# How is shelf deflection determined?

Question: How is shelf deflection determined? More specifically, how is the deflection determined when one or more edges of the plate have reinforcing edging applied to them?

I have been able to find plate deflection equations in a number of places, but of the resources I have found, they are limited to plates without any edging at all.

I have found a shelf sag calculator which happens to be well known and loved by woodworkers. But their calculator only handles a single side of edging, and they don't provide any explanation of the formulas that are used. And depending upon what deflections I find, I may need to be able to evaluate the deflection in a lattice structure with a supporting skin.

I would assume that the edging can be treated as simple beams supporting the plate, but I don't know how to apply that support to the plate.

Is there a standard set of equations and / or plate geometries that can be applied to estimate what the shelf deflection may end up being? Or is this getting into the non-trivial end of analysis and design?

Background: For a hobby project of mine, we need to build shelving to support a variety of items which cover a range of shapes and sizes. Some items are large and comparatively lightweight (e.g. a student's desk). Others are small, dense, and kind of heavy (e.g. concrete block). And of course, there are some in the middle for both size and either side of the weight spectrum.

We're planning on building several sets of shelving to accommodate the various sizes of things that need to be stored. And while we do have to design for the unknowns of the future, we're still trying to optimize costs through the amount of material used.

• Since you mention woodworkers I'm assuming you're asking specifically about wooden shelves? – Wasabi Sep 4 '15 at 18:24
• @Wasabi - "Maybe, but not necessarily." We might go with metal edging and a wooden skin, or we might build wooden lattices and then skin it with wood. I'm more interested in the design technique involved as I run into this problem fairly often. But general construction technique would be 1/4" plywood for the skin and some measure of supporting edging / lattice work. Support material to be determined based upon cost and amount of support provided. – user16 Sep 4 '15 at 18:28
• I want to give an answer, but I just need to confirm that I understood what you said. Is this what you mean by metal edging and a wooden skin and by wooden lattice (and then skinning with wood)? Ignore the support conditions, I just want to know if I understood the shelf itself. Also, are the wooden lattices usually made perpendicular and parallel to the wall (such as in the given image) or at an angle? – Wasabi Sep 5 '15 at 11:57
• @Wasabi - Yes to both examples. The lattices would be made perpendicular and parallel as that greatly simplifies construction. You could make an angled structure but I don't have the tooling to make that process very easy. – user16 Sep 5 '15 at 12:50

The exact deflection result is a function of the shelving's structural system. In the OP comments you mention that the two options would be either a wooden skin with metal edging or a wooden lattice skinned with wood.

Let's start with the wooden skin with metal edging. Structurally, this is equivalent to a slab with multiple supports. The sort of support depends on... the sort of support. If the shelf is fixed to a wall, then that shelf-face will have a fixed support (restricts vertical displacements and rotations). If the shelf is not fixed to a wall, then it must either rest on metal edging (which restricts the vertical displacements of the shelf but would allow for small rotations) which transmits the load to a column or there is no metal edging at all and the shelf itself needs to transmit the entirety of the load to the column.

For a rectangular shelf, you will have up to four supports, one on each face. Considering only more plausible configurations, you have the following possibilities:

• One face fixed on the wall*
• One face fixed on the wall, lateral faces with metal edging
• One face fixed on the wall, other faces with metal edging
• Two faces fixed on the wall (corner-shelf), other lateral face with metal edging
• Two faces fixed on the wall, other faces with metal edging
• All faces with metal edging
• All but one face with metal edging
• Both long faces with metal edging*
• Both short faces with metal edging*
• No edging, supported directly on columns.

As you can see, this is a long list and I've removed a few which I found to be implausible. For this reason, there is no simple equation that can be adopted. This is made worse by the fact that the behavior of slabs is defined by the following multivariate differential equation:

$$\dfrac{\partial^4w}{\partial x^4} + 2\dfrac{\partial^4w}{\partial x^2\partial y^2} + \dfrac{\partial^4w}{\partial y^4} = \dfrac{p}{D}$$

where $D = \dfrac{Eh^3}{12\left(1-\nu^2\right)}$. This is a pain to solve, so no one does.

There are two ways of getting around these equations. I found this article which shows a lot of the necessary math in each of the support configurations, but in practice one doesn't go to such lengths. I can't find any resources for the case of wooden slabs, but in reinforced concrete slabs there are different approaches obtained by smarter men that I (Marcus, Czerny, Nadai, etc) which result in dozens of tables of coefficients. These coefficients can then be used to figure out how much load is transmitted in the x-axis and how much in the y-axis. Knowing that, you can calculate the slab as two perpendicular beams, which should result in similar deflections. The biggest problem translating this to a wooden slab is wood's anisotropy.

The other way of getting around that ugly differential and the one I recommend is a finite element model. This is also probably the only one that can handle the "no edging, supported directly on columns" case and it can take into consideration the flexibility of the supports (in the case of the metal edging). It may seem like overkill but it's the best way to get a good estimate of what your actual deflections will be.

You'll notice that some of the options on the list are marked with an asterisk*. These cases are exceptions and can be quite trivially calculated, since they behave in a manner identical to that of beams. Also, long slabs (where one side is more than twice the other), can usually be assumed to be "infinite", meaning that they too can be considered equal to beams in the perpendicular direction (if $x \geq 2y$, then the slab works like beams with span $L=y$). For a beam or slab fixed on a wall with all other faces free, the maximum deflection is

\begin{align} \delta = \dfrac{PL^3}{3EI} && \delta = \dfrac{qL^4}{8EI} \end{align} for a concentrated load $P$ and a distributed load $q$, respectively, where $L$ is the slab's span from fixed to free face.

A slab supported at two opposing faces is equivalent to a simply supported beam of equal span.

\begin{align} \delta = \dfrac{PL^3}{48EI} && \delta = \dfrac{5qL^4}{584EI} \end{align}

for a concentrated load $P$ at midspan and a distributed load $q$, respectively.

Now onto the lattice method. If the lattice is very dense, it will behave very similarly to the slab above. If it is less dense, then the behavior may differ significantly. What the threshold is, I can't say.

Assuming the lattice doesn't act like a slab, it will behave like a grid. Once again, this can't be trivially solved and the best solution is an analysis tool, though in this case the tool can be much simpler (grids can be solved analytically without the need of FEM).

To add to Wasabi's answer - especially dealing with wood's anisotropy, here are a few things to consider:

1.) IF wood is end grain, plywood, oriented strand board, etc then for all design purposes, it is isotropic - as the grain is going in the "up" direction - where no real bending occurs. As a result, you can use the simpler equations.

2.) Under most conditions, as shown by Dr. Jack Vinson, where your longest length is in the primary grain direction (that are difficult to go into without a lot of partial differential equations), and a non-linear term that for the most part can be ignored, a shelf can be "squished" in the grain direction.

For example, let's say that your shelf is "A" wide and "B" long. And the wood has "a" modulus in the "A" direction, and "b" modulus in the "B" direction. If "B" is the grain direction, simply change the dimensions in the deflection calculator to:

"A" wide by (a/b)*"B", with modulus "a".

The calculated deflection, for woodworking purposes, will be within an order of magnitude of the actual deflection.

The first place I'd look, certainly for your flat plate design, is Roark's Formulas for Stress and Strain. This is a compendium of formulas build up over many years covering almost every structural element you'd ever want to design. I've used it many times for commercial structural projects. It includes formulas for the deflection of rectangular plates with various types of support, on any number of edges, and with various types of loading. Making 'stiffest case' assumptions about the dimensions and the supports will let you calculate the 'stiffest case deflection'. If that's too big, you'll have to rethink the design.

Paper copies of Roark aren't cheap so I'd try the library or a friendly structural engineer, who'll almost certainly know where to find one. You might even find it on-line. For a hobby project, Roark will be more than adequate. When it runs out of steam, it's usually time to consider computational analysis which, unless you've the time and inclination to learn or money to pay for, is a substantial overhead.

• While I don't have a copy of Roark's Formulas, I'm not certain it's as straightforward as you suggest. The edging will change the dimensions of the shelf and will apply a localized stiffness. I don't believe that it is as easy as declaring some edges as supported and some free. Please consider expanding upon your answer in order to show how it addresses the question. – user16 Sep 5 '15 at 21:19