Report - PEER - University of California, Berkeley

3.2.2 Design for Tolerable Mean Annual Frequency of CollapseA different way to express desired collapse performance is to target a tolerable meanannual frequency (MAF) of collapse, λ Coll . This performance target is more general(it permits the estimation of the probability of collapse over an expected life time),but it is more difficult to implement because the “accurate” computation of a MAFrequires integration over the S a hazard curve. An approximate implementation ispossible by means of the simplified closed form expression proposed by Cornell(Cornell 1996), which estimates the MAF of collapse as follows:1 2 2λ Coll = ∫ FC|S( c | sa)| dλS a(sa)| = λS( ηC)exp( k β RC )(3)aa2SaThe simplified expression on the right-hand side contains the MAF of the spectralacceleration associated with the median collapse capacity, λ Sa (η C ), and a term thataccounts, in an approximate manner, for the uncertainties inherent in the computationof the collapse capacity. This term contains the slope of the hazard curve at thereferenced spectral acceleration value, k, and the dispersion(s) in the collapse fragilitycurve, β (the σ of the log of the data if a log-normal distribution is assumed for theprobability of collapse given the spectral acceleration). In the example illustratedhere only record-to-record (RTR) variability if considered, which is explicitlycontained in the fragility curves shown in Figure 5 (these fragility curves are obtainedby using an “expected” structural model subjected to 40 ground motions). Thus, theterm β RC in equation (3) expresses the effect of RTR variability only (“Randomness incollapse Capacity”), and is found to be on the order of 0.4 to 0.5 (except for longperiod structures for which it is smaller because of the dominance of P-delta effects).As an example, let us target a tolerable annual frequency of collapse of 0.0002(i.e., a tolerable probability of collapse of approximately 0.0002x50 = 0.01 in a 50year life). This criterion could be used, together with median collapse capacityspectra of the type shown in Figure 6(b), to arrive at effective design solutions. Againusing the example of a 9-story frame structure, the following design alternatives couldbe explored. If a period of 0.9 sec. and a component ductility capacity of δ c /δ y = 4 areassumed, then the median [S a (T 1 )/g]/γ value from Figure 6(b) is 7.7. For the sitespecific hazard curve of the previously illustrated example the slope of the S a hazardcurve in the neighborhood of a MAF of 0.0001 to 0.0004 is about 2.2, and the β RCvalue is about 0.4 (from collapse fragility analyses, Ibarra 2003). Thus, fromequation (3), the MAF of the S a associated with the median collapse capacity, λ Sa (η C ),is equal to 0.0002/exp(0.5x2.2 2 x0.4 2 ) = 0.000136. From the S a hazard curve for thesite of the example problem, the corresponding S a is 2.8g, and the corresponding γvalue is 2.8/7.7 = 0.36. Again, this is a larger value than that based on loss-controlledperformance targets. Alternatives are to increase the component ductility capacity (ifit is increased from 4 to 6, [S a (T 1 )/g]/γ is 8.9, and for the same λ Sa (η C ) of 0.000136the γ value becomes 2.8/8.9 = 0.31), or to increase the structure period. If, for515