Keywords: reinforced-concrete, single-degree-of-freedom systems, non-linear dynamic analysis, performance based design, stiffness degradation, structural demands, inelastic deformation ratios.

This paper is focused on the single-degree-of-freedom (SDOF) systems, which are generally substituted to calculate the seismic demands of a multi-degree-of-freedom (MDOF) system as in nonlinear static analysis procedures described in ATC-40 [1] or FEMA-273 [2] and FEMA-356 [3]. It is widely agreed that nonlinear static analyses are satisfying when a target displacement is encountered and commonly, this target displacement is determined from the analysis of an equivalent SDOF system.

Recently, many researchers realized studies on SDOF systems, using different structural characteristics, to pronounce a few: yield strength, ductility, strain-hardening ratio, structural safety levels, etc. within a period range of SDOF systems under the effect of recorded strong motions. Despite the attractiveness of bilinear model's simple application, many studies have demonstrated the significance of stiffness degradation, especially for reinforced-concrete (RC) structures of low to moderate heights.

In this investigation, it is aimed to contribute the recent research by analyzing 156 different SDOF "stiffness-degrading" systems having eight different either ductility or yield-strength levels with four different strain-hardening ratios. For comparative purposes, all of the structures are investigated by both bilinear and stiffness-degrading (modified-Clough) hysteretic models. Excluding the three ensembles of NERHP site classes B, C, D and two small magnitude ensembles, the rest 130 ground motions of Chintanapakdee and Chopra's [4] strong motion database is used. The selected 130 motions, making a total of six ensembles represent large magnitude small or large distance earthquakes as well as near field motions with fault normal and parallel components either recorded on rock or soil. Computations are carried out for constant ductility or constant yield strength (reduction factor) SDOF systems having a damping ratio of and median and dispersion values of the inelastic deformation ratios in Equation (61)

are obtained for all six of the ground motion ensembles. Here, is the peak deformation of the inelastic system while is the peak deformation for the structure to remain elastic.

It is illustrated in the study that the significant difference in the values of the inelastic deformation ratios calculated by bilinear and stiffness-degrading models are observed within the acceleration sensitive region of the response spectrum. The median of or values in this region are higher for stiffness degrading systems hence the increments in the number of inelastic excursions. Inelastic deformation demand is found to be increasing when the structure becomes weaker (when increases). When the two structural models are compared with each other, it is clearly seen that in the acceleration sensitive region, stiffness-degrading systems demand more while they are almost the same in the velocity sensitive region. Bilinear systems seem to have higher inelastic deformation ratios in the displacement sensitive region. Besides, equations for the limiting values of and for or , which are established based on the structural dynamic theory for bilinear systems, are also valid for stiffness-degrading systems.

It is concluded that low to moderate rise RC structures in earthquake zones should be evaluated considering the stiffness degradation effect hence its significance in the acceleration sensitive region of the related spectrum. Finally, two empirical equations to calculate the inelastic deformation demands of stiffness degrading SDOF systems are proposed.

1

Applied Technology Council, "Seismic evaluation and retrofit of concrete buildings", Report No. ATC-40, Redwood City, California, 1996.

2

Building Seismic Safety Council, "NEHRP guidelines for the seismic rehabilitation of buildings", FEMA-273, Federal Emergency Management Agency, Washington D.C., 1997.

3

Building Seismic Safety Council, "Prestandard and commentary for the seismic rehabilitation of buildings", FEMA-356, Federal Emergency Management Agency, Washington D.C., 2000.

4

Chintanapakdee, C., Chopra, A.K., "Evaluation of the modal pushover analysis procedure using vertically 'regular' and irregular generic frames", EERC Report, EERC-2003-03, University of California, Berkeley, 2003.

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