Report - PEER - University of California, Berkeley

Acceleration-Displacement PlotYield Point SpectraSpectral AccelerationA eT sT > Tsµ=1 (elastic)A yA µ=3d∆ d∆ y=∆ u/3∆ u(a)Spectral Displacement, S dStrength CoefficientT sT > TsFigure 3. (a) ADRS format and (b) YPS format for response spectra, for µ=3.and viscous damping. Both approaches require calibration to the response of inelasticSDOF oscillators. While either approach may be used, it is often easier to developinelastic response spectra using displacement modification. Users of these approachesshould be aware of the large inherent variability in response amplitudes.Displacement modification relationships typically are derived on the basis ofanalyses in which oscillator strengths are adjusted to achieve constant ductility (µ)values or are determined using constant strength ratios (R factors). Regression of theresponse data leads to R-µ-T or R-C 1 -T relationships. Either form may be used, as C 1= ∆ u /S d and µ = ∆ u /∆ y = (∆ u /S d )(S d /∆ y ) = C 1 R, where ∆ u = peak displacement, ∆ y = yielddisplacement, and S d = spectral displacement (of the linear oscillator).Of particular relevance to design is the strength resulting in an expected ductility(or displacement) demand equal to a desired value. This is determined directly instudies in which constant R factors are used and the resulting displacement orductility demands are regressed. R-µ-T relationships, when derived on the basis ofconstant ductility responses, represent the mean of the R values associated with agiven ductility response. Oscillator strengths (V y ), however, are determined using theinverse of R (since V y = S a·m/R) and the mean of the 1/R values differs from the meanof the R values. Furthermore, slight nonlinearity in the R-µ relationship indicates thatthe expected ductility will tend to exceed the value of constant ductility that was usedto determine the R factor. Although the differences are small, it is preferable to userelationships derived on the basis of constant R values.Distinct from the relationship used to estimate inelastic response is the graphicaldepiction of inelastic demands. The Capacity Spectrum Method (e.g., ATC-40, 1996)provides engineers with an intuitive and easily visualized means for understanding therelationship between structural properties (initial stiffness and strength) and peakdisplacement. This graphical format was retained in the improvements suggested byFajfar (1999) and Chopra and Goel (1999). A similar visualization is available withthe Yield Point Spectra (YPS) format, which plots curves of constant ductilitydemand on the axes of yield displacement and yield strength coefficient. Figure 3compares these formats for oscillators having µ=3.C eµ=1 (elastic)C yC dµ=3∆ d ∆ y ∆ u=3∆ yYield Displacement, ∆ y(b)484