Ben C said:Quoted message said:Ben C said:Does the "thumb test" (squeezing a tyre to see if it's hard enough)
measure tyre pressure or casing tension?
[...]
Thank you to everyone for the explanations.
carlfogel> Thumb pressure measures casing tension.
carlfogel> [...]
carlfogel> Think of a trampoline.
carlfogel> Air pressure does not hold the trampoline up, any more than
carlfogel> it holds the trampoline down. What holds the tramoline taut
carlfogel> is the springs pulling it tight at the edges. Push down, and
carlfogel> the trampoline dents easily. Push a little further, and more
carlfogel> force is needed. The resistance is nicely progressive . . .
carlfogel> Just like a tire.
carlfogel> A tire is a doughnut-shaped trampoline. What stretches the
carlfogel> tire tight in all directions is the expanding spring of the
carlfogel> air pressure.
carlfogel> [...]
I am leaning towards this interpretation. Here is the explanation of
casing tension:
jobst.brandt> Casing stress is arrived upon by cutting across the
jobst.brandt> circular minor diameter of the tire (the tire is a
jobst.brandt> circular cross section having no structural belt as radial
jobst.brandt> tires do to change that) and take the two halves as solid
jobst.brandt> sections being pressed apart by inflation pressure. That
jobst.brandt> gives the lineal separation force which is the casing
jobst.brandt> tension.
Now, suppose I literally do cut across the minor diameter of my tyre, in
two places. I now have a piece of basically fabric hosepipe with no air
in it. It's a bit curved, but we'll pretend it's straight.
I now insert a piece of dowel into either end (to keep the hose
cylindrical). I attach clamps securely around the ends and attach the
clamps to strong springs such that they pull the hose out until it's in
the same tension as it was when it was on the bike and pumped up to
100psi. There's just atmospheric pressure in the hose (let's say I make
a small puncture somewhere, or the dowel is air-permeable, or it's a
clincher tyre anyway and therefore not a hose but more like a section of
guttering).
Now I squeeze it, against the force of the springs. We assume for the
sake of simplicity that the casing itself doesn't extend when stressed
(otherwise the length of the section I cut out has to be taken into
account).
Does it feel basically the same as it did when it was on the bike? If
so, it does seem that I'm measuring casing tension. The trampoline
springs are doing exactly the same job that the air was.
On the other hand I also believe this:
jobst.brandt> When a tire is pressed against a flat surface, the tire
jobst.brandt> flattens until the flat contact area times inflation
jobst.brandt> pressure equal the load.
If you think of the end of my thumb as a disk, with an area of half a
square inch, then to press that disk completely flat against either the
thin or the fat tyre will require the same force-- a force equivalent to
50lb.
But, at that point, the penetration of the disk along its normal into
the narrower tyre will be deeper, because of its narrower radius of
curvature.
If instead of asking, how hard do to I need to press to get the disk
flat against the tyre, I ask, how hard do I need to squeeze the tyre for
my thumb and forefinger to move 1/8 inch closer together, then for a fat
tyre and a thin tyre at the same pressure, I will have to squeeze the
fat tyre harder. This squares (at least qualitatively so far) with the
reasoning about trampoline springs.
Dear Ben,
Basically, an inflated elastic skin automatically occupies the lowest
possible tension shape. If you force a deformation anywhere locally,
the tension rises.
It might be easier to work through the geometry than devise spring
devices inside short sections. Air and rubber are sufficient.
The simplest tension figure that we can describe is a straight line,
say a rubber string stretched tight between two fixed points.
If we push the rubber string in any direction, the force increases the
tension.
Push the tight rubber string sideways at any point and the distance
obviously increases because a straight line is the shortest distance
between two points.
Instead of pushing the rubber string sideways, grab a point on the
rubber string and pull it toward either fixed point. Tension drops on
one side, but must rise to more than the original tension on the other
side--your pull has introduced a third fixed point and stretched a
section of the original rubber string further and tauter.
Let's get rid of those pesky fixed points.
An endless rubber string is just a rubber band. Imagine a rubber band
laid flat to form an air-tight seal between two plates.
Pump air into the sealed space and the rubber band will automatically
expand to form a circle with even pressure all around.
Push inward on either side of the rubber band circle, and the circle
will start to flatten.
The air pushed out of that section doesn't vanish. It's pushed into
the unflattened areas--which must bulge outward and cause the tension
to rise in the rubber band.
If there's just the right friction on the two points where the rubber
band is flattened, you could get a local tension lowering in the
chord, but that would just raise the tension even higher in the
unflattened sections of the rubber band.
A thumb press requires flattening two spots, one where our thumb is,
and one (or more) to oppose the thumb press. Locally, we can try to
lower tension by flattening a curve into a chord, but that relies on
friction to keep the flattened area from adjusting its tension as the
rest of the rubber band expands.
We could simplify to a single chord instead of two by grabbing two
spots on the rubber band and pulling them together (or apart). Either
way, the tension must rise in the rubber band.
If we pull the two points apart, the tension is obvious between our
fingers.
If we push the two points together, the rest of the rubber band has to
bulge as the more air is squeezed into what's left of the circle.
So much for one-dimensional rubber bands. Let's do a two-dimensional
surface, such as a trampoline.
Let's skip the ordinary flat trampoline and move straight to an
endless trampline--the two-dimensional surface of a sphere.
If we inflate a round balloon, every part of its surface automatcally
goes to the lowest possible even tension.
Mark a circle on the surface and label it "trampoline"--the tension
from inflation serves as the tension springs for an ordinary flat
trampoline.
We can push two (or more) spots inward on our balloon. The force will
cause of the balloon to bulge outward and its skin elsewhere to
increase in tension. If friction where we push our hands together
reduces the tension locally (a chord is shorter than a curve), then
there's even less balloon skin left elsewhere to constrain the same
amount of air and tension and pressure must rise even further.
Transform a sphere into a toroid and we have a tire that's still an
endless trampoline. The only way to reduce total tension is to press
inward everywhere by raising atmospheric pressure. Otherwise, the tire
must bulge out somewhere else whenever we push it inward (or outward)
locally with a thumb.
The most intuitive way to see what happens may be to imagine an inner
tube or tire force-filled with water. The skin of the toroid is
obviously tight. Push inward on it locally anywhere, and you know that
the water is bulging it outward elsewhere and raising the tension.
Less intuitively, pulling the water-filled tube outward anywhere must
also raise the tension. You have to pull at two points and apply force
to distort the tube. Pulling on the tube must increase its tension. As
soon as you let go and release the tension, the tube snaps back into
its original form, which is a slightly lower pressure.
Just like an air-filled tire--push it inward or outward locally, and
you raise the tension and pressure. As soon as you stop pushing, the
deforming tension is lost and the tire snaps back into its original
shape.
Cheers,
Carl Fogel