## Example 3613

Assuming that the load effect (i.e., maximum moment caused by the load on the member) on the retaining wall beam is normally distributed with a mean value of 250 kN m and a standard deviation of 25 kN m, assess the condition of the support beam from Example 36.12 in terms of probability of failure at years 10, 20, and 40.

### Answer

The probability of failure (pf) is the probability that the demand will exceed the capacity. In this case, the relevant failure mode is bending, so pf = p(Mdemand > Mcapacity). The limit state equation that defines the failure surface is g(X) = X1 — X2 = 0, where X1 = Mcapacity and X2 = Mdemand. Because both are assumed to be normally distributed, the results are as follows:

Year 0

Year 10

Year 20

Year 40

\J u Capacity ~ u Demand

V492 + 202

While the loss in plastic section modulus may not appear extreme, the probability of failure has increased dramatically with time.

This example shows how reliability can be computed at points in time for a deteriorating structure. The example was greatly simplified as it assumed that all distributions were normal and that the load did not change over time. It ignored extreme loading events and other failure modes, all of which must be accounted for in a real world problem.

### 36.6 Time-Dependent Reliability

The methods described above apply to computing the reliability of a structure at a specific point in time. When attempting to make decisions about a structure over its useful life, time becomes an important variable. If the future load and resistance of the structure can be predicted, the simplest approach is a point-in-time method where the reliability is computed at various specific times in the future. A trend is established, and the structure is planned for a repair when the reliability falls below an acceptable target reliability level. Loads tend to increase over time, and the resistance tends to decrease as the structure deteriorates, so the overall reliability can generally be expected to decrease over time. A weakness of this approach is that it fails to account for previous structural performance.

A more accepted, yet increasingly complex, approach to time-dependent reliability is to compute the probability that a structure will perform safely for a specified period of time (Enright and Frangopol 1998a,b). Whereas reliability is defined as the probability that an element is safe at one particular time, the survivor function S(t) defines the probability that an element is safe at any time t:

where the random variable T represents time and t > 0. The probability that a failure, pf(t), takes place over a time interval At is expressed as f (t)At = P[t1 < T< t1 + At]

where the PDF f (t) = — S'(t). It is assumed that the derivative S'(t) exists. The probability of failure between times ta and tb is

The reliability is often expressed in terms of a hazard function, H(t), also called the instantaneous failure rate, the hazard rate, or simply the failure rate. The hazard function expresses the likelihood of failure in the time interval t1 to t1 + dt given that the failure has not already occurred prior to t1 and can be expressed as f(t) _ —S'(t) _ 1 dps(t)

Thus, the hazard function is the ratio of f(t) to the survivor function S(t). All hazard functions must satisfy the nonnegativity requirement. Their units are typically given in failures per unit time. Large and small values of H(t) indicate great and small risks, respectively (Leemis 1995).

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