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 in advance for any input.
The secant modulus will be strain dependent, of course.
What you are contemplating is consistent with the visualization of the
bent steel specimen as a parallel array of tensile bars.
This seems like it should be well covered in published work, as it is a
problem that has probably been looked at many times in the last 100
years or so.
Apologies in advance for the long post...
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
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))
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
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.
You should use the E modulus, which is not changed during the
deformation. For any practical purpose the E modulus is equal for all
carbon and low alloy steels.
Kai Wøldike Sørensen