Calculating Bending Result of Volumetric Strain in a Beam

[ScarTissue]

I'm working on a problem involving the bending of a beam.

The bending is a result of an expansion within the beam, rather than an external load. Basically, there is an expansion in the volume of one side of the beam and a contraction on the other side. I have therefore calculated a volumetric strain distribution for a coss-section of the beam. What I need to know how to do, is calculate the resultant bending for that strain distribution. How would I go about this?

 

[drMS]

Hi-
for volumetric strain distribution I suppose that you mean the trace of the deformation tensor. But in principle it is not enough to reconstruct the displacement field, and then the bending. What about beam constraints? Is this volume expansion/contraction due to temperature variations?

M

 

[Studiot]

You haven't stated why there is an expansion on one side and a contraction on the other.

Whatever the reason, even if the beam is composed of a single material, for analytical purposes, it should be treated as though it were made of two different parts.

I have started you off with the attached sketch.

First we observe that at any section AA there is axial equilibrium. So the force in the tensile part is equal and opposite to the force in the compression part.

This pair of forces forms a couple which is balanced by bending moments developed within the material of each part. Each part develops a separate moment as shown.

So we get a second equation by summing these moments and equating them to this couple.

There are two equations and three unknowns (F, M₁, M₂) so we appeal to strain compatibility to supply a third equation. Then we can solve.

However to derive the third equation we need to know and factor in the effect of the source of the expansion.

So over to you