Reliable estimation of the mean annual frequency of collapse by considering ground motion spectral shape effects


Ground motion record (GMR) selection is an important issue in the nonlinear dynamic analysis procedure. Most of the current design codes recommend to use GMRs in which their mean spectrum be matched to a design spectrum e.g. uniform hazard spectrum. However recent research results have shown that the code methodology is neither robust nor realistic. On the other hand, GMR selection, based on spectral shape, is recently proposed in order to deal with this problem. \(\varepsilon \) and \(\eta \) are two powerful spectral shape indicators which are used for GMR selection purposes. A comparison between \(\varepsilon \) and \(\eta \) was made in order to access their capability to predict the linear spectral shape and the structural nonlinear response. The \(\eta \)-based conditional mean spectrum (E-CMS), which has been recently emerged as a new design spectrum, was also investigated in this study. The E-CMS formulation format is fully compatible with the existing CMS definition which makes E-CMS quite easy to be implemented. The resulted E-CMS was used as a target spectrum for the record selection. Analysis of a set of multi degree of freedom systems shows that the mean annual frequency of collapse is achievable, with more reliability, based on the new emerged \(\eta \) indicator. Therefore, the bias is decreased by employing the \(\eta \) concept into the record selection procedure. The bias reduction is more significant in higher hazard levels and in the case of structures with low natural periods or with significant higher mode effects.

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The authors are very grateful to Curt Haselton for providing us with the numerical models of the structures of Table 5. The authors are also very grateful to three anonymous reviewers for their important and valuable comments which helped to improve the paper.

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Correspondence to Alireza Azarbakht.



A proposed closed form solution, in order to obtain the correlation coefficient between \(\eta \) values, is presented in Eq. (12) for the purpose of practical applications. The genetic programming approach Banzhaf et al. (1998) was employed to derive (or evaluate) this relationship.

$$ \begin{aligned}&C_1 =I(T_{\min } +0.3592)+\cos (1.45\tan ^{-1}(T_{\max } )\times \cos (8.11T_{\max } )) \nonumber \\&C_2 =\tan ^{-1}\left( \frac{-I\cos (8.94T_{\max } )}{3.98}\right) \nonumber \\&C_3 =\cos \left( \cos \left( \frac{I+T_{\min } }{2T_{\max } }\right) \right) -\exp \left( \cos (T_{\min } )-4.054\right) -0.6114 \nonumber \\&C_4 =\cos \left( \max \left( \frac{I}{T_{\max } },-T_{\max } \right) \right) \nonumber \\&{\rho }^{\prime }_{\eta (T),\eta (T^{*})} =\left\{ {{\begin{array}{ll} C_1 +C_2 &{}\quad T_{\max } <0.3 \& T_{\min } <0.15 \\ C_3 +C_4 &{}\quad Otherwise \\ \end{array}}} \right. \end{aligned}$$

where \(T_{min}\) and \(T_{max}\) are, respectively, the small and large periods; \(I\) is the difference between two periods which is always negative (or with negative sign). The valid period range in Eq. (12) is between 0.01 to 5 s. A sample comparison is shown in Fig. 17 to compare the proposed relationship with the observed correlation coefficient values.

Fig. 17

a Correlations from empirical database. b Correlations obtained from predictive model

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Azarbakht, A., Shahri, M. & Mousavi, M. Reliable estimation of the mean annual frequency of collapse by considering ground motion spectral shape effects. Bull Earthquake Eng 13, 777–797 (2015).

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  • Uniform hazard spectrum
  • \(\eta \)-Based conditional mean spectrum
  • \(\varepsilon \) Indicator
  • \(\eta \) Indicator
  • Ground motion record selection
  • Seismic hazard