Monday, March 23, 2009

An Irregular Situation

I've made a variety of different sawhorses over the years, in fact, before I had a shop, I did almost all my work outdoors on a pair of sawhorses. The first ones I built were the result of following a plan in an old Fine Homebuilding article, which detailed a splay-legged horse using 2x6 top and legs, and 1x3 stretchers. This horse can be seen in the series of posts done recently on Germaine's deck and fence. They were fine, and sturdy, but held together with screws, so inevitably their lifespan is less than I would like. I gave this set of sawhorses to a friend a few years back, and he still uses them.

The next set I built were based on drawings in one of the Japanese carpentry layout books I have, and the first set I made didn't in fact go together so well. I was puzzled why the joinery didn't work out as I thought it should. Months later, I was re-reading the layout book and it suddenly dawned upon me what they were talking about with those funny drawings I hadn't paid enough attention to the first time around.

The key point to any structure with splayed legs/posts, like a sawhorse, is that when the leg splays out along a 45˚ line in plan (that is, slopes an equivalent amount in two directions at once, these directions being 90˚ in relation to one another), the leg takes a 'diamonded' aspect where it meets the floor. This is more easily explained in pictures.

Here is the leg, which is square in section - you can see that I have marked out a center-line on each face, and connected them across the bottom of the foot:

The grain pattern seen in the above picture is ideal for this application - indeed, for any post section.

Now, if I take that leg, and make a cut across the end that is 90˚ to the leg, and place it upon the floor, obviously it will produce a square outline:

Now I take the end and cut it at slope, both ways, which is effectively a cut along the 45˚ axis of the cross-section, arris to arris. Then I place that upon the floor and trace it's outline and mark the locations of the center-lines as well:

Note carefully in the photo above how the center-lines on the leg which has had the angular cut do not meet the floor at the same place as the leg with the 90˚ cut.

As you can see, the leg, square in section, produces a diamond-shaped profile where it meets the floor. The greater the slope is, the more pronounced the effect. That means, therefore, that any horizontal plane, such as is occupied by a top board, or a stretcher between legs, will also cut through a diamond-shaped section of leg.

Since the center-lines have also moved with the leg faces into a diamond shape, then it becomes more problematic when trying to connect other pieces to these legs. For one thing, these legs, if a tenon is cut into one end to fit into a horizontal top, must connect to a mortise in the top that is parallelogram-shaped (not the most convenient thing to cut). A stretcher which passes through such a leg as this cannot be centered on both entering and exiting faces. Either the tenon is off-centered in relation to the line of the stretcher center-line, or the stretcher and the mortise is off-centered at the entrance and centered at the exit, or some other combination. Leaving the legs square leads to all sorts of complications, and this is why you will rarely come across constructions of this form. Some people will make splay-legged bar stools with stretchers, but almost invariably with floating tenons. Though-tenons combined with splayed posts is a rare sight in western carpentry and woodwork - in fact, I've never seen it. It's a common form in Japan though - here's an example of a structure employing it:

And another:

The Japanese solution to the problem of the diamonding legs (this sounds a bit like a good title for a Hardy Boys novel!) is to re-shape the leg so that where a horizontal plane cuts through it, the profile is square in configuration. This is the most elegant solution in my view. I've written a detailed article on how to do this through a drawing technique (one of several methods actually), which was published in the 'Timber Framing: Journal of the Timber Framer's Guild' (#79) a few years back. That article was preceded by one on the hopper (TFJ #78), which is, in fact the logical starting place for an exploration of compound joinery. I make these articles available to interested persons, however I ask that people make the hopper before they tackle the sawhorse - I provide the articles one at a time upon the basis of completing one before the other and proving it with pictures).

So far, for every 10 people I send the hopper article to, about 6~7 complete it. However, for every 10 people to whom I send the sawhorse article, about 1~2 people (and maybe I'm being optimistic!) complete it - it's much more challenging to make than the hopper. And this is still considered 'basic' compound joinery - it's not an easy endeavor, not only in terms of coming to understand the layout techniques involved, but in the tighter tolerances for fit that arise out of non-orthogonal intersections between pieces.

Here's what the hopper looks like, in this case made in the form of a waterstone pond; the wood is Honduran Mahogany one of the most rot-resistant species on the planet:

The short sides fix to the long sides by means of wedged sliding dovetails, and the hopper proved to be completely watertight:

The hopper, though a simple form, can have various elaborations at the joints - here's the through-tenoned version with mitered return:

After making several sets of the regular sawhorse, I began to consider the whole form a little more deeply. First of all, some may wonder, why splay the legs at all? Many people construct sawhorses with either plumb legs, or with legs splayed only in one direction. The answer is that splaying greatly increases the stability of the sawhorse. A splayed form transmits imposed lateral loads efficiently into the legs directly and then down to the floor, whereas a non-splayed form loads the joints, and ultimately those joints begin to get sloppy. The bell tower in one of the above photos shows clearly the wisdom of a splayed structure in resisting loads that are top heavy - and is a sawhorse much different in respect to resisting top-heavy loads? No.

Thinking about it further, one can see that a sawhorse, by it's very configuration, tends to be most stable to loads that are applied parallel to the orientation of the top beam. Loads crosswise to the top beam, however, tend to have a greater propensity to tip the horse. One can of course increase the splay to make the horse more stable, but then you run into the problem that the increased splay means the feet are further apart and this can get in the way of working in and around the horse.

After some more thought about sawhorse design, I concluded that what would be ideal was a sawhorse that had more splay only in the direction in which it is least stable. It would be ideal if the splay was less in the direction that the sawhorse resists loads easily (parallel to the top beam), and since the pieces loaded atop the horse are in the way of moving around the horse in that direction, legs that splayed less out to the sides would be advantageous.

In short, the splay of the legs, when viewing the horse at one end, would be greater than the splay of the horse's legs when seen in the long-side view. When you have adjacent slopes that are different, you have what is called, 'regular plan, irregular slope'. By the way, the 5-legged stool I showed in a previous post on phi was 'regular polygonal plan, regular slope'.

So, that's that I wanted to do - an irregular sawhorse - however there are no plans or directions, even previous examples to stare at, to be found anywhere for something like this - I would have to figure out the geometry of the leg shape and other considerations, by myself.

I started working on the drawings and quickly got stuck. Getting the leg shape figured is the first crucial move. I decided to write to one of the pre-eminent Japanese professors of carpentry layout (it's called 'kiku-jutsu'), named Shinzo Togashi, and see if he could help me. And he did - in the most wonderful way, by not answering any of my questions.

Huh - how could not answering my questions be of any possible help? Well, instead of telling me the answer to any question, he'd simply tell me what aspect I needed to study in more detail. I'd do just what he said, hopefully gain a new insight, send him another drawing to see if I was on the right track, and he'd again tell me what I needed to consider, or look at again. Sometimes he told me to look at the same thing three times over - clearly I wasn't always getting something on the drawing truly in depth. But, stare at something long enough, and the truth tends to be revealed eventually.

It's great when someone doesn't give you the answer sometimes, providing it is something you really want to know, because it forces you to use yer noggin'. The knowledge gained this way, though hard-won, is the best kind to acquire I do believe.

After half a dozen exchanges like that over the space of a few months, I finally got the thumbs up to my drawing from Togashi-sensei. In the next post or three in this thread, I'll detail the making of that sawhorse - I hope the reader will find this of some interest.

--> Go to part II

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