# Low-Cost Assessment Method for Existing Adjacent Beam Bridges
## 2 Influence of Damage Joints on LLDFs Based on the Modified Hinge Connected Beam Method
### Hinge Connected Beam Method
\begin{array}{lcl}\delta_{11}=\delta_{11}=\cdots=\delta_{n-1,n-1}=2 \times\left(\omega+\frac{b}{2}\times\varphi\right)\\ \delta_{12}=\cdots=\delta_{n-2,n-1}=\delta_{21}=\cdots=\delta_{n-1,n-2}=- \left(\omega-\frac{b}{2}\times\varphi\right)\\ \delta_{ij}=0\quad\text{when $i<j-1$, $i>j+1$}\end{array} \right. \tag{2}\]  
where \(\omega\) represents the displacement at the center of the beam generated by \(g_{j}\) and expressed by Equation (3), and \(\varphi\) represents the rotation angle generated by \(g_{j}\) and expressed by Equation (4):  
\[\omega=\frac{pl^{4}}{\pi^{4}EI} \tag{3}\]  
Figure 2: (**a**) Force analysis diagram when external load \(p=1\) applied on beam-1. (**b**) Force analysis diagram when external load \(p=1\) applied on beam-\(n\). (**c**) Force analysis diagram when external load \(p=1\) applied on beam-\(n\). \[\varphi=\frac{pbl^{4}}{2\pi^{2}GI_{T}} \tag{4}\]  
where \(p\) represents external load, \(b\) represents the beam width, \(l\) represents the span length of simply supported beam, \(E\) and \(G\) represent elastic modulus and rigid modulus of the beam, respectively, and \(I\) and \(I_{T}\) represent the bending moment of inertia and the torsional moment of inertia of the beam, respectively. When the external load \(p=1\) is applied on the interior beams, here beam-\(k\), \(2\leq k\leq n-1\), \(\delta_{ij}\) is shown as Equation (5). \[\left\{\begin{array}{l}\delta_{11}=\delta_{11}=\cdots=\  
When the external load \(p=1\) is applied on interior beam-\(k\), where \(2\leq k\leq n-1\), the simplified form of Equation (1) is shown as Equation (7c):  
\[\left\{ \begin{array}{l}2(1+\gamma)g_{1}-(1-\gamma)g_{2}=0\\ -(1-\gamma)g_{1}+2(1+\gamma)g_{2}-(1-\gamma)g_{3}=0\\ \vdots\vdots\vdots\ddots\vdots\vdots\vdots\\ -(1-\gamma)g_{k-2}+2(1+\gamma)g_{k-1}+(1-\gamma)g_{k}=1\\ (1-\gamma)g_{k-1}+2(1+\gamma)g_{k}-(1-\gamma)g_{k+1}=1\\ \vdots\vdots\ddots\vdots\vdots\vdots\\ -(1-\gamma)g_{n-3}+2(1+\gamma)g_{n-2}-(1-\gamma)g_{n-1}=0\\ -(1-\gamma)g_{n-2}+2(1+\gamma)g_{n-1}=0\end{array} \right. \tag{7c}\]  
where \(\gamma\) is bending and torsional parameter and can be expressed by Equation (8) for reinforced concrete ARHB bridges:  
\[\gamma=\frac{\varphi_{2}^{b}}{\omega}\approx 5.8\times\frac{I}{I_{T}}\times \left(\frac{b}{I}\right)^{2} \tag{8}\]  
After obtaining \(g_{j}\) by solving Equation (7c), the lateral load distribution factor \(\eta_{ij}\) of each beam can be obtained by Equation (9a-c). \(i\) of \(\eta_{ij}\) is the number of the beam for which LLDF is being calculated, and \(j\) of \(\eta_{ij}\) is the number of the beam loaded by external load \(p=1\). \[\left\{ \begin{array}{l}\eta_{11}=1-g_{1}\\ \eta_{i1}=g_{i-1}-g_{i}\\ \eta_{n1}=g_{n-1}\end{array} \right. \tag{9a}\] \[\left\{ \begin{array}{l}\eta_{1k}=g_{1}\\ \eta_{ik}=|g_{i-1}-g_{i}|\\ \eta_{kk}=1-g_{k-1}-g_{k}\\ \eta_{nk}=g_{n-1}\end{array} \right.\] (9b) \[\left\{ \begin{array}{l}\eta_{1n}=g_{1}\\ \eta_{in}=g_{i}-g_{i-1}\\ \eta_{mn}=1-g_{n-1}\end{array} \right. \tag{9c}\]  
The accuracy of the hinge connected beam method in calculating the LLDFs of ARHB bridges has been verified by a large number of real bridge experiments and monitoring data in China.