How much load can a board support, if it is supported only at the ends?
At the moment my specific problem is hanging a tire swing between two trees, and depending on which trees I choose they could be 10, 12, or 14 feet apart (center to center). If I securely attach a 2×4 to the trees and hang the swing from the middle, would it be at risk of breaking from 2 children playing on it (so I'd guess around 300lbs max with them bouncing and jumping onto it)
But my bigger question is the general case: how do you calculate (or where do you look up) what kind of loads can be supported by the wood in question? (other examples would be a 1×10 shelf with supports 24" apart, or 1/2" plywood on a 2' x 4' frame). I'm not an engineer (well, software, but that doesn't count here), but I can do simple math (linear algebra, trig, calculus) and have a basic understanding of physics.
Best Answer
I'd go with @Aarthi's load bearing table resource for a general idea of what's reasonable.
If you're looking for equations though, you can start with these:
Beam Deflection Formulas
Beam Deflection and Stress Calculator
Area Moments of Inertia
Using the Parallel Axis Theorem
Wood Material Properties (Modulus of Elasticity (E) found in Table 4-3a)
For the dynamic loading, you'll want to do something similar to the fun I had on this question.
...and you may want to consult a good Mechanics of Materials book. (cheaper paperback international edition on Ebay)
As @Ian points out, the problem is not a simple one and is best solved by simply using what's worked for other people in the past. Go take a look at the swings at your local park and use the same size of beam, provided the span is comparable.
Also, if you're really worried, you could always make the rope into a 'Y' to eliminate bending stress on the beam, leaving it solely in shear. This way, the beam is bearing the compression load from the lateral tension on the 'Y', which will keep the trees from bowing toward each other.
Diagram: