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