The trouble with Splayed Legs

In last week’s post I shared a bunch of different designs for a 4-leg stool, with the plan being to design a stool that would be simple in design and fabrication to keep the overall cost down. That plan only sort of worked out. Of all the designs that I shared in last weeks post, the one that I kept coming back to again and again was a design with splayed legs. Initially I was hesistant to make a stool with splayed legs, because the process is a fair amount more involved than making a stool with straight vertical legs, and there are some subtle complexities involved that can really complicate fabrication.

The biggest issue that arises with splayed leg stools (or other structures) is what happens to the horizontal cross-section of the leg as it is tilted. To illustrate this, here’s a square leg standing perfectly straight, and 90 degrees to the plane it stands on.

Looking from top-down you can see the leg and plane are square relative to one another.

But if you take that leg and splay it in two directions, here around 45 degrees, the issue at hand starts to become clear….

Looking from top-down again, you can see how the faces of the leg are no longer parallel with the reference plane. In effect the steeper the angle the more pronounced the effect becomes. The real problem here in regards to something like a stool, is how a horizontal stretcher will intersect with the leg.

Bringing a stretcher in to meet the leg it’s clear that the wide shoulder of the stretcher will need to be cut at a angle to match the angle of the leg.

But looking from above you can see the stretcher will also need to be cut at an angle across the narrow face as well. In otherwords the stretcher requires a compound angle cut to cleanly meet the leg.

Below is the stretcher with a through tenon and shoulders cut to meet with the inner face of the leg. When you introduce joinery into the mix another issue arises which is the way a through tenon passes through the square splayed leg. You can see how even if the stretcher is centered on the back side of the leg, as the tenon passes through the leg it exits off-center…. not exactly an easy mortise to cut, let alone aesthetics and potential strength issues.

Heres a view of the leg and tenon projecting from the outside, you can see how the whole thing looks kind of messy with the tenon projecting off-center from the leg.

Here’s the same setup looking from the opposite side. Essentially you’re looking at the inside face of the leg and butt end of the stretcher, which is centered on the inner face of the leg.

That’s the basic gyst of the problem. To be fair the examples here with a leg at 45 degree splay are pretty extreme and most chairs, stools, or architecture would never be built at this kind of angle. Here in Japan this is a problem that is fairly common on certain structures like bell towers with splayed corner posts, and while a bell tower likely won’t have the most extreme splay to the posts, the size and width of the posts means that the problems illustrated above are still quite pronounced. In otherwords even with a less pronounced splay, as you scale up the size of a post, you also scale up the level of deviation from the square reference plane (horizontal cross-section).

So the solution in Japanese carpentry for dealing with splayed posts/legs, is to shape the leg into a non-square profile. In the image below I took the same leg splayed at 45 degrees, but adjusted the faces so that they are now square relative to the refernce plane. The result is that the leg becomes diamomd shaped in profile, and the tenon now projects evenly through the leg.

Looking again from the outside, you can see how the overall look of the leg and projecting tenon is much cleaner, and centered.

Taking the leg out of the picture and you can see more clearly the compound shoulder cut required on the tenon.

I’m speeding through this topic to some extent, but hopefully you can see that there are some issues you’ll run into when it comes to splayed leg stuctures. If you want to really take a deep dive into this topic I recommend taking a look at Chris Hall’s Japanese Carpentry Drawing essays. That’s where I first learned about all this, and I’m not aware of any other source that covers the topic in english.

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So where does that leave things for the 4-leg stool? It was all the above trouble that I had running in the back of my mind as I was thinking about the potential problems making a splayed leg stool. But thankfully after doing some modelling in Blender and Sketchup I realized that things weren’t going to be that bad. For one thing the legs on my stool are 30 x 30 mm square (1 3/16″ x 1 3/16″), and they only splay by 3 degrees in each direction.

Modelling a leg in Blender and looking at it from above you can see that the leg does start to deviate from the square white reference plane… but only by a really tiny amount.

For comparison sake, here’s how the same leg would look at a much steeper 45 degrees. Clearly the 3 degree splay leg only deviates from square by a tiny amount in comparison.

In my 3d model I measured the amount of deviation from square over the 30 mm width of the 3 degree splayed leg and it’s only 0.021mm (0.0008 in.). That’s a really tiny amount, and not enough to be concerned about in my opinion. So I decided that the legs of my stool could stay square in cross-section, with no need for jigs or any complicated milling to make them diamond-shaped in profile. That also meant I could avoid compound shoulder cuts on the stretchers. Instead I only needed to account for one angled shoulder cut at 3 degrees across the wide side of the stretcher; the shoulder cuts on the narrow faces could stay square.

My apologies if this sounds foggy. These are difficult issues to describe without having a model or example to look at in person. If you’re feeling adventurous trying drawing a model in Blender or Sketchup. Being able to spin around a model that illustrates these issues and seeing things from multiple vantage points gives a ton of clarity. Also please feel free to comment if you have questions.

I hope some of the complexities of the stretcher tenon and shoulder cuts will become more clear in the coming posts when I share some fabrication shots of the stool.

Thanks for reading.