# Influence Line-Based Design of Scissors-Type Bridge
## 2 The Equilibrium Theory  
### The Mechanism of the Scissors Structure Using the Cantilever Model  
The two-unit scissors model depends on its boundary condition and the position of the applied load to each nodal point (see Figure 4). This model is based on a cantilever that consists of a pinned support point on the left-hand side \(B_{1}^{L}\) and \(A_{1}^{L}\) and \(P\) at point \(B_{2}^{R}\). The rotation point of the scissor is denoted by \(C_{1}\), its length by \(\lambda\), and its height by \(\eta\)[11] and/or [17] (see Figure 5).  
Figure 4: Definition of the coordinate system and sectional forces in a scissors unit structure.  
Figure 5: Two-units scissors structure by cantilever model.  
Figure 3: Types of emergency bridges. (**a**) Double Warren truss (DWT). (**b**) The scissors-type bridge (SCI). (**c**) The smart bridge [20].  
In the equation, the external forces are on the left-hand side and internal forces (This implies that the force at the nodal point is \((\bullet^{*})_{\Phi}\). \((\bullet^{*})_{\Phi}\equiv\left[\begin{array}{cc}\bullet^{*}&|\mathrm{A}^{L}, \mathrm{B}^{L},\mathrm{C},\mathrm{A}^{R},\mathrm{B}^{R};*=\{L,R\};\Phi=\{x,y\} \end{array}\right]\), \(\bullet^{*}\) denotes nodal label.) are on the connection points \(B_{1,2}\) and \(A_{1,2}\) on the right-hand side. We obtain the following condition:  
\[\left\{\begin{array}{c}(B_{1,2})_{x}\\ (B_{1,2})_{y}\\ (A_{1,2})_{x}\\ (A_{1,2})_{y}\end{array}\right\}=\left\{\begin{array}{c}(B_{1}^{R})_{x}\\ (B_{1}^{R})_{y}\\ (A_{1}^{R})_{x}\\ (A_{1}^{R})_{y}\end{array}\right\}+\left\{\begin{array}{c}(B_{2}^{L})_{x}\\ (B_{2}^{L})_{y}\\ (A_{2}^{L})_{x}\\ (A_{2}^{L})_{y}\end{array}\right\} \tag{1}\]  
Therefore, the relevant equilibrium equations are as follows:  
\[\left\{\begin{array}{c}(B_{1,2})\\ (A_{1,2})\end{array}\right\} = \left\{\begin{array}{c}(B_{1}^{R})\\ (A_{1}^{R})\end{array}\right\}+\left\{\begin{array}{c}(B_{2}^{L})\\ (B_{2}^{L})\end{array}\right\} \tag{2}\] \[\left[L\right]\left\{\begin{array}{c}(B_{1}^{L})\\ (A_{1}^{L})\end{array}\right\} = -[R]\left\{\begin{array}{c}(B_{1}^{R})\\ (A_{1}^{R})\end{array}\right\}-\left\{\begin{array}{c}(C_{1})\\ \mathbf{0}\end{array}\right\}\] (3) \[\left[L\right]\left\{\begin{array}{c}(B_{2}^{L})\\ (A_{2}^{L})\end{array}\right\} = -[R]\left\{\begin{array}{c}(B_{2}^{R})\\ (A_{2}^{R})\end{array}\right\}-\left\{\begin{array}{c}(C_{2})\\ \mathbf{0}\end{array}\right\} \tag{4}\]  
Here,  
\[\left[L\right]=\left[\begin{array}{ccc|cc}1&0&1&0\\ 0&1&0&1\\ \hline-\eta&\lambda&0&0\\ 0&0&\eta&\lambda\end{array}\right]\text{,}\quad\left[R\right]=\left[\begin{array} []{ccc|cc}1&0&1&0\\ 0&1&0&1\\ \hline 0&0&\eta&-\lambda\\ -\eta&-\lambda&0&0\end{array}\right]\]  
At each nodal point, a solution is obtained by applying the equation for each nodal point in the following order. Then, we obtain their forces in matrix form based on the following equation [17] and/or [20].  
\[\left\{\begin{array}{c}(B_{1})\\ (A_{1})\end{array}\right\} = -([L]^{-1}[R])^{2}\left\{\begin{array}{c}(B_{2})\\ (A_{2})\end{array}\right\}-[L]^{-1}[R]\left\{\begin{array}{c}(B_{1,2})\\ (A_{1,2})\end{array}\right\} \tag{5}\] \[-[L]^{-1}[R][L]^{-1}\left\{\begin{array}{c}(C_{2})\\ \mathbf{0}\end{array}\right\}-[L]^{-1}\left\{\begin{array}{c}(C_{1})\\ \mathbf{0}\end{array}\right\}\]  
In this solution, the right-hand side is subject to \(P\) at point \((B_{2}^{R})_{y}=-P\) and the other nodal forces (\(=0\)) in Equation (5), the unknown reaction forces for the two-unit SCI can be expressed as \((A_{1}^{L})_{x}=-(B_{1}^{L})_{x}=-P\tan\theta\) and \((A_{1}^{L})_{y}=P,(B_{1}^{L})_{y}=0\).  
A comparison of the theoretical predictions for the one and two-unit SCI shows that the vertical reaction forces are identical in both; however, the horizontal reaction forces are double of those in the vertical direction. Therefore, the horizontal reaction forces may become very high if the number of scissors units is increased [21].