Report - PEER - University of California, Berkeley

Floor acceleration andinter-story drift aretherefore the structuralresponse probabilisticparameters consideredhere by the belldistribution. Theprobability that responseexceeds a specific limitstate can be directlycalculated from thevolume under the surfacedistribution exceeding thespecified limit. For agiven structural response,retrofit measures thatwould allow the nonstructuralcomponents toresist greater flooraccelerations (i.e., moveup the acceleration limitstate dotted line in Figure5) would directlytranslate into a smallervolume under theprobability distributionsurface, and thus asmaller probability ofPSa(Floor)PSa(Floor)Velocity Limit StateAcceleration Limit StateVelocity Limit StateAcceleration Limit StateLimit SPACESLimit SPACESSd (Floor)exceedence of the limit state. However, modifications to the structural system changethe probable structural response, which is equivalent to sliding the multidimensionalbell-curve within the limit space (i.e., moving along the dotted arrows in Figure 5).For example, stiffening the structural system in a manner that reduce interstory driftswould move the response surface to the left of the limit space of Figure 5, and couldalso move it upward or downward, depending on the initial structural period(although the former is more likely). Structural damage during an earthquake wouldweaken the structure, moving the response surface toward the right and possiblydownward (solid arrow in Figure 5), resulting in greater intersect with the driftcontrolledlimit states.Quantification of the seismic resilience curve is first presented for the case oflinear-elastic structural response. For this and all subsequent cases considered, thevertical axis of the resilience curves is in terms of “investment value” in the structuralsystem, or the non-structural system. The left part of Figure 6 illustrates that there isno structural loss (i.e., no drop in the value of structural investment) when theRelative Displacement Limit StateRelative Displacement Limit StateCracking Displacement Limit StateCollapse Displacement Limit StateCracking Displacement Limit StateCollapse Displacement Limit StateSd (Floor)Figure 5. Probability that response exceedslimit space: (a) non-structural limit states vs.structural limit states; (b) different sequence oflimit states.166