Equivalent Normal Concepts

Suppose that the statistical distribution for random variable vector,  is known. Then, the component  is dependent and continuous for the probability density function  and the probability distribution function . Also, it is assumed that the limit state function

is a continuous function of . Then, let’s define the probability of failure as

When the limit state function is linear, the probability of failure can be exactly evaluated without the multiple integrations. For all random variables , let’s consider that the mean value are  and the standard deviation are . Then, the normalized variable  is defined by

The probability of failure is given by

where  is ‘standard normal distribution function’ and the safety factor  can be calculated from the geometric relation shown in Figure 1. The safety factor  can be called as ‘Reliability Index’. Suppose that the linear limit state function is . Also, it is assumed that  and  are normal distributions. Then, the reliability index is denoted as

which is referred to as FOSM (First-Order Second Moment) method or MVFOSM (Mean Value First-Order Second-Moment) method.

Figure 1  The Safety Index for Linear Limit State Function

In order to extend the above concept to the general probabilistic distribution, the equivalent normalization concept is introduced. This uses transformation matrix to change the random variables into the equivalently normalized variables.

 or

where  depends on the distribution of . In the standard normal variable space shown in Figure 1, the probabilistic density function is decreased exponentially from the origin. Thus, the probability of failure  can be easily evaluated by obtaining the minimum distance() from the origin to the tangent plane of limit state function, which is graphically explained in Figure 2.

Figure 2  Graphical Representation of Reliability Index Analysis (RIA)