# Time-Variant Seismic Fragility of Offshore Continuous Beam Bridges Based on Collapse Analysis
## 2 Durability Damage Analysis of Materials  
### Concrete Carbonation  
Concrete carbonation refers to the reaction of CO\({}_{2}\) in the environment with the Ca(OH)\({}_{2}\) in the 83 cement paste. It leads to reduction of its alkalinity and it destroys the passivation film on the reinforcement bar surface and reinforces bar corrosion and concrete cracking. Finally, the durability damage is caused [3; 4; 5]. In this case, the carbonation rate (\(\overline{S}\)) is selected as the influence factor of concrete carbonation, and the effect of carbonation depth and section size are considered. As shown in Equations (1)-(5) [22].  
\[\left\{\begin{array}{l}A_{c}=\sum\limits_{i=1}^{n}L_{i}\cdot x_{c}\\ x_{c}=k\cdot\sqrt{t}\\ \overline{S}=A_{c}/A\end{array}\right. \tag{1}\]\[\sigma_{cp} =\left(1+0.619\cdot\overline{S}\right)\cdot\sigma_{p} \tag{2}\] \[\varepsilon_{cp} =\left(1-0.106\cdot\overline{S}\right)\cdot\varepsilon_{p}\] (3) \[\varepsilon_{cu} =\left(1-0.459\cdot\overline{S}\right)\cdot\varepsilon_{u}\] (4) \[E_{c} =\left(1+0.503\cdot\overline{S}\right)\cdot E \tag{5}\]  
where \(\overline{S}\) is the carbonation rate; \(A_{c}\) is carbonised area in mm\({}^{2}\); \(A\) is the sectional area in mm\({}^{2}\); \(L_{i}\) refers to the each side length of pier that the unit is mm; \(x_{c}\) refers to the depth of carbonation in mm; \(k\) refers to the coefficient of carbonation; \(t\) refers to the time of carbonation in year; \(\sigma_{cp}\), \(\varepsilon_{cp}\), \(\varepsilon_{cu}\), and \(E_{c}\) are the peak stress, peak strain, ultimate strain, and elastic modulus of carbonised concrete, respectively; \(\sigma_{p}\), \(\varepsilon_{p}\), \(\varepsilon_{u}\), and \(E\) are the peak stress, peak strain, ultimate strain, and elastic modulus of concrete before carbonation, respectively; \(\mu\) is [PERSON]'s ratio which is 0.2.  
According to the Equations (1)-(5), the changes of various mechanical parameters of C40 and C50 concrete over time during their service period and their comparison are shown in Figure 1. The case's design service life is 100 years, to discuss the seismic performance of the case after the completion of its service period. Hence, the service performance of each component and bridge system is analysed when the service period reaches 150 years. Therefore, the whole life cycle is divided into 0, 30, 50, 70, 100, and 150 years in the subsequent analysis of time-variant seismic performance.  
After considering carbonization effect, the peak stress, elastic modulus, and shear modulus of concrete all increase with the bridge service period, while the peak strain and limit strain both decrease  
Figure 1: The mechanical parameters of concrete: (**a**) Elastic modulus; (**b**) Peak stress; (**c**) Peak strain; (**d**) Ultimate strain.  
with the bridge service period increasing. Among them, C40 concrete changed more while C50 concrete changed less. Taking elastic modulus as an example, C40 and C50 concrete increased by 0.34%, 0.43%, 0.51%, and 0.61%; and 0.75%, 0.20%, 0.25%, 0.30%, 0.36%, and 0.44%, respectively, when they were in service at 30, 50, 70, 100, and 150 years compared with 0 years when they were in service.