# How do I calculate deflection in shelving boards?

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?

• Do you really mean 1.25 feet by 11.5 feet by 6 feet? This seems unlikely. Jan 12, 2016 at 13:09
• Regardless of the actual sag, it'll be noticeable. This is especially true if the shelf is mounted at eye level from a sitting or standing position. It would irritate my sense of feng shui (or OCD) to no end. Jan 12, 2016 at 16:43

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.)

• I get much less than that when I use the calculator (10 times less, in fact). My answer more closely matches AndyT's calculation. Please update your answer to list your parameters, or post a screen capture. Jan 12, 2016 at 14:06
• @isherwood: it really depends a lot on the parameters, which the original post did not specify. I picked worse-case for everything that was unclear: wood species (silver maple), shelf attachment (floating) load distribution (center), thickness (1", to account for final sanding/planing).
– Hank
Jan 12, 2016 at 14:43
• @isherwood: the post didn't say what the shelf was for, certainly didn't mention books. If this were for books the loads should be much higher. Anyway like I said in my answer, the tool is only useful if you have accurate inputs. "Garbage in, garbage out".
– Hank
Jan 12, 2016 at 15:07
• Sorry, not sure where I got books in my head. My mistake. Jan 12, 2016 at 15:27

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.

Notes:

1. 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.

2. 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.

3. Simply supported means the ends rest on supports but aren't restrained against rotation

• This seems fairly realistic considering that the load will be distributed evenly. I'd expect maybe twice that much based on real world experience, and maybe twice that again after a year or two. Jan 12, 2016 at 14:07