| Rim brake heat and spoke tension
Jim Smith points out that spoke tension is insufficient to
significantly restrain rim diameter from increasing due to thermal
expansion. While contemplating this on my evening walk, I wondered
whether the resulting spoke tension increase could be a significant,
and overlooked, factor affecting spoke life.
A few calculations are in order.
Let
dCr = (delta) increase in rim compressive force
dTs = (delta) increase in spoke tension
n = number of spokes = 36
R = wheel radius
As = spoke cross-sectional area ~ 2mm^2
ks = spoke elasticity ~ 30Mpsi
Fs = ks*As ~ 42,000kgf
a = coefficient of thermal expansion of rim ~ 25e-6/degC
Ar = rim cross-sectional area ~ 80mm^2
kr = rim elasticity ~ 10Mpsi
Fr = kr*Ar ~ 560,000kgf
dT = rim temperature increase
P = rim perimeter
dPt = change in P due to temperature increase in rim
dPc = change in P due to compressive force in rim
(1) dCr = n*dTs/2/pi
(2) dR/R = dTs/Fs
(3) dR/R = (dPt+dPc)/P
(4) dPt/P = a*dT
(5) dPc/P = -dCr/Fr
expanding (3) and plugging in (4) and 5 we get
(6) a*dT - dCr/Fr = dTs/Fs
Using (1) to eliminate dCr gives
(7) a*dT = dTs*(1/Fs + (n/2/pi)/Fr)
(7a) = dTs/Feff
where
1/Feff = 1/Fs + (n/2/pi)/Fr
~ 30,000kgF
From (7a) we find that
(8) dTs/dT = a*Feff ~ 0.75kgf/degC
So a 100degC rise in the rim temperature increases the spoke
tension by 75kgf (165lbf). This is not a trival amount, it
represents about 25% of the ultimate strength of a spoke.
Comments? I have ignored bending in the rim because
I couldn't easily compute its effect. Presumably it
significantly reduces the tension increase in the spokes.
Joe Riel
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