Published in Pacific Earthquake Engineering Research Center, April 1, 1999.
The ATC-40 and FEMA-274 documents contain simplified nonlinear analysis procedures to determine the displacement demand imposed on a building expected to deform inelastically. The Nonlinear Static Procedure in these documents, based on the capacity spectrum method, involves several approximations: The lateral force distribution for pushover analysis and conversion of these results to the capacity diagram are based only on the fundamental vibration mode of the elastic system. The earthquake-induced deformation of an inelastic SDF system is estimated by an iterative method requiring analysis of a sequence of equivalent linear systems, thus avoiding the dynamic analysis of the inelastic SDF system. This last approximation is first evaluated in this report, followed by the development of an improved simplified analysis procedure, based on capacity and demand diagrams, to estimate the peak deformation of inelastic SDF systems.
Several deficiencies in ATC-40 Procedure A are demonstrated. This iterative procedure did not converge for some of the systems analyzed. It converged in many cases, but to a deformation much different than dynamic (nonlinear response history or inelastic design spectrum) analysis of the inelastic system. The ATC-40 Procedure B always gives a unique value of deformation, the same as that determined by Procedure A if it converged.
The peak deformation of inelastic systems determined by ATC-40 procedures are shown to be inaccurate when compared against results of nonlinear response history analysis and inelastic design spectrum analysis. The approximate procedure underestimates significantly the deformation for a wide range of periods and ductility factors with errors approaching 50%, implying that the estimated deformation is about half the “exact” value.
Surprisingly, the ATC-40 procedure is deficient relative to even the elastic design spectrum in the velocity-sensitive and displacement-sensitive regions of the spectrum. For periods in these regions, the peak deformation of an inelastic system can be estimated from the elastic design spectrum using the well-known equal displacement rule. However, the approximate procedure requires analyses of several equivalent linear systems and still produces worse results.
Finally, an improved capacity-demand-diagram method that uses the well-known constant-ductility design spectrum for the demand diagram has been developed and illustrated by examples. This method gives the deformation value consistent with the selected inelastic design spectrum, while retaining the attraction of graphical implementation of the ATC-40 methods. One version of the improved method is graphically similar to ATC-40 Procedure A whereas a second version is graphically similar to ATC-40 Procedure B. However, the improved procedures differ from ATC-40 procedures in one important sense. The demand is determined by analyzing an inelastic system in the improved procedure instead of equivalent linear systems in ATC-40 procedures.
The improved method can be conveniently implemented numerically if its graphical features are not important to the user. Such a procedure, based on equations relating Ry and µ for different Tn ranges, has been presented, and illustrated by examples using three different Ry - µ - Tn relations.
Civil and Environmental Engineering
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