## 4 Methodology and Procedure
### M and C Matrices  
In the analysis of a steel building structure, the mass matrix is commonly obtained by assuming that the mass is concentrated at the nodes. Figure 4 shows the degrees of freedom associated to translations and rotations of a prismatic member with straight axes of a plane steel frame, while Equation (1) illustrates the corresponding mass matrix for an individual element (\(M_{LE}\)). In such an equation, \(\overline{m}\) and \(L\) represent the mass and the member length, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Event**} & \multirow{2}{*}{**Mw**} & \multirow{2}{*}{**R (km)**} & \multicolumn{3}{c}{**PGA (g)**} & \multicolumn{3}{c}{**Period (s)**} & \multicolumn{3}{c}{**PGV**} \\  & & & & **N-S** & **E-W** & **N-S** & **E-W** & **N-S** & **E-W** \\ \hline Imperial Valley, 1940 & 6.9 & 10 & 0.46 & 0.68 & 0.53 & 0.46 & 12 & 10 \\ \hline Imperial Valley, 1979 & 6.5 & 4.1 & 0.39 & 0.49 & 0.16 & 0.34 & 14 & 11 \\ \hline Landers, 1992 (g) & 7.3 & 36 & 0.42 & 0.43 & 0.73 & 0.33 & 7 & 10 \\ \hline Kern, 1952 & 7.3 & 25 & 0.52 & 0.36 & 0.25 & 0.23 & 3 & 3 \\ \hline Loma Prieta, 1989 & 7 & 12.4 & 0.67 & 0.97 & 0.21 & 0.2 & 9 & 15 \\ \hline Northridge, 1994, Newhall & 6.7 & 6.7 & 0.68 & 0.66 & 0.31 & 0.31 & 9 & 22 \\ \hline Northridge, 1994, Rinaldi & 6.7 & 7.5 & 0.53 & 0.58 & 0.39 & 0.29 & 58 & 29 \\ \hline Northridge, 1994, Sylmar & 6.7 & 6.4 & 0.57 & 0.82 & 0.31 & 0.36 & 36 & 35 \\ \hline North Palm Springs, 1986 & 6 & 6.7 & 1.02 & 0.99 & 0.17 & 0.21 & 8 & 22 \\ \hline Coyote Lake, 1979 & 5.7 & 8.8 & 0.59 & 0.33 & 0.15 & 0.21 & 8 & 5 \\ \hline Morgan Hill, 1984 & 6.2 & 15 & 0.32 & 0.55 & 0.18 & 0.16 & 7 & 8 \\ \hline Parkfield, 1966, Cholame SW & 6.1 & 3.7 & 0.78 & 0.63 & 0.37 & 0.3 & 4 & 4 \\ \hline Parkfield, 1966, Cholame SW & 6.1 & 8 & 0.69 & 0.79 & 0.17 & 0.21 & 3 & 3 \\ \hline North Palm Springs, 1986 & 6 & 9.6 & 0.52 & 0.38 & 0.13 & 0.21 & 11 & 26 \\ \hline Whititer, 1987 & 6 & 3.62 & 0.77 & 0.48 & 0.7 & 0.28 & 11 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Strong motion records. \[M_{LE}=\frac{\overline{m}L}{2}\left[ \begin{array}{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&0\end{array} \right] \tag{1}\]  
It must be noted that the nonzero values are associated to the DOFs 1, 2, 4 and 5 shown in Figure 4, which correspond to translational DOFs. The subscripts \(L\) and \(E\) in the symbol \(M_{LE}\) stand for the terms lumped and element, respectively. It is known that the consistent-type mass matrix is more appropriate than the concentrated-type one because the rotational effects [50,51] are considered in the former. By using the d'Alembert's principle, the virtual work concept and the appropriate interpolation functions, together with the Finite Element Method, it is shown that the element consistent mass matrix (\(M_{CE}\)) is:  
\[M_{CE}=\frac{mL}{420}\left[ \begin{array}{cccccc}140&0&0&70&0&0\\ 0&156&22L&0&54&-3L\\ 0&22L&4L^{2}&0&13L&-3L^{2}\\ 70&0&0&140&0&0\\ 0&54&13L&0&156&-22L\\ 0&-13L&-3L^{2}&0&-22L&4L^{2}\end{array} \right] \tag{2}\]  
Rayleigh damping model is expressed by Equation (3). In such an equation \(a_{0}\) and \(a_{1}\) are proportionality constants that are obtained by fixing modal damping ratios at two modes, namely the \(i\) (\(S_{i}\)) and \(j\) (\(S_{j}\)) modes. The damping ratio for the \(n\) mode is defined as shown in Figure 5, where \(\omega_{n}\) is the vibration frequency of that mode. It can be seen (Figure 5) that very large amounts of damping can result in the higher modes. It is argued [44,50,51] that if the \(i\) and \(j\) modes are adequately selected, the higher mode contribution will be effectively removed because of the very large amounts of damping; however, nothing is stated concerning the resulting very large damping forces. Results of using several combinations of the \(M\) (\(M_{L}\) and/or \(M_{C}\)) and \(K\) (\(K_{o}\) and/or \(K_{t}\)) matrices in Equation (3) can be found in the literature [3,34,40,41].