With regard to the tire question, I don't think we have had such a
discussion of zero-order (tension in bead) versus first order (pull of
the sidewall). I just looked back over all the messages in that thread,
and the only person using the words "tension in the bead" was you. It
would appear the zero-order first-order discussion was one you were
having with yourself. :)
If a flexible rope/wire/string/cable is involved, I view tension as
directed along the length of this. For the bead of the tire, assuming
the bead never pulls away from the rim, the bead tension is all
tangential. I always drew tension arrows radially, and I stated these
were the tension of the "sidewall" acting on the bead. The bead just
transfers these forces to the rim. Therefore I have always been saying
it is the sidewall tension (your first-order item) causing the support
of the wheel, with this tension being passed to the rim by the bead.
I don't see anyone actually talking about the tension in the bead except
you. When I first imagined the bead clinched by the rim, I assumed the
bead couldn't pull away from the rim because of the clinching. After
you corrected my rim picture I then thought maybe the bead might pass
the upper-sidewall tension to the wheel from underneath the wheel. But
I rejected that picture (at least from the first-order picture) because
I returned to the view that the bead is rigidly glued to the rim (which
is what you suggested we do). What changed for me was the mechanism by
which the bead is rigidly glued to the rim. So the bead just directly
couples the upper sidewall tension to the top of the rim (which is the
way I was picturing it all along).
Therefore, although this zeroth order versus first order post points out
that we should carefully determine the major factors, I don't see it as
applying to anything people have said about tire beads, other than you.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu