Report - PEER - University of California, Berkeley

4. DERIVATION OF THE VULNERABILITY FUNCTIONThe procedure of calculating the probability of the maximum response exceeding agiven limit state (L.S.) is described as follows:1. Define the coefficient of variation (C.O.V.) and median of the maximum responseasC.O.V.: δ=σ/µMedian:µxm=(2)21+δwhere, σ is the standard deviation and µ is the mean value.2. Determine the displacement limit state (L.S.) associated with a specific damagelevel.3. Calculate the conditional cumulative probability for the log-normal distribution asln( L.S.)− λ2Φ() , λ = ln xm, ξ = ln(1 + δ )(3)ξ4. Calculate the probability of maximum displacement exceeding a given limit stateasln( L.S.)− λProb (max. displ. > L.S.) = 1- Φ()ξ(4)Whilst the effect of stiffness and strength change on the vulnerability curves areconsidered by response using the Response Database, the effect of ductility change(∆u i ) can be implemented into the calculation procedure of conditional probability bychanging the limit states (L.S.), as shown in Fig. 6.60∆u i1Original L.S.Changed L.S.Base shear (kN)4020Original L.S. Changed L.S.00 50 100 150Displacement (mm)Prob. (Sd > L.S.)0.80.60.40.200 0.1 0.2 0.3 0.4 0.5PGA (g)Figure 6. Effect of ductility change on vulnerability functions.192