## 3 Research, Results, and Discussion
### Numerical Models of Steel Ordinary Moment Frames  
To evaluate the seismic performance of model OMFs, a numerical model is constructed using OpenSees software [33]. Figure 3 presents a nonlinear numerical model used for steel OMFs. The numerical model consists of columns, beams, and panel zones, and a leaning column installed to simulate the \(P-\Delta\) effect induced by gravity loads. The leaning column is modeled as a rigid truss element and connected to the ground by a pin. The leaning column is connected to the OMF using pins (Figure 3a). The gravity axial load applied on the leaning column is calculated with effective seismic weight (1.05 \(DL+0.25\)\(LL\)) [29], where \(DL\) and \(LL\) are the superimposed dead and live loads, respectively. The damping matrix is constructed using the commonly used Rayleigh damping matrix based on a damping ratio of 2% for the first and fifth modes [34; 35; 36]. It is noted that in NIST GCR 17-917-46v1 [34] and ATC 72-1 [35], it is recommended that the first-mode period and the third or higher \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Story** \\ **Number** \\ \end{tabular} & \begin{tabular}{c} **W0-OMF** \\ **(\(T_{1}\), s)** \\ \end{tabular} & \begin{tabular}{c} **Calculated** \\ **Period** \\ **(\(T_{1}\), s)** \\ \end{tabular} & \begin{tabular}{c} **Calculated** \\ **Period** \\ **(\(T_{1}\), s)** \\ \end{tabular} & \
\begin{tabular}{c} **Calculated** \\ **Period** \\ **(\(T_{1}\), s)** \\ \end{tabular} &
\begin{tabular}{c} **Upper Limit** \\ **of Period** \\ **(\(T_{1}\), s)** \\ \end{tabular} \\ \hline
9 & 09-story W0 & 3.72 & 09-story W44 & 3.72 & 09-story W51 & 3.57 & 1.92 \\ \hline
12 & 12-story W0 & 4.44 & 12-story W44 & 4.41 & 12-story W51 & 4.20 & 2.41 \\ \hline
15 & 15-story W0 & 5.05 & 15-story W44 & 4.91 & 15-story W51 & 4.65 & 2.88 \\ \hline
18 & 18-story W0 & 5.13 & 18-story W44 & 4.84 & 18-story W51 & 4.60 & 3.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated fundamental period (\(T_{1}\)) and upper limit of period (\(C_{u}T_{2}\)) according to ASCE 7-16 of the designed steel ordinary moment frames. period should be used to construct the Rayleigh damping matrix. In this study, the first and fifth modes were used. In order to avoid unrealistic damping forces, stiffness- and mass-proportional damping terms are constructed only using the elastic line elements for beams and columns, and nodes for lumped mass [36]. Individual beam and column elements are modeled using an elastic line element with rotational plastic hinge elements placed at both ends of the elastic element. The moment capacity of the column section should be reduced due to stress generated by axial force. The modeling parameters of the concentrated plastic hinges for steel columns are determined according to [PERSON] et al. [37]. The welded unreinforced flange-bolded web (WUF-B) connection is used for OMFs in this study, which is modeled as shown in Figure 3b. A panel-zone consists of four rigid line elements and one rotational spring element simulating the trilinear behavior of the panel zone [38]. To simulate the cyclic behavior of the WUF-B connection, the modified IMK model [39] is used. As shown in Figure 3b, \(\theta_{p}\) is the drift ratio (\(\theta\)) at capping strength, \(\theta_{g}\) is the \(\theta\) value at fracture, \(M_{c}\) is the capping moment strength, and \(M_{y}\) is the effective yield strength. The \(M_{c}\) and \(M_{y}\) values for beams are estimated according to [PERSON] and Krawinkler [19], and the \(\theta_{p}\) and \(\theta_{g}\) values are estimated according to ASCE 41 [40]. The residual strength is defined as 20% of the plastic moment strength for WUF-B connections according to [PERSON] et al. [41] and [PERSON] et al. [42]. The accuracy of the connection model was verified by [PERSON] et al. [43]. In Figure 3b, empty circles with black, blue, and red colors denote yielding, capping, and fracture, respectively. Panel zones are modeled according to [PERSON] and [PERSON] [38].