I started to write an entire how-to for Second Moment of Inertia, Bending Moment and Young's Modulus....
I gave up because even simple explanations become complex due to unit conversion. As you see, just explaining WHY this is not easy to explain will take many many paragraphs, let alone explaining the engineering itself.
OP and others. In general there are (3) things that we use to determine how much a beam will bend or deflect.
1 - the Second Moment of Inertia (Moment Area) which is derived from the cross section (shape) of the material. We use this (look up the equation in a table) to determine how a specific shape (in this case, a solid rectangle) resists bending. The issue here is what are the ACTUAL dimensions used for calculation vs the real world beam dimensions. Is the 2x8 1.5 x 7.5 or is it 1.43 x 7.325...
2 - The Bending Moment - the way that the beam is supported (one end or both and fixed or loose supports and where the load is placed (uniform or distributed and at what distance from the support(s).
3 - The Young's Modulus of the material itself. This is the materials properties as they relate to elongation or shortening when under tension or compression.
With these three things (all looked up) we plug in the values and determine the amount of deflection or bending. Where most people run into issues is conversion of the units of each of these. (imperial, metric, SI, etc.)
The biggest variable for our purposes is the Young's Modulus as it relates to wood. -We use a base value based on species and some assumptions for defects (grain, knots, heart or sapwood, etc). Two different pieces of the same size lumber can be vastly different with regard to strength or bending.
While I don't disagree with Rocket's numbers, they can easily be 100% off based on minor changes to the Young's Modulus used and the Moment of Inertia used. For 200 pounds per foot loading over the 19 foot perimeter, I come up with close to half the deflection that he does. But that does not mean that either of us are correct, it means that we made SLIGHTLY different assumptions using the same set of equations. He may have used point supported ends and I used fixed ends... neither is really correct or incorrect, as we don't know how stiff the side supports are to resist inward bending... that is another layer of calculations.
Also the base equation that we would use does not take twisting or torsional vectors into consideration. Tall beams under load want to twist, not just deflect. This is where blocking or other means to prevent twisting are important.
The point here is twofold. One, none of us can give you a 100% correct value. Two, safety factor, especially when there are so many variables in a design and no historical data or testing to consult, is important. That is where the 4x or more (over engineered in layman's terms) design advice comes in.
How do you add safety factor? Use real numbers and multiply the results... or inflate your assumptions and use the standard equations.
Example - Inflate the live load to 300 gallons at 10 pounds per gallon. So 3000 pounds. Make it 3500 pounds to account for your fat uncle and his neighbor leaning on the tank rim at your Halloween party. That increases the pounds per foot of perimeter load. Then run your calculation. Double the deflection value returned. If it is not reasonable then you need bigger beams, even if they are overkill. Is this overkill? Yes more than likely, but you are not an engineer trying to lighten a part in an airplane or reduce the span weight of a bridge on its piers. Likewise, none of us that understand the physics and engineering are certifying an end to end design, we are giving off the cuff advice that should be trusted as that.
