1. Introduction

In general, the reliability of a component is given by its capability to withstand the applied stress. Therefore, due to both the stress and the strength variables are random, then in the reliability field, the analysis is performed by using a probability density functions (pdf) to model the stress variable and a pdf to model the strength variable [1]. Moreover, because in general two types of failure modes exist, to know the fatigue and shock failures modes. And since fatigue occurs due to the repeated application of the load (cyclical load behavior), and the shock occurs when the applied stress (s) is higher than the product’s strength (S) (s>S). Then in both cases, the reliability of the analyzed component can be accessed by applying the stress-strength reliability analysis [2]. Unfortunately, if the stress and the strength variables follow both a Weibull distribution W~(β,η) with different shape parameter β , then a close solution of the Weibull/Weibull stress-strength function does not exist [3]. Therefore, in response to this problems researchers have being proposing some solutions as [4-12]. Unfortunately, their approaches are not efficient to determine the reliability of the analyzed product. This because of the solution for different shape parameter (β_S≠β_s) has been possible only in the case of (2β_S=β_s ), (β_S=2βs ). But a general solution has not been found yet. Even [7] mention that it is important to continue studying the Weibull/Weibull stress-strength behavior to generate a general solution for different shape parameter combinations.

Therefore, in this paper, based on the desired reliability index and the value of the estimated shape parameters η a common shape parameter β_c is derived. And then it is used to determine the reliability of the designed component. Moreover, based on the stress-strength analysis and the estimated Weibull parameters the safety factor which corresponds to the found reliability is also determined. And because its formulation is based only on the desired (or observed) R(t) index, given by the stress/strength Weibull analysis (see section 4.3), then the derived SF index could be considered as a probabilistic index also. In particular, it is highlighted that because the formulated SF_i index is based on the Weibull stress-strength analysis performed by using the addressed common shape parameter, the SF index also represents the corresponding stress-strength reliability R(t) index.

The structure of the paper is as follows: in section 2, the generalities of the Weibull distribution are given. Section 3 presents the stress strength of Weibull formulation. In section 4 the estimation of β_S =β_s , and β_S ≠β_s are both given. In section 5 an application of the methodology is sowed. Finally, section 6 presents the conclusion and references.

2. Generalities of the Weibull distribution

Due to its flexibility to model the design, production, and wear out or aging phases of a product, the Weibull distribution is widely used in reliability and lifetime analysis [2]. And since all products are subjected to a random (variant) environmental stress, and they have a strength to withstand the applied stress, which means that both the stress and the strength are modeling by a probability density function, then their reliability analysis is performed by using the stress-strength analysis given in the next section [13]. The compound Reliability Weibull/Weibull stress/strength probability density function is based on the Weibull distribution [14] given by:

with cumulative failure function and reliability function given by:

where the estimation of the β and η parameters is performed by using the linear form of eq. (2) as

which represents the linear model given by:

with Y_i =ln(-ln(1-F(t_i )),b₀ =-β ln(η), and x_i =ln(t_i ). In practice F(t_i )in Eq. (4) is estimated by the median rank approach to estimate the corresponding reliability of a data set and is given by [16]:

In eq. (7), form [2], the sample size n is determined to ensure the desired reliability index as:

Now based on the formulation from the above, let determine the corresponding Weibull/Weibull reliability strength function.

3. Stress-strength Weibull reliability formulation

The stress/strength function is the one we must use to determine the reliability of a product which is subjected to variant stress [8]. And since the Weibull/Weibull stress-strength function does not have the closure property [16,17], implying that it has not a close solution for β_s ≠β_S , then the purpose of this section is to present the formulation of the stress-strength Weibull/Weibull reliability function when (β_s =β_S ), as well as when (β_s ≠β_S ). In general, the stress/strength reliability function, as shown in Fig. 1, is given by the probability that the strength (S=Y) will be higher than the applied stress (s=X) R(t)=(P(Y>X)) [13]. Therefore, the inference region represents the corresponding cumulative failure function [18].

Figure 1
Stress-Strength Graphical Representation.
Source: The Authors.

Thus, based on the Weibull distribution defined in Eq. (1), the Weibull/Weibull stress/strength formulation for β_s =β_S is as follows.

3.1. Stress-strength Weibull function for ( 𝛃 𝟏 = 𝛃 𝟐 )

When the shape Weibull parameters are equal to β_s =β_S [19], the Weibull/Weibull stress/strength function is given as:

Unfortunately, because the stress behavior is determined by the environment on which the product is performing, and the product’s strength depends on the manufacturing process and material properties, then the stress and the product’s strength variables are independent each other, and as a consequence the probability that β_s =β_S is very low. Thus, now let present the stress-strength formulation when β_S≠β_s.

The formulation is as follows.

3.2. Stress-Strength Weibull Formulation for (β₁ ≠β₂)

Let first mentioning that for β_S≠β_s the Weibull distribution has not the closure property [19] given by:

It is to say, the closure property defined in Eq. (9) for the stress variable X₁=X and the strength variable X₂=Y holds only for β_s =β_S Therefore, the fact that the stress/strength Weibull/Weibull reliability function, for β_S ≠β_s has not a close solution is given as

Thus, in order to determine the reliability of a product when β_S ≠β_s , first a common β_c value must be determined and second, it has to be used in eq. (8) to determine the corresponding reliability. The common β_c value is determined as follows.

4. Common Beta parameter estimation (β_S≠β_S)=βc

The common β_c value which can be used to determine the reliability given by eq. (8), based on the desired known reliability R(tw), is given by:

In eq. (11) η_S and η_S are the Weibull strength and stress parameters respectively. Now, because the common approach consists of using a safety factor to deal with uncertainties, let derive the corresponding formula to relate the used safety factor with the desired reliability.

4.1. Probabilistic Weibull safety factor by using the common β value

The Weibull safety factor is based on the strength and stress scale parameters (η_S,η_s), of the two parameters Weibull distribution is given by:

Thus, the SF_w index for β_S =β_s is directly given by

And for β_S ≠β_s , because by using Eq. (10) in terms of eq. (13) with the common βc value

Then, the SF_w index for any desired R(w) index is given by

Or equivalently R(tw) in terms of known SF_w index is given by

Also, notice that from Eq. (11), η_S in terms of SF_w is given by

Finally, the corresponding mean and standard deviation σ_wS of the strength variable (or the mean , and stress standard deviation σ_wS of the stress, variable), by using the desired η_S (or η_S ) value is determined as follows.

The mean (or )is given by:

And the standard deviation σ_wS (or σ_wS) is given by

Now let present the steps to determine the common β_c value.

4.3. Steps to determine common β_C , SF_w and the stress-strength analysis for (β₁ ≠β₂ )

In this section the steps of the proposed methodology to estimate the reliability index R(t_w ) and the corresponding safety factor SF_w are given. However, it is important to notice that 1) the derived safety factor is unique to the used reliability index R(t_w ) 2) by using the common β_c value, the stress-strength analysis for (β₁ ≠β₂ ) can easily be performed. The steps are as follows:

Step 1. Determine the desired reliability R(t_w ) index to perform the analysis.

Step 2. By using the R(t_w ) index of step 1, estimate the corresponding cumulative failure probability as:

Step 3. By using the R(t_w ) index of step 1 in Eq. (7), determine the corresponding n value.

Step 4. Collect the set of n stress data and the set of n strength data.

Step 5. Determine the average (μ) of the stress and strength collected data.

Step 6. Determine the logarithm of each one of the stress and strength collected data and determine their log-average values as

Then determine the exponential of the average of the estimated logarithms as

Step 7. With the average of step 5 and step 6 calculate the corresponding Ratio as

Step 8. By using the result of step 7 estimate the maximum and minimum stresses σ_max and σ_min values of the stress and the strength data as

Step 9. By using the median rank approach estimate the average of the generated n y_i elements as

Step 10. Using the maximum and minimum stress σ₁ , σ₂ and μ_y values [13], determine the corresponding Weibull shape β value as

Step 11. By using the estimated β value of step 10, determine the corresponding Weibull scale η parameters as

The addressed β value of step 10, and the η value of step 11, are the Weibull stress distribution W_s (β(_s ,),η_s ).

Step 12. By using the strength σ₁ , σ₂ values of step 5 in Eq. (28 and 29) determine the Weibull strength family W_S (β(_S ,),η_S ).

Step 13. With the Weibull family of both the stress, strength and the desired reliability estimate the common β_c by using eq. (11).

Step 14. By using the βc, η_s and η_s values in eq. (8), determine the corresponding reliability index R(t_w ).

Step 15. By using eq. (13) calculate the corresponding probability safety factor SF_w

5. Numerical application

In this section the numerical application to determine the reliability R(t_w ).index and its corresponding safety factor SF_w is showed. In section 5.1 the reliability index for β_S =β_s , is presented. Section 5.2 presents the reliability index estimation besides to the corresponding probabilistic safety factor in the case of β_s ≠β_S

5.1. Numerical application for β_S =β_s

The stress in this application is the usage mileage distribution, the strength is the miles to failure distribution of the component. Table 1 gives the data for the distribution of stress and strength.

Table 1

Mileage distribution per year

Source: [20]

Since from this data the Weibull stress parameters are β_s=12.2171, and η_s=12791 and the Weibull strength distribution parameters are β_s=12.2171 and η_s=15041, then, from eq.(8) the reliability index is R(w)=0.878643, and from eq.(13) the corresponding Weibull safety factor is SF_w =1.1759. Now let present the stress-strength analysis for β_S ≠β_s .

5.1. Numerical application for β_S ≠β_s .

A shaft testing subjected to bending and torsion stress load is presented. The shaft is subjected to random variable stress, then its behavior must be represented by a probability density function. On the other hand, all elements or products possess an inherent variable strength to support the stress exposed, in such a way in Table 2 the corresponding test results are presented.

Step 1. The desired reliability to perform the analysis is 95%.

Step 2. From eq. (21), the failure probability is 5% i.e. F(t_w )=0.05.

Step 3. From eq. (7), the number of observations to be collected is n=19.49572575≅20.

Step 4. The experiment data is taken from Table 2.

Table 2

Stress-Strength Design Shaft

Source: The Authors

Step 5. From eq. (21), the average of the experiment data is μ_s =9142854.736 for the stress and μ_S =34847504.61 for the strength.

Step 6. Since the average of the log stress data is μ_Xs = 15.75, and the average of the log strength data is μ_Xs =17.3, then from eq.(22) μ_ys =6989990.705 and μ_yS=32633020,09.

Step 7. From eq.(23), the ratio of the stress and strength data are Ratio_s =5893371.078 and Ratio_S =12224343.63.

Step 8. From eq.(24 and 25), the maximum and minimum values of the stress are σ₁=15036225.81 and σ₂=3249483.65 and for the strength they are σ₁=47071848.24 and σ₂=22623160.98.

Step 9. From eq. (26), μ_yi =-0.5444.

Step 10. From eq. (27), the strength and stress shape parameters are β_s=1.44 for the stress and β_S=3.00 for the strength.

Step 11. From eq. (28), the strength and stress scale parameters are η_s=10201500.14 for the stress and η_S=39126161.24 for the strength.

Step 12. From eq. (11), the common shape parameter is βc=2.19038.

Step 13. From eq. (8) the corresponding reliability index R(t_w )=0,9499997.

Step 14. From eq. (13) the corresponding probabilistic safety factor is S_fw = 3.835.

6. Conclusion

Because all products or elements are subject to a random stress variable, and, the product or element possesses an inherent random strength to support the applied stress, then, the reliability of products must be determined by using the stress-strength reliability methodology. And because the Weibull distribution does not possess the additive property then the stress-strength analysis is not possible to perform, but by the formulated of a common shape βc parameter the stress-strength Weibull analysis can be performed. And because the βc value only depends on the desired reliability of the analysis, then for this reliability value it is unique. Since the derived SFw safety factor depends on the desired reliability, then it is a random variable, And because the SFw value is unique for the corresponding reliability index, then in practice the 𝑆𝐹𝑤 index can be used to represent the R(t) index and vice versa.

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Notes