## Ej1 wtF EN 1 W 2n

where Sa,n is the spectral acceleration for mode n, Sd,n is the spectral displacement for mode n, Vn is the base shear for mode n, W is the seismic weight of the building, w; is the seismic weight of the floor at level i, f ,-,n is the modal amplitude at level i for mode n, and Dc,n and Fc,n are the displacement and modal amplitude of control node for mode n.

The theoretical basis for making the conversion is outlined in the Appendix A.

Period T, s

FIGURE 21.12 Generating the ADRS demand and capacity spectrum.

Period T, s

Capacity curve

Capacity curve

Elastic spectrum

Reduced spectrum

Spectral displacement, Sd

Elastic spectrum

Reduced spectrum

Spectral displacement, Sd

FIGURE 21.12 Generating the ADRS demand and capacity spectrum.

### 21.5.3.2 Determining Demand

The next step in CSM is to convert the response spectrum (which represents the demand side of the equation) into ADRS format as well, thereby permitting a comparison of demand versus capacity. The design response spectrum is usually expressed in terms of spectral acceleration and period as shown in Figure 21.12a. Since a response spectrum results from the analysis of an SDOF system, the following relationships between pseudospectral acceleration and spectral displacement can be used:

SaT 2

Hence, it is possible to convert the spectral coordinates for every value of the period T, into spectral displacements. The resulting curve is in ADRS format and the radial lines shown in Figure 21.12b represent l values for corresponding T, values. The capacity curve obtained in the previous step can now be superimposed on the demand curve. If the elastic design spectrum is used to create the demand spectrum, the overlay is valid only if the structural response is also elastic. Hence, the next step in the process is to reduce the elastic response spectrum to an inelastic spectrum using the concept of equivalent damping. Using the fundamental principles of structural mechanics, the equivalent damping zd associated with dissipated energy during inelastic response is given by

where (fi/o) is the ratio of the forcing frequency to the natural frequency of the system, ED is the energy dissipated through hysteretic behavior, and ES is the strain energy at the maximum displacement. If it is assumed that the peak response is associated with the resonant frequency, then the ratio fi/o = 1.0.

The ATC-40 methodology for estimating the equivalent viscous damping is derived for a bilinear capacity curve, therefore, it is necessary to transform the capacity curve into bilinear form. Figure 21.13 shows a bilinear capacity curve and the energy dissipated in a single cycle given the maximum displacement demand and the corresponding spectral acceleration (Sdm, Sam). The equivalent damping corresponding to the dissipated energy can be computed using Equation 21.26 and reduced to the following form:

Sam Sdm

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