Here is a basic formula to help decide what size beams you need for a simple box bridge. The resulting bridge is not certified by a licensed engineer, but we have built many of these and not had structural problems. Here is a sample bill of materials and construction diagram. You may need to add railings if it is high above the mean high water.
S = W x (L squared) / 750
S - Section modulus of the beams
W - Weight per foot (pounds)
L - Length of the span (feet)
This formula uses strength as the benchmark, not deflection. Unlike sheetrock construction where deflection always dominates on long spans, moderate deflection is acceptable for foot bridges.
This formula is based on an allowable bending strength of 1200 psi for framing lumber. Depending on your source and interpretation that number can vary from about 1000 to 1500. The quality of the lumber is also variable, and individual pieces will have defects and imperfections. So any formula is by necessity an approximation, but this formula provides a reasonable value for the strength, including safety factor, for good quality boards without serious defects. The ultimate strength of boards without imperfections is far more.
The section modulus table for framing lumber is as follows:
3 - 2x4 (1.5 x 3.5)
7 - 2x6 (1.5 x 5.25)
13 - 2x8 (1.5 x 7.25)
20 - 2x10 (1.5 x 9)
30 - 2x12 (1.5 x 11)
Notice that the section modulus depends on the square of the depth, so a small increase in beam depth results in a large increase in beam strength. For example:
a 2x6 is a little better than two 2x4's
a 2x8 is about as good as two 2x6's
three 2x8's are about as good as two 2x10's
three 2x10's are about as good as two 2x12's
Wooden beams lose strength rapidly as they begin to deteriorate along their top or bottom surface, since that effectively decreases their depth. If for some reason pressure treated lumber won't be used for the beams, then the beams should probably be designed over sized to allow for some deterioration.
The actual dimension of wood changes as it dries out. 2x10s shrink to about 9", and the smaller sizes proportionately less. The smallest eventual size must be used for calculations, not the size at installation.
A conservative estimate of load for a medium sized bridge is one 200 lb person (including pack) every two feet, which is 100 lbs/ft. This is probably too much for short narrow bridges (under 2' wide), and probably too little for long wide bridges (3' wide and over) that might see two way traffic or be used as a picnic or group photo spot.
There's also the weight of the bridge itself. That's not much for short narrow bridges with small beams, but bigger bridges use a lot more wood. Snow load can probably be ignored, since snow on the bridge will discourage maximum hiker load.
Another factor is that hikers stomping across a bridge is considered a “dynamic” load, which requires more beam strength than a “static” load (hikers sitting or standing without moving much). Conservatively the dynamic load could double the effect of a static load, but hikers stomping across a bridge are likely to space themselves farther apart so this is probably a wash.
An 8’ bridge at 100 lbs/ft load plus 10 lbs/ft bridge weight can easily be handled by two 2x6s (S=14):
S = 110 x 8 x 8 / 750 = 9.4
But the same loading at 10’ would exceed the strength of two 2x6s:
S = 110 x 10 x 10 / 750 = 14.7