How to calculate 3D Principal stresses?

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To calculate 3D principal stresses from given stress tensors, one must first compute the three stress invariants: I1, I2, and I3. The relationship between these invariants and the principal stresses is essential for deriving the principal stress values. The individual equations for the principal stresses S1, S2, and S3 can be obtained from the characteristic polynomial of the stress tensor. While online calculators provide quick results, understanding the algorithm allows for manual calculations without reliance on software. Resources like Wikipedia and Wikiversity can offer additional guidance on the necessary equations and concepts.
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How to calculate 3D Principal stresses?

If I got the values of 6 stress tensors:
222.png


Then I can calculate the values for the 3 stress invariants I1, I2 and I3:
111.png


I also know the relationship between the stress invariants and principal stresses:
58798acd7ecee4275adddee10e6eef36.png


So how can I calculate the values for the 3 principal stresses?

I can find some simple stress calculator on-line to get the results straight away, but I need the algorithm step for the calculation. In the other word, I need the individual equations for S1, S2 and S3, then I can do the calculation myself without using any software calculator or speardsheet!

Thanks a lot!
 
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