# Low-Cost Assessment Method for Existing Adjacent Beam Bridges
## 4 Lateral Load Distribution Performance Rating Number LDN  
Damage on grouted joints and beams lead to the degradation of lateral load transmission performance, which can cause variations in the LLDFs of the bridges. In the service stage of the ARHB bridges, the grouted joints are always damaged before the beams. The degradation of lateral load transmission performance will increase the failure probability of adjacent rectangular hollow beams. According to \"the law of fives\", when damage occurs in the bridge, the earlier the repairs and maintenances are carried out, the lower the cost will be. Timely repairs and maintenance of damaged grouted joints can effectively avoid the degradation of beams. As joint damage is common in ARHB bridges, a low-cost assessment method that can be combined with conventional visual inspection is needed to evaluate the bearing performance of bridges and guide maintenance schedules. This paper proposes the lateral load distribution performance rating number \(LDN\), which is based on the concept of standard deviation, to assess the degradation degree of bearing capacity of ARHB bridges with damaged joints and generally intact beams, and is shown as Equation (18):  
\[\left\{ \begin{array}{l}LDN=S\times\sqrt{\frac{\left(\frac{\sum_{i=1}^{i=N}VA _{i}\right)^{2}}}{N}\times 100}\\ LDN_{pre}=S\times\sqrt{\frac{\left(\sum_{i=1}^{i=N}VA_{i,pre}\right)^{2}}{N} \times 100}\end{array} \right. \tag{18}\]  
where \(VA_{i}\) represents the variation in the LLDF of the beam being analyzed, i.e., beam-\(i\), in the bridge without transverse interior prestressed reinforcement; \(VA_{i,pre}\) represents the one of beam-\(i\) in the bridge with transverse interior prestressed reinforcement; \(S\) is the coefficient representing the design safety grade of the bridge and is determined by the _General Specifications for Design of Highway Bridges and Culverts (JTGD60-2015)_[22]; see Table 1 for values. \(N\) represents the total number of adjacent rectangular hollow beams of the bridge. The calculation formulas of \(VA_{i}\) and \(VA_{i,pre}\) are shown as Equation (19a,b):  
\[\text{VA}_{i}=\frac{\sum_{n=1}^{n=N_{i}}\left(S_{n}\times D_{n}\times P_{n} \right)}{\sum_{n=1}^{n=N_{i}}\left(S_{n,\text{ \emph{ref}}}\times D_{nL,ref}\times P_{n}\right)} \tag{19a}\]  
\[VA_{i,pre}=K_{1}\frac{\sum_{n=1}^{n=N_{i}}\left(S_{n}\times D_{n}\times P_{n} \right)}{\sum_{n=1}^{n=N_{i}}\left(S_{n,\text{ \emph{ref}}}\times D_{nL,ref}\times P_{n}\right)}+K_{2} \tag{19b}\]  
where \(K_{1}=0.7\) is the weight coefficient of joints and \(K_{2}=0.3\) is the weight coefficient of dowel action. The determinaiton of values of \(K_{1}\) and \(K_{2}\) is based on [29]. \(D_{n}\) represents the damage degree of joint-\(n\), and the value is presented in Table 2; \(D_{n,ref}\) is the corresponding maximum value. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Design Safety Grade** & \(S\) & **Applicable Objects** \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{1.1} & (1) Super large bridges, major bridges, and medium bridges on highways of all grades; \\  & & (2) Small bridges on freeways, Grade I highways, Grade II highways, national defense \\  & & highways, and busy roads near cities. \\  & & (1) Small bridges on Grade III and IV highways; \\
2 & 1 & (2) Culverts on expressways, Grade I highways, Grade II highways, national defense \\  & & highways, and highways with heavy traffic near cities. \\
3 & 1 & Culverts on Grade III and IV highways. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(S\) values according to design safety grade. \(N_{\gamma}\) is the number of the joints whose influencing extent on beam-\(i\) is considered, and the value is determined by the bending and torsional parameter \(\gamma\); when \(\gamma\leq\) 0.25, \(N_{\gamma}=5\), and when \(\gamma>\) 0.25, \(N_{\gamma}=3\).