## Pye and The Internet

I recently read David Pye’s book The Nature and Art of Workmanship.  I was introduced to this book through the frequent quoting of his catchphrase “the craftsmanship of risk,” which he uses to differentiate craftsman-made work from mass production. It’s an interesting read and is still relevant almost 50 years later. I found Pye’s book focused on developing a working definition of craft and individual workmanship that is focused on the refutation of definitions of craft that Morris and Ruskin  (particularly Ruskin), of the English Arts and Crafts movement, put forth. Academic invective is probably an appropriate description of his tone.

However, his final chapter is interesting in that it makes suggestions and predictions about individual craft in the future. The  four points in the final chapter are interesting and are still relevant today. Particularly point 1)

The workmanship of certainty has not yet found out, except in certain restricted fields, how to produce diversity and exploit it

I had never seen these before and these last 4 points he makes are far more important, in my opinion, than the notion of the “workmanship of risk”.  Some 50 years ago, Pye had already identified the problem that rapid  protyping and 3D printing is trying to solve. By designing in the virtual space (computers or CAD), you can marry the workmanship of certainty  to the diversity associted with the workmanship of risk. Now, this is far from perfect, anyone who has used a makerbot or similar home system knows that any given print is far from certain, but if you’ve had access to expensive production machines the world is different. Those machines just work. Almost without fail. Assuming you have some skill in designing items that work for printing.

This is all rosy and optimistic, especially since 3D printing is pretty limited in size and scope at this point. But the basic idea is there, great diversity is achievable within the framework of certainty.  Even if it is only doll furniture.

## Engineering Staked Furniture

After reading the Anarchist’s Design Book, I decided to make a three-legged stool as my first piece of staked furniture.  I made a quick sketch of the stool and a model to help guide my leg angles and placements.  Based on the model, I found a leg angle of 21 degrees to look nice, to me at least.

I’m an engineer and the things that I wondered was how much does the angle change the loads on the legs and how close was I to the failure point of the leg.  I googled around a little and didn’t find any clear guidance on the analysis of the legs of a chair, so I decided to go ahead and do it myself.

Like all good engineering problems, you start with a free-body-diagram (FBD).  This drawing catalogs all the forces acting on the leg and serves as the roadmap for an analysis. The thing we are most interested in is the moment at the point the leg enters into the seat $M_s$.  We need to make some assumptions to figure this out.  The first assumption is that the force on the leg due to the person sitting on it is distributed evenly.  That means

$F_s = \frac{W}{N}$

or the weight of the person $W$ is divided by the number of legs $N$. We also assume that the friction between the leg and the floor is zero, so $f_f=0$ in the FBD to the left.  Now we can sum up the forces and solve for the moments which gives us

$M_s = \frac{W}{N}\left(\frac{h}{\cos\theta}\right)\sin\theta$

$F_f = F_s = \frac{W}{N}$

This means that the moment at the leg mortise varies in a mostly linear way as the angle of the leg changes, assuming that the seat height remains fixed.  The graph below demonstrates that relationship.

While this is important, what we are really after is the stress in the leg.  Stress is simply the force in the leg divided by the cross-sectional area of the leg.  It is important because we can compare the calculated stress level to measurements made on representative wood specimens.  If our stresses in the leg are below the measured maximum stresses in the samples, then we can be sure that the leg won’t fail. Assuming our force analysis is correct.  When we do the stress analysis we get the following relationship, where $\sigma$ is the stress in the leg

$\sigma_{tensile} = \frac{W}{N}\left(\frac{2h}{\pi r^3}\tan\theta-\frac{1}{\pi r^2}\cos\theta\right)$

For the equation above, I’ve made the assumption that the leg is circular at the mortise, which is a pretty good assumption. When we do the same study as in the plot above and ask what the stress in the leg is for a given angle and a fixed seat height we get

From this plot I can see that my leg stress is approximately 1700 psi.  That’s actually not much and when you compare that stress level to measured failure stresses in typical leg woods we see that we have a lot of leeway

Species Modulus of Rupture (psi)
White Oak 18,400
Red Oak 18,100
Ash 15,000
Hard Maple 15,800
Hickory 20,200

Most of these woods have failure stresses 10 times higher than the stresses we see in the legs.  This is great and likely contributes to the longevity of the samples that Chris talks about in the book.

The next question is leg length.  I am currently making a stool that is about normal seat height, but will be making a counter height stool (23/24 inches) as the next one.  Modeling that seat gave me a leg angle of 17 degrees, which also happens to be the angle Mike Dunbar suggests in his stool article here.  The plot below shows the leg stresses for a range of seat heights from 17 to 27 inches for that 17 degree leg angle

Again, the results are good. For a standard 1.125 inch diameter mortise, the stresses are about 2000 psi. That gives me lots of head room without changing any of the mortise dimensions. If you want some more details about how I got these equations, check out this little document I put together chair_forces.

DISCLAIMER – this is a pretty “back of the envelope” calc and may have some errors. Use at you own risk.

## On the equivalency of dowels and dominos

I’m trying to keep my natural tendencies to dork out to a minimum here, but no one’s perfect.  So here’s an engineering post. Complete with equations, tables, and words like “stress” and “shear strength.”

I would love to own a festool domino system. Actually, i would love to own a festool anything, but they’re a little out of the current budget. While I was (and am) lusting after this tool, I started thinking about the differences between floating tenon and dowel construction. I have been reading Krenov’s book, The Art of Cabinetmaking, and he is all in on dowels, which is a little surprising.  It’s been drilled into my head that dowels are pretty inferior when it comes to joinery. And I buy this argument, if a joint depends entirely on glue then it probably isn’t the best choice. That being said, the engineer in me experiences a massive bout of cognitive dissonance when I read articles touting floating tenons from people that wouldn’t dowel joint to save their lives. From an engineering standpoint, dowels and floating tenons are the same.

Assuming that the dowels and dominos are made of the same material, there are two factors that determine the strength of the joint. The surface area and the cross sectional area. The surface area is related to glue bond strength. And more is, in general, better. Cross sectional area of the dowel or tenon is related to tensile and shear strength of the joint. Again, more is better. Using this insight, we can create a Dowel and Domino Equivalency ChartTM. The picture below gives us the formulas to calculate the surface areas and cross-sectional areas of an idealized dowel and domino.

CA in this image is cross sectional area, SA is surface area and L represents the length of the domino and the dowel (into the screen).  You can see pretty easily that the domino is gonna give you more cross sectional area and surface area for each individual domino.  So the question is, how many dowels do you need?  If we assume that the domino and dowel have the same diameter or thickness $D_{dm} = D_d = D$ and that they have the same length, then we can write the following

Cross Sectional Equivalency
$N_{CA} = 1+\dfrac{4W}{\pi D}-\dfrac{4}{\pi}$

Surface Area Equivalency
$N_{SA} = \dfrac{\pi - 1}{\pi} + \dfrac{W}{\pi D}$

These give the following results for typical domino sizes (off the website)

D (mm) W (mm) $N_{SA}$ $N_{CA}$ $N$
4 17 2 5 5
5 19 2 5 5
6 20 2 4 4
8 22 2 3 3
10 24 1 3 3

So we can see that the cross sectional area drives the number of dowels needed. Interestingly, you never need more than two dowels to get the same gluing area as a domino, but you need as many as 5 to get the same cross-sectional area. So while the “glue joint” should be just as strong if you are talking only about the shear strength of the glue joint, the mechanical strength of the joint will not quite match up. In other words, if you replace one domino with two dowels, the dowels will mechanically fail before the domino would fail. Even though the “glue joint” should be equivalently strong.

None of this takes into account the additional time you would need to drill five holes for dowels, so the domino is probably going to be significantly more efficient than dowels. There is also an underlying assumption that the mechanical fit of the dowels and the dominos are the same. I think this is easy to achieve with good drill bits that are matched to your dowels.

Tl;DR: Believe Krenov. If everyone thinks dowels are trash and dominos are not, then they are wrong. You just need to make sure you have enough dowels.