## 4 Case Study
### Element Maintenance Decision Model
#### 4.2.5 Duration of Each State  
The duration of bridge element states reflects the degradation of the technical condition in the natural environment and should be universally applicable. In this paper, based on the historical inspection data of bridges on the northbound section of a highway, 40 prestressed reinforced concrete continuous box girder bridges were selected for ease of calculation. The percentages of the elements in each state at different ages were calculated and are summarized in Table 6. The parameters for fitting the duration of each state could be obtained using the Weibull distribution, as shown in Table 7.  
State 5 represents the final state of the element, and its duration is a deterministic value (the bridge design service life minus the total duration of States 1 to 4). It does not follow the Weibull distribution. By substituting the Weibull parameters into Equation (11), the durations of the elements in each state can be obtained, as shown in the table.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Elements**} & \multirow{2}{*}{**Element**} & \multicolumn{4}{c}{**Cost (USD)**} \\ \cline{3-6}  & & **Repairing** & **Surface** & **Exposed Reinforcement Corrosion** & **Measure Number** \\ \hline R-1-1 & 4 & 23.45 & 0.00 & 1090.76 & al + b2 \\ R-2-1 & 3 & 0.00 & 0.00 & 1090.76 & b2 \\ R-3-1 & 1 & 0.00 & 0.00 & 0.00 & / \\ R-4-1 & 4 & 22.42 & 0.00 & 0.00 & al + a2 \\ R-5-1 & 3 & 36.94 & 0.00 & 0.00 & al + a2 \\ R-6-1 & 3 & 97.03 & 0.00 & 0.00 & al + a2 \\ R-7-1 & 4 & 326.30 & 0.00 & 1636.14 & al + a2 + b2 \\ R-8-1 & 2 & 0.00 & 195.18 & 0.00 & cl1 \\ R-9-1 & 4 & 239.57 & 321.16 & 0.00 & al + a2 + c1 \\ R-10-1 & 4 & 81.76 & 53.23 & 0.00 & a2 + c1 \\ R-11-1 & 5 & 76.30 & 21,048.78 & 299.96 & al + a2 + b2 + c2 \\ R-12-1 & 5 & 921.36 & 236.58 & 0.00 & cl + a1 \\ R-13-1 & 3 & 137.90 & 0.00 & 0.00 & al + a2 \\ R-14-1 & 4 & 345.70 & 7403.53 & 207.24 & al + a2 + b2 + c1 + c2 \\ R-15-1 & 5 & 173.42 & 501.37 & 545.38 & al + a2 + b2 + c1 + d1 \\ R-16-1 & 4 & 0.00 & 177.44 & 13.63 & b2 + c1 \\ R-17-1 & 4 & 101.47 & 211.84 & 0.00 & al + a2 + c1 + d1 \\ R-18-1 & 4 & 19.31 & 115.58 & 43.63 & al + a2 + b1 + c1 + d1 \\ R-19-1 & 4 & 50.25 & 266.15 & 0.00 & al + a2 + c1 \\ R-20-1 & 4 & 83.76 & 34.30 & 5.45 & al + a2 + b2 + c1 \\ R-21-1 & 4 & 12.56 & 0.00 & 32.72 & al + b2 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Calculation of the maintenance program costs.  
The durations in Table 8 represent the total time that bridge elements stay in multiple states without maintenance. The duration of State 1 was 1.36 years, thereby indicating that the element stays in State 1 for 1.36 years. The duration of States 1-2 was 3.31 years, meaning that the element stays in States 1 and 2 for a combined duration of 3.31 years. Therefore, the element stays in State 2 for 3.31 \(-\) 1.36 = 1.95 years. The duration of States 1-3 is 6.91 years, implying that the element stays in States 1, 2, and 3 for a combined duration of 6.91 years. Therefore, the element stays in State 3 for 6.91 \(-\) 3.31 = 3.6 years. The duration of States 1-4 is 30.01 years, thereby indicating that the element stays in States 1, 2, 3, and 4 for a combined duration of 30.01 years. Therefore, the element stays in State 4 for 30.01 \(-\) 6.91 = 23.1 years. Since the bridge on the highway is inspected and repaired every three years, the duration of each state cannot be exactly three years.  
\[T\big{(}s,s^{\prime},t^{\prime}\big{)}=\{1.36,1.95,3,3,3\} \tag{24}\]