## 7 Conclusions
The criterion is based on the following identity:  
\[\underline{\varepsilon}^{T}\underline{\underline{M}}\ \underline{ \varepsilon}=\left(\Gamma_{1}\right)^{2}+\left(\Gamma_{2}\right)^{2}+\ldots+ \left(\Gamma_{N}\right)^{2}=\sum_{i=1}^{N}(\Gamma_{i})^{2}, \tag{11}\]  
according to which the total inertia of the structure along the direction of excitation (\(\underline{\varepsilon}^{T}\underline{M}\underline{\varepsilon}\)) is equal to the sum of the squares of the modal participation factors. This result can be easily demonstrated by observing that the natural modes represent a base of the vector space (\(\mathbb{R}^{N}\)) and, therefore, the vector (\(\underline{\varepsilon}\)) may be expressed as a sum of its modal components (\(e_{1}\), \(e_{2}\), \(\ldots\), \(e_{N}\)):  
\[\underline{\varepsilon}=\underline{\Phi}_{V,1}e_{1}+\underline{\Phi}_{V,2}e_{2 }+\ldots+\underline{\Phi}_{V,N}e_{N} \tag{12}\]  
The components may be determined by invoking the orthogonality of the modal base. If we pre-multiply each member of Equation (12) by \(\underline{\Phi}_{V,i}^{T}\underline{\underline{M}}\), we obtain:  
\[\underline{\Phi}_{V,i}^{T}\underline{\underline{M}}\ \underline{ \varepsilon}=\underline{\Phi}_{V,i}^{T}\underline{\underline{M}}\ \underline{\Phi}_{V,1}e_{1}+\underline{\Phi}_{V,i}^{T}\underline{ \underline{M}}\ \underline{\Phi}_{V,2}e_{2}+\ldots+\underline{\Phi}_{V,i}^{T}\underline{ \underline{M}}\ \underline{\[PERSON]}_{V,i}e_{i}+\ldots+\underline{\[PERSON]}_{V,i}^{T}\underline{ \underline{M}}\ \underline{\[PERSON]}_{V,N}e_{N} \tag{13}\]  
Since it results in the following:  
\[\underline{\[PERSON]}_{V,i}^{T}\underline{\underline{M}}\ \underline{\[PERSON]}_{V,j}= \begin{cases}m_{G,i}&\text{if }j=i\\ 0&\text{if }j\
eq i\end{cases} \]  
we know, from Equation (13), that \(e_{i}=\frac{\underline{\[PERSON]}_{V,i}^{T}\underline{\underline{M}}\ \underline{ \varepsilon}}{m_{G,i}}\); therefore, Equation (12) may be rewritten as follows:  
\[\underline{\varepsilon}=\frac{\underline{\Phi}_{V,1}^{T}\underline{\underline {M}}\ \underline{\varepsilon}}{m_{G,1}}e_{1}+\frac{\underline{\Phi}_{V,2}^{T} \underline{\underline{M}}\ \underline{\varepsilon}}{m_{G,2}}e_{2}+\ldots+\frac{\underline{\Phi}_{V,N}^{T} \underline{\underline{M}}\ \underline{\varepsilon}}{m_{G,N}}e_{N}. \tag{14}\]  
Equation (11) may be derived from Equation (14) by pre-multiplying each member by \(\underline{\varepsilon}^{T}\underline{\underline{M}}\) and then by recalling the orthogonality of the modal base. If we select only \(\widetilde{N}\) modes for the evaluation of the seismic response of the structure, according to Equation (A10), we consider only a part of the actual inertia of the structure along the direction of excitation:  
\[\varepsilon^{T}\underline{\underline{M}}\,\varepsilon=I_{e}=\sum _{i=1}^{\widetilde{N}}(\Gamma_{i})^{2}+\sum_{i=\widetilde{N}+1}^{N}(\Gamma_{i} )^{2}=\sum_{i=1}^{\widetilde{N}}(\Gamma_{i})^{2}+I_{e,residual}\] (A14)  
The level of approximation of the actual seismic response is correlated with the amount of structural inertia that participates in the response; it follows that, the lower the residual inertia, the higher the precision of the obtained results. For slender metallic structures, such as the one considered in this paper, and irrespective of the direction of excitation (\(\varepsilon\)), the first 5-6 modes generally ensure that the residual inertia is lower than 3% of the actual one, which thus provides a more than satisfactory characterization of the seismic response.