Revista de Ciencias Tecnológicas (RECIT). Volumen 3 (1): 10-22
Revista de Ciencias Tecnológicas (RECIT). Universidad Autónoma de Baja California ISSN 2594-1925
Volumen 5 (3): e230. Julio-Septiembre, 2022. https://doi.org/10.37636/recit.v5n3e230.
ISSN: 2594-1925
1
Research article
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
Manuel Baro Tijerina1, Manuel Román Piña Monarrez2, Jesús Barraza Contreras2
1Industrial and Technology Department, Instituto Tecnológico Superior de Nuevo Casas Grandes, Casas Grandes,
México
2Industrial and Manufacturing Department at IIT Institute, Universidad Autónoma de Ciudad Juárez, Ciudad
Juárez, México
Autor de correspondencia: Manuel Baro Tijerina, Instituto Tecnológico Superior de Nuevo Casas Grandes, Casas
Grandes, México. Email: mbaro@itsncg.edu.mx. ORCID: 0000-0003-1665-8379
Recibido: 31 de Julio del 2022 Aceptado: 20 de Septiembre del 2022 Publicado: 22 de Septiembre del 2022
Keywords: Mechanical design; True stress-strain; Weibull distribution; Fatigue reliability analysis;
Stress/Strength, Reliability Engineering.
Resumen. – Basado en el estrés verdadero σ_t, la última resistencia del material S_ut, y la curva de fatiga b, la
curva S-N de material de acero dúctil es formulada. La distribución Weibull con parámetros β y η son usados para
determinar la confiabilidad del elemento y ambos son directamente determinados por la resistencia del material
que en este caso corresponde a 103 y 106 ciclos. Y como corresponde en la tabla de propiedades del acero A538
A (b) y recolectada esta información del libro de Ingeniería mecánica de Shigley: los autores presentan el estrés
verdadero, ultimo estrés y la curva de diferentes materiales. Entonces los parámetros Weibull β y η, así como los
percentiles de confiabilidad 95 y 5 % de la curva S-N son presentados. Se presenta una aplicación paso por paso
para el acero A538 A (b). Y basado en el máximo y mínimo estrés aplicado, la distribución Weibull correspondientes
es presentada. Por último, basado en el máximo y mínimo estrés, la distribución Weibull correspondiente fue
ajustada y usada con la resistencia de la distribución Weibull, en la función estrés-resistencia de confiabilidad con
el objeto de estimar la confiabilidad del elemento.
Palabras clave: Diseño mecánico; Estrés-resistencia; Distribución Weibull; Análisis de fatiga; Ingeniería de
confiabilidad.
Abstract. - 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 σt, 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.
Revista de Ciencias Tecnológicas (RECIT). Volumen 5 (3): e230
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1. 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
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)
() 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 cycles
and on the strength () value that corresponds to
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-
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 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-
provided. Finally, in section 7, the conclusions are
presented.
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2. Tensile Test Generalities
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
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 ().
S
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
mimics the normal distribution, and for β>5 [8], it
efficiently mimics the lognormal distribution.
However, before showing the Weibull distribution
values, let first present the generalities of a tensile
test formulation.
2.1 General Tensile Test Formulation
In a tensile test analysis, by defining the
engineering stress value as , and the
engineering strain value as
where F
is the applied force, is the initial area of the
tested element, and is the initial length, and L is
the final elongation of the tested element (see
Fig.1).
Figure 1. Test Specimen. Source: The Authors
strength , the true stress , and the true strain
values (see Fig. 2) on which the proposed
method is based, are as follows. Based on both F
and , the value is defined as
(1)
Therefore, based on the and values the true
stress value defined as the instantaneous applied
stress, at the coordinate, in terms of the and
values are determined as
(2)
And the true strain value at the coordinate is
given as
(3)
Figure 2. Stress-Strain representation. Source: The Authors
Thus, since now from Eq. (1) the value can be
determined, and from Eq. (2), the corresponding
value is given, then now let present how the b
value is determined.
Lo
Ao
Figure1. Test Specimen
Syt
Sut
Necking
Figure 2. Stress Strain Diagram
Flowcurve
T
= L/L0
=F/A
T
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2.2 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 and
cycles (see Fig.3). In this logarithm scale the
cycles-strength coordinates to determine b are [log
( and . Where
f
material presents after cycles, and
represents the corresponding fatigue strength limit.
Figure 3. S-N curve representation. Source: The Authors
Hence, since in this logarithm range the S-N curve
behavior is linear given as
for i=1,2 (4)
Where , ,
and , then the fatigue b
and parameters of the S-N curve are determined as
(5a)
(5b)
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
(5c)
However, when is unknown, then the fatigue b
value defined in Eq.(5a), based on the value is
given as
(6a)
Consequently, the cycles to failure defined in
Eq.(5c) based on the value is given as
(6b)
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.
3. 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
3.1 Generalities of the Weibull distribution
For the two parameter Weibull distribution [9]
given by
(7)
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
103104105106107
Se
Sf
S-N Diagram
Cycles
Figure 3. S-N Curve Representation
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is represented by either its cycles to failure N, or by
value, then by replacing t
in Eq. (7) with either or , the corresponding
Weibull reliability function is given as
(8)
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.
3.2 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
(9)
Note 1. 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 and standard deviation
values as
(10)
Note 2. Observe, once n was selected in step 2, the
and values computed from the elements
defined in Eq. (10) are both constant. For n=21 (or
R(n)=0.9535) they are and
. In this paper these two values
are used.
Step 4. Based on Eq.(6b), by using and
the and b values of step1, determine the
maximum strength value as
(11)
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 and the and b values
of step1 determine the minimum strength value
as
(12)
Step 6. By using the value from step 3, and the
and values, determine the strength Weibull
shape parameters as
(13)
Step 7. By using the addressed and values,
determine the Weibull scale parameters as
(14)
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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 and log-
standard deviation values used to formulate the
confidence interval of .
3.3 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
(15)
Thus, since from Eq. (15) , then we
need to determine its log-mean and its log-
standard deviation values. From [1] the
value is directly given by the strength scale
parameters as
(16a)
And from [13], based on both the value of step
3, and on the addressed β value, the value is
given as
(16b)
Thus, a confidence interval (CL) of is given as
(17)
Where is the th desired percentile given by the
normal distribution, (which for CL=0.95, is
).
Unfortunately, although from Eq. (16a)
, 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
, and R(t) represents the probability that a
observed (measured) 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
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
, in the analysis is the lower allowed
value that we can used to design the element.
Therefore, as shown in [14] if is
going to be monitored in a process, then in the
monitoring control chart the value must be set
us the lower allowed value.
Now based on the addressed and values, let
present the formulation to determine the reliability
percentile of the related S-N curve.
3.4 Reliability Percentiles for the S-N Curve
The efficiency of the proposed method is based on
the following two facts.
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1) Since from Eq.(14), is given as the square
root of the product of , and , then in logarithm
scale is the average between and
, implying that
or equivalently that the relation given in
Eq.(18) always holds
(18)
2) Because in logarithm scale the three values,
, and , all are in the same S-N
line, then this line represents the lower th-
reliability 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
(19)
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.
3.4.1 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
(20a)
Step 2. Determine the element that corresponds
to the desired lower reliability percentile of the S-
N curve as
(20b)
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
;
;
(21)
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
;
;
(22)
Step 5. Plot the upper and lower reliability
percentiles.
Now let present the numerical application.
4. Numerical Application
As an application let used data given in the first
row of Table A-
material is the steel grade (a) A538A (b). For this
material, the Weibull strength parameters of
section 3.2 are as follows.
4.1 Weibull Strength Parameters
Step 1. The corresponding strength data are
, and fatigue slope b=
0.065.
Step 2. Suppose R(n)=0.9535 is desired.
Step 3. The elements are given in Table 1. From
these data and
.
Step 4. The maximum strength is
.
Step 5. The minimum strength is
.
Step 6. The Weibull shape parameter is
.
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Step 7. The Weibull scale parameter is
.
Therefore the Weibull strength distribution to the steel
grade (a) A538A (b) material is W(4.909848,
806.7353MPa).
Now based on these parameters let determine the
corresponding log-mean and log-standard
deviation values mentioned in section 3.3.
Source: The Authors
Table 1. Elements of vector Y by using Eq.(10)
n1 2 3 4 5 6 7 8 9 10 11
Yi-3.403483 -2.491662 -2.003463 -1.6616459 -1.3943983 -1.1720537 -0.9793812 -0.807447 -0.6504921 -0.50450882 -0.366512921
n12 13 14 15 16 17 18 19 20 21 µy=-0.54562412
Yi-0.234122 -0.105285 0.0219284 0.1495258 0.279845 0.4159621 0.56250196 0.7276158 0.92931067 1.22965981 σy=1.17511694
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4.2 Log-mean and Log-standard Deviation
From Eq. (16a), the log-mean is
and from Eq.(16b)
the log-standard deviation is
, (observe both and were
determined without any observed failure time
data). Therefore, from Eq.(17), the 95%
confidence interval for is
; [
] or equivalently because from
Eq.(16a) , then by taking the
exponential, the 95% confidence interval for
is [ ],
unfortunately as shown next, this confidence
interval should not be used in reliability analysis.
For example, notice that although under
probabilistic point of view we can say with a
confidence level of 95% the lower expected
value of the Weibull scale parameter 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
Figure 4. Control Chart for x (logarithm Scale). Source:
The Authors
Figure 5. Control Chart for the Weibull scale parameter.
Source: The Authors
Unfortunately, as mentioned above in reliability,
monitoring (or using) the lower limit of is
not correct because in reliability the addressed
value (or nominal value) is the lower
allowed value. Thus, in the monitoring process,
the value (or equivalently the value) is the
one that must be set as the lower allowed limit in
the control chart (see Fig.6 and Fig.7).
Figure 6. Control Chart for x (logarithm Scale). Source:
The Authors
Figure 7. Control Chart for the Weibull scale parameter.
Source: The Authors
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.
Figure 8. Probabilistic Percentiles for the S-N curve.
Source: The Authors
µx
Figure 4. Control Chart for µx (logarithm Scale)
µxU
µxL Time
Nature log scale
()
Figure 5. Control Chart for Weibull scale parameter
(U)
(L) Time
MPa
Figure 6. Control Chart for µx (logarithm Scale)
µx Time
Nature log scale
Figure 7. Control Chart for Weibull scale parameter
()Time
MPa
103104105106107
Se
Sf
S-N Diagram
Cycles
Figure 8. Probabilistic S-N Curve Representation
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Seeing this observe that by using the upper and
lower limits of CL to determine R(), the
demonstrated reliability is not the desired one.
For the upper level ,
then with in Eq.(19), the
estimated reliability instead of be is
only
.
Similarly, if we use the lower confidence level
with
in Eq.(19), the estimated
reliability index instead of be , also
is only of
.
Therefore, the general conclusion is that by using
the CL limits in reliability analysis we sub-
estimate 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.
4.3 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
.
Step 2. From Eq.(20b) the lower element for
R(t)=0.05 is
.
Step 3. From Eq. (21) the upper strength values
are
.
and
.
Step 4. From Eq. (22) the lower strength values are
,
and
.
From the above data, notice because the value was
determined by using , then by using the
, and values in Eq. (19), the reliability
percentile is always .
For
,
, and
.
Similarly, since the value was determined by
using , then by using the ,
and values in Eq. (19), the reliability
percentile in all cases is always .
For
,
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and
.
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.
Figure 9. S-N curve in MPa values. Source: The Authors
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.
Figure 9. S-N curve in logarithm scale. Source: The
Authors
Table 2. Reliability Percentiles for the S-N curve of the A538A (b) steel
Limits Sf η(σ) Se ln(Sf) ln(η(σ)) ln(Se)
Upper 1849.08 1477.26 1180.20 7.5224 7.2979 7.0734
Mean 1009.79 806.74 644.51 6.9175 6.6930 6.4685
Lower 807.57 645.18 515.44 6.6940 6.4695 6.2450
Percentiles in Mpa Values
Percentiles in logarithm scale
500.00
700.00
900.00
1100.00
1300.00
1500.00
1700.00
1900.00
1 2 3
Figure 9. S-N curve in MPa values
6.00
6.20
6.40
6.60
6.80
7.00
7.20
7.40
7.60
1 2 3
Figure 10. S-N curve in logarithm scale
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Source: The Authors
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 , 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
. In Table 3,
by using the value defined in Eq. (10), the
corresponding value was determined as
(23)
2. 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
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
when the maximum applied
stress is constant at the value of
. It is to say
. In
Table 3, the value was determined as
(24)
From Table 3 also notice the rows where the
Weibull analysis reproduce the
and 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.
5. 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,
based on the distribution that represent the
applied stress, and on the distribution that
Table 3. Weibull Scale Analysis
nYi Yui σi
η(σi)R(t)
1 -3.4035 0.5000 403.35 1613.55 0.9673
-2.9702 0.5461 440.56 1477.26 0.9500
2 -2.4917 0.6020 485.66 1340.07 0.9206
3 -2.0035 0.6649 536.44 1213.23 0.8738
4 -1.6616 0.7129 575.11 1131.64 0.8271
5 -1.3944 0.7528 607.28 1071.69 0.7804
6 -1.1721 0.7876 635.42 1024.24 0.7336
-1.1023 0.7989 644.51 1009.79 0.7174
7 -0.9794 0.8192 660.85 984.83 0.6869
8 -0.8074 0.8484 684.40 950.94 0.6402
9 -0.6505 0.8759 706.63 921.02 0.5935
10 -0.5045 0.9023 727.96 894.04 0.5467
11 -0.3665 0.9281 748.71 869.26 0.5000
12 -0.2341 0.9534 769.17 846.14 0.4533
13 -0.1053 0.9788 789.62 824.22 0.4065
0.0000 1.0000 806.735 806.735 0.3679
14 0.0219 1.0045 810.35 803.14 0.3598
15 0.1495 1.0309 831.68 782.54 0.3131
16 0.2798 1.0587 854.05 762.04 0.2664
17 0.4160 1.0884 878.06 741.20 0.2196
18 0.5625 1.1214 904.66 719.41 0.1729
19 0.7276 1.1597 935.60 695.62 0.1262
20 0.9293 1.2084 974.84 667.62 0.0794
1.0972 1.2504 1008.74 645.18 0.0500
1.1023 1.2517 1009.79 644.51 0.0492
21 1.2297 1.2846 1036.33 628.00 0.0327
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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.
5.1 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
(=4.909848, ()=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 and the minimum principal
applied stress that generates a failure is
. ( are the principal stresses
given by the Mohr circle analysis).
Thus, with these two principal stress values, from
Eq. (14) the scale Weibull stress parameter is
, and from
Eq. (13) β=4.909848. Thus, the Weibull stress
distribution is Ws(=4.909848,
s=477.4935MPa). Consequently, from the
Weibull/Weibull stress/strength reliability
function [1] given as
(25)
Therefore, the reliability of the designed
component is
.
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, the estimated reliability is
Similarly, if the applied stress is s=536.44MPa, then
it is
. For detail of the given formulation see
[1].
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 log-
Revista de Ciencias Tecnológicas (RECIT). Volumen 5 (3): e230
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mean 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
6. Weibull/S-N analysis for Materials given in
Table A-23 of the Shigly’s book.
The analysis is presented in Table 4.
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15 ISSN: 2594-1925
Table 4. Weibull Strength Parameters, Log-Parameters and Reliability Percentiles for Tensile Test Data given in Table A-23 of the Shigly's book
Steel Ultimate True Fatigue Strength at Strength at
Grade Strength Stress Exponent N1=10^3 N2=10^6 Shape Scale Mean Stdev
(MPa) (MPa) b Sf Se
σ) Sf σ)Se Sf σ)Se
A538A (b) 1515 1655 -0.065 1009.79 644.51 4.909848 806.7353 6.6930 0.23934 1849.08 1477.26 1180.20 807.57 645.18 515.44
A538B (b) 1860 2135 -0.071 1244.59 762.12 4.494931 973.9233 6.8813 0.26143 2409.91 1885.82 1475.71 975.03 762.98 597.06
A538C (b) 2000 2240 -0.070 1315.76 811.29 4.559144 1033.1798 6.9404 0.25775 2524.12 1982.03 1556.36 1034.33 812.19 637.76
AM-350 (c) 1315 2800 -0.140 966.08 367.29 2.279572 595.6811 6.3897 0.51550 3555.35 2192.21 1351.71 597.01 368.11 226.98
AM-350 (c) 1905 2690 -0.102 1238.93 612.42 3.128824 871.0582 6.7697 0.37558 3201.28 2250.73 1582.43 872.47 613.41 431.27
Gainex (c) 530 805 -0.070 472.85 291.56 4.559144 371.2990 5.9170 0.25775 907.11 712.29 559.32 371.71 291.88 229.20
Gainex (c) 510 805 -0.071 469.27 287.36 4.494931 367.2170 5.9060 0.26143 908.66 711.05 556.42 367.63 287.68 225.12
H-11 2585 3170 -0.077 1765.55 1037.24 4.144676 1353.2559 7.2103 0.28352 3615.00 2770.82 2123.77 1354.92 1038.51 796.00
RQC-100 (c) 940 1240 -0.070 728.37 449.11 4.559144 571.9388 6.3490 0.25775 1397.28 1097.20 861.56 572.58 449.61 353.05
RQC-100 (c) 930 1240 -0.070 728.37 449.11 4.559144 571.9388 6.3490 0.25775 1397.28 1097.20 861.56 572.58 449.61 353.05
10B62 1640 1780 -0.067 1069.67 673.37 4.763285 848.6937 6.7437 0.24670 1995.54 1583.29 1256.20 849.60 674.08 534.83
1005-1009 360 580 -0.090 292.64 157.16 3.546001 214.4546 5.3681 0.33139 676.25 495.57 363.17 214.76 157.38 115.33
1005-1009 470 515 -0.059 328.89 218.80 5.409154 268.2541 5.5919 0.21725 569.53 464.54 378.90 268.51 219.01 178.63
1005-1009 415 540 -0.073 310.04 187.25 4.371782 240.9454 5.4846 0.26880 611.62 475.31 369.38 241.23 187.47 145.69
1005-1009 345 640 -0.109 279.49 131.63 2.927891 191.8084 5.2565 0.40135 770.79 528.98 363.03 192.14 131.86 90.49
1015 415 825 -0.110 357.55 167.24 2.901273 244.5348 5.4994 0.40503 995.30 680.69 465.53 244.96 167.53 114.58
1020 440 895 -0.120 359.50 156.93 2.659501 237.5196 5.4703 0.44186 1098.32 725.65 479.44 237.97 157.23 103.88
1040 620 1540 -0.140 531.34 202.01 2.279572 327.6246 5.7919 0.51550 1955.45 1205.72 743.44 328.36 202.46 124.84
1045 725 1225 -0.095 595.03 308.70 3.359369 428.5862 6.0605 0.34980 1440.53 1037.58 747.34 429.24 309.17 222.69
1045 1450 1860 -0.073 1067.92 644.97 4.371782 829.9229 6.7213 0.26880 2106.68 1637.19 1272.32 830.89 645.72 501.81
1045 1345 1585 -0.074 903.14 541.69 4.312704 699.4441 6.5503 0.27248 1798.27 1392.69 1078.59 700.27 542.33 420.01
1045 1585 1795 -0.070 1054.37 650.12 4.559144 827.9276 6.7189 0.25775 2022.68 1588.28 1247.17 828.85 650.84 511.07
1045 1825 2275 -0.080 1238.51 712.69 3.989251 939.5048 6.8454 0.29457 2607.67 1978.12 1500.56 940.70 713.60 541.32
1045 2240 2275 -0.081 1229.13 702.42 3.940001 929.1759 6.8343 0.29825 2612.12 1974.67 1492.77 930.38 703.33 531.69
1144 930 1000 -0.080 544.40 313.27 3.989251 412.9691 6.0234 0.29457 1146.23 869.50 659.59 413.50 313.67 237.94
1144 1035 1585 -0.090 799.72 429.47 3.546001 586.0525 6.3734 0.33139 1848.03 1354.28 992.45 586.89 430.09 315.18
1541F 950 1275 -0.076 715.54 423.28 4.199212 550.3410 6.3105 0.27984 1451.50 1116.40 858.65 551.01 423.80 325.95
1541F 890 1275 -0.071 743.25 455.13 4.494931 581.6169 6.3658 0.26143 1439.17 1126.19 881.28 582.27 455.65 356.56
4130 895 1275 -0.083 678.46 382.41 3.845061 509.3598 6.2332 0.30562 1468.94 1102.82 827.95 510.03 382.91 287.47
4130 1425 1695 -0.081 915.77 523.34 3.940001 692.2871 6.5400 0.29825 1946.18 1471.23 1112.20 693.18 524.02 396.14
4140 1075 1825 -0.080 993.53 571.72 3.989251 753.6687 6.6250 0.29457 2091.87 1586.84 1203.74 754.63 572.44 434.24
4142 1060 1450 -0.100 678.06 339.83 3.191401 480.0262 6.1738 0.36821 1719.72 1217.47 861.90 480.79 340.37 240.97
4142 1250 1250 -0.080 680.50 391.59 3.989251 516.2114 6.2465 0.29457 1432.79 1086.88 824.48 516.87 392.09 297.43
4142 1415 1825 -0.080 993.53 571.72 3.989251 753.6687 6.6250 0.29457 2091.87 1586.84 1203.74 754.63 572.44 434.24
4142 1550 1895 -0.090 956.13 513.47 3.546001 700.6748 6.5520 0.33139 2209.48 1619.16 1186.56 701.68 514.21 376.82
4142 1760 2000 -0.080 1088.80 626.54 3.989251 825.9382 6.7165 0.29457 2292.46 1739.01 1319.17 826.99 627.34 475.88
4142 2035 2070 -0.082 1109.91 629.92 3.891952 836.1532 6.7288 0.30194 2380.80 1793.59 1351.21 837.25 630.74 475.17
4142 1930 2105 -0.090 1062.09 570.37 3.546001 778.3221 6.6571 0.33139 2454.33 1798.59 1318.05 779.44 571.19 418.58
4142 1930 2170 -0.081 1172.40 670.00 3.940001 886.2909 6.7870 0.29825 2491.56 1883.53 1423.88 887.43 670.87 507.15
4142 2240 1655 -0.089 841.41 454.99 3.585844 618.7373 6.4277 0.32771 1926.36 1416.57 1041.69 619.61 455.64 335.06
4340 825 1200 -0.095 582.89 302.40 3.359369 419.8396 6.0399 0.34980 1411.13 1016.40 732.09 420.48 302.86 218.14
4340 1470 2000 -0.091 1001.47 534.12 3.507034 731.3685 6.5949 0.33507 2335.88 1705.89 1245.81 732.43 534.89 390.63
4340 1240 1655 -0.076 928.79 549.44 4.199212 714.3642 6.5714 0.27984 1884.11 1449.12 1114.57 715.23 550.10 423.10
5160 1670 1930 -0.071 1125.08 688.94 4.494931 880.4084 6.7804 0.26143 2178.52 1704.75 1334.01 881.40 689.72 539.73
52100 2015 2585 -0.090 1304.27 700.43 3.546001 955.8018 6.8626 0.33139 3013.98 2208.72 1618.60 957.17 701.44 514.03
9262 925 1040 -0.071 606.26 371.24 4.494931 474.4169 6.1621 0.26143 1173.92 918.62 718.85 474.95 371.66 290.84
9262 1000 1220 -0.073 700.46 423.04 4.371782 544.3580 6.2996 0.26880 1381.80 1073.85 834.54 544.99 423.54 329.15
9262 565 1855 -0.057 1202.78 811.31 5.598949 987.8368 6.8955 0.20988 2044.44 1679.09 1379.03 988.73 812.04 666.93
9050C (d) 565 1170 -0.120 469.96 205.15 2.659501 310.5005 5.7382 0.44186 1435.80 948.62 626.75 311.09 205.54 135.80
9050C (d) 565 970 -0.110 420.40 196.63 2.901273 287.5136 5.6613 0.40503 1170.23 800.33 547.36 288.02 196.98 134.72
9050X (d) 440 625 -0.075 353.43 210.52 4.255201 272.7739 5.6086 0.27616 710.31 548.21 423.10 273.10 210.78 162.68
9050X (d) 530 1005 -0.100 469.96 235.54 3.191401 332.7078 5.8073 0.36821 1191.94 843.83 597.39 333.24 235.91 167.01
9050X (d) 695 1055 -0.08 574.34 330.50 3.989251 435.6824 6.0769 0.29457 1209.27 917.33 695.86 436.24 330.92 251.03
Weibull Parameters
Log-Parameters
R(0.95), Yui=-2.970195249
R(0.05), YLi=1.0971887
Reliability Percentiles for the S-N Curve
Revista de Ciencias Tecnológicas (RECIT). Volumen 5 (3): e230
16 ISSN: 2594-1925
7. Conclusions
1. Although the relation holds, the
confidence interval CL limits of a S-N curve
defined in Eq. (17), should not be used to perform
a reliability analysis. They sub-estimate the
reliability index.
2. From Eqs. (21 and 22) the upper and lower ,
, and values to determine any desired
reliability percentile for a S-N curve are given by
using only the corresponding and values.
3. Observe that although here the Weibull
strength parameters were both determined for
and , any other desired
values between these two values can be used.
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].
6. Although the Weibull analysis performed in
Table 3 is for constant stress values, and that
given by the stress/strength methodology is for
variable stress behavior, for high reliability
. 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.
8. Authorship acknowledgements
Manuel Baro Tijerina: Conceptualización; Ideas;
Metodología; Análisis formal; Investigación;
Borrador original. Manuel R. Piña
Monarrez: Conceptualización; Ideas;
Escritura. Alberto Jesús Barraza
Contreras: Análisis de datos; Escritura; Borrador
original; Revisión y edición.
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