Designing Capable and Reliable Products Episode 1 Part 9 pps
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of the 3-parameter Weibull distribution. The procedure for the 3-parameter Weibull
distribution is more complex, as you would expect, due to the distribution being
modelled by three rather than two parameters. Essentially it requires the determina-
tion of the expected minimum value, xo, a location parameter on the x-axis. As shown
in Appendix X, the linear recti®cation equations are a function of lnx
i
ÿ xo where
xo < x
min
, the minimum variable value on the data. We don't know the value of xo
initially, but by searching for a value of xo such that when lnx
i
ÿ xo is plotted
against ln ln1=1 ÿ F
i
, the correlation coecient is its highest value, will give a
reasonably accurate answer. The process can be easily translated to computer code
to speed up the process.
Determining the parameters for the common distributions can also be done by
hand using suitably scaled probability plotting paper, a straight line through the
data points being determined `by eye', as described earlier. See Lewis (1996) for
examples of probability plotting graph paper for some of the statistical distributions
mentioned.
Further improvement in the selection of the best linear recti®cation model can be
performed by comparing the uncertainty in model ®t as determined from the standard
error of the data (Nelson, 1982). Also, the use of con®dence limits in determining the
uncertainty in the estimates from linear regression is useful for assessing the nature of
the data, particularly when small samples are taken and/or when outlying data points
control the gradient of the regression line. Con®dence limits are generally wider than
some inexperienced data analysts expect, so they help avoid thinking that estimates
are closer to the true value than they really are. A discussion of their application in
data analysis can be found in Ayyub and McCuen (1997), Comer and Kjerengtroen
(1996), Nelson (1982), and Rice (1997).
Example ± ®tting a Normal distribution to a set of existing data
We will next demonstrate the use of the linear recti®cation method described above by
®tting a Normal distribution to a set of experimental data. The data to be analysed is
in the form of a histogram given in Figure 4.9. It shows the distribution of yield
strength for a cold drawn carbon steel (SAE 1018). The data is taken from ASM
(1997a), a reference that provides data in the form of histograms for several important
mechanical properties of steels. Data collated in this manner has been chosen for
analysis because a designer may have to resort to the use of data from existing
sources. Also, the analysis of this case raises some interesting questions which may
not necessarily be met when analysing data collated in practice and displayed using
the methods described in Appendix I.
After a visual inspection, it is evident that the SAE 1018 yield strength data has a
distribution approaching the Normal type, although there is an abnormally high fre-
quency value around the mid-range of the data. Further analysis of the data as shown
in Table 4.2 using the cumulative frequency modelling approach yields Figure 4.10.
Note that the mean rank equation is used to determine the plotting positions on the
y-axis, F
i
, the x-axis plotting positions being the mid-class values for the yield strength
in MPa. The class width w 13:8 MPa. The values determined graphically for the
mean and standard deviation are also shown.
The estimated cumulative frequency ®ts the data well, where in fact the curve is
modelled with a ®fth order polynomial using commercial curve ®tting software.
Statistical methods for probabilistic design 145
(Commercial software such as MS Excel is useful in this connection being widely
available.) Omissions in the ranked values of F
i
in Table 4.2 re¯ect the omissions
of the data in the original histogram for several classes. As can be judged from
Figure 4.10, inclusion of the cumulative probabilities for these classes would not
follow the natural pattern of the distribution and are therefore omitted. However,
when a very low number of classes exist their inclusion can be justi®ed.
Linear recti®cation of the cumulative frequency, F
i
, is performed by converting to
the Standard Normal variate, z. The linear plot together with the straight line
equation through the data and the correlation coecient, r, is shown in Figure
4.11. From Figure 4.11, it is evident that the mean is 530 MPa because the regression
line crosses the Standard Normal variate, z, at 0 representing the 50 percentile or
median in the non-linearized domain. The mean and standard deviation can also
be found from the relationships given in Appendix X. For the Normal distribution
Figure 4.9
Yield strength histogram for SAE 1018 cold drawn carbon steel bar (ASM, 1997a)
Table 4.2 Analysis of histogram data for SAE 1018 to obtain the Normal distribution plotting positions
Mid-class (MPa) Frequency ( f ) Cumulative freq. (i) F
i
i
N 1
z È
ÿ1
SND
F
i
(x-axis) (N 52) ( y-axis) ( y-axis)
431.0 1 1 0.01887 ÿ2.08
444.8 0 1
458.6 2 3 0.05660 ÿ1.59
472.4 3 6 0.11321 ÿ1.21
486.2 2 8 0.15094 ÿ1.03
500.0 3 11 0.20755 ÿ0.82
513.8 4 15 0.28302 ÿ0.57
527.6 5 20 0.37736 ÿ0.31
541.4 13 33 0.62264 0.31
555.2 5 38 0.71698 0.58
569.0 5 43 0.81132 0.88
582.8 5 48 0.99566 1.32
596.6 3 51 0.96226 1.78
610.4 0 51
624.2 0 51
638.0 1 52 0.98113 2.08
146 Designing reliable products
we can calculate the mean and standard deviation from:
ÿ
A0
A1
ÿ
ÿ11:663
0:022
530:14 MPa
1 ÿ A0
A1
A0
A1
1 11:663
0:022
ÿ11:663
0:022
45:45 MPa
The conclusion is that the Normal distribution is an adequate ®t to the SAE 1018
data. A summary of the Normal distribution parameters calculated from Figures
4.10 and 4.11 and other values for the mean and standard deviation from various
sources (including commercial software and a package developed at Hull University
called FastFitter
Ã
) are given in Table 4.3. The frequency distributions derived from
the Normal distribution parameters from source are shown graphically overlaying
the original histogram in Figure 4.12 for comparison.
It can be seen from Table 4.3 that there is no positive or foolproof way of determin-
ing the distributional parameters useful in probabilistic design, although the linear
recti®cation method is an ecient approach (Siddal, 1983). The choice of ranking
equation can also aect the accuracy of the calculated distribution parameters
using the methods described. Reference should be made to the guidance notes
given in this respect.
The above process above could also be performed for the 3-parameter Weibull
distribution to compare the correlation coecients and determine the better ®tting
distributional model. Computer-based techniques have been devised as part of the
approach to support businesses attempting to determine the characterizing distributions
Figure 4.10
Cumulative frequency distribution for SAE 1018 yield strength data
Ã
The FastFitter software is available from the authors on request.
Statistical methods for probabilistic design 147
from sample data. As shown in Figure 4.13, the users screen from the software, called
FastFitter, is the selection of the best ®tting PDF and its parameters representing the
sample data, here for the yield strength data for SAE 1018. The software selects the
best distribution from the seven common types: Normal, Lognormal, 2-parameter
Weibull, 3-parameter Weibull, Maximum Extreme Value Type I, Minimum Extreme
Value Type I and the Exponential distribution. Using the FastFitter software, it is
found that the 3-parameter Weibull distribution gives the highest correlation
coecient of all the models, at r 0:995, compared to r 0:991 for the Normal
distribution. The mean and standard deviation in Table 4.3 for FastFitter are
calculated from the Weibull parameters, the relevant information is provided in
Appendix IX.
4.2.3 The algebra of random variables
Typically, if the stress or strength has not been taken directly from the measured
distribution, it is likely to be a combination of random variables. For example, a
Table 4.3 Normal distribution parameters for SAE 1018 from various sources
Source of Normal
distribution parameters
Mean, Standard
deviation,
Reference (Mischke, 1992) 541 41
FastFitter 532 44
Moment calculations 537 41
Normal linear recti®cation 530 45
Cumulative. freq. graph 534 40
Commercial software 545 38
Average 537 42
Figure 4.11
Normal distribution linear recti®cation for SAE 1018 yield strength data
148 Designing reliable products
failure governing stress is a function of the applied load variation and maybe two-
or three-dimensional variables bounding the geometry of the problem. The
mathematical manipulation of the failure governing equations and distributional
parameters of the random variables used to determine the loading stress in particular
are complex, and require that we introduce a new algebra called the algebra of random
variables.
We need this special algebra to operate on the engineering equations as part of
probabilistic design, for example the bending stress equation, because the parameters
are random variables of a distributional nature rather than unique values. When these
random variables are mathematically manipulated, the result of the operation is
another random variable. The algebra has been almost entirely developed with the
application of the Normal distribution, because numerous functions of random
variables are normally distributed or are approximately normally distributed in
engineering (Haugen, 1980).
Engineering variables are found to be either statistically independent or correlated
in some way. In engineering problems, the variables are usually found to be unrelated,
for instance a dimensional variable is not statistically related to a material strength
(Haugen, 1980). Table 4.4 shows some common algebraic functions, typically with
one variable, x, or two statistically independent random variables, x and y. The
mean and standard deviation of the functions are given in terms of the algebra of
Figure 4.12
Normal distributions from various sources for SAE 1018 yield strength data
Statistical methods for probabilistic design 149
random variables. Where the variables x and y are correlated in some way, with
correlation coecient, r, several common functions have also been included.
When a function, , is a combination of two or more statistically independent
variables, x
i
, then equation 4.5 can be eectively used to determine their combined
variance, V
(Mischke, 1980).
V
%
n
i 1
@
@x
i
2
Á
2
x
i
1
2
n
i 1
@
2
@x
2
i
23
2
Á
4
x
i
4:5
To determine the mean value, , of the function :
%
x
1
;
x
2
; FFF;
x
n
ÿÁ
1
2
n
i 1
@
2
@x
2
i
Á
2
x
i
4:6
Equation 4.5 is exact for linear functions, but should only be applied to non-linear
functions if the random variables have a coecient of variation, C
v
< 0:2
(Furman, 1981; Morrison, 2000). If this is the case, then the approximation using
just the ®rst term only diers insigni®cantly from using higher order terms
(Furman, 1981). However, for a function whose ®rst derivative is very small, the
higher terms cannot be ignored (Bowker and Lieberman, 1959). Approximate solu-
tions for the mean and standard deviation,
, are provided by omitting the higher
order terms, for example equation 4.5 is often written as:
%
n
i 1
@
@x
i
2
Á
2
x
i
23
0:5
4:7
Data
Correlation
coefficient
Distribution
parameters
Figure 4.13
FastFitter analysis of SAE 1018 yield strength data
150 Designing reliable products
and
%
x
1
;
x
2
; FFF;
x
n
ÿÁ
4:8
Equation 4.7 is referred to as the variance equation and is commonly used in error
analysis (Fraser and Milne, 1990), variational design (Morrison, 1998), reliability
Table 4.4 Mean and standard deviation of statistically independent and correlated random variables x and
y for some common functions
Function () Mean (
) Standard deviation (
)
x
x
x
x
2
2
x
2
x
2
x
Á
x
Á
1 0:25
x
x
2
!
x
3
3
x
3
2
x
Á
x
3
x
Á
2
x
Á
1
x
x
2
!
x
4
4
x
6
2
x
Á
2
x
4
x
Á
3
x
Á
1
9
4
x
x
2
!
x
n
n
x
Á
1 0:5nn ÿ 1
x
x
2
!
n Á
x
Á
n ÿ1
x
Á
1 0:25n ÿ 1
2
x
x
2
!
x
0:5
0:5
x
1 ÿ
1
8
x
x
2
!
x
Á
0:5
x
2
x
1
1
16
x
x
2
!
1
x
1
x
1
x
x
2
!
x
2
x
1
x
x
2
!
1
x
2
1
2
x
1 3
x
x
2
!
2
x
3
x
1
9
4
x
x
2
!
1
x
3
1
3
x
1 6
x
x
2
!
3
x
4
x
1 4
x
x
2
!
x Æy
x
Æ
y
2
x
2
y
0:5
2
x
2
y
Æ 2r Á
x
Á
y
0:5
x Á y
x
Á
y
x
Á
y
r Á
x
Á
y
2
x
Á
2
y
2
y
Á
2
x
2
x
Á
2
y
0:5
2
x
Á
2
y
2
y
Á
2
x
2
x
Á
2
y
1 r
2
0:5
x
y
x
y
2
y
Á
x
3
y
1
y
2
x
Á
2
y
2
y
Á
2
x
2
y
2
x
0:5
x
y
y
Á
x
2
y
y
y
ÿ r Á
x
x
x
y
Á
2
x
2
x
2
y
2
y
ÿ 2r Á
x
Á
y
x
Á
y
0:5
Statistical methods for probabilistic design 151
analysis (Haugen, 1980) and sensitivity analysis (Parry-Jones, 1999). Most impor-
tantly in probabilistic design, through the use of the variance equation we have a
means of relating geometric decisions to reliability goals by including the dimensional
and load random variables in failure governing stress equations to determine the
stress random variable for any given problem.
The variance equation can be solved directly by using the Calculus of Partial
Derivatives, or for more complex cases, using the Finite Dierence Method. Another
valuable method for `solving' the variance equation is Monte Carlo Simulation.
However, rather than solve the variance equation directly, it allows us to simulate
the output of the variance for a given function of many random variables. Appendix
XI explains in detail each of the methods to solve the variance equation and provides
worked examples.
The variance for any set of data can be calculated without reference to the prior
distribution as discussed in Appendix I. It follows that the variance equation is
also independent of a prior distribution. Here it is assumed that in all the cases the
output function is adequately represented by the Normal distribution when the
random variables involved are all represented by the Normal distribution. The
assumption that the output function is robustly Normal in all cases does not strictly
apply, particularly when variables are in certain combination or when the Lognormal
distribution is used. See Haugen (1980), Shigley and Mischke (1996) and Siddal
(1983) for guidance on using the variance equation.
The variance equation provides a valuable tool with which to draw sensitivity
inferences to give the contribution of each variable to the overall variability of the
problem. Through its use, probabilistic methods provide a more eective way to
determine key design parameters for an optimal solution (Comer and Kjerengtroen,
1996). From this and other information in Pareto Chart form, the designer can
quickly focus on the dominant variables. See Appendix XI for a worked example
of sensitivity analysis in determining the variance contribution of each of the
design variables in a stress analysis problem.
4.3 Variables in probabilistic design
Design models must account for variability in the most important design variables
(Cruse, 1997b). If an adequate characterization of these important variables is
performed, this will give a cost-eective and a fairly accurate solution for most
engineering problems. The main engineering random variables that must be ade-
quately described using the probabilistic approach are shown in Figure 4.14. The
variables conveniently divide into two types: design dependent, which the designer
has the greatest control over, and service dependent, which the design has `limited'
control over. Typically, the most important design dependent variables are material
strength and dimensional variability. Material strength can be statistically modelled
from sample data for the property required, as previously demonstrated; however,
diculties exist in the collation of information about the properties of interest.
Dimensional variability and its eects on the stress acting on a component can be
great, but information is typically lacking about its statistical nature and its impact
on geometric stress concentration values is rarely assessed.
152 Designing reliable products
Important service dependent variables are related to the loading of the component
and stresses resulting from environmental eects. These are generally dicult to
determine at the design stage because of the cost of performing experimental data
collection, the nature of overloading and abuse in service, and the lack of data
about service loads in general. Also, the eect that service conditions have on the
material properties is important, the most important considerations arising from
extremes in temperature, as there is a tendency towards brittle fracture at low
temperatures, and creep rupture at high temperatures. To this end, it has been
cited that the quality control of the environment is much more important than quality
control of the manufacturing processes in achieving high reliability (Carter, 1986).
Among the most dramatic modi®ers encountered in design of those mentioned
above are due to thermal eects on strength and stress concentration eects on
local stress magnitudes in general (Haugen, 1980). As seen from Figure 4.14, there
are several important design dependent variables that terms of an engineering analysis
are then:
. Material strength (with temperature and residual processing eects included)
. Dimensional variability
. Geometric stress concentrations
. Service loads.
4.3.1 Material strength
The largest design dependent strength variable is material strength, either ultimate
tensile strength (Su), uniaxial yield strength (Sy), shear yield strength (
y
) or some
Figure 4.14
Key variables in a probabilistic design approach
Variables in probabilistic design 153
other failure resisting property. For de¯ection and instability problems, the Modulus
of Elasticity (E) is usually of interest. Shear yield strength, typically used in torsion
calculations, is a linear function of the uniaxial yield strength and is likely to have
the same distribution type (Haugen, 1980).
With mass produced products, extensive testing can be carried out to characterize
the property of interest. When production is small, material testing may be limited to
simple tension tests or perhaps none at all (Ayyub and McCuen, 1997). Material
properties are often not available with a sucient number of test repetitions to
provide statistical relevance, and remain one of the challenges of greater application
of statistical methods, for example in aircraft design (Smith, 1995). Another problem
is how close laboratory test results are to that of the material provided to the customer
(Welling and Lynch, 1985), because material properties tend to vary from lot to lot
and manufacturer to manufacturer (Ireson et al., 1996). However, this can all be
regarded as making the case for a probabilistic approach. Ideally, information on
material properties should come from test specimens that closely resemble the
design con®guration and size, and tested under conditions that duplicate the expected
service conditions as closely as possible (Bury, 1975). The more information we
have about a situation before the trial takes place and the data collected, the more
con®dence there will be in the ®nal result (Leitch, 1995).
One of the major reasons why design should be based on statistics is that material
properties vary so widely, and any general theory of reliability must take this into
account (Haugen and Wirsching, 1975). Material properties exhibit variability
because of anisotropy and inhomogeneity, imperfection, impurities and defects
(Bury, 1975). All materials are, of course, processed in some way so that they are
in some useful fabrication condition. The level of variability in material properties
associated with the level of processing can also be a major contribution. There are
three main kinds of randomness in material properties that are observed (Bolotin,
1994):
. Within specimen ± inherent within the microstructure and caused by imperfec-
tions, ¯aws, etc.
. Specimen to specimen ± caused by the instabilities and imperfections of the
manufacturing processes with the batch.
. Batch to batch ± natural variations due to processing, such as material quality,
equipment, operator, method, set-up and the environment.
Other uncertainties associated with material properties are due to humidity and
ambient chemicals and the eects of time and corrosion (Farag, 1997; Haugen,
1982b). Brittle materials are aected additionally by the presence of imperfections,
cracks and internal ¯aws, which create stress raisers. For example, cast materials
such as grey cast iron are brittle due to the graphite ¯akes in the material causing
internal stress raisers. Their low tensile strength is due to these ¯aws which act as
nuclei for crack formation when in tensile loading (Norton, 1996). Subsequently,
brittle materials tend to have a large variation in strength, sometimes many times
that of ductile materials.
Strain rate also aects tensile properties at test. An increasing strain rate tends to
increase tensile properties such as Su and Sy. However, a high loading rate tends to
promote brittle fracture (Juvinall, 1967). The average strain rate used in obtaining a
154 Designing reliable products
stress±strain diagram is approximately 10
ÿ3
ms/m, and this should be kept in mind
when performing experimental testing of materials (Shigley, 1986).
It has been shown that the ultimate tensile strength, Su, for brittle materials
depends upon the size of the specimen and will decrease with increasing dimensions,
since the probability of having weak spots is increased. This is termed the size eect.
This `size eect' was investigated by Weibull (1951) who suggested a statistical func-
tion, the Weibull distribution, describing the number and distribution of these ¯aws.
The relationship below models the size eect for deterministic values of Su
(Timoshenko, 1966).
Su
2
Su
1
v
1
v
2
1=
4:9
where:
Su
1
ultimate tensile strength of test specimen
Su
2
ultimate tensile strength of component
v
1
effective volume of test specimen
v
2
effective volume of component design
shape parameter from Weibull analysis of test specimen data:
As can be seen from the above equation, for brittle materials like glass and ceramics,
we can scale the strength for a proposed design from a test specimen analysis. In a
more useful form for the 2-parameter Weibull distribution, the probability of failure
is a function of the applied stress, L.
P 1 ÿ exp
ÿ
L
v
2
=v
1
4:10
where:
P probability of failure
L stress applied to component
characteristic value;
and for the 3-parameter Weibull distribution:
P 1 ÿ exp
ÿ
L ÿ xo
ÿ xo
v
2
=v
1
4:11
where:
xo expected minimum value:
A high shape factor in the 2-parameter model suggests less strength variability. The
Weibull model can also be used to model ductile materials at low temperatures
which exhibit brittle failure (Faires, 1965). (See Waterman and Ashby (1991) for a
detailed discussion on modelling brittle material strength.)
Variables in probabilistic design 155
Several researchers and organizations over the last 50 years have accumulated
statistical material property data. However, property data is still not available for
many materials or is not made generally available by the companies manufacturing
the stock product. This is a problem if you want to design with a speci®c material
in a speci®c environment. For example, it is not adequate just to say the statistical
property of one particular steel is going to be close to that of another similar steel.
Approximate values for the mean and standard deviation of the ultimate tensile
strength of steel, Su, can be found from a hardness test in Brinnel Hardness (HB).
From empirical investigation (Shigley and Mischke, 1989):
Su
3:45
HB
4:12
Su
3:45
2
Á
2
HB
0:152
2
Á
2
HB
0:152
2
Á
2
HB
0:5
4:13
Unfortunately, the statistical data in references such (ASM, 1997a; ASM, 1997b;
Haugen, 1980; Mischke, 1992) is the best available to the designer who requires
rapid solutions. An example of such data was shown in Figure 4.9. Although the
property data strictly applies to US grade ferrous and non-ferrous materials, conver-
sion tables are available which show equivalent material grades for UK, French,
German, Swedish and Japanese grades. However, their casual use could make the
answers obtained misrepresentative of the problem. They should be treated with
caution as a direct comparison is questionable because of small deviations in com-
positions and processing parameters. Material properties for UK grade materials in
statistical form would be advantageous when using probabilistic design techniques.
However, there are no immediate plans by the British Standards Institute (BSI) to
produce materials property data in a statistical format, and all data currently
published is based on values (pers. comm., 1998).
Table 4.5 shows the coecient of variation, C
v
, for various material properties at
room temperature compiled from a number of sources (Bury, 1975; Haugen, 1980;
Haugen and Wirsching, 1975; Rao, 1992; Shigley and Mischke, 1996; Yokobori,
1965).
Further insight into the statistical strength properties of some commonly used
metals is provided by a data sheet in Table 4.6. Again caution should be exercised
in their use, but reference will be made to some of these values in the probabilistic
design case studies at the end of this section.
Table 4.5 Typical coecient of variation, C
v
, for various materials and mechanical properties
(Su ultimate tensile strength)
Steel, C
v
Other materials, C
v
Su 0:05 Su of cast iron 0.09
Yield Strength Sy0:05 to 0.08 Su of wrought iron 0.04
Endurance limit Se0:08 Su of brittle materials 0:3
Brinell Hardness (HB) 0.05 Su of glass 0.24
Mod. of Elasticity E0:01 to 0.03 Fracture toughness K
c
of metallic materials 0.07
Mod. of Rigidity G0:02 to 0.04 Mod. of Elasticity E of nodular cast iron 0.04
Fracture toughness K
c
0:05 to 0.1 Mod. of Elasticity E of titanium 0.09
Poisson's ratio 0:02 to 0.26 Mod. of Elasticity E of aluminium 0.03
156 Designing reliable products
Finally, it is worth investigating how deterministic values of material strength are
calculated as commonly found in engineering data books. Equation 4.14 states that
the minimum material strength, S
min
, as used in deterministic calculations, equals
the mean value determined from test, minus three standard deviations, calculated
for the Normal distribution (Cable and Virene, 1967):
S
min
ÿ 3 4:14
For example, the deterministic value for the yield strength, Sy, for SAE 1018 cold
drawn steel for the size range tested is approximately 395 MPa (Green, 1992).
Table 4.6 gives the mean and standard deviation as Sy $ N540; 41MPa. The
lower bound value as used in deterministic design becomes:
Sy
min
540 ÿ 341417 MPa
The values are within 5% of each other. If deterministic values are actually calculated
at the negative 3 limit from the mean, 1350 failures in every million could be
expected for an applied stress of the same magnitude as determined from SND
theory. It is evident from this that reliability prediction and deterministic design
are not compatible, because as the factor of safety is introduced to reduce failures,
the probability aspect of the calculation is lost. (Note that the ASTM standard on
materials testing suggests setting the minimum material property at ÿ2:33 from
the mean value (Shigley and Mischke, 1989).)
Table 4.6 Material strength data sheet
Material Condition Ultimate tensile
strength (MPa)
Yield strength
(MPa)
Su
Su
Sy
Sy
Free cutting carbon steel
BS 220M07
Cold drawn 517 27 447 36
Mild steel
BS 070M20
Normalized 506 25 ± ±
Low carbon steel
SAE 1018
(BS 080A17)
Cold drawn 604 40 540 41
Medium carbon steel
SAE 1035
(BS 080A32)
Hot rolled 594 27 342 26
Medium carbon steel
SAE 1045
(BS 080M46)
Cold drawn 812 49 658 45
Low alloy steel
SAE 4340
(BS 817M40)
Cold drawn
annealed
1 10 mm
803 9 ± ±
Structural steel
BS Grade 43C
Hot rolled
t 16 mm
± ± 324 16
Stainless steel
BS 316S16
Sheet annealed
t 3mm
579 20 ± ±
Aluminium alloy
7075-T6
Sheet aged 555 27.5 484 22
Titanium alloy
Ti-6Al-4V
Bar 934 46 900 50
Variables in probabilistic design 157
Material properties and temperature
A number of basic material properties useful in static design depend most notably on
temperature (Haugen, 1980). For example, Figure 4.15 shows how high temperatures
alter the important mechanical properties of a low carbon steel, and the variation that
can be experienced. Temperature dependent materials properties are sometimes
available in statistical form, as shown in Figure 4.16 where the 3-parameter Weibull
distribution is used to model the tensile strength of an alloy steel over a range of
temperatures (Lipson et al., 1967). This type of information is strictly for high tem-
perature work where the application of the load lasts approximately 15 to 20 minutes
(Timoshenko, 1966).
Experiments at high temperatures also show that tensile tests depend on the duration
of the test, because as time increases the load necessary to produce fracture decreases
(Timoshenko, 1966). This is the onset of the phenomenon known as creep. All materials
begin to lose strength at some temperature, and as the temperature increases, the defor-
mations cease to be elastic and become more and more plastic in nature. Given su-
cient time, the material may fail by creep usually occurring at a temperature between
30 and 40% of the melting temperature in degrees Kelvin (Ashby and Jones, 1989).
In carbon steels, for example, design stresses can be solely based on short-term proper-
ties up to an operating temperature of about 4008C, while at temperatures greater than
this, creep behaviour is likely to overrule any other design considerations.
Creep stresses used for design purposes are usually determined based on two
criteria: the stress for a given acceptable creep deformation after a certain number
of hours, which ranges from 0.01 to 1% deformation in 1000 hours; and the nominal
0 100 200 300 400 500
Temperature (°C)
700
600
500
400
300
200
100
0
Stress (MPa)
Modulus
of
Elasticity
(GPa)
250
200
150
100
50
0
Figure 4.15
Mechanical properties of a low carbon steel as a function of temperature (adapted from Water-
man and Ashby, 1991)
158 Designing reliable products
stress required to produce rupture after a speci®ed time or at the end of the required
life. The creep rupture stress for several steels at 1000 hours is shown in Figure 4.17. It
is evident that a large variation exists in the rupture stress values for a given tempera-
ture, and in general creep material properties tend to have a coecient of variation
much greater than static properties (Bury, 1974). For example, C
v
0:7 has been
cited for the creep time to fracture for copper (Yokobori, 1965). Although little
statistical data has been found on the properties highlighted, creep strength data
and properties at high temperatures for various materials can be found in ASM
(1997a), ASM (1997b), Furman (1980) and Waterman and Ashby (1991).
Many mechanical components also operate at temperatures far lower than room
temperature. As the temperature is reduced, both the ultimate tensile strength,
Su, and tensile yield strength, Sy, generally increase for most materials. However,
temperatures below freezing have the eect of altering the structure of some ductile
Figure 4.16
Short-term tensile strength Weibull parameters for an alloy steel at various temperatures (Lipson
et al., 1967)
Variables in probabilistic design 159
metals so they fracture in a brittle manner. The temperature at which this occurs is
called the ductile-to-brittle transition temperature. Tests to determine this usually
involve impact energy tests, for example, Charpy or Izod, which measures the
energy to break the specimen in joules. The plot of the results for a structural steel
are shown in Figure 4.18, showing the regions of brittle and ductile behaviour.
Figure 4.17
1000 hour creep rupture stress as a function of temperature for various steels (Waterman and
Ashby, 1991)
Figure 4.18
Ductile to brittle transition diagram for a structural steel (Mager and Marschall, 1984)
160 Designing reliable products
The ductile-to-brittle transition temperature for some steels can be as `high' as 08C,
depending on the composition of the steel (Ashby and Jones, 1989). However, there is
no way of using the data directly from impact tests quantitatively in the design
process. Design speci®cations do usually state a minimum impact strength, but
experience suggests that this does not necessarily eliminate brittle failure (Faires,
1965). The Robertson test can yield more information than either the Charpy or
Izod tests because the transition temperature is statistically correlated with the tem-
perature at which the actual structure has been known to fail in a brittle manner
(Benham and Warnock, 1983; Ruiz and Koenigsberger, 1970). The test uses a severely
notched specimen tested under static tension, and a plot showing the variation of the
nominal stress at fracture with the test specimen temperature drawn. The test gives
useful results from which design calculations can be based; however, the test is
more expensive and complex compared to other methods. In general, it is dangerous
to use a material below its transition temperature because most of its capacity to
absorb energy without rupture has been lost and careful design and analysis is
required.
Residual stresses and processing
In a component of uniform temperature not acted upon by external loads, any
internal stresses that exist are called residual stresses. The material strength, therefore,
is dependent not only on the basic material property, but on the residual stresses
exhibited by the manufacturing process itself, for example by forging, extrusion or
casting. This could be further aected by secondary processing in production such
as welding, machining, grinding and surface coating processes, from deliberate or
unavoidable heat treatment, and assembly operations such as fastening and shrink
®ts, all of which promote residual stresses. Many failures result from unsatisfactory
welding and joining of parts of engineering components (Heyes, 1989), and this can
only be attributable to residual stresses aecting either the static or more commonly
the fatigue properties. Additionally manufacturing processes result in variations in
surface roughness, sharp corners and other stress raisers (Farag, 1997). It is also
evident that the pattern of residual stresses in a component may not be permanent
and could change over time due to changing environmental conditions in service
(Faires, 1965).
The methods used to measure residual stresses in a component are performed after
the manufacturing process, and are broadly classed into two types: mechanical (layer
removal, cutting) and physical (X-ray diraction, acoustic, magnetic). Further
reference to the methods used can be found in Chandra (1997), Juvinall (1967),
and Timoshenko (1983).
To remove residual stresses usually requires some manner of stress relieving
(usually involving heat treatment) or promotion/inducement of opposing stresses
by shot peening, or similar methods. Designers need to consider carefully the in¯u-
ence of the residual stresses on component behaviour which may be induced during
manufacture, and decide whether stress relief is appropriate (Nicholson et al.,
1993). If stress relief is not possible, the problem of how to quantify residual stresses,
which will either be detrimental or advantageous to the strength of the material,
becomes a dicult one. Signi®cant residual stresses tend to be bene®cial if compres-
sive, and detrimental if tensile. For example, residual stresses in brittle materials are
Variables in probabilistic design 161
problematic if tensile because they have low toughness and this could accelerate
catastrophic brittle fracture. The presence of residual stresses are generally detrimen-
tal to the product integrity in service and should be eliminated if expected to be
harmful (Chandra, 1997).
Theoretically, the eects of the manufacturing process on the material property
distribution can be determined, shown here for the case when Normal distribution
applies. For an additive case of a residual stress, it follows that from the algebra of
random variables (Carter, 1997):
S
So
V
4:15
S
2
So
2
V
0:5
4:16
where:
S
mean of the final strength
S
standard deviation of the final strength
So
mean of the original strength
So
standard deviation of the original strength
V
additive quantities of strength from the process
V
standard deviation of the additive strength from the process:
For a proportional improvement in the strength, the product of a function of random
variables applies:
S
So
Á
V
4:17
S
2
So
Á
2
V
2
V
Á
2
So
2
So
Á
2
V
0:5
4:18
As can be seen from the above equations, the standard deviation of the strength
increases signi®cantly with the number of processes used in manufacture that are
adding the residual stresses. This may be the reason for the apparent reluctance of
suppliers to give precise statistical data about their product (Carter, 1997).
A practical diculty using the above approach is that there are too many processes
and therefore variables involved. However, in the nuclear industry the additional
material factors are often taken into consideration (Carter, 1997). The diculties
associated with the measuring of the variables must also be an inhibitory factor in
their use. Residual stresses have been measured and modelled for a number of
manufacturing processes, such as welding, cold drawing and heat treatment processes
(ASM, 1997a; ASM, 1997b; Osgood, 1982), but their application is so speci®c that
transfer to the general case could be misleading. Statistical descriptions of materials
behaviour should take into account the in¯uences of variable metallurgical factors,
such as heat treatment and mechanical processing, and this returns to an earlier
important statement (Haugen, 1982a). Information on material properties should
come from test specimens that closely resemble the design con®guration and size,
and are tested under conditions that duplicate the expected service conditions as
closely as possible.
162 Designing reliable products
4.3.2 Dimensional variability
The manufacturing process introduces variations in that absolute dimensional
accuracy cannot be attained and variations within speci®ed tolerances are a necessary
feature of all manufactured products (Carter, 1986). Proper tolerances are crucial to
the proper functioning, reliability and long life of a product and the act of assigning
tolerances in fact ®nalizes reliability (Dixon, 1997; Rao, 1992; Vinogradov, 1991).
Large tolerances and/or large variance can result in signi®cant degradation of relia-
bility because the failure probability is a function of the magnitude of dimensional
variability and tolerance allocated, and aects load induced stress in a component
(Haugen, 1980; Kluger, 1964).
Component reliability will vary as a function of the power of a dimensional vari-
able in a stress function. Powers of dimensional variables greater than unity magnify
the eect. For example, the equation for the polar moment of area for a circular shaft
varies as the fourth power of the diameter. Other similar cases liable to dimensional
variation eects include the radius of gyration, cross-sectional area and moment of
inertia properties. Such variations aect stability, de¯ection, strains and angular
twists as well as stresses levels (Haugen, 1980). It can be seen that variations in toler-
ance may be of importance for critical components which need to be designed to a
high reliability (Bury, 1974).
The measures of dimensional variability from Conformability Analysis (CA) (as
described in Chapters 2 and 3), speci®cally the Component Manufacturing
Variability Risk, q
m
, is useful in the allocation of tolerances and subsequent analysis
of their distributions in probabilistic design. The value q
m
is determined from process
capability maps for the manufacturing process and knowledge of the component's
material and geometry compatibility with the process. In the speci®c case to the ith
component bilateral tolerance, t
i
, it was shown in Chapter 3 that the standard devia-
tion estimates were:
H
i
t
i
Á q
2
mi
12
4:19
i
t
i
Á q
4=3
mi
12
4:20
The
H
in equation 4.19 relates to the fact that this is not the true standard deviation,
but an estimate to measure the process shift (or drift) in the distribution over the
expected duration of production. Equation 4.20 is the best estimate for the standard
deviation of the distribution as determined by CA with no process shift.
A popular way of determining the standard deviation for use in the probabilistic
calculations is to estimate it by equation 4.21 which is based on the bilateral tolerance,
t, and various empirical factors as shown in Table 4.7 (Dieter, 1986; Haugen, 1980;
Smith, 1995). The factors relate to the fact that the more parts produced, the more
con®dence there will be in producing capable tolerances:
t
Factor
4:21
Historically, in probabilistic calculations, the standard deviation, , is expressed
as t=3 (Dieter, 1986; Haugen, 1980; Smith, 1995; Welling and Lynch, 1985), which
Variables in probabilistic design 163
relates to a maximum Process Capability Index, C
p
1. This estimate does not take
into account process shift, typically Æ1:5 from the target during a production run
due to tooling and production errors (Evans, 1975), and relies heavily on the
tolerances being within Æ3 during inspection and process control. Unless there is
100% inspection, however, there will be some dimensions that will always be out
of tolerance (Bury, 1974).
In equations 4.19 and 4.20, improved estimates for the standard deviation are
presented based on empirical observations. This is shown in Figure 4.19 for a
Æ0:1 mm tolerance on an arbitrary dimensional characteristic, but with an increasing
q
m
, as would be determined for less capable design schemes. It shows that increasing
risk of allocating tolerances that are not capable, increases the estimates for the ,
Figure 4.19
Standard deviation estimates based on q
m
and t= 3
Table 4.7 Empirical factors for determining
standard deviation based on tolerance
No. of parts manufactured Factor
4to5 1
10 1.5
25 2
100 2.5
500 to 700 3
164 Designing reliable products
