# Reliability Assessment of RC Bridges Subjected to Seismic Loadings
## 5 Conclusions  
An approach was proposed to obtain both the reliability index and the probability of failure that considers epistemic uncertainties. Demand hazard curves are also obtained based on numeric and closed-form expressions. The probability of failure is proposed in a closed-form expression format, which has the following advantages: (a) it can be used for different kinds of structures; (b) it can be adapted for different environmental loads; (c) it calculates the probability of failure with and without the consideration of the epistemic uncertainties; and (d) it is familiar to structural engineers. The approach was illustrated in a continuous bridge designed to comply with a drift threshold equal to 0.004. Uncertainties related to mechanical and geometric properties were considered, together with uncertainties for seismic loadings. Prior to obtaining demand hazard curves, different fragility curves were generated considering different thresholds. Demand hazard curves were determined for drift thresholds between 0.001-0.012. Considering the stipulations in the AASHTO design code [39], the lifespan of bridges must be guaranteed up to 75 years. Based on the results, the serviceability limit state could be exceeded 17 years before reaching the structure lifespan, which means that the design drift threshold is expected to be exceeded at 58 years. Therefore, the use of a design drift threshold equal to 0.004 is not recommended. A drift threshold exceedance does not mean that the structure is unable to resist seismic loadings; such exceedance means that the structure could present undesirable reliability levels before the recommended interval of serviceability. The reliability index target for bridges is to overcome \(\beta\) = 3.5, considering the structure as new [39]. Considering such target, the bridge under study presented a reliability index 18.24% lower than the recommendation given by [38] when epistemic uncertainties are considered; if such uncertainties are not considered, the reliability index is 16.28% lower. Thus, the value of 0.004 of design drift threshold is not recommended for this type of topology. It is recommended to explore lower values of design drift threshold between 0.001 to 0.003. The reliability index and its probability of failure give certainty about both the safety level that a structure has under design loads, and the capacity of the structure to present serviceability levels during a certain time interval. The exceedance demand rate provides the \begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{E(v_{F})}\) & \(\mathbf{P_{F}}\) & \(\mathbf{\beta}\) \\ \hline With epistemic uncertainties & \(1.531\times 10^{-3}\) & \(1.529\times 10^{-3}\) & 2.96 \\ Without epistemic uncertainties & \(1.318\times 10^{-3}\) & \(1.317\times 10^{-3}\) & 3.01 \\ \hline \end{tabular}
\end{table} 
Table 6: Reliability indicators. Figure 9: Demand hazard curves. instant when the structure reaches a certain level of damage. If the damage level compromises the structural integrity, decisions need to be made about inspection or maintenance actions with the aim to extend the lifespan of the system. Conceptualization, D.T. ; methodology, D.T., D.H. and G.V. ; software, D.H. and G.V. ; validation, D.T., D.H. and G.V. ; data curation, D.H. and G.V. ; writing--original draft preparation, D.H. and G.V. ; writing--review and editing, D.T., D.H. and G.V. ; funding acquisition, D.T. All authors have read and agreed to the published version of the manuscript. This research was funded by Consejo Nacional de Ciencia y Tecnologia Grant A1-S-8700. Not applicable. Data Availability Statement: Data is contained within the manuscript. The first and second author gratefully acknowledge economic support from CONACyT during their PhD studies. They also appreciate the support of Universidad Autonoma Metropolitana (UAM). The third author appreciates the support given by both Consejo Nacional de Ciencia y Tecnologia through the Ciencia Basica Project CB 2017-2018 A1-S-8700, and Universidad Autonoma Metropolitana.