Does the concept of Mohr's circle extend to bending and torsion in plates? 1

[bugbus]

This is more of a theoretical question, which I hope has a simple answer. Please refer to my sketch below.

Does the Mohr's circle concept extent to bending and torsion for a plate?

For example, if we consider a plate subject to principal bending moments of M1 and M2, is it the case that there will always be a maximum torque of (M2 - M1)/2 acting on a plane oriented 45 degrees to the axes of the principal moments?

Then, for the case of M1 = M2, would we expect there to be no torsion acting on any plane?

 

[SWComposites]

No, Mohr's circle applies to stresses at a single point, not to loads/moments.

 

[271828]

I agree with SWComp.

You can find clear non-theoretical discussions about the bending and twisting moments in reinforced concrete textbooks in the chapter on yield line analysis.

 

[Italo01]

I'll disagree with the answers above.

The analogy to the Mohr's Circle is possible but require other assumptions, and since we are talking about a question from a theoretical point of view, there's no problem with these assumptions.

The assumptions are:

- Pure Bending, which appears to be assumed on the drawing;
- Bending forces uniformly distributed on the edges of the plate;
- Plate of uniform thickness.

Timoshenko shows in his book "Theory of plates and Shells" (Chapter 2-Section 10) that if these assumptions are valid, the moments will behave accordingly with Mohr's circle. Notice that he even uses the term Mohr's Circle.

 

[EdStainless]

I worked for an engineer whose PhD thesis was developing the equations for strength and stiffness of truss plates.
We would often do this as a first pass in a design.
The issue is that in the real world you quickly get combinations of forces in all planes and the errors start piling up.


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P.E. Metallurgy, consulting work welcomed

 

[bugbus]

Thanks all for the responses. I should have been clearer in my original post - in my mind I suppose I was thinking of a rather small element of a slab, so that the assumption of uniform bending and uniform torsion would be valid.

I was thinking a bit more about this over the weekend and I think it's pretty clear now.

As Italo01 mentioned, assuming pure bending that is uniformly distributed along the plate edges and a plate of constant thickness, then it's straightforward to determine the principal stresses sigma_1 and sigma_2 at some arbitary point from the principal bending moments M1 and M2. And since the stresses at a minute point obey Mohr's circle (as SWComposites said), it follows that the bending moment and torsion should too if we're looking at the scale of the plate element (i.e. considering the plate thickness).

Looking at a tiny element, I believe we would get the stresses shown below.