# Homework Help: Re: Stress on an Axially Loaded Beam

1. Aug 20, 2009

### paxprobellum

1. The problem statement, all variables and given/known data
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.

2. Relevant 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.

3. 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?

2. Aug 20, 2009

### 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.

3. Aug 20, 2009

### paxprobellum

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.

Last edited: Aug 20, 2009
4. Aug 20, 2009

### nvn

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