I'm currently looking into the theoretical stiffness of steel sections after they have been deformed into a circular shape.
The radius of curvature induces strains that are well beyond the elastic region, though only just into the region of strain hardening (around about 0.04, with a low carbon steel) - so I'll ignore the effect of strain hardening for the time being.
The traditional stiffness equation relates E and I and the diameter. Determining I (the moment of inertia) is fine, using poisson's ratio to determine the change in section dimensional properties and hence also a change in the neutral axis depth. The elastic modulus is fine (for the elastic core of the steel section), however...
Do I use the secant modulus for the effective modulus for fibres at a given strain? Or am I completely off the mark?
Thanks for the encouragement Jim - that was the way I was trying to visualise the problem. Unfortunately all of the results of investigations conducted over the last 100 years don't appear to be too well documented on the web . Any references (either on the web or in textbooks) would be greatly appreciated.
Quoting from pg 21 of Roark and Young's "Formulas for Stress and Strain", 5th Edition: "The laws governing plastic deformation are less amenable to mathematical statement than those assumed to govern elastic behavior, but a mathermatical theory of plastic action is being developed".
Foolish me needs to give it a shot on an incredibly simple rectangular section.
Utilising tensile test data to determine the Secant Modulus according to strain for different fibres seems to indicate that the modulus plummets rapidly outside of the elastic region (same yield stress on double the yield strain halving the secant modulus). Using a relationship that ignores strain hardening, ie the yield plateau stays at the yield stress:
ep = stress x ((1/Es)-(1/E))
Where: ep = plastic strain (obtained through geometry of flexed steel and accounting for the elastic strain of 0.0012) Es = Secant modulus E = Elastic Modulus
For a particular example the stress ranges linearly from 0 (at neutral axis) to 0.04 (at the extreme fibres) resulting in an extremely small elastic core, where the Elastic Modulus applies.
This results in a secant modulus of about 6.5GPa in the outer fibres, or an average secant modulus across the section of about 35GPa, implying that the effective rigidity of the section is about 17.5% of a similar 'unyielded' specimen.
This is why my original request was made - this seems like a somewhat large effect, and doesn't seem to be born out in testing (which results in an effective E of around 80GPa).
Also, the strain hardening effect has very little effect in secant modulus. An increase from 240 to (for an unlikely example) 300MPa at the 0.04 strain point results in a secant modulus of 7.5GPa (instead of 6.5GPa).
Something here seems amiss and I'd appreciate any pointers into what it is (or may be).
If you are doing 3 point bending, instead of pure moment 4 point bending, then you probably need to explicitly consider the cross section shear in resolving the problem. That would raise the effective stiffness above that from merely the axial stresses and resulting bending moments. Which sounds like what you need....
PLasticity of Beams is a subject I recall existing from my undergraduate work many years ago. There were books on it, I believe.
Learning to search is as important as inventiveness and curiosity.