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Strength of Materials: Endurance Limit

When determining the fatigue limit there are number of factors that are normally considered. They are represented in the equation below.

endurance limit (1)

Where ka represents the surface condition modification factor, kb represents the size modification factor, kc represents the load modification factor, kd represents the temperature modification factor, ke represents the reliability factor, kf represents any other miscellaneous effects, and S'e is the endurance limit determined during the rotary beam test used to create the S-N diagram. The reason why modification factors should be included instead of the just using the value obtained from test data is because the test data does not take in consideration of any variations that could occur to the specimen. The modification factors on the other hand can be used to improve the estimation of the endurance factor. The data obtained during testing can be used to represent the S'e value, however, it is also acceptable to use the data seen in equation 2 for steel.

test endurance limit (2)

The surface factor will take in consideration a variety of different surface conditions. The reason why this is important is because for the beam testing the specimen is highly polished to remove any surface defects. If these surface defects are left, cracks are more likely to start at those points, which in effect will lower the overall endurance limit. Equation 3 and the table below would be used to determine the surface factor modification for different surface finishes on steel.

surface finish modification (3)

surface finish table

In addition the surface conditions the size of the specimen also needs to be considered. The reason for this is because as the size of a specimen increases the statistical imperfections in the sample. So as the size of the specimen increases, the endurance limit will start to decrease. The size factor needs to only be considered for bending and torsional loads. If the load is axial then the size factor can be set equal to 1. If it is the bending or torsional loading then equation 4 can be used to determine the size factor.

load  modification (4)

Make note that the variable d in equation 4 represents the diameter of a rod. If the specimen happens to be rectangular the equation 5 could be used to compute an equivalent diameter.

equivalent diameter (5)

S'e is determined using a bending force on the test specimen. However there are different types of loads besides bending that can be place on the specimen, and these can affect the actual endurance limit. To determine what the load factor would be for bending, axial force, and a torsional force refer to equation 6.

load factor (6)

Testing is normally done at room temperature. However, it is known that temperature can change how a material is expected to behave. For example the lower the temperature the more brittle the material will normally become, while at higher temperatures that material will normally become more ductile. Temperature will also have an effect on the predicted endurance limit. The temperature factor is determined by taking a ratio of tensile strength at operating temperature (ST) over the tensile strength at room temperature (SRT).

temperature factor (7)

Refer to the table below to see the temperature factor for steel at different temperatures.

temperature factor table

The test data taken during testing will have some statistical variation between samples. Due to this statistical variation a reliability factor needs to be included to adjust the endurance limits according to how reliable you want your prediction to be. This is because the test endurance limit is essentially calculated by a best fit line through a scatter plot. To calculate the reliability faction equation 8 and the table below would be used.

reliability factor (8)

reliability factor table

Finally, there are miscellaneous effects that can affect the endurance limit. An example of a miscellaneous effect would be how a stress concentration would affect the sample. There are also other residual stresses that you may or may not know about that can affect the endurance limit. This is something that the engineer has to keep in mind even if he or she cannot determine what these values actually are, which is why safety factors are used.

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