Re: Stress on an Axially Loaded Beam

[paxprobellum]

See: see: https://www.physicsforums.com/showthread.php?p=2314761

Homework Statement

Consider a hollow beam of length L where a force F is applied in compression at the bottom of the beam. (An off-center axial point load.) Determine the stress at the top and bottom of the beam at x=L/2. The inside radius is r1, the outside radius is r2.

Homework Equations

There is both compressive and bending stress in the beam. The compressive stress is -F/A and the bending stress is My/I, where F is the force, A is the cross sectional area, M is the applied bending force, y is the distance to the neutral axis, and I is the second moment of area.

The Attempt at a Solution

Applying this force is the same as a pure bending moment, except you gain additional compressive stress. Thus, at all points in the beam, the stress is:

S = -F/A + My/I

Most of the beam will be in compression, and a smaller part of the beam will be in tension.

M is a constant over the whole beam, not a function of x, and is equal to F*r2/2.

y, however, is a bit tricker to solve. By definition, y is the distance to the neutral axis. In other words, the stress should be zero at y=0.

So, I set the stress S = 0 = -F/A + M(y-y0)/I

where

y0 = distance from the midpoint to the shifted neutral axis due to compressive stress

and solved for y0

(which is -1/2r2 * ((r2^4 - r1^4)/(r2^2 - r1^2)))

So now I have:

S = -F/A + F*r2*c/2I

where c = y-y0
and y0 is a constant
and y is the distance from the midpoint of the beam

Basically I have moved my coordinate system from the midpoint of the beam to some other point, so "c" (y-y0) is nonzero at zero stress.

Thoughts?

 

[nvn]

paxprobellum: The applied axial load does not shift the neutral axis for the bending stress calculation. Try it again. Also, don't you have a mistake in the location of your axial load, when you computed M? Double check that value.

 

[paxprobellum]

paxprobellum: The applied axial load does not shift the neutral axis for the bending stress calculation. Try it again. Also, don't you have a mistake in the location of your axial load, when you computed M? Double check that value.

The applied axial load does shift the neutral axis. Consider it qualitatively -- pure bending produces compression on one side, tension on the other, and the neutral axis at the midpoint of the beam. If you apply a compressive load on top of that, tensile stress less than the compressive load will be compressive, and the neutral axis will shift towards the tensile side of the beam.

I think M is computed correctly. I'm not sure if I understand your question though.

 

[nvn]

paxprobellum: Yes, the axial load shifts the neutral axis after the load is applied, but not for the bending stress calculation. Does your textbook recommend the method you are using for computing compound stresses? Instead of me reviewing your derivation just yet, how about if we assign numeric values to your parameters? Then if your derivation produces correct numeric answers (i.e., the correct stresses), then that would mean your derivation is correct. E.g., let r1 = 26 mm, r2 = 30 mm, and F = 27 000 N. Therefore, what stress do you get at the top and bottom of the round tube at x = 0.5*L, using your method?

r2 is a radius. Why do you not have F applied at the bottom of the beam?