## 1030

Lognormal

Normalb Normalb

Normalb

Lognormal a Not part of the PONTIS definition — created to quantify the observed corrosion. The normal distributions are truncated at zero.

Source: Estes and Frangopol 2003. Reprinted with permission from the American Society of Civil Engineers.

For example, the bridge manager discovers that when an inspector classifies a girder as CS 2, there will be somewhere between 0 and 5% section loss. Due to the training that the inspectors receive, they can be expected to classify the bridge correctly 98% of the time. The bridge manager observes the inspection results for three different girders. The first girder is given a rating of CS = 2. It was rated as CS = 1 during the last inspection and has just made the transition to CS = 2. The second girder has been classified as CS = 2 for the fourth consecutive inspection. The third girder has been classified as CS = 2 for the past eight inspections. The bridge manager uses historical data on many similar bridges to determine that a bridge typically remains in CS = 2 for 10.8 years before transitioning to CS = 3. What values for thickness loss might the bridge manager use to conduct the reliability analysis and update the maintenance plan for this bridge?

It is assumed that a girder enters a CS for the first time at the mean value of section loss associated with this CS. CS 2 is assumed to be normally distributed and is correctly classified 98% of the time. The girder enters this CS at mean section loss value m = 2.5%. Assuming that the incorrect inspector classifications (2%) are evenly divided between 1% too low and 1% too high, the standard deviation of section loss is computed using the cumulative distribution function of the standard normal variate F(s). Therefore, if the girder is classified as CS = 2, the probability that all values of section loss are less than 5% is 0.99. Therefore,

/loss — u\ (5 .0% — 2 . 5%' p(loss < 5%) = 0 . 99 = F(-^-j = F(-s-

The standard deviation is therefore

The standard deviation would have been higher if the quality of the inspectors had been lower and vice versa. For the first girder that has just entered the CS, the percent section loss parameters are N[2.5, 1.074]. As the girder remains longer in a given CS, it is assumed to deteriorate. Assuming linear transition through the CS, the mean value of the distribution is expected to shift progressively higher until it reaches 5% section loss as follows:

Girder 2 has been at CS = 2 for 6 years (i.e., four consecutive inspections that represent three inspection intervals, each lasting 2 years). The mean value for the Girder 2 section loss is m = 2 . 5% + 0 . 23% per year (6 years) = 3 . 89%

Assuming the same rate of mean section loss for all girders and a constant standard deviation of the loss, the percent section loss distribution for Girder 2 is therefore N[3.89,1.074]. Finally, Girder 3 has remained at CS = 2 for 14 years (seven inspection intervals), which exceeds the expected 10.8-year transition period. It would be unreasonable to assume a mean value of section loss greater than 5% because an inspector would have logically classified such a girder as CS = 3. The percent section loss distribution for Girder 3 is therefore N[5.0, 1.074] and will remain at that rating until an inspector rates the girder into a higher CS.

This example illustrates just one approach to the problem. The bridge manager could have assumed a constant coefficient of variation rather than a constant standard deviation throughout the CS. The object is to make conservative assumptions and determine how a structure is actually behaving in relation to the deterioration model that may have been postulated several decades earlier. While this approach is not a substitute for a specified NDE inspection, it is better than no additional information. At least, it will identify those structures where a more detailed inspection is needed. This approach has been used on an actual highway bridge (Estes and Frangopol 2003) and on the miter gate for a navigational structure (Estes et al. 2002, 2004).

### 36.7.3 Inspection Optimization

Because targeted NDE inspections are time consuming and expensive, they should be planned and timed for when they will produce the most benefit. Such an inspection will typically trigger a decision on whether or not to make a repair. The timing of the inspection should coincide with when the information is most needed.

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