Through verification, 40 test data under 5 load levels meet the requirements of lognormal distribution. Therefore, the distribution function f (XP) of the variable can be obtained by integrating the probability density function, that is, as shown in the formula, P (x < XP) means the probability that the variable x is less than XP:

The meaning of formula F (XP) = P (x < XP) on the coordinate axis is the area enclosed by the distribution function curve from – ∞ to XP of the abscissa axis and – ∞ to XP of the abscissa axis, that is, the shadow area in Figure 1. When the logarithmic fatigue life satisfies the normal distribution, f (XP) represents the failure rate.

If the normal probability density function is known, it can be seen from the formula that f (XP), that is, the value of probability, is determined by the upper integral limit XP. Then let XP and f (XP) be the abscissa and ordinate respectively to obtain the normal distribution function curve, as shown in Figure 2.

As can be seen from Fig. 2, f (XP) increases with the increase of XP, because when XP increases, the shadow area surrounded by the left curve of XP expands. At that time, the area of this part should be equal to 0.5, that is, f (XP) = 0.5. When XP tends to – ∞ or ∞, f (XP) is limited to 0 and 1, respectively.

The over value cumulative frequency function is obtained by integrating the density function, that is, the integration of the value XP to – ∞ to the density function represents a greater probability than the value ‘, which plays an important role in fatigue reliability:

It can be seen from the formula that when the variable x is logarithmic fatigue life, the function p (x < XP) can be used to represent the reliability P, that is, the survival rate. P (x < XP) is also the distribution function of XP normal distribution, and P (x < XP) is the opposite event of failure rate. They have the following relationship:

The Supervalue cumulative frequency curve also shows that in Fig. 2, P (x < XP) decreases with the increase of XP, which is opposite to the change of P (x < XP) curve.

The formula is standardized and the variables are replaced before integration, so that:

Using the above relationship, the formula can be written as:

At this time, the lower integral limit becomes:

According to the formula, the integrand function is transformed into a standard normal probability density function:

Thus, P (x < XP) can be expressed not only by the area surrounded by the normal probability density curve (the shaded area in Fig. 3 (a)), but also by the area surrounded by the standard normal probability density curve (the shaded area in Fig. 3 (b)).