3 of the spokes on my wheel are broken. How can I tell if the wheel is still good to use?
Can I just replace the spokes or do I need to buy a brand new wheel?
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The spokes support the wheel under compressive load. The rim in itself is relatively weak in compression. Of course, losing a spoke means that the remaining spokes have to carry increased load and you risk breaking more spokes or buckling the rim beyond repair.
(Note: minor buckles can be removed through "truing" the wheel, a process of adjusting spoke tension to pull the rim in both the axial and radial directions so it is round and straight again)
Have a bike shop assess whether the rim has been damaged from riding on 3 less spokes. If they deem it is still usable, then they can replace your spokes (possible more than just the 3 which broke) and true the wheel if necessary.
There are some interesting websites that consider analysis of a spoked wheel:
I just talked to an experienced wheel builder -- Rich, the guy who builds for Rivendell -- and he said that his rule of thumb is after three spokes have broken, it's time to respoke or rebuild the wheel.
That assumes you've replaced the spokes as they broke, and re-trued the wheel if necessary from any time riding with missing spokes.