# Time-Variant Seismic Fragility of Offshore Continuous Beam Bridges Based on Collapse Analysis
## 2 Durability Damage Analysis of Materials
### Corrosion of Reinforcement Bar  
Based on the \"_Standard for durability assessment of concrete structures_\" [24], the surface chloride ion concentration of concrete and the time of initial corrosion of steel reinforcement considering chloride-induced corrosion are calculated as shown in Equations (6)-(9) [15,17]. \[t_{i}^{\prime}=\left(\frac{c}{K}\right)^{2}\times 10^{-6} \tag{6}\]  
\[K=2\sqrt{D}erf^{-1}\left(1-\frac{M_{cr}}{M_{s}}\right) \tag{7}\]  
\[t_{i}=t_{i}^{\prime}+0.2t_{1} \tag{8}\]  
\[M_{s}=M_{s}{}^{\prime}\cdot k \tag{9}\]  
where \(t_{i}^{\prime}\) is the time when reinforcement bar begins to rust without considering chloride ion diffusion coefficient in year; \(c\) is the concrete cover thickness in mm; \(K\) is the chloride erosion coefficient in m\({}^{2}\) per year; \(t_{i}\) is the time when the reinforcement bar begins to rust in the offshore environment in year; \(t_{1}\) refers to the time of cumulative when the chloride reaches the stable value on the concrete surface in year, \(t_{1}\) is 12.5 year in this case; \(D\) is the diffusion coefficient of chloride ion in m\({}^{2}\) per year, \(D\) is 12.5 m\({}^{2}\) per year in this case; _erf_ is the error function; \(E_{\textit{rf}}\) is the error function; \(M_{cr}\) is the critical chloride ion concentration of reinforcement bar corrosion in kg per m\({}^{3}\); \(M_{s}\)' is the concentration of chloride ion on the concrete surface at 0.1 km from the coast; \(k\) is the correction coefficient of the distance from the coast. Based on the \"_Standard for durability assessment of concrete structures_\" [24], and the research on the performance of reinforced concrete materials in the whole life cycle [25], yield strength, diameter deterioration, elastic modulus, and corrosion rate of reinforcement bar is shown in Equations (10)-(20) [24]. \[t_{cr}=t_{i}+t_{c} \tag{10}\]  
\[t_{c}=\frac{\delta_{cr}}{\lambda_{cl}} \tag{11}\]  
\[\delta_{cr}=0.012c/d+0.00084f_{cuk}+0.018 \tag{12}\]  
\[\lambda_{cl}=0.0116i \tag{13}\]  
\[\ln i=8.  
\[E=(1-1.166\rho)\cdot E_{s} \tag{20}\]  
where \(t_{cr}\) refers to the time of concrete cracking, which unit is year; \(t_{c}\) refers to the time of the reinforcement bar begins to rust until the concrete cracking in year; \(\Delta_{cr}\) refers to the depth of reinforcement bar corrosion when the concrete is cracking, which unit mm; \(\lambda_{cl}\) refers to the average annual rate of reinforcement bar corrosion before the concrete cracks, which unit is mm per year; \(d\) refers to the diameter of reinforcement bar in mm; \(f_{cuk}\) is the standard compressive strength of concrete cube in MPa; \(i\) is the corrosion current density in \(\mu\)A per cm\({}^{2}\); \(M_{sl}\) is the concentration of chloride ion on the reinforcement bar surface in kg per m\({}^{3}\). \(M_{s0}\) is the chloride ion content added in concrete preparation in m3 per kg; \(t_{cr}\) is concrete cracking time in year; \(d\) is the diameter of reinforcement in mm; \(c\) is the thickness of the concrete protective layer in mm. \(K\) is atmospheric temperature; \(\rho\) is concrete resistivity in K\(\Omega\cdot\) cm; \(m_{cl}\) is the local environmental coefficient; \(M_{s0}\) is the content of chloride ion mixed into concrete during preparation in kg per m\({}^{3}\); when the water-cement ratio is 0.3-0.4, or the concrete is C40 or C50, \(k_{\rho}\) is 11.1; \(M_{cl}{}^{u}\) is the average concentration of chloride ion on the concrete cover in kg per m\({}^{3}\); \(RH\) is the environmental relative humidity; \(\lambda_{cll}\) is the average annual corrosion rate of reinforcement bar after concrete cracking in mm per year; \(X_{c}\) is the thickness of concrete protective layer in mm; \(i_{corr}\) is corrosion current density of reinforcement in cm\({}^{2}\) per \(\mu\)A; \(f_{yc}\) and \(f_{y}\) are the yield strength of corroded and uncorroded reinforcement bars respectively; and \(E\) and \(E_{s}\) are the elastic modulus of corroded and uncorroded reinforcement bars, respectively.