What is the ideal distance between supports of a shelf?

Question

I've got 122x25x1cm and 61x25x1cm plywood boards that I want to use to make shelves for books and tools.

I'll use 2 supports for each size of board and I'd like to know how far apart they should be. Perhaps someone has developed a program that tells the answer.

Picture¹ of the other parts (not sure how you call them):

_{¹ Image from: http://patentimages.storage.googleapis.com/EP0404699B1/imgf0001.png}

Answer 1

Enter The Sagulator - it's a free online calculator for sag of shelves which is a wonderful tool exactly for these questions.

Running your 2 shelf sizes, the larger shelf span (122 CM) won't hold more than about 2 KG overall without noticeably sagging. The shorter shelf (61 CM) can hold about 10 KG overall.

As you can see, 10 MM plywood isn't that stiff for shelf use. You can add a wood support under the shelf. Even a 1X2 (20 MM by 40 MM) attached vertically will increase the load capacity of the longer shelf to about 18 KG, and the short shelf to well over 100 KG.

You might consider using three support brackets for the longer shelf so that each unsupported span is 61 CM. This, together with the wood bracing under the plywood should provide decent support.

Alternatively, if you don't want to use wood bracing under the plywood, you might increase the number of supports so that the unsupported span is shorter. With a span of 30 CM the plywood can support about 35 KG. This would mean 3 supports for the 61 CM shelf, and 5 supports for the 122 CM shelf.

Edit: as the original poster and Henry Jackson suggested, the Sagulator cannot help directly with optimizing the position of the supports for the shelf - it only calculates the sag of a given length of shelf, and cannot provide the sag for a shelf that is only supported at one end. In the following diagram, the Sagulator can help with determining B, but not with determining A:

This is due to the mechanical formula used by the Sagulator. Searching a little through the reference provided by the Sagulator, we can see that the actual formula used (for uniform load with the shelf fixed to the supports) is this: Structural Beam Bending Stress & Deflection Equations / Calculation - Fixed at Both Ends with Uniform Loading. Indeed, punching the numbers gives the same result, if the Sagulator "Apply WoodBin lab correction?" is not checked - i.e. only the mechanical formula is used (based only on the dimensions and the properties of the wood).

This is all well and good, but what about dimension A for the shelf? Here comes the following formula: Structural Beam Bending Stress & Deflection Equations / Calculation - Cantilevered Beam with Uniform Load. This is the formula for measuring the maximal deflection at A. Comparing the two formulas ("Critical Deflection" in the first vs "Deflection at the unsupported end" in the second), one notices that the calculation is the same (Wl^3 / x EI) except for the fixed denominator x - 384 in the first formula and 8 in the second. This would mean that the maximal deflection for the unsupported end would be 384 / 8 = 48 times greater than the maximal deflection for the shelf supported at both ends. So, if you have a figure of 100 KG for a supported shelf span (B) of 96 CM, the maximal length of the unsupported shelf (A) that will still be able to support 100 KG is 2 CM (96 / 48 = 2).

Naturally a 2 CM shelf won't need to support 100 KG. Here some tinkering is required to get meaningful results. Using the 122 CM shelf and ignoring the width of the supports, to support a total load of 60 KG (typical for a 122 CM bookshelf fully loaded with books), we'll get about 0.5 KG per 1 CM. A span of 16.5 CM with a load of 8.25 KG will give a sag of 0.01 MM per running foot. Converting this to a shelf supported only on one end by multiplying by 48 gives us 0.48 MM per running foot, as suggested by the Sagulator for a maximal deflection visible by the human eye (0.51 MM per running foot). This will leave us with a supported shelf span of 89 CM (122 - (16.5 * 2)). This supported span can't support the needed load of 43.5 KG (60 - (8.25 * 2)). Adding a third support at the middle of the supported shelf gives us two spans of 44.5 CM, each easily carrying the 21.75 KG load:

Two points in conclusion:

1. All of these calculations are theoretical. Your exact shelf material and load distribution will probably behave differently. It is always better to add a safety margin for the expected load (so if you think you will load the shelf with 60 KG, design it to support 120 KG (or even more)).
2. The edge of the shelf supported only on one end can support 48 times less weight than the same span supported at both ends. Again, err on the side of caution and don't expect the shelf to support exactly what the calculations showed.

Answer 2

If the shelves are going to be evenly loaded, for two supports, you would want them placed 1/4 of the way in from each end. This way, the cantilevered end(s) take the rotation that would otherwise be caused by sag in the middle.

This of course assumes that you have stiff enough material to support the span that results in, and that you can anchor to the wall in those points.

Answer 3

1/4 from the end is not the answer. Open end bends more and thus the right distance is 1/6.

Btw, is it 1/8 if there is 3 supports?