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...
| | | | |
| | | | |
| | ----- | |
| | / ___ \ | |
| | | / \ | | |
| | | \___/ | | |
| | \ / | |
| | ----- | |
Plywood will not sag over time if you intend to use it for the door faces. I recommend birch plywood for not only its hardness but looks. However your hinges and what they are attached to are the big concern in this project. It looks like you've used a low grade pine to build the unit. This is a soft wood and a couple of cabinet hinges screwed into this are going to loosen quickly. Your best chance for longevity is to use a piano hinge so that you have a continuous hinge from top to bottom with 15 screws or more on each hinge. I highly recommend ordering a stainless steel or flat nickel piano hinge for this. What they have in stock at the box stores is of the "bright" variety and looks cheap to me.
I'd also recommend that you not attempt to build the door to be perfectly flush with the "lip" at the bottom. Leave yourself 1/4 of an inch gap. Units like these always sag at least to some degree and you wind up with the door scratching the "lip" and its going to look really bad with time.
Also take a look at ball catches to use either at the top or bottom of the doors to hold them in place once closed.
Best Answer
I think your two choices are
An example of the former would be to make the shelf into a T-beam (or use keshlam's suggestion of a torsion box)
An example of the latter might be support wires up to the top of the wall or to the ceiling, or a glass-brick pillar, or a wooden arch support.
I would think about the myriad of ways bridge builders have tackled the same structural problem over hundreds of years.