The method's steps to estimate the Weibull shape (β) and scale (η) parameters, based only on the ratio of the maximal and minimal principal stresses (σ 1 /σ 2 ) and on the designed reliability (R(t)) are given in Section 4.1. The method's efficiency is based on the following facts: (1) The square root of σ 1 /σ 2 represents the base life on which the Weibull lifetimes are estimated (see Equation 61).(2) The mean of the logarithms of the expected lifetimes (g(x)) is completely determined by the determinant of the analyzed stress matrix (see Equation 13).(3) The Weibull distribution is a circle centered on the arithmetic mean (μ), and it covers the whole principal stresses' span (see Figure 5). (4) σ 1 /σ 2 and g(x) completely determine the σ 1i and σ 2i values, which correspond to any lifetime in the Weibull analysis (see Equation 54). And (5) σ 1 /σ 2 and η completely determine the minimal and maximal lifetime, which corresponds to any σ 1i and σ 2i values (see Equation 57). Additionally, by using the addressed stress β and η parameters, when the stress is either constant or variable, the formulation to estimate the designed R(t) index is given. The steps to determine both the material's strength average (μ M ) for a desired R(t) index and the R(t) index, which corresponds to a used μ M value, are given.

Because the normal process capability indices (PCIs) C p , C pu , C pl , and C pk represent the times that the process standard deviation is within the specification limits; then, based on and by using the direct relations among the parameters of the Weibull, Gumbel (minimum extreme value type I) and lognormal distributions, the Weibull and lognormal PCIs are derived in this paper. On the other hand, because the proposed PCIs P p , P pu , P pl , and P pk were derived as a function of the mean and standard deviation of the analyzed process, they have the same practical meaning with those of the normal distribution. Results show that the proposed PCIs could be used as the standard C p , C pu , C pl , and C pk if a short-term variance is analyzed. An application to a set of simulated data is presented.

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Because in Weibull analysis, the key variable to be monitored is the lower reliability index (R(t)), and because the R(t) index is completely determined by both the lower scale parameter (η) and the lower shape parameter (β), then based on the direct relationships between η and β with the log-mean (μ x ) and the log-standard deviation (σ x ) of the analyzed lifetime data, a pair of control charts to monitor a Weibull process is proposed. Moreover, because the fact that in Weibull analysis, right censored data is common, and because it gives uncertainty to the estimated Weibull parameters, then in the proposed charts, μ x and σ x are estimated of the conditional expected times of the related Weibull family. After that both, μ x and σ x are used to monitor the Weibull process. In particular, μ x was set as the lower control limit to monitor η, and σ x was set as the upper control limit to monitor β. Numerical applications are used to show how the charts work.

In this paper, the formula to estimate the sample size n to perform a random vibration test is derived only from the desired reliability (R(t)). Then, the addressed n value is used to design the ISO16750‐3 random vibration test IV for both normal and accelerated conditions. For the normal case, the applied random vibration stress (S) is modeled by using the Weibull stress distribution [W(s)]. Similarly, for the testing time (t), the Weibull time distribution [W(t)] is used to model its random behavior. For the accelerated case, by using the over‐stress factor fitted from the W(t) and W(s) distributions, four accelerated scenarios are formulated with their corresponding testing's profiles. Additionally, from the W(s) analysis, the stress formulation to perform the fatigue and Mohr stress analysis is given. Since the given Weibull/fatigue formulation is general, then the formulas to determine the W(s) parameters, which correspond to any principal stresses values and/or vice versa, are given. Although the application is performed to demonstrate R(t) = 0.97 by testing only n2 = 6 parts, the guidelines to use the values given in columns n, S, and t of the Weibull analysis table to generate several accelerated testing plans are given.

In Weibull accelerated life test analysis (ALT) with two or more variables (<em>X₂, X₃, ... X_k</em>), we estimated, in joint form, the parameters of the life stress model r{X(t)} and one shape parameter β. These were then used to extrapolate the conclusions to the operational level. However, these conclusions are biased because in the experiment design (DOE) used, each combination of the variables presents its own Weibull family (β_i, η_i). Thus the estimated β is not representative. On the other hand, since (β_i, η_i) is determined by the variance of the logarithm of the lifetime data σ_t² , the response variance σ_y² and the correlation coefficient R², which increases when variables are added to the analysis, β is always overestimated. In this paper, the problem is statistically addressed and based on the Weibull families (β_i, η_i) a vector Y_η is estimated and used to determine the parameters of r{X(t)}. Finally, based on the variance σ_y² of each level, the variance of the operational level σ_op² is estimated and used to determine the operational shape parameter β_op. The efficiency of the proposed method is shown by numerical applications and by comparing its results with those of the maximum likelihood method (ML).

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.

Applying Goodman, Gerber, Soderberg and Elliptical failure theories does not make it possible to determine the span of failure times (cycles to failure-Ni) of a mechanical element, and so in this paper a fatigue-life/Weibull method to predict the span of the Ni values is formulated. The input’s method are: (1) the equivalent stress (σeq) value given by the used failure theory; (2) the expected Neq value determined by the Basquin equation; and (3) the Weibull shape β and scale η parameters that are fitted directly from the applied principal stress σ1 and σ2 values. The efficiency of the proposed method is based on the following facts: (1) the β and η parameters completely reproduce the applied σ1 and σ2 values. (2) The method allows us to determine the reliability index R(t), that corresponds to any applied σ1i value or observed Ni value. (3) The method can be applied to any mechanical element’s analysis where the corresponding σ1 and σ2, σeq and Neq values are known. In the performed application, the σ1 and σ2 values were determined by finite element analysis (FEA) and from the static stress analysis. Results of both approaches are compared. The steps to determine the expected Ni values by using the Weibull distribution are given.