# Seismic Design Procedure for Low-Rise Cold-Formed Steel-Special Bolded Moment Frames
## 5 Nonlinear Analysis
### Archetype Frames
#### 5.2.2 Dynamic Analysis  
It was also essential to determine the maximum storey drift ratio at the DBE level of the seismic intensity. ASCE 7 provides the storey drift ratio limit at the ultimate limit state (ULS), which is 2.0%. Figure 21 shows the results for the maximum storey drift ratio with 44 ground motions (i.e., 22 earthquake events with two components) at the DBE level. The ground motion sets stipulated in FEMA P695 [13] were normalised and scaled to match the maximum considered earthquake (MCE) level; therefore, all ground motions were scaled to two-thirds to be equal to the DBE level [2]. The ground motions stipulated in FEMA P695 are summarised in Appendix A. Table 7 summarises the average values of the maximum storey drift ratio at each storey. As the results show, no storey exceeded the 2.0% limitation. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Frame Name** & **S0202** & **S0203** & **S0302T** & **S0302** & **Avg. of \(\Omega\)** \\ \hline \(V_{\text{DBE}}\) (kN) & 47.15 & 110.3 & 56.05 & 60.50 & \\ \(V_{\text{max}}\) (kN) & 132.4 & 391.7 & 137.5 & 152.7 & \\ \(\Omega\) & 2.808 & 3.552 & 2.454 & 2.524 & 2.835 \\ \hline \end{tabular}
\end{table} 
Table 6: Design base shear and system overstrength factor. As shown in Table 5, a deflection application factor of \(C_{d}=5.5\) was assumed as the design value. Table 8 summarises the storey drift ratios. The deflection application factors calculated from the dynamic analysis results, \(C_{d,cal}\), ranged from 2.38 to 4.03, with an average of 3.26. This value was smaller than the assumed design value of 5.5; therefore, it can be concluded that the designed deflection application factor of \(C_{d}=5.5\) was on the conservative side. These conservative results might be due to the significant strength-hardening effects due to the bolt bearing, which was different from that in conventional steel moment-resisting frames [3]. To assess the collapse margin ratio (\(CMR\)) of the archetype frames, incremental dynamic analysis (IDA) was performed. Ground motions were normalised and scaled to adjust the seismic intensity equal to the maximum considered earthquake (MCE) level according to FRMA P695. The exceedance probability was 2% in 50 years for the MCE level, which was equal to 1.5 times the DBE level [3]. FEMA P695 defines the system collapse as being evaluated when the collapse ratio reaches 50% of the ground motions, i.e., twenty-two ground motions will take down the frame. The \(CMR\) is defined as follows:  
\[CMR=\frac{\hat{S}_{MT}}{S_{MT}} \tag{9}\]  
where \(S_{MT}\) is the ground motion intensity corresponding to the MCE level, and \(\hat{S}_{MT}\) is the ground motion intensity where the ground motion intensity is equal to a collapse ratio equal to 50%. Figure 22 shows the results from the IDA; the vertical axis is the collapse ratio, and the horizontal axis is the acceleration factor (\(AF\)). The \(AF\) is defined as follows:  
\[AF=\frac{S_{a}}{S_{MT}} \tag{10}\]  
where \(S_{a}\) is the input ground motion intensity. Therefore, the \(AF\) is equal to 1.0 when the input ground motion intensity is equal to the MCE level. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frame Name** & **S0202** & **S0203** & **S0302T** & **S0302** \\ \hline
1F & 1.226 (\%) & 1.301 (\%) & 1.141 (\%) & 1.389 (\%) \\
2F & 1.146 (\%) & 0.9087 (\%) & 1.184 (\%) & 1.219 (\%) \\
3F & - & - & 0.5977 (\%) & 0.3595 (\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average maximum storey drift ratio at the DBE level. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frame Name** & **\(\Delta_{\text{max}}\)** & **\(\Delta_{\text{DBE}}\)** & **\(K_{0}\) (\(\times\)10\({}^{2}\) kN/rad)** & **\(C_{d,cal}\) (=\(\Delta_{\text{max}}\)/\(\Delta_{\text{DBE}}\))** \\ \hline S0202 & 1.226 (\%) & 0.304 (\%) & 155.1 & 4.03 \\ S0203 & 1.301 (\%) & 0.365 (\%) & 302.2 & 3.56 \\ S0302T & 1.184 (\%) & 0.498 (\%) & 112.6 & 2.38 \\ S0302 & 1.389 (\%) & 0.454 (\%) & 133.3 & 3.06 \\  & & & Avg. & 3.26 \\ \hline \hline \end{tabular}
\end{table}