Second-order Logarithmic formulation for hazard curves and closed-form approximation to annual failure probability

• Annual failure probabilities are needed in performance-based engineering.
• Current solutions use a linear approximation for the hazard curves.
• A linear approximation is found to deviate from the actual seismic hazard curves.
• A linear approximation can result in significant errors.
• A novel Second Order Logarithmic Formulation (SOLF) is proposed.

Closed-form solutions to compute annual failure probabilities are helpful in the implementation of performance-based engineering. Closed-form solutions to compute the annual failure probability have been proposed in the context of seismic hazard using a linear approximation in the logarithmic scale for the hazard curves. However, a linear approximation is found to significantly deviate from the actual seismic hazard curve and hence can result in significant errors in the computation of the annual failure probability. This paper develops a closed-form solution to compute the annual failure probability of systems using a novel Second-order Logarithmic Formulation (SOLF) to model the hazard curves. Thereafter, we illustrate the proposed closed-form solution by computing the annual failure probabilities for reinforced concrete (RC) bridges subject to the seismic hazard of San Francisco, CA and Memphis, TN. We demonstrate that SOLF yields accurate estimates of the annual failure probabilities whereas the existing linear logarithmic formulation can significantly over or under estimates such probabilities. The proposed formulation is general and is expected to yield accurate estimates for hazards other than the seismic hazard (e.g., hurricanes and floods).