## 2 Ground Motion Intensity Measures
### Spectral Shape IMs  
As discussed earlier, _Sa(T)_ considers the response of a building with fundamental frequency \(T\) to a given ground motion record. However, due to the presence of higher modes, the response is also affected by ground motion contributions at other periods. Previous studies observed that the addition of spectral information in IMs reduces variability in the EDP-IM relationship significantly ([PERSON] and [PERSON] [4], [PERSON] and [PERSON] [20], [PERSON] and [PERSON] [21]). [PERSON] and [PERSON] [4] reported an increase in efficiency for tall buildings by including spectral values at the second and third periods. Studies accommodate spectral shape by combining spectral values over a variety of periods [22; 23; 24; 25; 26]. Based on time history analyses of single degree of freedom structures (SDOF) with different hysteretic behaviors, [PERSON] and [PERSON] [22] found the use of _Sa_ at an elongated natural period to be more efficient. To account for period elongation, [PERSON] et al. [23] proposed a new IM (Equation (2)), with optimal values of \(c=2\) and \(\alpha=0.5\) obtained by calibrating the response of four moment frame structures subjected to 16 near- and far-field ground motion records. Later, [PERSON] [24] replaced the multiplier _/c_/ in Equation (2) with a relative lateral strength parameter (\(R\)). The efficiency was tested by subjecting 80 records to many SDOF systems with different hysteretic behaviors and values of \(R\). This provides flexibility to the IM to adapt for buildings with different non-linear behaviors. \[IM_{Cordova}=S_{a}\Big{(}T_{f}\Big{)}\cdot\Bigg{[} \begin{array}{c}S_{a}\Big{(} cT_{f}\Big{)}\\ S_{a}\Big{(}T_{f}\Big{)}\end{array} \Bigg{]}^{\alpha} \tag{2}\]  
[PERSON] and [PERSON] [17] showed that for structures with small contributions from higher modes, any reasonable choice for period elongation results in smaller dispersion. However, when higher modes are significant, two new IMs are proposed, listed in Equations (3) and (4), in which \(T_{a}\), \(T_{b}\), \(T_{c}\) are empirically defined periods whose values were computed from time history analysis of 5, 9 and 20 story frames subjected to 30 ground motion records. It was observed that any reasonable value higher than the first three elastic periods for \(T_{a}\), \(T_{b}\) and \(T_{c}\) provides smaller dispersion [17]. Later, [PERSON] [25] and [PERSON] et al. [26] proposed modifications to Cordova [23] and Vamvatsikos and Cornell [17] by using three different frame configurations (4, 10 and 16 stories). The frames were subjected to 30 scaled and unscaled ground motion records. Equation (5) for short periods (\(T_{f}<1.5\) s) considers period elongation, and Equation (6) for long periods (\(T_{f}>1.5\) s) accounts for higher mode effects [25]. \[IM_{Vamvatsikos}=S_{a}(T_{a})\cdot\Bigg{[} \begin{array}{c}S_{a}(T_{b})\\ S_{a}(T_{a})\end{array} \Bigg{]}^{\beta} \tag{3}\]  
\[IM_{Vamvatsikos}=S_{a}(T_{a})\Bigg{[} \begin{array}{c}S_{a}(T_{b})\\ S_{a}(T_{a})\end{array} \Bigg{]}^{\gamma} \tag{4}\]\[IM_{Lin,short}=S_{a}(T_{1})^{a}\cdot S_{a}(CT_{1})^{1-\alpha} \tag{5}\]  
\[IM_{Lin,\;long}=S_{a}\Big{(}T_{f}\Big{)}^{\beta}\cdot S_{a}(T_{2})^{1-\beta} \tag{6}\]  
The above IMs only consider spectral information at certain periods and lack information of spectral shape. [PERSON] et al. [27] proposed an alternative IM (Equation (7)) based on spectral values computed at 10 logarithmically spaced values between \(T_{f}\) and \(T_{n}\). The efficiency of this IM when compared to \(Sa(T_{f})\) and \(PGA\) was better for frames with higher mode effects. [PERSON] et al. [28] modified Equation (7) to incorporate P-delta effects (secondary effects when axial and transverse loads are applied simultaneously) by including a period band \(\delta T\) (Equation (8)). Similar IMs were put forward by [PERSON] and [PERSON] [29; 30], [PERSON] et al. [31; 32], and [PERSON] et al. [33; 34]. For near-field ground motions, the spectral frequency content around the first mode of vibration is more efficient as IM than \(Sa(T_{f})\)[19].