# Durability of Subsea Tunnels under the Coupled Action of Stress and Chloride Ions
## 5 Engineering Application
### Chloride Ion Concentration Distribution  
We chose the vault (maximum positive bending moment) section and the section (maximum negative bending moment) with an angle of about 45 degrees to the vault for analysis. We used the mean value of the parameters, using MATLAB software (developed by MathWorks company, US), to find the numerical solution of Equation (13) and determine the concentration distribution of the chloride ions under the action of the coupling effect at any time. The results show that the trend in the chloride ion concentration distribution in the two sections are about the same because their total cross-section are pressurized, with a comparable and smaller stress ratio (\(\sigma_{\mathrm{c}}\)/\(\sigma_{\mathrm{c}}\)). The stress ratio of the vault section was 0.36 and 0.22, and 0.36 and 0.23 for the section with an angle of about 45 degrees to the vault. This means that the assumptions in Sections 3.1 and 3.3, in which the concrete is in the linear elasticity working stage, are reasonable and the derived formula is applicable. Figure 6 shows the relationship between chloride ion concentration and calculation depth, measured from the inside and outside of the lining separately, at some critical time. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Mean Value** & **Coefficient of Variable** & **Type of Probability Distribution** \\ \hline \(f_{\mathrm{c}}\) (kN/m\({}^{2}\)) & 39,500 & 0.11 & Normal distribution \\ \(f_{\mathrm{t}}\) (kN/m\({}^{2}\)) & 3220 & 0.11 & Normal distribution \\ \(A_{\mathrm{s}}\) (m\({}^{2}\)) & 0.011 (0.023) & 0.01 & Normal distribution \\ \(D_{0}\) (m\({}^{2}\)/year) & \(1.61\times 10^{-4}\) & 0.7 & Logarithmic normal distribution \\ \(C_{\mathrm{s}}\) (\%) & 0.57 (0.15) & 0.5 & Logarithmic normal distribution \\ \(C_{cr}\) (\%) & 0.46 (0.065) & 0.2 & Logarithmic normal distribution \\ \(C\) (m) & 0.05 (0.06) & 0.3 & Normal distribution \\ Density of concrete (kN/m\({}^{3}\)) & 26.5 & 0.07 & Normal distribution \\ Density of soil (kN/m\({}^{3}\)) & 17.11 & 0.01 & Normal distribution \\ The buried depth of the top of lining (m) & 49.5 & 0.05 & Normal distribution \\ Coefficient of static earth & 0.06 & 0.15 & Normal distribution \\ pressure & & & Logarithmic normal distribution \\ Coefficient of strata & 300,000 & 0.15 &
\begin{tabular}{c} Logarithmic normal distribution \\ \end{tabular} \\ resistance & 300,000 & 0.05 & Normal distribution \\ Water depth (m) & 50 & 0.05 & Normal distribution \\ \hline \hline \end{tabular} Note: The values in brackets are the parameters for the inside of the lining.  
\end{table} 
Table 1: Random parameters. The chloride ion concentration curve can be divided into three stages according to the distribution form. In the early period, the C value decreased with the increase in calculation depth. At this point, the actual depth that chlorine ions reached by convection was small, diffusion played a dominant role. In the middle period, the C value first rose and then fell with the increase in calculation depth. When the time was 30 years, the C value reached the maximum value, and convection was about to stop (convection stopped when \(t=31\) years). After 31 years, only diffusion was acting. Therefore, the C value of the coupling area and its vicinity reduced gradually, and the C value of the diffusion zone increased gradually. At this stage, convection played a major role. In the late period, the effect of convection weakened, only diffusion worked, and the C value decreased with the increase in the calculation depth. Figure 6: The relationship between concentration of the chloride ion and the calculation depth in the (**a**) inside and (**b**) outside of the lining. The designed service life of the tunnel is usually 100 years. Figure 7 provides a comparison of the concentration distribution of the chloride ions in the inside and outside of the lining under the coupled action and the action of diffusion alone. Figure 7 shows that the C value decreases with increasing calculation depth.