## S

Note that the elastic design spectrum already incorporates 5% damping, hence the equivalent damping (Equation 21.27) from inelastic behavior must be added to the elastic viscous damping. For behavior other than bilinear hysteresis, a modification factor k is introduced. The final damping value incorporating elastic damping, equivalent inelastic damping, and general hysteretic behavior is given by

Finally, the elastic spectrum is transformed into a reduced spectrum for the damping ratio given by Equation 21.28. ATC-40 provides the following spectral reduction factors, which are derived using the well-known Newmark-Hall relationships:

where SRA is the reduction factor in the constant acceleration region of the spectrum and SRV is the reduction factor in the constant velocity region. There are imposed limits on the above reduction factors depending on the expected shape of the hysteresis loops.

### 21.5.3.2.1 Performance Point

The CSM methodology attempts to predict the expected peak displacement given a demand response spectrum. A trial displacement value dt (Figure 21.12b) and the corresponding spectral acceleration is selected on the capacity curve. The equivalent viscous damping associated with these spectral magnitudes can now be estimated using Equation 21.27. The 5% damped elastic spectrum is transformed into a reduced inelastic spectrum using Equations 21.20 and 21.21. The demand and capacity curves in ADRS format are overlaid. If the trial displacement is within 5% of the displacement at the intersection of the demand and capacity curves, the performance point has been located. Figure 21.12b shows a case where the trial displacement is the performance point.

### 21.5.3.3 Performance Assessment

Once the performance point has been established, the acceptability of the design can be assessed by comparing the demand to the acceptance criteria. ATC-40 global acceptance criteria for different performance levels are listed in Table 21.7. Component acceptance criteria are essentially similar to FEMA-356 specifications.

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