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−0.5n

was used.

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 ( xi)=

i−0.5n

was used.

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Fig.8.1-2 Probability distributions of tensile strength of pure iron and carbon steels

(normal probability paper)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 was used.

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

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Fig.8.1-4 shows the hardness data of steels. P=F ( xi)=

i−0.5n

was used.

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. Relatively large number of hardness data of 0.45% carbon steel is plotted in

Fig.8.1-5. The median rank was used. A straight line is well

fitted to the data. We can consider hardness follows the normal distribution.

Fig.8.1-4 Probability distributions of hardness of carbon steels(normal

probability paper)

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Vickers hardness

Vickers hardness

Fig.8.1-4 Probability distributions of hardness of 0.45% carbon steel(normal probability paper)

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 was used.

Fig.8.1-6 Probability distribution of notched fatigue life of 2024 aluminum alloy

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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.

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.

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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.

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.

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Fracture time

The logarithmic normal distribution represents well creep fracture life.

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.

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Fracture time

Stainless steel

Stainless steel

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 a normal distribution.

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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Peak

Freq

uenc

y

Bend

ing st

rain

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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Freq

uenc

y

Freq

uenc

y

Peak Amplitude

Torsi

onal

strain

Fig.8.2-5 Load distributions observed on components of automobiles

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