Engineering Shelves

Moving to Australia entailed all the ordinary dramas of moving to a new house, amplified by about 1,000.  As an engineer, some of my challenges would naturally need to be of a technical nature. 

For readers who would like a window into the murky mind of an engineer, I've dug this piece out of the archives, dusted it off, corrected the math, and present for your consideration, what happens when you mix engineering and home improvements. 

January 2002

Engineering Shelves

I put up some shelves in my home office this week.  It would never do for an engineer's office to have shoddy shelving.  This called for a bit of planning & forethought.  Perhaps even some scale drawings. and a detailed bill of materials.  Of course, I'd need to do some calculations based on simplifying assumptions, and solve an equation or two.  A guy might as well have some fun, right?

The shelves were to be approximately 800.10 mm long (that's somewhere between two and three feet, converted into metric), and supported by those long metal brackets that attach vertically to the wall.  In the U.S., I've always put these kinds of brackets directly on the wall studs because sheet rock won't hold anything heavier than a framed picture.  But this house, like most Australian houses (that are still standing*), is all brick, inside and out.  So I can put the brackets anywhere I want.  Ah, but where? 

Engineers are different from normal people, because they wonder if there is a "best" way to do things.  A normal person would say, "I can put them anywhere I want?  Fine." He then puts them wherever he feels like and moves on to something else.  But an engineer says, "What is the optimal placement of supports on a two-support beam with distributed loading?"  That's pretty much how an engineer talks, too, even when talking to himself. 

Let us humor the engineer for a moment.  Suppose the brackets were too far apart.  What would happen?  The shelves would sag unnecessarily in the middle, of course.  Too close together, and the edges would droop needlessly.  The shelves would probably still hold the same number of books, but they wouldn't be "optimal."  Perfection is so rare in life, you must seek it where it can be found.  And this is one thing I think I can get exactly right.  No extra expense or special parts, just drilling the holes in EXACTLY the right places.  How could I resist trying?

First, the engineer makes some basic assumptions about the project.  Assumption #1 is that the shelves will be supporting a fairly uniformly distributed load (a.k.a. books).  Assumption #2 is that books are all approximately the same linear density (weight per thickness).  Thicker books are heavier, but take up more space as well.  So it all averages out.  Assumption #3 is that since there would be loading on either side of each support, the shelf could be treated as if it were composed of three separate beams: two cantilevered beams representing the overhanging ends and one beam fixed at both ends with non-pivoting supports, representing the span between the brackets.  Oh - pardon me - I was just talking to myself again.  Engineer talk, you know.

Next, the engineer reasons.  What if the supports could be placed such that the middle and ends of the shelf sagged by an equal amount when full of books?  Now move the supports further apart.  The middle would begin to sag more, would it not?  And, if you move the supports closer together, the ends would sag more.  So, is it not the case that when the middle and end deflections are equal, then a loaded shelf has the absolute minimum deflection possible?  (Here's a tip:  you can save us both about four pages of calculus by just saying "yes."  Excellent. Thank you.)

Finally, the engineer calculates.  From inside one of those books of average linear density, I found formulas for the maximum deflections of cantilevered beams (w*[L1^4]/[8EI]) and two-support beams (w*[L2^4]/[384EI])   I knew I was keeping all those books for something: to help me make shelves to keep my books on.  Setting the deflection of the midspan equal to the deflection of the overhanging ends, one finds that the midspan length L2 is 2.63 times the overhang length L1.  Therefore, for optimal support the overhanging lengths should be 21.6% of the total length of a 2-support shelf with uniform distributed load.

The three spans together add up to 800 mm, so the overhangs must each be 173 mm, and the distance between the supports is what's left, namely 454 mm.  And that's precisely where I marked and drilled the holes for the brackets.  Problem solved.

The next problem was, how many screws were needed to keep the brackets attached to the wall when all the shelves were fully loaded?  I remember having to do a problem like that in school.  I remember it took a long time, and it was hard.  Based on that valuable educational experience, I decided to put in as many screws as I could.  No sense "screwing around" with that sort of thing, I say.  (Engineers all enjoy a good pun once in a while.)  Half of engineering is knowing which problems are worth solving.

The last problem was to determine how thick and wide the shelves needed to be to support books of average linear density.  I approached this problem by going to the hardware store and picking out some 12" x 1/2" wood that "looked about right."  It was on sale, too.

That might not have been very scientific, but I was getting tired, and figured what the heck - it's only some shelves, right?  They may not be perfect in all ways, but By Golly, those brackets are spaced absolutely perfectly.  It's a good feeling do have done something - even something not very important - just right.

Now, when I'm in my office and the rest of my life is in the crapper, I can look at those shelves and think, "Those brackets are absolutely the correct, optimal distance apart.  So my life isn't ALL wrong, after all."






PS  Here are some of the books I referenced for this project:


Roark's Formulas for Stress and Strain


Engineering Formulas by Gieck and Gieck

* Footnote:  Australia is famous for its legions of very hungry, hard-working and persistent termites.  In no time at all, the only thing left of a wood-frame house would be the mortgage.