Stress on an Axially Loaded Beam

[paxprobellum]

Homework Statement

Consider a cantilever 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.

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.

My difficulty is determining what M is. In cantilever problems with transverse loads, M = M(x) = F*(L-x). I have a feeling M is not a function of x, but is constant over the whole beam. I think its F*d, where d is the thickness of the beam, but I'm not really sure why this is the case.

 

[Mapes]

I have a feeling M is not a function of x, but is constant over the whole beam.

Agreed. Could the lever arm that produces M have something to do with the nature of the off-axis load...?

 

[paxprobellum]

Agreed. Could the lever arm that produces M have something to do with the nature of the off-axis load...?

Yes. The moment is produced due to the lever arm. I suppose the lever arm appears as the force drifts from the center. So the lever arm must be the distance to the center of the beam, or half the thickness.

Good?

 

[Mapes]

Sounds good.

 

[paxprobellum]

Sounds good.

Thanks ;)