# Durability of Subsea Tunnels under the Coupled Action of Stress and Chloride Ions
## 3 Effect of Stress on Chloride Ions Transfer
### The Stress of a Section of a Subsea Tunnel's Lining
#### 3.3.1 Normal Stresses of the Lining Section Corresponding to Bending Moment  
During the normal service process, the internal forces experienced by the subsea tunnel lining are relatively small. Therefore, assuming that the concrete is in linear elasticity work, each section is consistent with the plane cross-section assumption, as shown in Figure 4.  
It can be seen from the strain figure (Figure 4):  
\[\begin{split}\varepsilon_{2}&=\varepsilon_{1}\cdot \frac{h-x_{0}}{x_{0}}\\ \varepsilon_{3}&=\varepsilon_{1}\cdot\frac{x_{0}- \varepsilon_{s}}{x_{0}}\\ \varepsilon_{4}&=\varepsilon_{1}\cdot\frac{h-x_{0}- a_{s}}{x_{0}}\end{split} \tag{14}\]  
where \(\varepsilon_{1}\) is the strain on the edge of compression zone, \(\varepsilon_{2}\) is the strain on the edge of the tensile zone, \(\varepsilon_{3}\) is the strain of the compressive reinforcement, \(\varepsilon_{4}\) is the strain of the tension reinforcement, \(h\) is calculated height of the section, \(x_{0}\) is the height of the compression zone of the section, and \(a_{s}\) and \(a_{s}\)' are the distance from the center of tension reinforcement and the center of compressive reinforcement to the edge of the section, respectively.  
Figure 4: The section strain and stress corresponding to bending moment.  
Figure 3: The loading and inner force distribution of the tunnel lining section.  
The stress or internal force at each point are shown by Equation (15):  
\[\begin{split}\sigma_{1}&=E_{c}\varepsilon_{1}\\ \sigma_{2}&=E_{c}\varepsilon_{2}=E_{c}\varepsilon_{1} \frac{h-x_{0}}{x_{0}}\\ F_{3}&=E_{s}\varepsilon_{3}A_{s}^{\prime}=E_{s} \varepsilon_{1}\frac{x_{0}-a_{s}^{\prime}}{x_{0}}A_{s}^{\prime}\\ F_{4}&=E_{s}\varepsilon_{4}A_{s}=E_{s}\varepsilon_{1} \frac{h-x_{0}-a_{s}}{x_{0}}A_{s}\end{split} \tag{15}\]  
where \(\sigma_{1}\) is the stress on the edge of the compression zone, \(\sigma_{2}\) is the stress on the edge of the tensile zone, \(F_{3}\) is the internal force of the compressive reinforcement, \(F_{4}\) is the internal force of the tension reinforcement, \(E_{s}\) is the elastic modulus of the reinforcement, and \(A_{s}\) and \(A_{s}{}^{\prime}\) are sectional area of tensile reinforcement and compression reinforcement, respectively.  
The static balance equations were established based on Figure 4:  
\[\begin{split} 0&=\frac{1}{2}\sigma_{1}x_{0}b+F_{3}- \frac{1}{2}\sigma_{2}(h-x_{0})b-F_{4}\\ M&=\frac{1}{2}\sigma_{1}x_{0}(\frac{h}{2}-\frac{x_ {0}}{3})b+F_{3}(\frac{h}{2}-a^{\prime}{}_{s})+\frac{1}{2}\sigma_{2}(h-x_{0}) \Big{[}\frac{h}{2}-\frac{1}{3}(h-x_{0})\Big{]}b+F_{4}(\frac{h}{2}-a_{s})\end{split} \tag{16}\]  
where \(b\) is the width of the calculated section.  
Simultaneous equations:  
\[\begin{split} x_{0}&=\frac{E_{c}hh^{2}-2E_{s}(A_{a} \theta_{s}-A_{s}^{\prime}a_{s}^{\prime}-A_{s}h)}{2E_{c}h+2E_{s}(A_{s}+A_{s}^{ \prime})}\\ \varepsilon_{1}&=\frac{M}{A+B}\end{split} \tag{17}\]  
Among them:  
\[\begin{split} A&=\frac{1}{2}E_{c}b[x_{0}(\frac{h}{2} -\frac{x_{0}}{3})+\frac{(h-x_{0})^{2}}{x_{0}}(\frac{h}{6}+\frac{x_{0}}{3})]\\ B&=E_{s}[A_{s}\frac{h-x_{0}-a_{s}}{x_{0}}(\frac{h}{2} -a_{s})+A_{s}{}^{\prime}\frac{x_{0}-a_{s}^{\prime}}{x_{0}}(\frac{h}{2}-a_{s}^ {\prime})]\end{split}\]  
where \(\sigma_{1}\) and \(\sigma_{2}\) can be obtained by substituting Equation (17) into Equation (15).