Weibull strength distribution and reliability S-N percentiles for tensile tests Análisis de resistencia Weibull para los percentiles S-N y su nivel de confiabilidad en test de tensión

. - Based on the true stress, the ultimate material’s strength, and the fatigue slope b values, the probabilistic percentiles of the S - N curve of ductile materials are formulated. The Weibull β and η parameters used to determine the product’s reliability are determined directly from the material’s strength values corresponding to 103 and 106 cycles. And since in Table corresponding to the properties of this A538 A (b) steel and collected by table 23 - A of Shigley Mechanical Engineering Design book; authors present the σ t , Sut, and b values of several materials, then the Weibull parameters for each one of these materials as well as the 95% and 5% reliability percentiles of their S N curves are given. A step - by - step application to the steel A538 A (b) material is presented. And based on the maximum and minimum applied stress values, the corresponding Weibull stress distribution was fitted and used with the Weibull strength distribution, in the stress/strength reliability function to determine the element’s reliability.


Introduction
Since the reliability of a mechanical component depends on the applied stress value and on the strength that the used material presents to overcome the applied stress, then because both the applied stress and the material's strength are random variables, then researchers have been proposing to use a probabilistic stress-cycles S-N curves. However, because the probabilistic percentiles of the S-N curves are based on the common confidence interval (CL) of the expected average, as shown in section 3.3, then the proposed formulations are inefficient to perform a reliability analysis. Thus, in this paper based on the theory given in [1], a Weibull methodology to determine the strength distribution and the reliability percentiles of the S-N curve are both given. In the proposed Weibull/tensile test methodology, the only needed inputs are 1) the ultimate material's strength [2] ( ) value, (which is a measure of the maximum stress that an object/material/structure can withstand without being elongated, stretched or pulled). 2) the true stress ( ) [2] value, (which measures the change in the area with respect to the time while the specimen is loading), and 3) the fatigue slope b value of the S-N curve. With these three inputs, the corresponding strength Weibull shape β and scale ( ) parameters used to determine the reliability percentiles of the S-N curve, are both determined based on the = strength value that corresponds to 1 = 10 3 cycles and on the strength ( ) value that corresponds to 1 = 10 6 cycles. The validation that the addressed strength β and ( ) parameters completely represent the and values, is demonstrated by showing that by using the β and ( ) parameters we always can reproduce the and values.
And because in the Table A-23 of the Shigly's  book, for several steel materials, authors present  their  , and b values, then in this paper by using the proposed methodology, their corresponding strength β and ( ) parameters, the log-mean µ and log-standard deviation ( ) values, as well as the 95% and 5% reliability percentiles of their S-N curves are all given in section 6. The novelty of the given reliability percentiles is that they do not represent a confidence interval CL of the S-N curve, instead they represent a reliability confidence interval for the S-N curve. But more importantly notice that because the S-N reliability percentiles are the reliability percentiles of the strength ( ) parameter, then because in any Weibull analysis the reliability percentiles of ( ) are always determined, then automatically we can use these ( ) percentiles as the corresponding S-N percentiles. Consequently, any Weibull strength analysis can be seeing as a representation of the reliability percentiles of the related S-N curve [3], [4]. Additionally, because the reliability of the component depends on the applied stress and on its strength, then in section 5, the Weibull strength parameters that represents the desired S-N reliability percentiles, and the Weibull parameters that represents the applied stress, are both used in the stress/strength methodology [5] to determine the reliability of the designed element.
The structure of the paper is as follows. Section 2 presents the generalities of a tensile test. In section 3, the steps of the proposed Weibull/Tensile/Reliability percentiles methodology are given. In section 4, a step-by-step application of the proposed method is given. In section 5, the stress/strength analysis to determine the reliability of the component is presented. In section 6 the Weibull β and ( ) parameters, the 95% and 5% reliability percentiles and the corresponding log-mean and log-standard deviation for each one of the steel materials given in the Table A-23 of the Shigly's book are provided. Finally, in section 7, the conclusions are presented. ISSN: 2594ISSN: -1925 In general, in a tensile test the material properties are directly measured from a sample that is tested at controlled tension force (F) until failure. The most general material's properties [2] are the ultimate tensile strength , (it is a measure of the maximum stress that an object/material/structure can withstand without being elongated, stretched or pulled), the true stress , (it measures the change in the area with respect to time while the specimen is loaded), the maximum elongation (L), and the reduction in the initial area ( 0 ).

Tensile Test Generalities
Since these material's properties are random variables, then in the analysis a probability density function (pdf) must be used [6] pg. 10. In the analysis, the most used pdfs are the normal, lognormal and Weibull distributions. Fortunately, as demonstrated in [7], for mechanical stress the best distribution is the Weibull distribution, and from [1] we have that from the Weibull analysis we always can reproduce the analyzed principal stresses (or strength) values. Therefore, in this paper the Weibull distribution is used. Also notice that for β≈3.4 the Weibull distribution efficiently mimics the normal distribution, and for β>5 [8], it efficiently mimics the lognormal distribution.
However, before showing the Weibull distribution completely reproduce the used material's strength values, let first present the generalities of a tensile test formulation.

General Tensile Test Formulation
In a tensile test analysis, by defining the engineering stress value as = / 0 , and the engineering strain value as ɛ = where F is the applied force, 0 is the initial area of the tested element, and 0 is the initial length, and L is the final elongation of the tested element (see Fig.1).

Figure 1. Test Specimen. Source: The Authors
The relationships among the ultimate material's strength , the true stress , and the true strain ɛ values (see Fig. 2) on which the proposed method is based, are as follows. Based on both F and 0 , the value is defined as Therefore, based on the and ɛ values the true stress value defined as the instantaneous applied stress, at the coordinate, in terms of the and ɛ values are determined as And the true strain value at the coordinate is given as

Fatigue Slope Formulation
In the analysis, the fatigue slope b value of the S-N curve is the exponent that let us to determine the strength range that corresponds to a desired pair of life cycles values [1]. The common approach in the S-N analysis consists in determining b in the logarithm range given by 1 = 10 3 and 2 = 10 6 cycles (see Fig.3). In this logarithm scale the cycles-strength coordinates to determine b are [log (10 3 ), log ( )] and [log(10 6 ) , log( )]. Where f represents the strength's percentage that the material presents after 10 3 cycles, and represents the corresponding fatigue strength limit. Hence, since in this logarithm range the S-N curve behavior is linear given as Where 1 = ( ), 2 = ( ), 1 = (10 3 ) and 2 = (10 6 ), then the fatigue b and parameters of the S-N curve are determined as Therefore, based on Eqs. (5a and 5b) the relation between the applied stress and its corresponding cycles to failure is given by the Basquin formula given as However, when is unknown, then the fatigue b value defined in Eq.(5a), based on the value is given as Consequently, the cycles to failure defined in Eq.(5c) based on the value is given as Now that from Eq. (5a and 6a) we can determine the b value, let present the methodology to determine the strength Weibull β and ( ) parameters directly from the and values.

Weibull/Tensile Test/Reliability Methodology
This section is structured to present 1) the steps to determine the strength Weibull β and ( ) parameters directly from the maximum = ( ) = and the minimum ( ) = tensile strength values. 2) how to use the derived β and ( ) parameters to determine the reliability percentile of the related S-N curve. And 3) how to determine the log-standard deviation value directly from the β value. Let start given the Weibull's generalities.

Generalities of the Weibull distribution
For the two parameter Weibull distribution [9] given by Where t represents the desired life time, β is the shape parameter and η is the scale parameter. However, since in this paper the life of the element is represented by either its cycles to failure N, or by its material's strength value, then by replacing t in Eq. (7) with either or , the corresponding Weibull reliability function is given as From Eq. (8), notice that 1) although to determine the reliability of the element we can use either or , the corresponding ( ) and ( ) values are different ( ( ) ≠ ( ) ). And 2) the ( ) and ( ) values are related by the life/stress model, as can be the Arrhenius, the inverse power law model and the Basquin equation defined here in Eq.(5c). Also notice that because in Weibull analysis, by supposing the failure mode remains constant, then in the analysis the β value is considered to be constant [10]. Consequently, as shown in Eq. (8), in any Weibull analysis, we always have two Weibull families. One representing the cycles to failure W(β, ( ) ), and the other representing the material strength W(β, ( ) ). Here the analysis is performed based on the W(β, ( ) ) family. Now let present the steps to determine the β and ( ) parameters directly form the tensile = ( ) = and ( ) = values.

Steps to Determine the Weibull Strength Parameters
Step1. From the used material determine the corresponding , and fatigue slope b values. Step2. Determine the desired reliability R(n) index to perform the analysis. In practice, it is R(n)=0.9535. And it corresponds to test a set of n=21 parts [11]. From [11], the relation between R(n) and n is given as Here observe R(n) is not the reliability of the element, instead R(n) is just the reliability on which the analysis will be performed. R(n) is alike the confidence interval CL used in the quality field.
Step3. By using the n value of step 2 in Eq. (10), compute the elements [12] and its corresponding arithmetic mean µ and standard deviation values as Note 2. Observe, once n was selected in step 2, the µ and values computed from the elements defined in Eq. (10) are both constant. For n=21 (or R(n)=0.9535) they are µ = −0.54562412 and = 1.17511694. In this paper these two values are used.
Step 4. Based on Eq.(6b), by using 1 = 10 3 and the and b values of step1, determine the maximum strength value as Note 3. Observe that because = * , then from Eq. (11) the f value is directly given as = ( )/ .
Step 5. If the value is unknown, then based on Eq.(6b), by using 2 = 10 6 and the and b values of step1 determine the minimum strength value as Step 6. By using the µ value from step 3, and the and values, determine the strength Weibull shape parameters as Step 7. By using the addressed and values, determine the Weibull scale parameters as ( ) = � * 2 (14) ISSN:  The β and ( ) parameters determined in steps 6 and 7 are the parameters of the Weibull strength distribution.

Note 4. Notice if ,
, and are known then from Eq.(5a) b can be estimated, implying the true stress value is not necessary. It is to say, as shown in Eqs. (13 and 14), the Weibull strength parameters only depends on the and values. Now based on the β and ( ) parameters let determine the corresponding log-mean µ and logstandard deviation values used to formulate the confidence interval of µ .

Steps to Determine the Log-mean and the Log-standard Deviation
The analysis is based on the linear form of the reliability function [2] defined in Eq.(9) given as Thus, since from Eq. (15) = ln( ), then we need to determine its log-mean µ and its logstandard deviation values. From [1] the µ value is directly given by the strength scale ( ) parameters as And from [13], based on both the µ value of step 3, and on the addressed β value, the value is given as Thus, a confidence interval (CL) of µ is given as Where /2 is the th desired percentile given by the normal distribution, (which for CL=0.95, is Unfortunately, although from Eq. (16a) µ = ln ( ), the CL limits defined in Eq. (17) cannot be used to determine a confidence interval for ( ) . Consequently, Eq. (17) cannot be used to determine the reliability percentiles of the S-N curve neither. This fact occurs because there is not a direct relationship between CL and R(t). CL represents an instantaneous probability that the strength of n identical components behaves around µ , and R(t) represents the probability that a observed (measured) µ value stay around this value through the time. It is to say, while the CL value depends only on the lack of homogeny of the material, the R(t) index depends also on the applied stress, the desired time t, and on the observed µ value. Thus, Eq. (17) should not be used to determine the S-N percentiles that represents the desired R(t) index. Numerically, the deficiency of using CL in reliability analysis is given in section 4.2.
Here notice that in contrast to Eq. (17), in reliability analysis we are interested only in the upper limit. Consequently, since from Eq. (8) the R(t) index depends only on the ( ) value, then because µ = ln ( ( ) ), in the analysis µ is the lower allowed value that we can used to design the element. Therefore, as shown in [14] if µ = ln ( ( ) ) is going to be monitored in a process, then in the monitoring control chart the µ value must be set us the lower allowed value. Now based on the addressed µ and values, let present the formulation to determine the reliability percentile of the related S-N curve.

Reliability Percentiles for the S-N Curve
The efficiency of the proposed method is based on the following two facts.
2) Because in logarithm scale the three values, ln ( ), ln ( ) and ln ( ), all are in the same S-N line, then this line represents the lower threliability percentile for which it is expected the product present the desired R(t) index. Consequently, from Eq. (18) and Eq. (8), we have that the following reliability relationship always holds Eq. (19) implies that in practice, the derived reliability percentiles of the S-N curve can also be used as the minimum strength ( ) value that the used material must present to have the desired reliability. Now based on the above two facts, the steps to determine the reliability percentiles of the S-N curve are as follows.

Steps to Determine the Reliability Percentiles for the S-N Curve
Step 1. Determine the element that corresponds to the desired upper reliability percentile of the S-N curve as Step 2. Determine the element that corresponds to the desired lower reliability percentile of the S-N curve as Step 3. By using the value of step1, determine the upper values of , , and that corresponds to the upper reliability percentile of the S-N curve as Step 4. By using the value of step 2, determine the lower value of , ( ) , and that corresponds to the lower reliability percentile of the S-N curve as Step 5. Plot the upper and lower reliability percentiles.
Now let present the numerical application.

Numerical Application
As an application let used data given in the first row of Table A
Step 3. The elements are given in Table 1. From these data µ = −0.54562412 and = 1.17511694. Now based on these parameters let determine the corresponding log-mean µ and log-standard deviation values mentioned in section 3.3.

Log-mean and Log-standard Deviation
From Eq. (16a), the log-mean is µ = ln(806.7353) = 6.692995 and from Eq.(16b) the log-standard deviation is = , and then it should be monitored in the production process in logarithm scale as in Fig.4 and/or in natural scale as in Fig.5   Unfortunately, as mentioned above in reliability, monitoring (or using) the lower limit of ( ) is not correct because in reliability the addressed ( ) value (or nominal µ value) is the lower allowed value. Thus, in the monitoring process, the ( ) value (or equivalently the µ value) is the one that must be set as the lower allowed limit in the control chart (see Fig.6 and Fig.7).   Additionally, it is shown that although by using the CL limits defined in Eq. (17), the 95% confidence for the S-N curve plotted in Fig.8 is possible, they do not the 95% reliability confidence interval for the S-N curve. Consequently, because the CL confidence interval is not a reliability percentile, then by using the CL values in Eq. (19), the estimated reliability is not the desired R(t)=0.95 index. Therefore, the general conclusion is that by using the CL limits in reliability analysis we subestimate the real R( ) index (0.8653<0.95) of the element, and consequently the CL limits should not be used in the reliability analysis.
Now we know the CL values should not be used, let determine the reliability percentiles for the S-N curve that we can use in any reliability analysis. Following section 3.4.1, the analysis is as follows.

Reliability Percentiles for the S-N Curve
The reliability percentile analysis for the S-N curve is as follows Step 1. From Eq.(20a) the upper element for R(t)=0.95 is = (−( (0.95))) = −2.970195249.
Step The corresponding percentiles of the S-N curve in MPa and in logarithm scale are all given in Table 2.
Source: The Authors Here it is very important to notice from either Table 2 or Figure 9 that data in MPa do not fall in a right line with the ( ) value.
In contrast observe from Fig. 10 that in logarithm scale they are in line with the ( ) value. Also notice from Fig.9 and Fig.10 that the upper and lower percentiles are not symmetric around the ( ) value, and that this fact is due to in Weibull analysis, the ( ) does not represent the 0.50 percentile, instead it represents the 0.6321 failure percentile, implying the limits around the ( ) value never will be symmetric around the ( ) value. Additionally, remember that as shown in Eq. (18), the symmetrical behavior around ( ) occurs only for the and values from which the ( ) value was determined. In order to clarify the mentioned facts, in Table 3 the Weibull analysis for the expected values of ( ) are given.  The practical interpretation of data given in Table  3 is as follows.
1. The values of the column in Table 3 represent the maximum applied stress values for which a product that has the ( ) strength value, will present the reliability R(t) index given in the row of Table 3 that corresponds to the selected value. For example, if a component (material) with strength of ( ) = 806.7353 , is subjected to constant stress of =403.35MPa, then as shown in Table 3, it is expected the element will present a minimum reliability of  Table 3, by using the value defined in Eq. (10), the corresponding value was determined as The values of the column ( ) in Table 3, represent the strength value that a product should has to present the given reliability R(t) index when the applied stress is constant at the ( ) value. For example, the ( ) = 1613.55 value given in the first row of Table 3, represents the minimum strength value that a product (material) must have to presents a reliability of ( ) = 0.9673 when the maximum applied stress is constant at the value of ( ) = 806.7353 .  Table 3, the ( ) value was determined as From Table 3 also notice the rows where the Weibull analysis reproduce the = 1009.79 and = 644.51 values, as well as the upper 95% and lower 5% percentiles of ( ) were also added. Also from Table 3, notice that as shown in Fig. 9 and in Fig. 10 the behavior around the ( ) value is not symmetrical. Now let determine the reliability of a component by using the stress/strength analysis.

Stress/Strength Analysis
Since all mechanical element is subjected to an applied stress and it has an inherent strength to overcome the applied stress, then because both the stress and the strength are random variable, the element's reliability must be determined based on the distribution that represent the applied stress, and on the distribution that represent the inherent strength. Therefore, the right reliability function to be used in the analysis of a mechanical element is the composed reliability function known as a stress/strength reliability function [15]. In this stress/strength approach any pair of combination of stress and strength functions is possible. However, the most common combinations are the normal/normal, the log-normal/log-normal, the Weibull/Weibull and any pair of combination among these three distributions [16]. But because here the analysis is a stress-based analysis which is efficiently modeled by the Weibull distribution, then the Weibull/Weibull approach is used as follows.

Numerical Analysis
In this section, the strength Weibull distribution data addressed in section 4.1 of the steel grade (a) A538A (b) material is used. From this section the addressed Weibull strength family is W (β=4.909848, η(σ)=806.7353MPa). Therefore, to apply the stress/strength analysis the corresponding stress Weibull distribution must be addressed. Doing this, suppose from an application the maximum principal applied stress is 1 = 600 and the minimum principal applied stress that generates a failure is 2 = Finally, it is important to observe because the reliability index given in Table 3 and that given from Eq. (25) tends to be the same for high reliability indices, (say a reliability above 0.90), then the reliability of an element can be determined directly by using the Weibull strength parameters as in Table 3, or by using the stress and strength distributions in Eq. (25).
Seeing this numerically, suppose that in an application the used material is subjected to reversible stress with Weibull stress parameter ηs=403.35MPa. Therefore, from Eq. (25), as shown in Table 3 Consequently, for high reliability indices, the column of any Weibull Strength analysis can be used as the maximum allowed constant stress value that we can apply, in order the component presents the desired reliability. Similarly, the ( ) column of any Weibull Strength analysis can be used as the minimum allowed strength value that the used material must present, in order the designed element present the desired reliability when it is subjected to a maximum stress value represented by the strength scale ( ) value. Now by using the proposed Weibull/S-N methodology, the Weibull parameters, the logmean and log-standard deviation parameters and the 0.95 and 0.05 reliability percentiles of each one of the steel materials given in Table A-23 of the Shigly's book are all given in Table 4. Table A-23 of the Shigly's book.

Weibull/S-N analysis for Materials given in
The analysis is presented in Table 4. ISSN:   4. As shown in Table 3, the lower reliability percentiles of the S-N curve are the minimum strength values given in the column ( ) of Table   5. Due to the column ( ) of Table 3 represents the minimum strength values that the designed element must have to present the desired reliability, then the reliability percentiles of the S-N curve can be used as the accelerated levels in and ALT test to demonstrate the product presents the intended reliability [17]. Table 3 is for constant stress values, and that given by the stress/strength methodology is for variable stress behavior, for high reliability = 0.9678. indexes, the estimated reliability indexes are both similar [18] [ ( ) ≅ � , , ( ) �]. Formal formulation why this fact occurs is an open issue on which more research must be undertaken.