A Mean Value Reliability Method for Bimodal Distributions

In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. For example, the random load of a bridge may have two peaks, with a distribution consisting of a weighted sum of two normal distributions, suggested by traffic load data. When binomial variables are involved, traditional reliability methods, such as the First Order Second Moment (FOSM) method and the First Order Reliability Method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation for bimodal variables and then employs a mean value reliability method to accurately predict the reliability. A limit-state function is at first approximated with the first order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The saddlepoint approximation is then applied to estimate the reliability. Examples show that the new method is more accurate than FOSM and FORM.