Fatigue Analysis of a Gondola Ski Lift (Part 4)

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Once one knows the operating point using the mean and alternating stress, a fatigue model must be used to determine the life expectancy of the component. The fatigue model is developed from an SN-curve. An SN curve compares the alternating stress to the number of cycles at failure. The SN curve can be determined from experimental data or empirically derived using fatigue factors. For this example, we will empirically derive the curve because we don’t have the luxury of breaking several Gondola bent arms to acquire the fatigue data.

STEP 6: Gather the data necessary to build an SN Curve empirically

Before one begins empirically deriving an SN-curve one needs to know three things : 

1. The loading scenario (Axial, Bending, Torsion, Biaxial)
2. The material
3. The Ultimate tensile strength of the material

For our scenario we have an element that is (1) axially loaded, (2) made up of steel, and (3) we will assume an ultimate tensile strength () equal to 120 ksi. This is illustrated in the figure below:   

Recall from Part 2 that the worst case stress location on the inside of the bent arm is axially loaded. The axially loaded situation is used to define fatigue strength factors. Knowing that the material is steel, one can acquire empirical relations to derive the ideal SN curve. Finally, knowing the ultimate tensile strength allows one to apply the empirical relations to derive the SN curve. 

STEP 7: Empirically derive the ideal SN-Curve

The ideal SN curve is an SN curve that is not reduced by fatigue strength factors. For steel, the ideal SN curve can be computed with the following relations. Essentially they give points at cycles and cycles on the SN curve. The points are calculated below from the empirical relations for steel:

Using the two data points above, one can construct the ideal SN-curve as shown below:

As one can observe from the figure above, represents the y-intercept at cycles and represents the “knee” or endurance limit for steel which occurs at cycles. The curve compares the alternating stress () to the number of cycles () at which the part breaks and is usually plotted on a log plot. If one is satisfied with this curve, they can develop a fatigue diagram from it. However, the ideal case is not sufficient for practical engineering applications. To improve the estimation of the fatigue life of a part, one usually applies fatigue strength factors. These can be found in a textbook for the different loading cases. The reference I used is listed at the end of this section.

STEP 8: Empirically derive the SN-Curve from fatigue strength factors

The Fatigue reduction factors reduce the values shown in the ideal SN curve. This means that the part will fail at a lower number of cycles. The empirical relationships used to derive SN curve based off of fatigue strength factors is given below at the endurance limit () and the y intercept () as defined on an SN-curve:

(1)  

(2)  

Without going into too much detail, the empirical fatigue factors take into account loading condition, temperature, reliability, the surface finish, and size of the part being evaluated. These values are up to criteria specified based on the application and engineering judgement. I would reference a book to determine these values. And really it is not too difficult to get these values. In the meantime the values I calculated are shown below:

The values calculated above can be used to develop a more accurate SN curve of the component in question. Plotting the values results in the following SN-curve:

If you compare the ideal SN curve to the SN curve derived from fatigue reduction factors, it should be clear that the ‘knee’ has dropped from 60 psi to 34 psi i.e. a significant amount!! The SN curve derived from fatigue factors will be used to develop a fatigue model in the upcoming section.

REFERENCES:

Juvinall, R. and Marchek, K. Fundamentals of Machine Compnent Design. 5th edition.