Inelastic Spectrum-Based Approach for Seismic Design Spectra

A consistent approach is proposed for deriving inelastic design spectra and estimates of maximum displacement directly through statistical studies of inelastic response spectra, without the need to resort first to elastic spectra. The main finding is that the seismic response coefficient, or dimensionless yield strength, $C_y$ , that will maintain the mean ductility ratio, $\overline{\mu }$ , equal to a prescribed value can be estimated by an expression of the form $C_y(T,\overline{\mu })=C\left(T\right){\overline{\mu }}^{-n\left(T\right)}$ . $C$ is interpreted as a mean unreduced inelastic spectrum, and $n$ depends only on the elastic natural period, $T$ , of the structure. Explicit formulas for both $C$ and $n$ are obtained for a set of 87 accelerograms recorded in California. $C$ differs from the mean 5% damped elastic spectrum. Another significant result is that $C_y(T,\overline{\mu }=2)$ can be closely approximated by a highly damped mean elastic spectrum, i.e., $\overline{C_e}(T,\xi )$ , with $\xi$ , the critical damping ratio taken to be 30%. Based on these two results, $C_y$ can be conveniently written as $C_y(T,\overline{\mu })=\overline{C_e}(T,30){(\overline{\mu }∕2)}^{-n\left(T\right)}$ . This means that the seismic coefficient of an inelastic system can also be expressed approximately in terms of a highly damped mean elastic spectrum divided by a reduction factor that depends only on $\overline{\mu }$ and $T$ . With $C_y$ determined, it is straightforward to approximate consistently the normalized mean maximum relative displacement of the structure as the product of $C_y$ and $\overline{\mu }$ ; the approximate results differ by less than 10% from the corresponding exact values, for elastic natural periods between 0.1 and $3.0\mathrm{s}$ and mean ductility ratios up to 5.