Re: leaf spring energy

[brian whatcott <inet@intellisys.net>]

At 12:25 4/19/98 -0700, you wrote:
> brian whatcott says:
> > I cannot do better than remind the readership of an excellent paperback -
> > actually two of them - written by (Professor) James E Gordon.
> > "Structures" and "The New Science of Strong Materials".
> > This is material more gripping than a mystery, more relevant than watching
> > "Titanic", and engaging to the attention.
>
> Thanks, I'll definitely look these up.
>
> > There, one finds in a slim appendix, the royal road to reasonable stress
> > calculations - and I will echo his notes on Beam Theory here:
> >
> > The basic formula for stress s at a point P distant y from the neutral axis
> > of a beam is
> >
> > s/y  =  M/I  = E/r
> >
> > Where: s is tensile or compressive stress
> >       y is distance from the neutral axis
> >       I is second moment of area of cross-section about the neutral axis
> >       E is Young's modulus ( or stiffness or inverse springiness, if you
> > will)
> >       r is radius of curvature of the beam at the section we are examining.
> >       M is 'moment' or force times perpendicular distance from section of
> >          interest.
>
> I hesitate to ask this before looking up the reference, but...
> You speak of a neutral axis, where I naively would have expected a neutral
> plane:  tension above and compression below.  Is it necessary to have
> inhomogeneity 'across' the beam, i.e., along the axis perpendicular to both
> the beam length and the bending force?
>
> -- 
> --James McLean

Gordon is speaking, in the usual way, of forces and moments as seen at a
cross-section. In the case of your spring or metal rule - this section
would be a rectangular area. In a line bisecting the thickness of the rule,
there is no tension or compression; at the top surface (which you are
depressing,) there is a tension, and at the bottom surface there is a
compression - for this analytical purpose, shear is altogether ignored. 
It is this bisector which he calls the neutral axis.

 I should also mention that Gordon's books are entertainments of a kind
that insinuate interesting facts on our attention. If you want a text ( but
I am sure you have access to many) the opening chapters of Shanley's
Strength of Materials would be a more serious-minded approach.

But to answer your question, the transverse variation in stress is also
ignored
for the usual (engineering) treatment, but the neutral plane you have in
mind does cut the neutral axis in the cross section of interest, if I
understand your point.

Whatcott    Altus OK