This is a fairly complex problem to answer starting from scratch as it has multiple components so I will just summarise the calculations that will need to be done.
In terms of the stresses in the plank, typically as a minimum you will need to calculate the following forces:
- Bending moments
- Shear forces
- Bearing stresses
These will need to be calculated for different load cases including different locations for the weights on the beam as different positions for the load will give different worst case results. The method of calculation for the stresses will vary depending on the structural details you adopt at the supports but in the case you describe will probably be based on what is known as a simply supported beam.
Having calculated the forces in the beam, you will need to calculate some geometric properties of the beam in order to calculate the stresses. Typical geometric properties will be Second Moment of Area (for bending moments), Shear Area (for shear force), and Bearing Area (for bearing stresses). Again the calculation of these properties will depend on the detailing you choose, as will the use of these properties to calculate the stresses.
The final calculations you will have to do will be the stresses that the wood can withstand. Again, this is somewhat complex because wood, being an organic material, has different strengths in different loading conditions with factors such as grain direction, load type, load duration, wood type, etc. all affecting the calculation. You will also need to include an appropriate factor of safety in the calculations.
Having said all of that, this is over the top for most domestic applications and for the most part basing the size on what has worked in similar circumstances before is normally adequate.