Wood – Can you support a 20ft 2×6 only at the ends

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...
|      |           |          |      |
|      |           |          |      |
|      |         -----        |      |
|      |       /  ___  \      |      |
|      |      |  /   \  |     |      |
|      |      |  \___/  |     |      |
|      |       \       /      |      |
|      |         -----        |      |

This should be a good starter project, here are a few tips I've picked up over the years:

• Even pressure treated wood will deteriorate over time if it's exposed or in direct contact with the ground/moisture. Do what you can to protect it from the elements and get it raised off of the soil. Use a moisture barrier between wood and concrete.
• Take care to perfectly level the area you are locating this.
• Make sure your cuts are straight, preferably using a square to mark your lines.
• Measure twice, cut once.
• Look for the direction that the wood bows, mark the bowed up side on the wood, and layout all the wood with the bowed side up. You don't want the floor to be wavy or to have a dip in the center for water/dirt to collect.
• Assemble the sub-assembles first, driving your nails/screws from the side of the joists into the spacers.
• Make sure all connections stay perfectly vertical since they lose their strength when they start to tip over. You can use a clamp to get leverage and untwist a warped joist, but if the warp is too significant, I'd cut it into pieces for the short spacers.
• Pre-drill when you are within 2" of the end of a board and going in from the side (e.g. all of the sub-assembly nails/screws and the 4 corners). Other than that, you can avoid pre-drilling, but it may help you keep a straight line if this is your first build.
• Metal brackets to hang the joists will give you a stronger connection. They should include directions for the type of fasteners, which will likely be shorter but very thick nails.
• Nails will have more shear strength, but screws will be less likely to pull out. If you have a sub-floor going on top of this, securing everything together, then I'd go with nails. I'd be lying if I didn't say I'd put in at least a few deck screws in each corner.
• 3 1/2" is a good length for any nails/screws. That gets roughly half of the fastener in each piece of lumber. When in doubt, get a longer/thicker fastener.
• Any untreated nails or screws will react with the chemicals in pressure treated lumber and quickly corrode. Hot dipped galvanized nails or treated deck screws should be ok, but check the label on the box to be sure it will work with the chemicals in your lumber.
• For each joint, a 2x4 would take 2 nails, a 2x6 would take 3, etc. Stay about 1" away from the edge of the board and space the nails about 1 1/2" - 2" apart.