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 legleg_forces 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.

force_angle
Moment at the seat for h=19 in.

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

stress_angle
Stress at the leg mortise for h=19 in, r=1.125 in

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

seat_height
Mortise stress for a 17 degree leg that is 1.125 in in diameter.

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.

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