Originally Posted by
desconhecido
If you add a 40 unit load to the hub shouldn't the sum of the vertical components of all the tensions change by 40 units? That is, shouldn't the sum of all the forces on the hub be zero?
yes. In the diagram I posted, if you do an integration between the graphs and the zero-line, it does appear that the surface-area above zero is roughly equal to surface-area below zero.
Originally Posted by
desconhecido
An interesting thing about the graph, if I am interpreting it correctly. There seems to be a large deformation of the rim at the bottom, as expected, but the deformation at the top (and therefore, the change in tension in the upper spokes) appears to be the lowest of all the spokes. The spokes from about 70 to about 110 degrees seem to elongate (and therfore increase in tension) more than the other spokes and yet changes in the tension of these spokes can do little to support the load because the vertical component of tension is small.
As mentioned before, the rim deforms to maintain constant circumference, but it doesn't deform uniformly. It actually deforms more into an oval shape. But the total tension-changes are balanced so that the amount of tension in the contact-area equals the tension changes amongst all the other spokes above.
You can't separate the spoke-pull into vertical & horizontal components because the hub doesn't "hang" from the top-spokes, but rather it hangs "within" the rim. The rim is pulling outwards at all times and any change in the shape at one spot causes a change everywhere else.
It's more like how a pneumatic tyre works. The skin of the tyre is what holds the car up, but you'll notice that none of the tyre-casing is stretched vertically. it's stretched outwards all around. At the bottom, the casing is actually unstretched and the deformation causes an increase in air-pressure due to the decreased volume. This increase air-pressure acts on ALL parts of the tyre ABOVE the load zone. It's this increased air-pressure pushing on the tyre casing that holds the car up on the tyre.
Same thing with a wheel, the increased tension on the non-loaded spokes is what keeps the hub from moving down and the total sum of increased tension balances the load at the hub.