## 4 Examples
### Static and Dynamic Response Analysis of Steel Frames with Linear Semi-Rigid Connections  
Figure 9 displays the configuration and loading of a single-bay two-story steel frame with W14 \(\times\) 48 steel beams and W12 \(\times\) 96 steel columns. The elastic modulus is 200 GPa. A vertical concentrated load \(P\) was applied to four joints, meanwhile a horizontal load 0.01\(p\) was added to consider the geometric imperfection or wind load. Several studies ([PERSON] and [PERSON] [9], [PERSON] and [PERSON] [12]; [PERSON] et al. [20]) carried out stability analysis of the structure in the two cases of rigid and semi-rigid connections using different simulation methods. In the above analysis, the bending-resistant stiffness of semi-rigid beam-column joints was taken to be 3.48 \(\times\) 10\({}^{7}\) kN-m/rad. The MDEM discretized the single-bay two-story steel frame into 28 particles (i.e., 28 contact elements). Of these, each column was represented by 4 contact elements while every beam by 6 contact elements. The bending-resistant stiffness of the contact elements at the beam ends was modified to be 3.48 \(\times\) 10\({}^{7}\) kN-m/rad according to Table 1. The horizontal displacement of steel frames with linear semi-rigid joints is shown in Figure 9. The figure shows that the load-displacement relationship of the modified MDEM is in a good agree with the published results. Additionally, the effect of semi-rigid connections on the structural behavior is great, and structural loading capacity may be overestimated if semi-rigid connections are ignored. Another single-bay two-story steel frame with linear semi-rigid joints was adopted to analyze the dynamic response. The configuration and loading of this structure are shown in Figure 10. All the beams and columns are W8 \(\times\) 48 and the Young's modulus is 205 GPa. The material of all the frame members was assumed to be perfectly elastic and the viscous damping was ignored. The initial geometric imperfection \(\psi\) was taken to be 1/438 rad. The lumped masses of 5.1 T and 10.2 T were set at the top of the columns and in the middle of the beams, respectively. The stiffness of semi-rigid connections was 23,000 kN-m/rad. In this study, each beam was simulated by 9 particles (i.e., 8 contact elements), while every column was discretized into 4 particles (3 contact elements). As the stiffness of each frame member is same, the bending-resistant stiffness of the contact elements at the semi-rigid joints can be determined to be 11,500 kN-m/rad by Equation (13) not Table 1. The vertical static loads of 50 KN and 100 KN were first applied on the beam-column joints and in the middle of beams to consider the second-order effects, and then the horizontal forces were added suddenly at each floor during 0.5 s until the calculation ended. The horizontal displacement on the top is shown in Figure 11. The comparison indicates the displacement responses of the MDEM and existing results ([PERSON] and  
Figure 9: Horizontal displacement of steel frames with linear semi-rigid joints. [PERSON] [13], [PERSON] and [PERSON] [18]) are consistent, and the influence of semi-rigid connections on the periods and the amplitudes of displacement responses of steel frames is more significant as well as the amplitude increase is up to 60%. Snap-Through Buckling Analysis of the Williams Toggle Frame with Linear Semi-Rigid Connections and Supports  
The Williams Toggle frame is frequently taken as a typical example for snap-through buckling analysis. The structure has the obviously nonlinear bifurcation instability, that is, each bifurcation point may have one or more equilibrium paths and each path represents the extreme point instability or snap instability [12; 33]. Many researchers ([PERSON] and [PERSON] [12], [PERSON] et al. [20]) performed static stability analysis of the steel frame with linear semi-rigid connections and supports, and the bending-resistant stiffness of each semi-rigid connection or support \(k\) is 10 EI/L.