## Stock Purchase

For me, the scariest part of building a chair, or stool in this case, was choosing the leg stock. Watching you tube, TV, or the internet told me that I was going to have to rive some wood to make sure that I didn’t have weak legs. That’s a challenging prospect for an urban woodworker.  Logs are hard to come by and sourcing appropriate wood to rive is difficult, to say the least.  Where I live there are a couple of hardwood stores in town, but they tend not to stock a lot of material, preferring to order as the customer needs it. Which leaves me combing the small lumber racks at woodcraft most of the time.  Kinda sad.  Even though, I’ve had pretty good success with this approach and was able to score enough stock for a couple of 3-legged stools.

I thought I’d just show a few pictures of what I looked for in the leg stock.   I was following some suggestions made by Brendan Gaffney of burnHeart Made about how he approached choosing stock for his tools.  After I started this post, he posted a fantastic informational video on what he looks for.

I pretty much followed his advice, looking for nearly perfectly flatsawn growth rings on the end as shown below.

Then I went to the side of the board and tried to make sure that the grain didn’t run out. This is about as good as I could get in the 8/4 stock that was available. There is a little run out at the end, but if you put that toward the floor then you should be safe (I know that might seem counter intuitive, but the highest stresses are at the top of the leg).

So while I couldn’t get riven stock for this build, I was able to get some leg stock that I feel comfortable sitting on. The hardest part was getting a good look at how the grain ran along the length of the board. The rough-sawn texture makes the grain hard to see and so you have to spend a bit of time angling it around in the light before it reveals itself.  The other option I had thought of involved finding 12/4 stock and just trying to rive the oak dry.  I am pretty certain that would be at least moderately successful.

## Laying out the seat

Apparently the stool is on everyone’s list at the moment.  I like it because it is a sorta halfway point between a chair and, well, nothing.  Sitting on the ground?  Maybe.  Plus there are only three legs, so I only have to worry about one drilling angle once I laid out the mortise locations. To locate the leg positions, I played with the wireform model I put up a couple of posts earlier and decided on a 4 inch diameter leg circle on a 12 inch seat.   The next thing to do was to lay out an equilateral triangle on the seat bottom so I could get the legs equidistant from one another.  Just a little bit of simple geometry  construction was all it took.

I have some pictures of me constructing that triangle that I will post a little later.  Good old high-school geometry reminders from the teacher in me. After laying this out, I drilled the 5/8 in holes with a wood owl bit and an 8 inch brace with a holdall chuck.   On instagram I joked that the leg layout lines looked like the cover of a metal album, it definitely has that archaic magic symbol look to it. For this stool seat, I beveled the underside with a jack plane and then cleaned up the bevel with a new, to me, stanley M151.  I fell in love with the spokeshave almost immediately.  Get the thing sharp and it is a blast to use.  It’s definitely an ongoing lesson in reading grain though.  The spokeshave is like the chisel in that regard.

After I beveled the underside, I reamed the 5/8 holes with the large veritas reamer, checking my leg angle as I went.  You don’t have to have a fancy sliding bevel like the Blue Spruce Toolworks one below, but man, it makes it nicer.  Everything I have ever picked up from David’s company has been ultra nice and super functional.  You can see my 16 oz Blue Spruce mallet in the background of this photo, which is acrylic infused and indestructible.  After this step, I started in on shaping the legs.

## 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.