Report - PEER - University of California, Berkeley

3.2.1 Design for Tolerable Probability of Collapse at a Specific Hazard LevelDesired performance at the collapse prevention level could be expressed in terms of atolerable probability of collapse at a specified hazard level, as for instance, a tolerableprobability of collapse of 0.1 at the 2/50 hazard level. For a 9-story frame structurewith T 1 = 0.9 sec. the corresponding ([S a (T 1 )/g]/γ values (R µ values) for severalcombinations of system parameters are presented in the collapse fragility curvesshown in Figure 5. More general design aids are collapse capacity spectra of the typeshown in Figure 6, which show the effect of component ductility capacity on thecollapse capacity (the R µ value causing collapse), for a 10% and 50% probability ofnon-exceedance, and assuming α c = -0.1 and no cyclic deterioration (γ s,c,k,a = ∞).[Note that the R µ value causing collapse is strongly period dependent and decreases toa low value for long period structures because of P-delta effects.][S a,c (T 1 )/g]/γ654321010% Collapse Capacity Spectra for BH FramesN=Var, T 1=Var, BH, Peak Oriented Model, LMSR-N, ξ=5%α s=0.03, δ c/δ y=Var, α c=-0.10, γ s,c,k,a=?, λ=0δSeries1c/δ y = 2δSeries2c/δ y = 4δSeries3c/δ y = 60 1 2 3 4Period (sec)[S a,c (T 1 )/g]/γ109876543210Median Collapse Capacity Spectra for BH FramesN=Var, T 1=Var, BH, Peak Oriented Model, LMSR-N, ξ=5%α s=0.03, δ c/δ y=Var, α c=-0.10, γ s,c,k,a=?, λ=0δSeries1c/δ y = 2δSeries2c/δ y = 4δSeries3c/δ y = 60 1 2 3 4Period (sec)Figure 6. Collapse capacity spectra for frames with plastic hinges at beam ends(strong-column designs); (a) 10% probability of non-exceedance, (b) median.For a tolerable probability of collapse of 10% in a 2/50 event, data of the typeshown in Figure 6(a) provides the necessary design decision support (similar spectraare available for other combinations of system parameters). For instance, if T 1 isselected as 0.9 sec. and the component ductility capacity is 4.0, the R µ is 4.6, whichfor the 2/50 hazard of the example problem of Section 3.1.1 results in a required baseshear strength coefficient of γ = 1.7/4.6 = 0.37. This is larger than estimated from thedesign for acceptable direct losses. Thus, collapse prevention would control therequired strength, unless a larger ductility capacity (better detailing) is utilized or amore flexible structure is used. For instance, for T 1 = 0.9 sec. and δ c /δ y = 6, the R µ is5.4, which would result in γ = 1.7/5.4 = 0.31. Alternatively, a more flexible structurecould be selected (albeit this would be a poor solution based on monetary losses, seeFigure 3). For T 1 = 1.8 sec. and δ c /δ y = 4, the R µ is 3.5, which for the 2/50 S a value of0.86g at 1.8 sec. results in a required base shear strength coefficient of γ = 0.86/3.5 =0.25. These are the kind of trade-offs that can be evaluated through the use ofcollapse capacity spectra, presuming that a tolerable probability of collapse isspecified at a specific hazard level.514