# Low-Cost Assessment Method for Existing Adjacent Beam Bridges
## 2 Influence of Damage Joints on LLDFs Based on the Modified Hinge Connected Beam Method
### Influence of Damaged Joints at Different Locations on the LLDF  
By combining Equation (9a-c) and Equation (14a-c), LLDFs containing the damage degree of damaged joints can be obtained and shown as Equation (15):  
\[\eta_{ij}=\eta_{ij,intact}+\lambda_{1}d_{1}+\cdots+\lambda_{i}d_{n}+\cdots+ \lambda_{n}d_{n} \tag{15}\]  
where \(\eta_{ij}\) is the LLDF of beam-\(j\), \(d_{n}\) is the damage degree of joint-\(i\), and \(\lambda_{i}\) is the influence degree of joint-\(i\) on the LLDF of beam-\(j\). The influence of damaged joints at different locations on LLDFs can be analyzed by comparing \(\lambda_{i}\). A simple example is used here to illustrate the relationship between the influence degree of joint-\(i\) on beam-\(j\) and the relative position of joint-\(i\) with beam-\(j\). Let the number of beams of an ARHB bridge be \(n=7\), and the bending and torsional parameter be \(\gamma=0.1\). When the external load \(p=1\) loads onto the exterior beam (beam-1), the shear force \(g_{i}\) at joint-1 to joint-6 and the LLDFs of beam-1 to beam-7 are shown as Equation (16a,b):  
\[\left\{ \begin{array}{l}g_{1}=0.577-0.577\mathrm{d}_{1}-0.299\mathrm{d}_{2}-0.155\mathrm{d}_{3}-0.079\mathrm{d}_{4}-0.039\mathrm{d}_{5}-0.016\mathrm{d}_{6 }\\ g_{2}=0.299\mathrm{d}_{1}-0.732\mathrm{d}_{2}-0.379\mathrm{d}_{3}-0.194\mathrm{ d}_{4}-0.095\mathrm{d}_{5}-0.039\mathrm{d}_{6}\\ g_{3}=0.155-0.155\mathrm{d}_{1}-0.379\mathrm{d}_{2}-0.771\mathrm{d}_{3}-0.395 \mathrm{d}_{4}-0.194\mathrm{d}_{5}-0.079\mathrm{d}_{6}\\ g_{4}=0.079-0.079\mathrm{d}_{1}-0.194\mathrm{d}_{2}-0.395\mathrm{d}_{3}-0.771 \mathrm{d}_{4}-0.379\mathrm{d}_{5}-0.155\mathrm{d}_{6}\\ g_{5}=0.039-0.039\mathrm{d}_{1}-0.095\mathrm{d}_{2}-0.194\mathrm{d}_{3}-0.379 \mathrm{d}_{4}-0.732\mathrm{d}_{5}-0.299\mathrm{d}_{6}\\ g_{6}=0.016-0.016\mathrm{d}_{1}-0.039\mathrm{d}_{2}-0.079\mathrm{d}_{3}-0.155 \mathrm{d}_{4}-0.299\mathrm{d}_{5}-0.577\mathrm{d}_{6}\end{array} \right. \tag{16a}\]\[\left\{ \begin{array}{l}\eta_{11}=0.423+0.577\mathrm{d}_{1}+0.299\mathrm{d}_{2}+0. 155\mathrm{d}_{3}+0.079\mathrm{d}_{4}+0.039\mathrm{d}_{5}+0.016\mathrm{d}_{6}\\ \eta_{21}=0.278-0.278\mathrm{d}_{1}+0.433\mathrm{d}_{2}+0.244\mathrm{d}_{3}+0. 115\mathrm{d}_{4}+0.056\mathrm{d}_{5}+0.023\mathrm{d}_{6}\\ \eta_{31}=0.144-0.144\mathrm{d}_{1}-0.353\mathrm{d}_{2}+0.392\mathrm{d}_{3}+0. 201\mathrm{d}_{4}+0.099\mathrm{d}_{5}+0.04\mathrm{d}_{6}\\ \eta_{41}=0.076-0.076\mathrm{d}_{1}-0.185\mathrm{d}_{2}-0.376\mathrm{d}_{3}+0. 376\mathrm{d}_{4}+0.185\mathrm{d}_{5}+0.065\mathrm{d}_{6}\\ \eta_{51}=0.040-0.040\mathrm{d}_{1}-0.099\mathrm{d}_{2}-0.201\mathrm{d}_{3}-0. 392\mathrm{d}_{4}+0.353\mathrm{d}_{5}+0.144\mathrm{d}_{6}\\ \eta_{61}=0.023-0.023\mathrm{d}_{1}-0.056\mathrm{d}_{2}-0.115\mathrm{d}_{3}-0. 224\mathrm{d}_{4}-0.433\mathrm{d}_{5}+0.278\mathrm{d}_{6}\\ \eta_{71}=0.016\mathrm{d}_{1}-0.016\mathrm{d}_{1}-0.039\mathrm{d}_{2}-0.079 \mathrm{d}_{3}-0.155\mathrm{d}_{4}-0.299\mathrm{d}_{5}-0.577\mathrm{d}_{6}\\ \end{array} \right. \tag{16b}\]  
When the external load \(p=1\) loads onto the interior beam (from beam-2 to beam-4), the LLDF of the beam with external load applied on is shown as Equation (17):  
\[\left\{ \begin{array}{l}\eta_{22}=0.290+0.278\mathrm{d}_{1}+0.433\mathrm{d}_{ 2}+0.224\mathrm{d}_{3}+0.115\mathrm{d}_{4}+0.056\mathrm{d}_{5}+0.023\mathrm{d} _{6}\\ \eta_{33}=0.255+0.144\mathrm{d}_{1}+0.353\mathrm{d}_{2}+0.392\mathrm{d}_{3}+0. 201\mathrm{d}_{4}+0.099\mathrm{d}_{5}+0.040\mathrm{d}_{6}\\ \eta_{44}=0.248+0.076\mathrm{d}_{1}+0.185\mathrm{d}_{2}+0.376\mathrm{d}_{3}+0. 376\mathrm{d}_{4}+0.185\mathrm{d}_{5}+0.076\mathrm{d}_{6}\\ \eta_{55}=0.255+0.040\mathrm{d}_{1}+0.099\mathrm{d}_{2}+0.201\mathrm{d}_{3}+0. 392\mathrm{d}_{4}+0.353\mathrm{d}_{5}+0.144\mathrm{d}_{6}\\ \eta_{22}=0.290+0.023\mathrm{d}_{1}+0.056\mathrm{d}_{2}+0.115\mathrm{d}_{3}+0. 224\mathrm{d}_{4}+0.433\mathrm{d}_{5}+0.278\mathrm{d}_{6}\\ \end{array} \right. \tag{17}\]  
It can be concluded from Equations (16b) and (17) that:  
1.