# Climate Change Impact on Corrosion of Reinforced Concrete Bridges and Their Seismic Performance
## 3 Structural Modelling and Analysis Method
Basically, in this approach, the corrosion penetration in a generic concrete volume is described by a parabolic law, as follows:  
\[s=k\cdot t^{\frac{1}{n}}, \tag{3}\]  
where \(s\) is the carbonated concrete layer thickness, \(k\) represents the penetration rate coefficient, and \(t\) indicates the time. Considering that the analyzed viaduct was built in the 1970s, \(n=2\) can be fixed, considering that it was built using normal compacted concrete [65]. The steel reinforcement reduction diameter and area are calculated using Equations (4) and (5), respectively:  
\[d(t)=d_{0}-2P(t)=d_{0}-2i_{corr}k(t-t_{i}), \tag{4}\]  
\[A_{s}(t)=\pi[d_{0}-2i_{corr}k(t-t_{i})]^{2}/4, \tag{5}\]  
where \(d_{0}\) is the initial diameter of the steel reinforcement and \(t_{i}\) is the time of corrosion initiation. Equation (4) explains the variation over time of the steel reinforcement diameter \(d\), which depends on the corrosion penetration \(P(t)\). Equation (5), instead, regulates the related variation over time of the cross-section area of the steel rebar. In this work, three different corrosion levels are considered, following the indications reported in [53; 54] and discussed in the previous Section: (i) slight, (ii) moderate, and (iii) high. For each analyzed corrosion level, two different assumptions were made, to obtain the \(i_{corr,20}\) value. The first assumption is based on [53], where \(i_{corr,20}\) values equal to 0.172 \(\mu\)A/cm\({}^{2}\), 0.345 \(\mu\)A/cm\({}^{2}\), and 0.431 \(\mu\)A/cm\({}^{2}\) represent slight, moderate, and high corrosion scenarios, respectively. In the second assumption, the \(i_{corr,20}\) values are estimated considering the indications reported in [54]. In this case, \(i_{corr,20}\) values equal to 0.1 \(\mu\)A/cm\({}^{2}\), 1 \(\mu\)A/cm\({}^{2}\), and 5 \(\mu\)A/cm\({}^{2}\) represent slight, moderate, and high corrosion levels. The variation over time of the steel reinforcement diameter is evaluated by considering the difference between \(d_{0}\), which represents the initial diameter, and \(d(t)\), which indicates the diameter calculated at the time \(t\), which is obtained using the corrosion effects. It can be highlighted that the variation of the steel reinforcement diameter depends on the initiation time (\(t_{i}\)), considered to be a random variable, and on the mean corrosion current density (\(i_{corr}\)) value, which is evaluated, in this work, by considering the previously discussed different assumptions reported in [53; 54]. With an initial value of concrete cover thickness equal to 25 cm, the procedure is repeated iteratively using different values of concrete cover  
Figure 7: Connections between the elastomeric bearings and the pier cap. thickness. The other parameters considered constant are illustrated in Table 1, where \(f_{ck}\) is the concrete compressive strength, \(k\) is the penetration rate coefficient, \(\varepsilon_{u,0}\) is the steel ultimate deformation, \(w/c\) is water/cement ratio, and \(t_{i}\) is the initiation time. The effects of corrosion processes over time are calculated with the following time steps: when the bridge was built, and 13.5 years, 50 years, 75 years, and 100 years after its construction. To obtain the seismic performance of the viaduct considering the above-mentioned time steps, a multi-modal pushover analysis is performed, using the capacity spectrum method (CSM) to determine the performance point [66; 67]. For each vibration mode, with a participating mass at least equal to 1%, a capacity curve is calculated using the related modal load profile. The performance point defined by each curve is obtained using the seismic demand spectrum. Consequently, the internal actions acting on each pier of the viaduct are calculated, with regard to the performance point evaluated by means of the intersection between the above-mentioned demand spectrum and the capacity curve in the ADRS plane. The obtained internal actions are combined by means of the CQC (complete quadratic combination) rule, in order to perform the structural checks.