_{c}\) denotes the elastic modulus of the concrete segments; \(\varphi\) denotes the position of the neutral axis, which is turned into a positive angle counterclockwise and a negative angle clockwise by the horizontal centre line, as shown in Figure 2.  
By combining Equations (1) and (2), the expression of the \(M\) can be obtained as follows:  
\[M=E_{c}t\frac{\varepsilon_{c}R_{0}{}^{2}}{1-in\varphi}\Big{[}\Big{(}\frac{\pi }{2}-\varphi\Big{)}-\cos\varphi\sin\varphi\Big{]}+E_{c}t\frac{\varepsilon_{t}R _{0}{}^{2}}{1+\sin\varphi}\Big{[}\Big{(}\frac{\pi}{2}+\varphi\Big{)}+\sin \varphi\cos\varphi\Big{]} \tag{3}\]  
The tensile stiffness of bolts is equivalent to the uniformly distributed stiffness along the centre-line of the tunnel lining [24], as shown in Figure 1:  
\[k_{j}=\frac{n\cdot E_{b}A_{b}}{2\pi R_{0}l_{b}} \tag{4}\]  
where \(k_{j}\) denotes the linear stiffness of the bolts; \(n\) denotes the number of bolts; \(A_{b}\) denotes the cross-sectional area of a bolt; \(E_{b}\) denotes the elastic modulus of the bolt material; \(l_{b}\) is the bolt length. In the concrete tension zone [12], the force equilibrium equation between the bolts and concrete is established as follows:  
\[2\int_{-\frac{\pi}{2}}^{\varphi}E_{c}t\frac{\varepsilon_{t}}{R_{0}+R_{0}\sin \varphi}(R_{0}\sin\varphi-R_{0}\sin\alpha)R_{0}d\alpha= 2\int_{-\frac{\pi}{2}}^{\varphi}\frac{l_{b}\delta_{t}\cdot k_{j}}{R_{0}+R _{0}\sin\varphi}(R_{0}\sin\varphi-R_{0}\sin\alpha)R_{0}d\alpha \tag{5}\]  
where \(\delta_{f}\) denotes the tensile strain of the bolt. The deformation equilibrium equation is established based on the plane assumption, as shown in Figure 1:0  
Figure 1:1  
Figure 1:2  
According to the Code for Design of Concrete Structures, the maximum elastic strain of concrete under axial compression conforms to the following equation:  
Figure 1:3  
where \(f_{cu\cdot k}\) denotes the standard compressive strength of concrete. By combining Equations (5)-(7), the expression of the \(\varepsilon_{t}\) can be obtained as follows:  
Figure 1:4  
The solution of \(\theta\) can be obtained by rewriting Equation (7).