# 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  
Despite the fact that computer technology and numerical simulation methods have been widely used in bridge design and analysis, simplified analysis methods are still favored by engineers. That is because the simplified methods are time-efficient and can be used to verify the results of numerical simulation [18]. At present, many bridge design codes, such as the _AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications (9 th edition, 2020)_[19], the _Canadian Highway Bridge Design Code (2019)_[20], and the _General Specifications for Design of Highway Bridges and Culverts (2015)_[21], adopt simplified methods. That is, they regard the lateral and longitudinal effects of live loads as uncoupled phenomena to transform the spatial stress problem into the plane stress problem by lateral load distribution factors [22; 23]. The lateral load distribution factor LLDF is defined as the ratio of the live load carried by the individual beam and the applied live load, and expressed as the ratio of the load effect of the individual beam and the total load effects of all beams. The hinge connected beam method is a frequently used simplified method for calculating LLDFs of adjacent rectangular hollow-beam (ARHB) bridges in China. This method is derived from the force method and is applicable to fabricated T-beam bridges without intermediate diaphragms, connected only by steel plates or reinforcements between flanges, or fabricated bridges that are connected only by cast-in-situ bridge decks, or adjacent rectangular hollow-beam (ARHB) bridges that are connected by cast-in-situ joints and without diaphragms and transverse prestressed ties. The basic assumptions of the hinge connected beam method are as follows [24; 25]:  
1. The adjacent rectangular hollow-beam (ARHB) bridge is simplified as several beams in parallel and horizontally hinged with each other. Each beam is a Euler-Bernoulli uniform beam and simply supported at the end;
2. Only analyzing shear force transmission between adjacent beams, leaving the transfers of normal stress, bending moment, and longitudinal shear force out. Assuming that each beam has only two degrees of freedom that are vertical displacement and longitudinal rotation. Ignoring torsional effect, lateral deflection deformation, and section deformation;
3. The adjacent beams have no relative displacement at joints, and the stiffness of joints is infinite. As shown in Figure 2a-c, for an ARHB bridge with \(n\) beams and \(n-1\) joints, when applying the dimensionless external load \(p=1\) on the central axis of one of the beams, a pair of vertical shear forces with equal size and opposite direction will be generated in each joint. According to the principles of force balance and the deformation compatibility condition that the vertical relative displacement of two adjacent beams at joints is zero, the canonical equation can be obtained, as shown in Equation (1):  
\[\left\{ \begin{array}{lclclcl}\delta_{11}g_{1}&+&\delta_{12}g_{2}&+&\cdots+\delta _{1,n-1}g_{n-1}+\delta_{1p}=0\\ \delta_{21}g_{1}&+&\delta_{22}g_{2}&+&\cdots+\delta_{2,n-1}g_{n-1}+\delta_{2p}= 0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ \delta_{n-1,1}g_{1}+\delta_{n-1,2}g_{2}+\cdots+\delta_{n-1,n-1}g_{n-1}+\delta _{n-1,p}=0\end{array} \right. \tag{1}\]  
where \(g_{j}\) represents the vertical shear force at joint-\(j\), \(\delta_{ij}\) represents the vertical displacement at joint-\(i\) caused by the vertical shear force \(g_{j}\), and \(\delta_{ip}\) represents the vertical displacement caused by external load \(p\) at joint-\(i\). \(\delta_{ij}\) is composed of the displacement at the center of the beam and the rotation angle generated by \(g_{j}\). When the external load \(p=1\) is applied on the exterior beams, \(\delta_{ij}\) is shown as Equation (2). \[\left\{