# Collapse Analysis of a Two-Span Reinforced Concrete Bridge Model
## 4 Analysis of Collapse Mode
### Influence of Structural Systems
#### 4.5.2 Shock Mitigation System  
Damping is the friction and other hindrances that attenuate free vibration, and the special components placed in the structural system that can provide movement resistance are dampers [33; 34]. The commonly used dampers on bridges are fluid viscous dampers, which have been widely used in structural engineering. They have no initial stiffness and will not affect other calculations of the structure (such as period, vibration mode, etc.). It can not only reduce the displacement of dynamic response but also control the force, so as to decrease the structural cost. 1. Parameter Design  
If the structural system was a damping system rather than an isolation system, the collapse modes would be different. The commonly used fluid viscous damper was chosen, and the damping system is described below. Dampers are usually located at the maximum displacement of the structure, so the dampers were symmetrically set in two simply supported ends of the two-span continuous beam bridge model, with a total of two dampers (see Figure 20). A viscous damper is a velocity-dependent damper whose damping force can be expressed as  
\[F=Cv^{\alpha} \tag{1}\]  
where \(C\) is the damping coefficient, \(v\) is velocity, and \(\alpha\) is the velocity index. When the local vibration input is certain, \(v\) is the determined value and the damping force is related to the values of \(C\) and \(\alpha\). \(F\) increases with \(C\). If \(\alpha\) decreases, the hysteresis curve of the damper becomes square, the area increases, and the energy dissipation effect is improved. However, when \(\alpha\) is close to 0, \(v^{\alpha}\) is close to 1 and the damping force is basically only related to \(C\), which will lose the basic characteristics of the velocity correlation of the viscous damper and violate the original intention of the design. Therefore, the value of the velocity index must not be too small and is usually selected between 0.2 and 1. According to the characteristics of the research bridge in this paper, the optimization range of damper parameters is determined as shown in Table 9. The damping coefficient increases from 1 kN(s/m)\({}^{\alpha}\) to 12 kN(s/m)\({}^{\alpha}\) when the velocity index varies from 0.2 to 1. The 42 structures were subjected to El Centro seismic waves with PGA of 0.4 g. The velocity index and damping index of the damper directly affected the actual output, output rate, and maximum stroke of the damping force, as shown in Figures 21 and 22. When the velocity index is determined, the larger the damping coefficient, the larger the damping force peak value generated by the damper and the smaller the piston rod stroke. When the damping coefficient is constant, the velocity index decrease will result in higher nonlinearity, a larger actual output of the damper, and a smaller required stroke. When the damping coefficient is large enough, the velocity index can be increased and the damping force is similar to that generated by a small damping coefficient and small velocity index. For example, if \(\alpha=0.4\), \(C=10\) kN(s/m)\({}^{0.4}\), then \(F_{\max}=6.40\) kN, \(D_{\max}=9.04\) mm; if \(\alpha=0.2\), \(C=8\) kN(s/m)\({}^{0.2}\), then \(F_{\max}=6.48\) kN, \(D_{\max}=8.18\) mm. The damping forces of the two dampers are equivalent, but the latter has a small stroke. The damper control of the structure is reflected in two aspects: force and deformation control. Considering the characteristics of seismic damage and the collapse failure of continuous beam bridges,  
Figure 20: Layout diagram of fluid viscous damper. displacement control is important. Sometimes the substructure has a large displacement response under the condition of small shocks, which leads to serious consequences of falling beams. Therefore, the relative displacement of the pier and the beam at the simply supported end of the two-span continuous beam model bridge are analyzed.