Walls – How to support this shelf

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:

|      |______________________|      |
|      |  |                 | |      |
| tree |  |                 | | tree |
|      |__|_________________|_|      |
|      |   \               /  |      |
|      |    \             /   |      |
|      |     \           /    |      |
|      |      \         /     |      |
|      |       \       /      |      |
|      |        \     /       |      |
|      |         \   /        |      |
|      |          \ /         |      |
|      |           Y          |      |
|      |           |          |      |
|      |           |          |      |
|      |           |          |      |
      ...more rope and trees...
|      |           |          |      |
|      |           |          |      |
|      |         -----        |      |
|      |       /  ___  \      |      |
|      |      |  /   \  |     |      |
|      |      |  \___/  |     |      |
|      |       \       /      |      |
|      |         -----        |      |

It is highly likely that your beams are made of pressure treated wood. If so, you should try to use the same type of wood in repairing them.

SHIM - When there is a slight gap between a beam and a joist, that can often be cured with shims, slim pieces of wood (preferably of the same type, or if not available, a rot resistant type such as cedar). Shims should be forced in (tapped in with a hammer) so they are tight. Often you can drive shims from opposite sides. Since they are tapered, opposing shims can make a tight level gap filler.

LEVEL - If you cannot shim them adequately and need to move the joists to meet the beams, you need to make sure they remain reasonably level. They may have a slight pitch down away from the house, and that would be ok. You do not want to cause them to shift so that they slope toward the house. In some circumstances, that could lead to a drainage problem.

FILL - The holes in the beams can be filled with slivers of pressure treated wood that are coated with exterior glue. Force several narrow wedges in to pack the holes. Let the glue dry and then saw or chisel the surface flat. You can then drill new holes even close to the old ones.