## Info

Total expected cost of repair $172,130

Source: Estes amd Frangopol 2001c. Reprinted with permission from Elsevier.

Total expected cost of repair $172,130

Source: Estes amd Frangopol 2001c. Reprinted with permission from Elsevier.

The expected cost of repair E(Crep) is the sum over all branches of the discounted replacement costs associated with an individual branch Crep|Branch. multiplied by the probability of taking that branch Pb..

= (TTj^ = (1 + rf where nr is the total number of repairs. Table 36.6 shows the repair costs associated with each inspection and the expected cost associated with each path. The total expected cost of repair is $172,130. The total expected cost is therefore

None of the eight paths as shown will actually be taken. After the first inspection, a decision to repair or not repair will be made. Half of the branches on the tree will be eliminated, and the optimization will be run again using the updated information. Frangopol and Estes (1999) and Estes and Frangopol (2001c) provide more detail on this bridge deck example and methodology.

### 36.7.4 Updating

Once an inspection has been completed (whether it is NDE or visual), the information should be used to update the assessment of structural condition, the load and deterioration models, and the future repair and maintenance strategies. Bayesian updating techniques are very useful when faced with two sets of uncertain information and a planner needs to know which to believe. Bayesian updating uses both the prior information and the new inspection information to account for the relative uncertainty associated with each (Enright and Frangopol 1999a).

Assume that prior to an inspection, a random variable 0 was believed to have a density function f'(0), where 0 is the parameter of that distribution (i.e., the deterioration model). During an inspection, a set of values x1, x2,..., xn representing a random sample from a population X with underlying density function f(x) are observed and are fit to a new density function f(x) (i.e., the inspection results). The updated or posterior density function f"(0), which uses both sets of information and provides the best use of both, can be expressed as (Ang and Tang 1975)

where L(0) is a likelihood function and k is a normalizing constant. For the case where both f '(0) and f(x) are normally distributed, the posterior function f''(0) is also normally distributed and has the mean value and standard deviation, respectively, as

β" )2 + m'(s)2 and β 2(s)2 m =-=-=β and s =\ -n-ΓΆ

where m, m', and m'' represent the mean values of the inspection results, the prior distribution, and the posterior distribution, respectively. The values a, a', and a" represent the standard deviations of those same distributions.

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