Principal Stress, Mohrs Circle

[sweetness2]

So I understand how to use Mohrs circle and the transformation equations to find principal stresses and stresses for a given plane, but what is the point? Is there a purpose to knowing this other than finding stresses for a given direction, the stress invariants or that shear does not occur on the principal directions? How are these things used in the real world / design?

Don't people usually use average stress when designing? I guess I just don't see a practical application.

 

[Mapes]

Hi sweetness2, welcome to PF. In metals, failure occurs due to shear. That doesn't mean a rod under normal axial load, for example, won't ever fail; it means that it will fail along a plane 45° from the load axis, because this is the angle of maximum shear. The transformation equations (and their graphical equivalent, Mohr's circle) can be used to analyze this and more complicated loading configurations.

People absolutely do not rely on average stress when designing. They need to know the location and magnitude of the maximum characteristic failure stress for that material (in metals, the von Mises stress).

 

[sweetness2]

Thank you for your answer Mapes. A follow up question: I have seen von Mises stress represented several ways,

1) vonMises = sqrt (((S1 - S2)^2 + (S2 - S3)^2 + (S1-S3)^2)/2)

where S# represents principal stresses (found from mohr's circle or transformation equations)


2) vonMises = sqrt (Sx^2 + 3*Txy^2)

where Sx is normal stress and Txy is shear stress.

Are these two equations identical or is one an approximation of true von Mises criteria? I suspect my answer would be solved by substituting expressions from Mohr's circle into the first equation...but I would also would like confirmation.

 

[Mapes]

They're both exact, but the first equation applies to all possible load states, while the second assumes a more simplified loading. Can you tell which stresses have been assumed to be zero?