# Numerical Analysis of Reinforcing Effect for Scissors-Type Bridge with Strut Members
## 2 Theoretical Evaluation for Strut Reinforcement
### Example Calculation for a Unit Model
#### 2.2.2 Calculation Results  
Let us consider the reinforcing effect using specifically sectional properties of \(L_{0}=1000\) mm; \(A=564\) mm\({}^{2}\); and \(I=3.3\times 10^{5}\) mm\({}^{4}\) based on the pedestrian type of the MB [23].  
The change in the bending stress depends on the angle \(\theta\), which acts on the concentrated load (\(P=1\) kN) at the end of the system, as shown in Figure 4. The abscissa of the graph shows the angle \(\theta\) of the scissor structure, and the ordinate shows the absolute values of the maximum bending stresses in the model. In both results, the maximum bending stress is obtained at the pivot in the member \(\overline{B_{1}^{L}A_{1}^{R}}\), and the stress values are increasing according to expansion of the scissor structure. When we focus on the design angle of 60 degree of the MB in Figure 4a, the maximum bending stress without reinforcement, that is \(\alpha=0\), is over 45 MPa. However, it can be seen that the maximum bending stress at the pivot is largely decreased by applying the upper reinforcement in comparison with the initial state. Even if the value of \(\alpha\) is small, e.g., \(\alpha=0.1\) and \(0.2\), we can obtain a good reinforcing effect. In other words, there is an optimal size of the reinforcing member which shows the stiffening effect with light weight. Similarly, Figure 4b shows the reduction of the maximum bending stress approximately half from the initial state, when the value of \(\alpha\) is over 0.05. Thus, it is considered that the vertical reinforcing member equally distributes the force to each scissor member if it has enough extensional rigidity.  
The change in the maximum bending stress in the scissor member \(\overline{B_{1}^{L}A_{1}^{R}}\) and the axial stress in the upper horizontal strut member \(\overline{A_{1}^{L}A_{1}^{R}}\) and in the right vertical strut member \(\overline{A_{1}^{R}B_{1}^{R}}\) depends on the ratio of extensional rigidity \(\alpha\), as shown in Figure 5. The abscissa of the graph shows \(\alpha\), and the ordinate shows the absolute values of stresses in the design angle of 60 degrees. From Figure 5a, both stresses in the scissor member and the strut member depend on \(\alpha\). When \(\alpha\) is smaller than 0.1, the bending stress in the scissor member increases suddenly. On the other hand, the bending stress is almost constant after \(\alpha\) is 0.3. From Figure 5b, the bending stress in the scissor member is decreased by the value of \(\alpha\). The stress value is almost constant when the value of \(\alpha\) is over 0.05. As well as the result in Figure 4b, it is thought that the vertical member is capable of dispersing the force.  
Figure 4: Relationship between the bending stress and expansion angle \(\theta\): (**a**) model with upper horizontal strut member; (**b**) model with right vertical strut member.  
Based on the above examples, we found the reduction of the stress in scissor member by applying the reinforcing member between each hinge part. The reinforcing effect showed the nonlinear solution depending on the deployment angle and cross section of the scissors and strut members. In addition, this tendency differed according to the reinforcing layout. Hence, in order to provide feedback to the design of the reinforced scissor structure, it is important to seek the optimal reinforcing layout and allowable size of the reinforcing member.