How do I calculate the deflection of a hard maple board 5/4' x 11-1/2' x 6' supported at ends only that will carry a load of 125 pounds?
Short answer: it will sag a lot.
Using The Sagulator I get a deflection of about 0.9" at the center, which will clearly be noticeable. (I'm assuming your "5/4 maple" is actually going to end up 1" thick when the finishing is done.)
Also I'm not sure what you are going to be using the shelving for but if this is going to be for books you should probably assume 30 lbs per foot, which would give you a deflection of 1.3".
(You can use that calculator to estimate sag for a variety of shelving configurations, but make sure you pick accurate values for all the inputs. Also note the footnote that says that long-term sag will be 50% higher than the calculated value.)
A simply supported beam under a load W, with span L has a midspan deflection of:
- WL^3/EI * 5/384 if the load is distributed along the length
- WL^3/EI * 1/48 if the load is applied at midspan
I for a rectangular section is breadth times depth cubed, all divided by 12 (I = bd^3/12). Your dimensions aren't clear, but I'll assume b = 11.5 inches and d = 1.25 inches, giving I = 1.8717 inches^4. I'll assume your span L is 6 foot = 72 inches.
E is the Young's Modulus / Elastic Modulus and depends on the material. For hard maple it is apparently 1.83 * 10^6 lbf/inch^2.
Plug this all in and we get deflection = 125 * 72^3 / (1.83 * 10^6 * 1.8717) * 5/384 = 0.18 inches.
I haven't included the weight of the wood itself. At a quick calc the wood weighs 26 lb, so that's in increase of 20%, if you haven't already included it.
I haven't taken account of long term vs short term. I know little about wood - I am applying general beam theory. My professional engineering experience is with concrete and steel.
Simply supported means the ends rest on supports but aren't restrained against rotation