> Wouldn't your beams start cracking at the bottom, where they are subject to traction?

You have a few mistakes here. I’m not trying to demean you, but I’m going to number them just for clarity, as it can get confusing when there are many misunderstandings.

1. You are intending to ask about tension (which the rebar helps with), not traction (the force your tires exert against a road).

2. Tension is not only experienced at the bottom of beams, the location with the most tension will depend on the geometry. For a vertical beam, I think tension will probably be pretty even through the whole beam in most “normal” designs and loading configurations. But it will really depend on the geometry and on the loads being applied.

3. I think when you say concrete beams you’re meaning columns (apologies if I’m wrong about this). Concrete columns are remarkably good at holding up without rebar, because they experience almost exclusively compression! And indeed, ancient Roman designs did not use rebar at all :). It’s certainly possible.