Wood – How to reduce the thickness of a board except in one narrow area

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

I'd make a template in the shape of a square donut out of 1/2" plywood by using a table saw to cut out the square hole in the 1/2" plywood. For example, if ...

i) the recess in the desktop needs to be 8"x12"

ii) the diameter of the base of the router is 6"

iii) the diameter of the router bit is 1/2"

... then I'd ...

1) start with a rectangular piece of plywood that measures 24"x28"

2) set the rip fence on the table saw to 5.25"

3) plunge cut on the table saw (advanced technique) along all four edges, but not all the way to the corners so as to leave the cutout solidly attached to the donut at all four corners.

4) make the final cuts in the corners with a jigsaw or handsaw or ...

At this point I ought to have a square donut template of the proper size. If the router is not high-end, then I'd mark a spot on the top of the router's base that I'd always keep rubbing against the template (this is important for precision work).

To route out the recess in the desktop, I'd secure the plywood template to the desktop. Then start excavating the recess at its center, progressing in an outward spiral until the final passes are done with the router's base bearing against plywood template.

Hope this helps.