PontifEx Libris

Shelving is often conceived as a piece of furniture, but as I learned woodworking after I learned engineering, and considering that the chief source of my shelving needs were my dad's old civil engineering books which were sitting around the basement... I decided to approach the problem of how to design bookshelves as an engineering problem. How best to design a long-span shelf that minimized materials in some way? Traditional shelving made from MDF or wood was either over-engineered (ie stronger than it needed to be), or followed the rule heavier==more expensive.

There are 3 ways to engineer a bookshelf

1. A simply supported beam. That is how 99.999% of bookshelves in existence are. A 3/4" thick piece of solid material. Simple beams have large bending forces on them when they flex so they have to be made thick and/or rigid.

2. A truss supporting a thin skin. Due to the difficulty of fabricating this, not many people have attempted this. The torsion box design is the closest analogue to a truss used in shelving today.

3. An arch. Properly designed arches only experience compressive forces and no bending forces (which are the main reason shelves have to be made so strong) so they can make very efficient use of the materials at hand. In other words, an arch can be made much thinner and lighter than a regular bookshelf. The one I built is 42" long and can be lifted with a pinky finger. The shelf load does have to be rather symmetrical, since arches like to have continuous loading.

There is no free lunch in nature though. An arch experiences less bending, but to do this it has to have a larger vertical rise and horizontal forces. The arch shelf I built was mounted in a wall recess and supported on 2 sides. You could butt arches side-to-side and span any length you wanted (the arches will support each other on the side, like 2 people standing shoulder-to-shoulder) but there will need to be vertical supports. This can be provided by a simple piece of wood standing upright on the floor, wedged between 2 arches.

Note that the shelf above does not sag, even with a full load of books. It weights only a few ounces.

I chose a parabolic arch shape, though any curved shape will do. The diagrams show the basic thought process I went through.

The shelf resembles a "deck arch bridge" with open spandrels (spandrels are the spaces between the vertical columns coming up from the arch). This space will not go to waste since every shelf usually collects knick-knacks and small objects in front of the books anyways. These things can be stuffed inside the arch spandrels now. The central hollow is large enough to fit a 8.5x11" sheet of paper sideways.

In the end, I chose the following dimensions

1. span = 42"
2. rise = 5"
3. depth=7.5"
4. number of vertical columns (aka ribs) = 6
5. column locations = 2 x 6" from center, 2x 12" from center, 2x 21" from center (ie at the ends). This is done so that the gap in the middle is 12" and you can put a sheet of paper inside.
6. The gap between the arch at the shelf will be 1" at the middle.

If you use less columns, make sure the top deck will not buckle in between the columns. Substituting 1/2" plywood might help.

The columns will be made out of ordinary 1x4 SPF lumber trimmed on the table saw. You could use plywood.

Mathematical model of the arch

I modelled the arch numerically so I could cut the columns to size beforehand, and cut the end at the tangent angle of the parabola. This will make it easier to glue up. You could also do it by direct measurement of a curved piece of wood.

The equation of a parabola is

y(x) = a* x*x + b*x + c

Here we chose the x-axis to be along the span of the shelf, y being the vertical axis, and the origin being in the center of the shelf. Our design challenge is to find the equation of a parabola with the following boundary conditions

1. It has to go through 2 points (0,1") , (21",5")

2. It has to have continuity in the middle i.e. slope or y'(0) = 0

differentiating once, we get y'(x) = 2 * a * x + b

1. tangent y'(0) = 0 so b = 0
2. y(0) = 1 so that means c = 1
3. y(21") = 5" so a = 0.00907029478458

So for any point x, the height of the arch above the top shelf is

y = 0.00907*x*x + 1 (inches)

Here is a table of data

• x, y, tangent angle

2x Piece 1, 6" away from center on either side, height h=1.33", angle=6.2 degrees

2x Piece 2, 12" away from center on either side, height h=2.31", angle=12.3 degrees

2x Piece 3, 21" away from center on either side, height h=5", angle=20.9 degrees

Python code to generate this automatically is here

import math

c=1.0
span=42 # inches
height=5
x,y=(span/2,height)
a=(height-c)/(span/2)**2

X=[0,6,12,21]

tangents=[math.degrees(math.atan(2*a*x)) for x in X]

Y=[a*x**2+c for x in X]

print X
print Y
print tangents

distance from center where to place columns (in) [0, 6, 12, 21]
height of column (in) [1.0, 1.32, 2.30, 5.0]

angle to rip one face at (deg) [0.0, 6.2, 12.2, 20.8]

On the 42" plywood, mark

1. 1. The center line
2. 2. 6" on either side
3. 3. 12" on either side

Mark the line across both sides, and the edges.

On the solid wood pieces, mark the width needed. Also mark the ends of the boards along their centers. Since they are 3/4" thich, mark 3/8".

Materials required

I used a width of 7.5" throughout.

1. 1/4" luaun plywood, 42" long and ripped to width.
2. 1/4" luaun plywood, longer than 42" by about 10". It will be trimmed to length. This is because calculating the length of a curved piece of plywood used in the arch can be tricky, so it's safer to trim it flush after gluing.
3. 1x4 SPF lumber, cut to width
4. a 5" piece of lumber, cut to width

Trim each piece of solid wood to the exact width specified in the previous step and at the exact angle. One face should be flat 90degrees, the opposite face should be ripped on the table saw at an angle below

Innermost piece == 1.33" high, angle = 6.2 degrees

Middle piece = 2.31", angle = 12.3 degrees

Outer piece = 5.0", angle = 20.9 degrees

You can pick the angles by measuring them directly. I calculated them so that they would fit nicely. Set the angle on the table saw to the angle above, and run the piece through it as shown.

Glue each piece, positioning the center of the solid boards along the lines marked on the flat-portion plywood. This requires clamping pieces at 90 degree angles, so it should be easy using a corner jig, or clamping a speed-spare on the plywood and butting the solid wood ribs up alongside it. glue and nail with a brad nailer, or clamp. If desired screw it in using #6 screws.

When all of the pieces have been glued, spread glue along the angled faces and drop another piece of 1/4" plywood which has been cut to the same width as before but not cut to length. Glue and nail it to the solid wood ribs.

Note that the shelf will have a tendency to bow inwards giving it two curved faces. You should clamp the first (flat) face to the workbench at this point.

Use a flush-cutting saw, or a Japanese pull-saw, or a router with a flush cutting bit, or block plane. Note you cannot use a table for this, since the face of the plywood you want to cut is curved.

The shelf must be installed between two abutments, supports which can give horizontal force as well as vertical. As the arch is loaded, it tends to bend outwards. It requires the supports to "push back", keeping it straight.

The easiest way to do it is to anchor it in a bookshelf carcase strong in tension. In this case I designed it to fit a wall nook with framing lumber behind the drywall. The thinner the shelf (ie the smaller the vertical columns at the ends are), the more horizontal pushing force is required. A taller shelf requires less side force.

Place the shelf, shim the sides, level it and screw it into the studs using 2.5" screws.

I left it unfinished since winter is already upon us...and it's too cold to finish anything in the garage. However it can be painted, stained, and/or trim added to the front face.

When loading books, try to load them symmetrically, from the ends inwards. This shelf has no visible buckling when when loaded from one side and even though the glue had not dried yet (i.e. it was only held up with the brads.) The strength of the shelf comes from the design of the arch itself, and the horizontal forces applied by the wall itself. Yet the entire shelf is light enough to be lifted with one finger.