# Risk-Cost Optimized Maintenance Strategy for Steel Bridge Subjected to Deterioration
## 7 Working Example  
_Configuration of the structural system._ The proposed methodology was applied to a railway bridge as a working example. Maintenance had just been carried out for the bridge and it was assumed that the bridge would last for a further 60 years. Thus, \(t_{L}=60\) (years). The railway bridge had one span, and the length of the span was 12 mm. The bridge contained two supported girders, as shown in Figure 0(a). The geometry of girder is shown in Figure 0(b). The flange width and thickness were 229 and 19.6 mm, respectively. The web depth and thickness were 572 and 11.9 mm, respectively. These parameters were treated as normally distributed variables to consider the changes in the cross-section along the girders. The coefficient of variance (COV) for these parameters was 0.02 based on Li [2]. The girders were made of G250 low carbon steel with average yield strength (\(\sigma_{yo}\)) of 320 MPa before corrosion and an elastic modulus of 210 GPa. They were normally distributed random variables, with both COVs equal to 0.02 to consider the uncertainties due to steel manufacturing tolerances [2]. The deck slab was 2.5 m wide and 305 mm thick with a specific concrete strength of 34.5 MPa. Reinforcing steel was Grade 40. Diffusion coefficient \(D_{f}\) for chloride was determined as 31.5 mm\({}^{2}\)/year [38]. The coefficient followed lognormal distribution with COV of 0.20. The cover thickness of slab reinforcement \(c\) was 30 mm, which followed normal distribution and had a COV of 0.02 [38]. Surface chloride content \(C_{0}\) was 0.8% with COV of 0.5, following lognormal distribution [38]. _Bridge load modeling._ The loading needed to be determined for the bridge girders. Based on the Department of Railway Transportation data, the superimposed dead load on the girder (including the self-weight of the deck, the railway ballast and track, and the weight of overhead structures) was determined to be 19.2 kN/m. Trains operating on the viaduct had four motor cars and two trailer cars. The axle load was 118 kN for a motor car (\(P_{1}\)) and 108.5 kN for a trailer car (\(P_{2}\)) for the tracks, as shown in Figure 5. The frequency of the train was 80 per day with COV of 0.05 [2]. Based on the above information, \(M_{s}\) and \(V_{s}\) were determined as follows:  
\[M_{s}=\frac{ql^{2}}{8}+\frac{1}{4}P_{1}l+\frac{P_{1}}{2}\left(\frac{l}{2}-x_{1} \right)+\frac{P_{2}}{2}\left(\frac{l}{2}-x_{2}\right)+\frac{P_{2}}{2}\left( \frac{l}{2}-x_{3}\right), \tag{28}\]  
\[V_{s}=\frac{ql}{8}+P_{1}+P_{1}\left(\frac{l-x_{1}}{l}\right)+P_{2}\left(\frac{l -x_{2}}{l}\right)+P_{2}\left(\frac{l-x_{3}}{l}\right) \tag{29}\]  
where \(l\) is the length of the girder, \(q\) is the uniform disturbed load calculated based on the self-weight of the girder and superimposed dead load (19.2 kN/m). \(x_{1}\), \(x_{2}\) and \(x_{3}\) were determined based on Figure 5, and were 2.2, 6.9,and 9.1 m, respectively. Also, \(M_{max}=M_{s}\) and \(M_{min}=\frac{ql^{2}}{8}\) to determine the mean stress effect in Equation (20) for fatigue damage calculation. For deflection failure, \(\Delta_{S}\) was calculated as follows:  
\[\Delta_{S}(t)=\frac{P_{1}l^{3}}{48 EI(t)}+\frac{P_{1}\left(\frac{l}{2}-x_{1} \right)\times[3l^{2}-(\frac{l}{2}-x_{1})^{2}]}{48 EI(t)}+\frac{P_{2}\left(\frac {l}{2}-x_{2}\right)\times[3l^{2}-(\frac{l}{2}-x_{2})^{2}]}{48 EI(t)}+\frac{P_{ 1}\left(\frac{l}{2}-x_{1}\right)\times[3l^{2}-(\frac{l}{2}-x_{1})^{2}]}{48 EI(t )} \tag{30}\]  
where \(E\) is the elastic modulus of steel and \(I(t)\) the second moment of inertia of the girder cross-section at time \(t\). All the basic variables are summarized in Table 1.  
_Probability of failure prediction._ Based on the basic variables summarized in Table 1, the probability of failure of each failure mode was determined. In detail, for flexure failure, \(M(t)\) was determined by Equations (11), (23) and (24) based on the geometry and yield strength of the girder summarized in Table 1. \(M_{s}\) was determined by Equation (28).