## 4 Examples
### Dynamic Analysis of the Two-Span, Six-Story Vogel Steel Frame with Nonlinear Semi-Rigid Connections  
Figure 21 shows the configuration and loading of the two-span, six-story Vogel steel frame. Each span and each floor height were 6 m and 3.75 m, respectively. The section types of beams were IPE240, IPE300, IPE360 and IPE400, while the section types of columns were HEB160, HEB200, HEB220, HEB240 and HEB260. An initial geometric imperfection of 1/450 was taken into account. The distributed loads of 31.7 kN/m and 49.1 kN/m applied on the beams were converted into lumped masses at the joints. The elastic modulus and the Poisson ratio were 205 GPa and 0.3. In order to more intuitively study the energy consumption of semi-rigid connections, the material of all the frame members was assumed to be perfectly elastic, and the viscous damping was ignored.  
Figure 19: Load-Displacement curves at each floor of the steel frame.  
Figure 20: Load-Moment curves at the right joints and support.  
In this study, there were 10 contact elements for each column while 16 contact elements for every beam. The nonlinear behavior of the semi-rigid connection was simulated by the Richard-Abbott model. The parameters of the model were as follow: \(R_{\mathrm{ki}}=12336.86\,\mathrm{kN}\cdot\mathrm{m}/\mathrm{rad}\); \(R_{\mathrm{kp}}=112.97\,\mathrm{kN}\cdot\mathrm{m}/\mathrm{rad}\); \(M_{0}=180\,\mathrm{kN}\cdot\mathrm{m}\); \(n=1.6\). The instantaneous stiffness of the contact elements at semi-rigid connections was determined by Equations (13) and (15). The frequency of the dynamic force \(\omega\) is \(1.66\,\mathrm{rad}/s\), which was close to the fundamental natural frequency of this structure. The dynamic response analysis of the frame was performed at three cases of rigid connection, linear semi-rigid connection and nonlinear semi-rigid connection, as shown in Figure 22. The comparison of the numerical results obtained using the modified MDEM and those presented by [PERSON] and Chan [PERSON] and [PERSON] (2013) illustrates that the two match well, and the modified MDEM can accurately capture the nonlinear behavior of the semi-rigid connection. In addition, it is observed that resonance occurs in the frames with rigid or linear semi-rigid connections, but not in the frame with nonlinear semi-rigid connections, the reason is that the hysteresis damping of the nonlinear connection causes energy dissipation (Figure 23).  
Figure 21: Configuration and loading of the two-span, six-story Vogel steel frame.  
In order to simulate the collapse behavior of this structure, the ultimate strain was taken as the fracture criterion. The material was elastic during the calculation, and the beam-column joints were assumed to be linear semi-rigid connections. The sine wave with the frequency of 1.66 rad/s was applied. The failure processes of steel frames with different connections are shown in Figures 24 and 25. It can be seen from the figures that the displacement at the top of the frame with semi-rigid connection is obviously larger than that of the frame with rigid connection, and the fracture time of the frame with rigid connection is earlier. It also indicates that the ductility of the semi-rigid connection is better than that of rigid connection. Steel frames with semi-rigid connections have more anti-collapse ability than those with rigid connection.  
Figure 23: Hysteresis loops at the semi-rigid connection \(G\).  
Figure 22: Horizontal displacement response of the top joint.