## Example 3619

A highway bridge is expected to have a service life of 45 years. The concrete bridge deck is not expected to last that long and will potentially have to be replaced several times during the life of the bridge. The reinforcing in the bridge deck is expected to corrode as chlorides from the deicing salts penetrate the deck and reach a critical concentration at the reinforcing steel. The tensile forces resulting from the corrosion expansion cause the bridge deck to spall. Based on the deterioration model provided in Estes and Frangopol (2001c), the corrosion initiation time is a normally distributed variable with a mean value of 19.6 years and a standard deviation of 7.51 years. The bridge deck will be replaced when there is active corrosion over 50% of the bridge deck (see Example 36.17). The bridge manager plans for three lifetime inspections using the half-cell potential test. The present value cost of replacing the bridge deck is \$225,600, and the cost of conducting the half-cell test is \$1,072. Using a discount rate of 2% along with the assumptions provided in Estes and Frangopol (2001c), what are the optimal inspection times for the half-cell tests? What is the expected total cost of the inspections and deck replacements?

Every time an inspection is made, there will be a decision to replace the deck or not replace the deck. With three lifetime inspections, there are eight possible repair/no repair paths available as shown in Figure 36.11. The inspection times are determined by optimizing the minimum expected total lifetime cost E(Qot). The constraints are that the expected damage at each inspection time (t1, t2, and t3) and at the end of service life (t4 = 45 years) will be less than the maximum allowable damage of 50% active corrosion. Additional constraints were added so that inspections must be at least 2 years apart, but not more than 20 years apart. Using the optimization program ADS (Vanderplaats 1986), Figure 36.12 shows that the optimum inspection times were at t1 = 10.05 years, t2 = 19.76 years, and t3 = 35.45 years (Estes 1997; Frangopol and Estes 1999; Estes and Frangopol 2001c). The expected percentage of damage, which is a weighted average of the eight possible paths in Figure 36.11, is shown for each time, t. The probability of taking any given path is a function of the probability of failure at the time of inspection, as shown in Figure 36.11. The most likely path, for example, was Branch 6, where the likelihood was 58.9%. The path involves a single repair after the second inspection and no repair after the first and third inspections. This reflects that the deck was 10.2% damaged at t1, 50.9% damaged at t2, and 30.1% damaged at t3. Conversely, the least likely paths (0.1%) were Branch 1 and Branch 8, which are associated with repair after every inspection and after none of the inspections, respectively. The expected damage at any point in time E(Damage)(t) is equal to the sum over all possible branches of the event tree given that the particular branch is taken as E[Damage(t) | Branch,] multiplied by the probability of taking that branch Pb..

where m is the number of inspections.

The minimized expected total cost E(Ctot) is the sum of the expected repair costs E(Crep) and the inspection costs Cinsp. The inspection costs are the discounted actual costs of inspection:

°insp h (1 + (1+0. 02Ï10'06 + (1.02V9'76 + (1.02Ï35' 45 ^

+ Replace deck

- No deck replacement

% Damage to deck at time of inspection if path is taken

- (77.4%) Branch 2 — 0.3% + (9.0%) Branch 3 — 5.8%

- (98.0%) Branch 4 — 0.5% + (9.0%) Branch 5 — 16.0%

- (77.4%) Branch 6 — 58.9% + (9.0%) Branch 7 — 18.3%

ranch 8

 t0 1 t1 1 t2 t3 t4
0 0