Single Angle Bending Stresses. Geometric and Principal Axis 1

[Ussuri]

I have a puzzle which must have a reason which I am not seeing.

Consider you have a symmetrical angle subject to an applied moment about one geometric axis. You can calculate the bending stresses (using M/S with S calculated with respect to the geometric axis) for each of the stress points on the section.

Now consider the same angle with the applied moment resolved into the principal axis, giving you a biaxial moment. Now if I calculate the bending stresses (again M/S but now with S with respect to the principal axis) I get a stress component for each of the applied moments, which I then sum to give the total.

I had expected the stresses calculated in either method to work out the at the same value. I found they didn't.

Can someone shed some light on this? I am missing something somewhere.

 

[JAE]

Is it the fact that you are using Z instead of S?



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[Ussuri]

I used the elastic modulus in the calcs (which is Z = I/y for me) but i used S in the text above for our mainly US audience.

 

[rb1957]

I think it's because using the geometric axes is not strictly correct.

but some things look "odd" in your s/sheet ...

1) Zu at points 2 and 5 look "odd"
2) Zv should be close to zero for points 2 and 5
3) in the principal axes there will be a Zmax and a Zmin, in the geometry axes Z is the same for both axes.
4) shouldn't u and v be co-ords with respect to the centroid ?


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[Denial]

Your first method is fundamentally incorrect in theory.[ ] When you have a bending moment applied about an axis that is not one of the principal axes, you HAVE TO resolve that moment into two components, one about each of the principal axes.[ ] You then calculate the stresses from each of these components separately, and sum the results.

 

[KootK]

1) Your geometric axis case stress distribution would produce a x-direction offset between the centre of compressive stress acting on the section and the centre of tensile stress acting on the section. This creates a y-axis moment on the section.

2) Because of #1, your geometric axis case implies the existence of an My acting on the cross section concurrently with Mx to balance the moment just discussed above. You could not have Mx and the calculated stress distribution without it.

3) If you add the My from #2 to your principal axis case, I suspect that the stresses would then be the same. Of course, calculating My from the geometric axis case stresses would be a bit of work.


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