Since the designed bearing’s reliability of 90% was determined in a lab environment, it does not represent the actual used environment. In this paper, a new methodology to determine the actual reliability that corresponds to the use conditions is offered. This new method is based on the standard method used to select the ball bearing. The proposed method is based on the two parameters of Weibull distribution, where the shape (β) and scale (η) parameters are both determined from the Hertz contact stresses values, which are generated under the surface of the motionless outer race, and by the forces transmitted between the ball and the outer race. Therefore, the derived reliability is different from the 90% index offered by manufacturers.

Since products are subjected to a random variable stress-strength, their reliability must be determined using the stress-strength analysis. Unfortunately, when both, stress and strength, follow a Weibull distribution with different shape parameters, the reliability stress-strength has not a close solution. Therefore, in this paper, the formulation to perform the analysis stress-strength Weibull with different shape parameters is derived. Furthermore, the formulation to determine the safety factor that corresponds to the designed reliability is also given. And because the relationship between the derived safety factor and the designed reliability is unique, then because reliability is random, the derived safety factor is random.

In this paper, based on the Hertzian contact analysis, we determine the minimum dynamic load of a ball bearing using a Weibull probabilistic approach. The lower L₁₀ life limit was determined considering a confidence interval, the sample size that corresponds to the L₁₀ life (90% reliability), and the standard deviation of the related Weibull scale parameter η. Then, we determine a functional relationship between the normal and the Weibull distributions to determine the corresponding $1\mathsf{\sigma }$ , $2\mathsf{\sigma }$ , and $3\mathsf{\sigma }$ Six Sigma indices of the ball-bearing life. Finally, a standard method was established to determine the minimum dynamic load rating required for a ball bearing in such a way that it achieves 90% reliability at the lower bound of its ${\mathrm{L}}_{10}$ life. In the case study, the SKF 6009 ball bearing was initially selected, with a dynamic load rating of 22.1 kN, exhibiting a lower bound reliability of 0.86 under a nominal Weibull scale parameter of strength of 910 MPa. After applying the proposed analysis, it was replaced with the SKF 71909 ball bearing, which provides a nominal Weibull scale parameter strength of 1174.54 MPa, which ensures 90% reliability at the lower bound of its L₁₀ life. Its reliability indices at the $1\mathsf{\sigma }$ , $2\mathsf{\sigma }$ , and $3\mathsf{\sigma }$ Six Sigma levels were $\mathrm{R}\left(\mathrm{t}\right)=0.9174$ , $\mathrm{R}\left(\mathrm{t}\right)=0.8880$ , and $\mathrm{R}\left(\mathrm{t}\right)=0.8534$, respectively.