How much load can a 2×4, 2×6, and 2×8 each hold is laying horizontally, yet turned on it's side? We have found numbers for wood turned upright, but it wasn't what we were looking for.
Wood – find load ratings for two-by lumber laid flat
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I know you mentioned that you want to build the bike rack out of wood (and 2x4s) but...
You could probably build nearly that exact bike rack out of PVC pipe or even a slightly simpler design:
http://www.utahmountainbiking.com/goodies/TruckbedBikeRack.htm
You will not have problems mounting the TV if it is under 150 lbs and into 2 or more studs. Use proper screws and mounting system/plates.
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Best Answer
The reason you can't find tabulated information on dimension lumbar laid flat (usually called "weak axis bending") is because the strength is so small that there is virtually no structural application. The only time you would use 2x lumber like that is something like flooring where the boards are almost continuously supported and are carrying very little weight.
However if you want to get a sense of the strength you can compare the moment of inertia of the various cross-sections and orientations. Bending strength is proportional to that value.
I have done that math for you:
You can see that especially for the deeper joists you lose practically all of the strength.