Report - PEER - University of California, Berkeley

3.2.1 A DCF Displacement-Based FormatUnder certain assumptions about the analytical forms of the distributions andrelationships in the framework equation (e.g., Cornell (2002), Jalayer (2003)) it ispossible to obtain an closed form solution to the PEER framework equation for thecase when DV is a binary (i.e., limit state) variable, the IM is a scalar such as S a , theEDP or demand is a scalar such as MIDR, and the (random) capacity is measured inthe same terms (e.g., MIDR):2k(2)2 2λLimit State= λS( Cˆ)exp[( β )]2 D|S+ βaa C2bin which Ĉ is the median displacement capacity, the β’s are dispersions of MIDRgiven S a and of capacity as indicated, the b and k are parameters reflecting thedependence of drift on Sa and λ Sa on S a respectively, and λ Sa is the S a hazard curve.For purposes of assessment of safety compliance (or “checking”), this limit statefrequency result can be set equal to the allowable limit frequency (e.g., 1/2500 peryear), and the result inverted to provide a LRFD-like code checking inequality. Wecall this a Demand and Capacity Factor (DCF) format as here demands and capacitiesare measured in dynamic displacement, not force terms, as is preferred for explicitlynonlinear problems. Several equivalent alternative formats are available. Forexample, the tolerable limit state frequency, λ o, is satisfied if:(3)γ ⋅ Dˆ≤ φ Cˆin whichDˆS λ oais the median drift displacement demand at ground motion leveloS λa, which is in turn the Sa level associated with hazard level λ o , and the capacityand demand factors have forms such as k φ2C = exp[ − βC]. Note that DˆS λcan beoa2bfound by Option C above by analyzing n’ records at simply one S a level, implying atotal sample size of only 10 or less. A format such as this amplified to includeepistemic uncertainty effects is the basis of the FEMA SMRF guidelines (FEMA(200X)).3.2.2 Global Collapse Assessment via an IM-based ProcedureIn this section we demonstrate an IM-based version of the limit state global dynamicinstability collapse. Applying the same assumptions alluded to just above (Section3.2.1) a formula for the collapse limit state frequency can be developed from thereduced IM or fragility form of the PEER framework equation stated above (Section3.1), λ ( DV = 1) = ∫GDV| IM(0 | IM)| dλ(IM)| :1 2 2λLimit State= FC( sa) | dλS ( sa) | = λS( Cˆ)exp(k βRC)(4)∫aa2SaD | Sλa S o C⋅a49