# Low-Cost Assessment Method for Existing Adjacent Beam Bridges
## 2 Influence of Damage Joints on LLDFs Based on the Modified Hinge Connected Beam Method
### Modified Hinge Connected Beam Method Considering Joint Damage  
The hinge connected beam method shown in Section 2.1 can only be used to calculate LLDFs of intact bridges with undamaged joints. This method assumes that there is no vertical relative displacement between adjacent beams at joints, and the stiffness of joints can be regarded as infinite. However, when damage exists in joints, adjacent beams will have a vertical relative displacement \(\Delta\) at the damaged joint. The right side of Equation (1) established according to the deformation compatibility condition at the joint will no longer be zero. The canonical equations considering the relative displacement \(\Delta_{i}\) caused by joint damages are shown in Equation (10):  
\[\left\{ \begin{array}{l}\delta_{11}g_{1}+\delta_{12}g_{2}+\cdots+\delta_{1,n -1}g_{n-1}+\delta_{1p}=\Delta_{1}\\ \vdots\vdots\vdots\ddots\vdots\vdots\vdots\\ \delta_{11}g_{1}+\delta_{22}g_{2}+\cdots+\delta_{i,n-1}g_{n-1}+\delta_{ip}= \Delta_{i}\\ \vdots\vdots\ddots\vdots\vdots\vdots\\ \delta_{n-1,1}g_{1}+\delta_{n-1,2}g_{2}+\cdots+\delta_{n-1,n-1}g_{n-1}+\delta_{ n-1,p}=\Delta_{n-1}\end{array} \right. \tag{10}\]where \(\Delta_{i}\) represents the relative displacement at joint-\(i\), \(\Delta_{i}=0\) when joint-\(i\) is intact and \(\Delta_{i}\
eq 0\) when joint-\(i\) is damaged. The simplified form of Equation (10) is shown as Equation (11a-c) for external load \(p=1\) applied on exterior beam-1, exterior beam-\(n\), and interior beam-\(k\), respectively:  
\[\left\{ \begin{array}{l}2(1+\gamma)g_{1}-(1-\gamma)g_{2}=1+\frac{\Delta_{1}}{ \omega}\\ -(1-\gamma)g_{1}+2(1+\gamma)g_{2}-(1-\gamma)g_{3}=\frac{\Delta_{2}}{\omega}\\ \vdots\vdots\ddots\vdots\vdots\vdots\\ -(1-\gamma)g_{n-3}+2(1+\gamma)g_{n-2}-(1-\gamma)g_{n-1}=\frac{\Delta_{n-2}}{ \omega}\\ -(1-\gamma)g_{n-2}+2(1+\gamma)g_{n-1}=\frac{\Delta_{n-1}}{\omega}\\ \end{array} \right. \tag{11a}\]  
\[\left\{ \begin{array}{l}2(1+\gamma)g_{1}-(1-\gamma)g_{2}=\frac{\Delta_{1}}{ \omega}\\ -(1-\gamma)g_{1}+2(1+\gamma)g_{2}-(1-\gamma)g_{3}=\frac{\Delta_{2}}{\omega}\\ \vdots\vdots\ddots\vdots\vdots\vdots\\ -(1-\gamma)g_{n-3}+2(1+\gamma)g_{n-2}-(1-\gamma)g_{n-1}=\frac{\Delta_{n-2}}{ \omega}\\ -(1-\gamma)g_{n-2}+2(1+\gamma)g_{n-1}=1+\frac{\Delta_{n-1}}{\omega}\\ \end{array} \right. \tag{11b}\]  
\[\left\{ \begin{array}{l}2(1+\gamma)g_{1}-(1-\gamma)g_{2}=\frac{\Delta_{1}}{ \omega}\\ -(1-\gamma)g_{1}+2(1+\gamma)g_{2}-(1-\gamma)g_{3}=\frac{\Delta_{2}}{\omega}\\ \vdots\vdots\ddots\vdots\vdots\vdots\\ -(1-\gamma)g_{k-2}+2(1+\gamma)g_{k-1}+(1-\gamma)g_{k}=1+\frac{\Delta_{k-1}}{ \omega}\\ (1-\gamma)g_{k-1}+2(1+\gamma)g_{k}-(1-\gamma)g_{k+1}=1+\frac{\Delta_{n}}{ \omega}\\ \vdots\vdots\vdots\ddots\vdots\vdots\vdots\\ -(1-\gamma)g_{n-3}+2(1+\gamma)g_{n-2}-(1-\gamma)g_{n-1}=\frac{\Delta_{n-2}}{ \omega}\\ -(1-\gamma)g_{n-2}+2(1+\gamma)g_{n-1}=\frac{\Delta_{n-1}}{\omega}\\ \end{array} \right. \tag{11c}\]  
Many scholars and specialists have proposed various models of the relative displacement \(\Delta_{i}\). [PERSON] [26] utilized Equation (12) to express the relative displacement \(\Delta_{i}\). As shown in Equation (12), \(g_{i}\) represents the shear force at joint-\(i\) generated by external load \(p\), \(\omega\) represents the maximum displacement of adjacent beams at the completely damaged joint when the dimensionless shear force \(g_{i}=1\), and \(a\epsilon[0,1)\) represents the damage rate of joint-\(i\). \[\Delta_{i}=-g_{i}a_{i}\omega \tag{12}\]  
[PERSON] [27] used a spring connection model to simulate the shear force transmission in grouted joints of ARHB bridges and expressed the joint damage degree by reducing the joint rigidity. The author considered that the relative displacement \(\Delta_{i}\) is inversely proportional to the rigidity of the shear keys at joint-\(i\), and proposed a linear elastic model, which is shown as Equation (13):  
\[\Delta_{i}=-\frac{g_{i}}{k_{i}} \tag{13}\]  
where \(g_{i}\) represents the shear force at the joint-\(i\), and \(k_{i}\) represents the stiffness of joint-\(i\).