Re: leaf spring energy

[David Dockstader <DRDOCK00@UKCC.UKY.EDU>]

On Sun, 19 Apr 1998 12:17:14 -0700 (PDT) James Mclean said:
> >
> If I understand this correctly, you are saying that the stress tensor can
> always be diagonalized.  Is that correct?

If there is no hydrostatic stress or if you pull that out and treat it
separately, yes it is correct.

> Please elaborate on the ellipsoid visualization.  I'm not seeing how
> rotating a diagonal matrix can be so represented.  Do you mean a genuine
> ellipsoid, or just an oval-oid like shape?
They are real ellipses.

> I think that in these terms, my 'misconception' could be restated as:
> In a bent leaf spring, the principle stress planes are never perpendicular
> to the surfaces of the spring.
Unless your spring is in a really strong wind there will be no shear stresses
on the surfaces.  This makes the normal stress on the surface one of the
principle stresses and the other two will be mutually perpendicular.

I have usually only concerned myself with what is happening in the plane
containing the max and min principle stress, because this determines
type and direction of failure.  2D is nicely visualized using a graphical
representation known as a Mohr Circle.

I have not had occasion to give this much thought for about 15 years so I'd
have to think a bit to give you a lucid treatment of the 3D case.  I hate it
when people give me references that I can not readily access.  I learned this
in the context of structural geology and rock mechanics, but I would expect
the same basic treatment to show up in any engineering texts on stress
analysis.  The treatment of bending beams is a classic that should be in
any text (kind of like the freely falling ball or the field around a point
charge)  The other favorite is the stress around an elliptical hole in an
infinite sheet.

If you have a good geology library look for a book by Arvid Johnson.  If you
want more specific references I can dig them up when I'm at school.  I can
also ask my son, who is a mechanical engineer for some engineering references.
Find someone with a program called "Pro Engineer" and they should be able to
run stress analysis for any shape (3D as well as 2D) if you really want to
see what's going one.

Back to the bending beam/leaf spring problem.  On the outside of the bend the
min normal stress is parallel to the surface (I always take extension as
neg and compression as pos, but some engineering texts reverse this.  It's
like falling bodies you can pick up or down as pos, just stick with it) the
max normal stress is perpendicular to the surface, max shear is +/- 45 deg.
On the inside of the bend max and min normal stresses reverse position.
Bending beams generally fail under extension on the outside of the bend, but
shear failure is possible on the inside and will come at +/- 30 deg to the
surface (a tradeoff on getting the shear stress big and the normal stress
small.

> --James McLean
> jmclean@chem.ucsd.edu
> post doc
> UC San Diego, Chemistry