Bending a patterned beam (FOSDT)

[Cue]

Say I have a wooden beam with an angled pattern in it if the beam is flexed (and assuming First order shear deformation theory holds: parallel lines remain parallel after flexing) what would the flexed pattern look like?

I think it would look like the pattern in the middle picture since the strain is linear but I'm being told by somebody else that it will look something like the bottom picture when flexed.

Can somebody with some knowledge on deformation confirm one or the other.

http://img858.imageshack.us/i/beamflexed.png/

 

[AlephZero]

You can answer the question yourself by drawing a grid of vertical and horizontal lines on the beam and drawing your sloping lines through the intersections of those lines.

For simple beam theory, if the beam is in pure bending (i.e. there is a bending moment applied to each end, no shear forces, so the neutral axis of the beam bends into an arc of a circle), then the "vertical" lines will remain straight and at right angles to the neutral axis. So the grid will deform into a set of concentric circles and radial lines. You can plot the shape of the diagonal lines from that information. They will be slightly curved.

Actually it is more complcated than this if Poisson's ratio for the material is not zero. When the beam is bent, the neutral axis is not mid way between the top and bottom because of the strains caused by Poisson's ratio. Google "anticlastic curvature" for more on this if you are interested.

If there are shear forces on the beam, the "vertical" lines also bend into S-shaped curves, with different amounts of curvature as the shear varies along the length of the beam. A good book on Timoshenko beam theory should explain that in detail.

 

[Cue]

Thanks Alephzero, are there any particular Timoshenko beam theory books that you recommend?

 

[AlephZero]

Why not go straight to the source and look at Timoshenko's own books? They were first published in about 1950, but still in print.