# Quantification of Energy-Related Parameters for Near-Fault Pulse-Like Seismic Ground Motions
## 6 Closed-Form Approximation of Energy-Related Spectra  
Closed-form approximations for the spectra of relevant energy-related parameters are now proposed. Particularly, the following functional form is proposed to approximate the spectra of input energy and input energy reduction factor:  
\[\mathcal{F}_{1}=b_{1}+a_{1}\left(\frac{T}{\overline{T}_{1}}\right)\left[ \frac{n_{1}}{n_{1}-1+\left(\frac{T}{\overline{T}_{1}}\right)^{n_{1}}}\right] \tag{2}\]  
On the other hand, it is proposed to approximate the spectra of the hysteretic energy reduction factor using the following functional form:  
\[\mathcal{F}_{2}=a_{2}\left(\frac{T}{\overline{T}_{2}}\right)\left[\frac{n_{2} }{n_{2}-1+\left(\frac{T}{\overline{T}_{2}}\right)^{n_{2}}}\right] \tag{3}\]  
Finally, the spectra of the cumulative plastic deformation ratio is approximated using a modified version of the functional form employed in [28]:  
\[\mathcal{F}_{3}=\left[a_{3}+\left(\frac{1}{\mu}-1\right)e^{-b_{3}\ T^{3}\ \mu^{-n_{3}}}\right]^{-1}. \tag{4}\]  
These closed-form approximations involve some constants, the numerical values of which are determined through a regression analysis intended to minimize the sum of squared residuals. In order to provide an effective support for the development of energy-based design spectra, mean and standard deviation values were considered in the regression analysis. Additionally, there was also the need to look for a compromise between simplicity and accuracy. To this end, several preliminary regressions analyses were conducted by employing different combinations of the considered seismological parameters and hysteretic behaviors. In so doing, the best compromise was found by considering the earthquake magnitude only together with the type of hysteretic behavior (i.e., non-degrading or degrading) and the associated ductility level. This agreed with the outcomes of the sensitivity analysis. In fact, the earthquake magnitude was found to be the most influent parameter, since it directly affected both pulse period and maximum seismic demand value. The influence of soil type was found to be rather modest, perhaps because of a statistical distortion in the available database, since selected records belong to classes B and C according to the Eurocode 8 soil classification. Since near-fault earthquakes were herein considered, it seems reasonable that the less influential seismological parameter was the site-to-source distance. Furthermore, the sensitivity analysis highlighted that large differences in the spectral values can occur when comparing non-degrading and degrading behaviors (regardless of the specific hysteretic model, at least for typical behaviors observed in ordinary civil constructions made of steel, masonry or concrete) for different ductility levels. After parameters calibration, it was found that the proposed closed-form approximations provided very good predictions of the energy demand values (see, for instance, the results in Figures 10-13). It was pointed out that the regression analysis in Figures 11 and 13 was valid for \(T>0.1\) s, whereas in general, a better accuracy was achieved for \(T\leq 4\) s. The greater accuracy for low-mid period values did not depend on the regression model or the parameter calibration. Conversely, this was because there were more spectral values at low period values, whereas just a few data points were available for mid-long period values. This, in turn, implied that the regression analysis was forced to achieve a better accuracy level for low-mid period values rather than for mid-long ones. This is a desired output, since low period values are of interest for most ordinary structural applications. An attempt was made to correlate the constants involved in Equations (2)-(4) with the earthquake magnitude. In doing so, satisfactory closed-form correlations were obtained for the constants involved in the regression formula of the spectra of the hysteretic energy reduction factor (see Figures 14 and 15).