# Improvement of Performance Level of Steel Moment-Resisting Frames Using Tuned Mass Damper System
## 6 Modeling and Verification
Moreover, the parameters \(c_{unit}^{1}\) and \(c_{unit}^{2}\) are the two coefficients for unit conversion, while \(E\) depicts Young's modulus, and \(I\) shows the moment of inertia. The parameters \(\theta_{p}\), \(\theta_{pc}\), \(\theta_{y}\), exhibit pre-capping plastic rotation in monotonic loading, post-capping plastic rotation, and yield rotation, respectively. The parameter \(L\) is the beam shear span (distance from plastic hinge location to the point of inflection); \(\Lambda\) shows the capacity of the reference cumulative rotation, and \(M_{y}\) is the effective yield moment [56]. The mass of all stories and also the values of the damping ratio for all modes are assumed to be equal to \(m=300\),000 kg, and 3%, respectively. The Young's modulus is 200 MPa. The maximum allowable stroke for TMD is considered to be equal to 1000 mm. The values for the first six natural frequencies of the six-story steel frame are calculated numerically and compared with the results presented by [PERSON] in [39] and [PERSON] et al. in [46] to validate the model (see Table 1). It can be seen from Table 1 that the differences are negligible. It confirms the accuracy of the modeling approach. Figure 2: Six-story steel moment-resisting frame. Additionally, the time history of cumulative hysteresis energy, \(E_{h}\), (see Equation (23)) for the structure with (W) and without (W/O) TMD are compared with [46] (see Figure 3) subjected to the Northridge earthquake (see Figure 4). The values of mass, stiffness, and damping ratio of the TMD system are considered to be equal to 180,000 N-s\({}^{2}\)/m), 5,264,000 N/m, and 0.05 based on [46], respectively. Figure 3 also shows the verification of the result obtained by this study and described in [46]:  
\[E_{h}=\sum_{k=1}^{n_{e}}E_{h,k} \tag{23}\]  
where \(E_{h,k}\), \(k\), and \(n_{e}\) present the summation of work done by internal forces (e.g., moment, axial, shear), an integer counter, and the number of elements, respectively. In this paper, to optimize and analyze the structure dynamically, several real earthquakes and one artificial earthquake are selected (see Table 2). Figure 5 shows the acceleration time history of the artificial earthquake. The artificial earthquake is calculated using the Gaussian White Noise process and based on the Kanai-Tajimi filter and power spectral density function (PSDF) [58; 59; 60]: \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode Number**} & \multicolumn{3}{c}{**Natural Periods of Vibration (rad/s)**} \\ \cline{2-4}  & **[PERSON] [39]** & **[PERSON] et al. [46]** & **Present Study** \\ \hline
1 & 5.15 & 5.07 & 5.07 \\
2 & 14.28 & 13.96 & 14.22 \\
3 & 25.13 & 25.13 & 25.13 \\
4 & 34.91 & 34.91 & 34.91 \\
5 & 44.88 & 44.88 & 44.88 \\
6 & 57.12 & 57.12 & 57.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The benchmark Special Moment Resisting Frame’s (SMRF’s) natural frequencies of vibration. Figure 4: Acceleration time history of the Northridge 1994 earthquake. Figure 3: The cumulative hysteresis energy of the controlled and uncontrolled six-story steel frame building. \[S_{Kanai-Tajimi}(\omega)=S_{0}[\frac{\omega_{g}^{4}+(2\omega_{g} \times\xi_{g}\times\omega)^{2}}{\left(\omega^{2}-\omega_{g}^{2}\right)^{2}+(2 \omega_{g}\times\xi_{g}\times\omega)^{2}}] \tag{24}\] \[S_{0}=\frac{0.03\times\xi_{g}}{\pi\omega_{g}\times(4\times\xi_{ g}^{2}+1)}\]  
where \(S_{0}\), \(\omega_{g}\), and \(\xi_{g}\) are the intensity of the PSDF, frequency, and damping of the soil, respectively. In this paper, \(\omega_{g}\) and \(\xi_{g}\) are considered to be equal to 25.13 rad/s and 0.8 rad/s, respectively, based on [61], which indicates that the structure has been located on the stiff soil. The strong ground motion of the artificial earthquake is big enough, and it can be used in the dynamic analysis of the structures based on Uniform Building Code 97 [62]. It should be noted that [PERSON]'s classification has been considered in the selection of the earthquake to cover all existing categories for the earthquakes (see [8; 63]).