Chapter 8 Examples of probability distributions of material strength and loads
8.1 Examples of probability distributions of material strength
As mentioned in Section 3.3, there are two models for material strength, the
weakest link model and the averaged strength model, which are expected to be
closely related to the extreme value distribution and the normal distribution,
respectively.
Let’s investigate probability distributions for various kinds of material strength.
1. Yield strength
Fig.8.1-1 shows examples of probability distributions of yield stress or 0.2%
offset proof stress of pure iron, and carbon steels well used as mechanical
structural material. The data are from the book (1998), “ Fatigue reliability design
databook of metallic materials” published by the Society of Materials Science,
Japan.
The data was plotted on a normal probability paper and the symmetric sample
cumulative distribution method P  F ( x i ) 
i  0.5
was used.
n
It can be said that the data fall on straight lines. It implies the yield stress follows
the normal distribution.
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Fig. 8.1-1 Probability distributions of yield stress of pure iron and carbon steels
(normal probability paper)
2. Tensile strength
Fig.8.1-2 shows the data of tensile strength. P  F ( x i ) 
i  0.5
was used.
n
Fig.8.1-2 Probability distributions of tensile strength of pure iron and carbon steels
(normal probability paper)
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The data are well represented by straight lines on a normal probability paper.
Fig. 8.1-3 shows the tensile strength data of structural alloy steels. Here, median
rank P  F ( xi ) 
i  0.3
was used.
n  0.4
Fig. 8.1-3 Probability distributions of tensile strength of structural alloy steels
(normal probability paper)
Theses alloy steels also follow well the normal distribution.
Generally, we can say the tensile strength of most metallic materials follows the
normal distribution.
3. Hardness
Fig.8.1-4 shows the hardness data of steels. P  F ( x i ) 
i  0.5
was used.
n
Although the number of hardness data is smaller compared with the yield stress
or tensile strength, hardness also can be said to follow the normal distribution.
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Relatively large number of hardness data of 0.45% carbon steel is plotted in
Fig.8.1-5. The median rank P  F ( xi ) 
i  0.3
was used. A straight line is well
n  0.4
fitted to the data.
We can consider hardness follows the normal distribution.
Vickers hardness
Fig.8.1-4 Probability distributions of hardness of carbon steels
(normal probability paper)
Vickers hardness
Fig.8.1-4 Probability distributions of hardness of 0.45% carbon steel
(normal probability paper)
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4. Fatigue strength
Notched fatigue life data of an aluminum alloy is plotted on a logarithmic and
Weibull probability papers in Fig.8.1-6. The median rank P  F ( xi ) 
i  0.3
was
n  0.4
used.
Fig.8.1-6 Probability distribution of notched fatigue life of 2024 aluminum alloy
The logarithmic normal distribution seems to be fitted to the data better than the
Weibull distribution. The probabilistic features of fatigue strength have been long
and widely investigated due to practical importance.
The logarithmic normal distribution represents the probability distribution of
fatigue strength and life well and also frequently well assumed.
However, as can be found in Fig.8.1-7, as fatigue life is prolonged at low
stresses, the life probability distribution becomes curvilinear and complicated.
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Fig.8.1-7 Fatigue life distributions for various stresses
(log-normal probability paper)
Even when the data is well approximated by a straight line on a logarithmic
probability paper, the upper and lower tail portions of the data often deviate from
the straight line. This tendency should be considered in estimating reliability.
5. Stress corrosion cracking
As one type of fracture, there is the stress corrosion cracking phenomenon
occurring in corrosion environment.
Fig.8.1-8 shows stress corrosion cracking life data of 18-8 stainless steel
plotted on a Weibull probability paper. The median rank was used.
The stress corrosion cracking life follows the Weibull distribution for high stress
levels, while at low stress levels the data plot consists of two lines.
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Fracture time
Fig.8.1-8 Probability distributions of stress corrosion cracking life of
18-8 stainless steel (Weibull probability paper)
6. Creep fracture
As one type of fracture, there is the creep fracture occurring at high
temperatures.
Fig.8.1-9 shows creep fracture life data of 18-8 stainless steel plotted on a
log-normal probability paper. The median rank was used.
The logarithmic normal distribution represents well creep fracture life.
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Stainless steel
Stainless steel
Fracture time
Fig.8.1-9 Probability distributions of creep fracture life of 18-8 stainless steel
(log-normal probability paper)
Conclusively speaking from the results hitherto mentioned, the probability
distribution of material strength may be approximated by the normal, or log-normal
or Weibull distribution.
However, considering the normal distribution, the material strength does not
follow the normal distribution in the strict sense because the normal distribution
ranges from -∞ to +∞, while the material strength has no negative value.
Despite of it, if the coefficient of variation CV(=standard deviation/ mean) is
smaller than 0.3, the probability for negative values can be ignored. So, practically
the normal distribution is well used.
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8.2 Examples of probability distributions of loads
As already noted, the detailed knowledge of probability distribution of loads is
beyond the undergraduate level. So, here, several examples of load distribution are
only shown.
The Fig. 8.2-3 shows the bending strain data induced in a girder of a main body
of a container by sea waves. The peak strain distribution may be approximated by
Bending strain
Frequency
a normal distribution.
Peak
Fig. 8.2-3 Bending strain induced in a girder of a main body of a container
by sea waves
Fig.8.2-4 shows torsional strain data induced in the axle of a rail road rolling
stock. The strain distributions are too complicated to be approximated by specific
distributions.
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Frequency
Torsional strain
Frequency
Peak
Amplitude
Fig.8.2-4 Torsional strain data induced in the axle of a rail road rolling stock
Fig.8.2-5 shows examples of load distributions observed on components of
automobiles.
The load distributions are very complicated.
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Fig.8.2-5 Load distributions observed on components of automobiles
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