Question 17.2: Construct response spectra for the NS component of E1 Centro......
Construct response spectra for the NS component of E1 Centro earthquake (see Fig. 17.5a). Consider damping factors 0, 0.02, 0.05, 0.1 and 0.2.
The spectral displacement S_{d}, spectral velocity S_{v} and spectral acceleration S_{a} are determined from Eq. 17.9a, b and c respectively.
s_d=|u(t)|_{\max } 17.9a
s_v=|\dot{u}(t)|_{\max } 17.9b
s_a=\left|\ddot{u}_t(t)\right|_{\max } 17.9c
The responses are evaluated numerically by direct integration of the equation
\ddot{u}+2~ \rho ~\omega_n ~\dot{u}+\omega_n^2~ u=-\ddot{u}_g 17.10
for ρ = 0.02, 0.05 and 0.1 from which maximum responses are determined. The response spectral S_d, S_v and S_{a} are presented in Fig. 17.5b, c and d due to the El Centro earthquake (NS) shown in Fig. 17.5a.
Let us define pseudo-spectral velocity and pseudo spectral acceleration as S_{pv}, S_{pa} as S_{p a}=\omega_n^2~S_d=\omega_n ~S_v ; S_{p v}=\omega_n~ S_d. Usually the parameters S_{pv} and S_{pa} have certain characteristics that are of practical interest. The pseudospectral velocity S_{pv} is close to spectral velocity S_{v} for short period structure. The comparison between S_{pv} and S_{v} for the NS component of the El Centro earthquake for ς = 0.05 is illustrated in Fig. 17.6. For zero damping, the pseudo-spectral acceleration S_{pa} is identical to spectral acceleration. However, for damping other than zero, the two are slightly different. Nevertheless for damping levels encountered in most engineering applications, the two can be considered practically equal.
S_{p a}=\omega_n^2 ~S_d 17.11a
S_{p v}=\omega_n ~S_d 17.11b
Hence the spectral relationship significantly expedites the construction of earthquake response spectra. Evaluation of spectra displacement s_{d} after numerical integration to obtain time history response, the corresponding pseudo spectral velocity S_{pv} and pseudo spectral acceleration S_{pa} can readily be established and we see later how s_{d}, s_{pv} and S_{pa} can then all be plotted on a four-way logarithmic paper.
Then for a given frequency or for a given period all the spectral quantities can be read simultaneously for the same tripartite plot. A tripartite plot of s_{d}, s_{pvr}, S_{pa} for the NS component of the El Centro earthquake for various damping factors is presented in Fig. 17.7.
