For fatigue life calculation you need stress range=Max stress-Min stress. E.g. If the cantilever beam is subjected to 1kN to 10kN alternating load range, the stress corresponding to 1kN will be minimum stress and stress corresponding to 10kN will be max stress. Then Stress range /2 will give you amplitude or alternating stress. You can use von Mises or principal stress range for calculating the alternating stress.

Check fatigue analysis calculation basics.

The use of principle stresses allows you to differentiate between tensile and compressive stress fields, while von Mises stress does not. Note that the modified Goodman diagram includes behavior for σ_m<0. This would suggest that the principal stress is a better choice. However fatigue damage is the accumulation of plastic strain, and von Mises stress is typically used to predict plastic flow, so perhaps von Mises stress is the better choice. What we did was use a signed von Mises stress. At any given point, evaluate principle stress to determine if the point is in a tensile or compressive field, and if compressive, multiple the von Mises stress by -1 and proceed to your fatigue calculations.

Rick Fischer
Principal Engineer
Argonne National Laboratory

Slightly off topic-

Use of max principal or von Mises or Tresca as stress for multi-axial HCF depends on the biaxiality ratio as per MSC fatigue(thread727-123897: von Mises or principle stress for fatigue analysis?). Lot of criteria are used for HCF life calculation for stress based approach- Sines, Findley.

If there is mean compressive stress, the fatigue life is higher than mean tensile stress which is obvious. Since the crack formation(till the point we calculate the " fatigue life") will be delayed. How this will affect when we use signed von Mises is not understandable.

Anyway, I am not fully familiar with multi-axial fatigue and hence my comments may appear primitive. But would appreciate any comments regarding the same.

Von Mises stresses should not be used for fatigue, since - as mentioned above - you can not see if the stress is compressive or tensile, as Von Mises stresses are always positive values. A common and (in my opinion, the most) simple method is using the Absolute Maximum Principal Stresses (sigma_amps), defined as:
sigma_amps = sigma3 if |sigma3| > |sigma1|, otherwise sigma_amps = sigma1. To be clear, the principal stresses sigma1, sigma2 and sigma3 are numbered in such a way that sigma1 > sigma2 > sigma3.
When you have multi-axial loading and especially non-proportional stresses, the fatigue calculation becomes much more complex when not using specialized software like nCode. When the stresses are non-proportional, the direction of the principal stresses change during the load cycles, which is hard to extract manually from FEA.
A note on the S-N curve: you should always check if the y-axis is in terms of stress amplitude or stress range, as both types of S-N curves are used.

I stumbled on this thread just now. Apologies for being late to the party.

If the assessment is restricted to high-cycle fatigue, then maximum principle stress is the way to go. In a bi-axial stress field, if there are multiple load cases, the orientation of maximum principle stress can change with each different load case. In this situation, one way of proceeding is to perform a 180° sweep around the design detail under consideration, typically a hole in a plate-like structure, and resolve the bi-axial stress to a tensile stress oriented in the angular position under consideration. One angular position will result in the lowest calculated fatigue life. That fatigue life is the one to be quoted.

The appropriate angular increment is the responsibility of the engineer. It might be that one could start with a very course increment, perhaps 30°, and then use a much smaller angular increment in the region of relatively low calculated fatigue life.

Performing this assessment in an FEA post-processor can be arduous. It might be more convenient to extract the stress components and do the calculation outside of the post-processor using Excel or a programming language, or some other appropriate tool.

The resulting stress-time history may or may not need to be processed via a cycle-counting methodology, such as the rainflow method originating with Endo & Matsuishi. The engineer must assess his or her data and make the decision on the way forward. The resulting stress cycles will probably have differing mean stresses (or stress ratios) and the empirical SN data must allow this to be taken into account when performing the cumulative damage summation.