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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):

enter image description here

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

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Depends on the loads, where your studs are, the type of shelf supports you are using, whether or not you will face the shelf with a horizontal piece, etc. –  DA01 Jul 22 '13 at 20:24
Everything is in mm? –  DMoore Jul 22 '13 at 20:33
The triangular pieces are brackets; the pieces that you attach to the wall are called "uprights" or "standards" depending on manufacturer. –  Niall C. Jul 22 '13 at 20:37
I didn't realize you had selected a particular hanging system. Apologies. In that case, contact the manufacturer of the hanging system. They will tell you exactly what their ideal requirements are. –  DA01 Jul 22 '13 at 22:45
You will need supports at every stud unless you stiffen the plywood by rabbeting it into vertical wood strips front and back. Then support every other stud should be OK. I'm thinking strips about 15mmx30mm in section, not little batten strips. –  bcworkz Jul 23 '13 at 1:16
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2 Answers

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:

Shelf diagram

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:

122 CM shelf supports

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.
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I forgot to mention that I was going to use only 2 brackets per shelf and the sagulator does not take this into account. For a uniform load of 20,75Kg it says the ideal span is 33cm which would make both ends 44,5cm wide - way too long since nothing supports the tips. –  user2534 Jul 23 '13 at 12:45
I don't quite understand what you are saying. The Sagulator works by you providing the distance between the supports - so if you have a 61 CM shelf with a bracket at each end, you'd get a span of about 59 CM. –  Eli Iser Jul 24 '13 at 12:14
Is that you would do ? cause this way the sagging would be maximum. I just noticed that I typed the wrong dimensions, it's 122cm x 25cm x 1cm and 61x25x1cm. I was commenting about the 122cm ones, if I put 2 brackets separated by 33cm as sagulator says, sagging will be too important on the left and right. –  user2534 Jul 24 '13 at 15:55
I agree with @user2534, I don't think that calculator is capable of optimizing the location of the supports. It's just calculating the deflection of a shelf supported at both ends. –  Henry Jackson Jul 24 '13 at 16:59
I see. I will update the answer to deal with this. –  Eli Iser Jul 25 '13 at 5:38
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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.

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