Saturday, July 11, 2009

Spoked Wheel - Part 4 - Elbow Strength

From previous post , the tensile strength and cross section areas are
233 ksi (2.34 mm) 0.0066658 in*in
240 ksi (2.0 mm) 0.0048695 in*in
245 ksi (1.8 mm) 0.0039443 in*in

Then it follows that:
The ultimate tensile load of 2.34mm spoke is 1553 lbf
The ultimate tensile load of 2.0 mm spoke is 1169 lbf
The ultimate tensile load of 1.8 mm spoke is 966 lbf

The spoke is predicted to break in tension at those forces. But for infinite spoke life, what we are interested in is is fatigue strength. For ductile steel, fatigue strength (or endurance limit) is approximately 0.5 of tensile.
Taking the elbow as the limiting factor, which we assume to be in shear we want the shear fatigue strength. Using distortion energy (von mises) failure theory, this is 0.58*tensile fatigue strength.
To summarize the max force that should be put into a spoke:
2.34mm: 233*0.5*0.58*0.0066658 = 450 lbf (2004 N, 204 kgf)
2.0 mm : 240*0.5*0.58*0.0048695 = 339 lbf (1508 N, 154 kgf)
1.8 mm : 245*0.5*0.58*0.0039443 = 280 lbf (1247 N, 127 kgf)
What this means is that for a spoke that has 1.8mm elbows, if the force in the spoke is kept under 280 lbf, the spoke should have infinite life.
In the elbow, the force transitions from an axial force into a shear force . Someday I may model this with FEA to see the stress contours. If there is bending stress in the elbow, which is definitely possible from not adequately stress relieving the spokes, then the stress in the elbow could be above the fatigue strength, and thus infinite spoke life will not be achieved.

Also note that just because this much force can be put into the spoke, doesn't mean it should be. The rim and nipple strength might not match. In the days of box section rims, the amount of spoke tension was determined by tightening the wheel's spokes to within a half turn of the onset of rim buckle. With today's deeper profile and/or sturdier rims, buckling is not the limiting factor. I have only experienced it with a very light box section rim, long ago...

Also of course, in the future, will look into the nipples and rim. The goal is to have as highly tensioned spokes as possible with infinite fatigue life of the spokes, nipples, and rim.
2014 comments:  Ahh this is good stuff. I still use this, my own blog post,  as reference :)