Sunday, March 7, 2010

Backed into a Corner

One of the many comments to my previous posting (by Mark) asked,

"One question - what does "backed" mean, as in "all four faces of the compound sloped legs will be backed"?

There I was, blithely assuming that readers know what I was talking about, and, as with many things, assumptions can be a killer. Since the vast majority who read blogs do not comment, even if they don't understand some point, or even when they disagree with it, I tend to think there must be other readers who don't know what backing means in reference to carpentry work. There's certainly no sin in not knowing about this matter - so I will do my best to explain.

I will explain it using two way, the first being an example using a roof.

Here's a view of a hipped roof, a form of prism:

The type of hip roof in the above illustration is often referred to as a pyramidal roof, for obvious reasons. It is a sub-type of hip roof, a hip roof having no ridge.

In the next picture, I have stripped the roofing off, and placed a hip rafter in the scene, on top of the triangle which produces the 'hip' of the prism:

The centerline of the upper face of the rafter coincides exactly with the hip corner of the geometrical prism, the faces of the prism having been removed in the above picture for clarity's sake. The hip rafter itself is a 3" x 6.5" rectangular section.

Next, I replace the roof planes of our prism and zoom in to look at how the hip rafter fits in this prism:

Notice how it doesn't exactly fit to the prism - the arrises on both sides of the top stick out above the plane of the roof.

Just to be clear, I will reiterate that the centerline of the top face of the rectangular rafter is coincident with the meeting of the roof planes in the prism - in the following picture, I have made the interface more obvious by making the hip rafter portions that stick up a translucent red:

Obviously, if we left the hip rafter just like that, the portions that stick out above the roof planes would cause some problems for applying roof sheathing, etc. We need to remove this excess wood, and the removing of this excess wood is termed backing the hip rafter, or taking the backing of the hip rafter:

A close up again, with the roof planes made somewhat translucent reveals that our hip rafter upper surface has been profiles so as to be coincident with the roof planes meeting upon it:

The rectilinear hip rafter we started with, by the way, would be termed an un-backed hip rafter.

One added point is that people who frame with standard dimensional 2x lumber often do not bother with a backing cut since that adds the dreaded time and money to the race, and the backed surfaces if done are rather minimal for nailing. Thus the problem is solved in that case by dropping the hip rafter down so that it's outside arrises meet the planes of the prism instead of the rafter's centerline, and there will therefore be a small triangular gap from the arris's to the intersection of the roof planes.

Example Two.

The sawhorse is a very similar form to the hip roof, with the exception that the hip rafter is now a leg or post, and is rotated so that its corner is aligned to meet the intersection of the planes of the prism. Let's look at that situation a little more to see how it compares. First, here's the leg I'm going to work with, and I have laid a square on it to confirm that it is a square section (90˚) of material:

I've also taken the liberty of marking centerlines on the faces of the leg and across the end grain - more on that in a bit.

Here's one end of the leg, which I have cut square, and a floor plan which shows the outline of that leg along with its respective centerlines:

If the leg were stood up on end, it would stand plumb and fit the outline on the floor plan exactly, and the centerlines would be perfectly coincident.

However, if I want to splay the leg, the situation changes a bit. It's going to get a little weird in fact. In the next photo, I have taken the leg, cut a slope across the foot and superimposed the leg upon the plan - notice that the sides of the leg do not meet our original outline, not even remotely:

Notice too in the above picture how the centerline on each face no longer meets the centerline on the floor.

In the next picture, I have removed that leg after having traced the outline of it on the floor plan:

One can see that the leg, once at a compound splay (that is, sloping two directions at once), produces a diamond-shaped footprint at the floor. Which means this, if you consider the prismatic form the legs are fitting within: any stick of wood that is in plane with the prism and is horizontal (i.e., parallel to the ground) will meet the leg at the corner unevenly, because it is meeting a leg in the form of a diamond. It will meet the leg and only touch at the very outside corner in fact and there will be a space left between the surface of the board and the surface of the leg. This may be counter-intuitive for some of you, after all, isn't this just a rather ordinary form? Why/how could something so simple looking be more complicated???

Take a look - here I have placed a board in plane with the prism - observe, if you will, that it meets the arris of the leg perfectly:

If we rotate the camera around though to the backside, however, we can see that the board does not in fact meet cleanly:

If the leg only sloped out a little, the gap would be small and we might be able to let this slight mis-fit go, (as I've heard some carpenters say, "why, from 10 feet away no one will notice", or, in another popular variant, "a lick of paint and that will disappear my friend"), however if you screwed the board firmly to the face of the leg, the effect would be that the board would be stressed so as to bow inwards (at the mid-point of it's run between legs), if not immediately, then over time. "Nice work - how come the boards are bowed like that?"....

There are further ramifications. Remember the matter of the centerlines not being coincident? If I connect the dots, as it were, of the intersections of the centerlines on each face of the leg where they meets the floor, and connect them, we have the following:

See how the centerline marks, when connected, are swiveled out of alignment with the floor plan of the prism? What's the deal with this, you might ask? Okay, well let's look at the case where we have a horizontal piece of wood, aligned to the plan axis, the same thickness as the leg and meeting the leg like this:

It looks okay from the above view, however if we extend that stick of wood through the leg, the following situation develops:

Notice how the horizontal stick emerges out the other side of the leg in a non-flush position. That means too that if we put a tenon on that horizontal stick, and entered the leg with the tenon centered, the point of emergence could not also be centered. Thus in order to have a centered tenon, we have some less-than-desirable options: either the tenon must be cut so as to be bent, which makes assembly impossible if trying to do that all the way around the sawhorse, or the mortise must be off center on at least one face, which also means the the mortise is not aligned with the leg and is therefore more of a PITA to cut out. And, this effect only becomes more pronounced with increased leg slope. What is hard to notice in a leg that slopes 2~3˚ becomes much more apparent with a 30˚ slope of leg.

While we cannot extend an actual piece of wood right through another one as shown in the drawing (ah, the wonders of 3D), if one made the horizontal stick a ground sill, say, and cut the leg to sit on top of it, it is readily apparent that the leg would not meet the sill cleanly. No way José. Even if you made the sill wider to accommodate the leg, it would be apparent that the lines of the feet and the lines of the sill arrises would be out of whack with one another.

The French sawhorse I completed recently demonstrates the various connections of the splayed from when the legs are left square in section - namely: the mortises become non-congruent with the axis of the leg, and some of those pieces require barbes on the outside of the connecting pieces to keep in plane with the prism. You need to keep everything in plane with the prism, particularly in roof work, when other layers will be added on top of the hip rafters (legs), like roofing boards and so on, and in order to have a clean transition between pieces.

You can see in the above example that I have kept the leg exactly aligned with the diagonal axis on the floor plan, which means that the gap we saw with a board fitting to one side of the leg, as shown above, would be also present were we to fit a board along the other axis in plan. The French deal with that by turning the leg to one side of the plan or another, thus making the barbe of a connecting piece a little more pronounced on one side, and not required on the other. It's a reasonable solution.

The Japanese approach is more subtle and refined I think and leads to a cleaner result without the hassle of weird mortises and barbes. Looking back to the first example I gave for the backing cut, with the hip rafter, with its sides plumb to the floor, we took the backing by removing the arrises which protruded above the roof plane. With a sawhorse, the situation is a little different because the leg is not in the same rotational position as is a hip rafter - looking again to the following photo, it would appear in fact that we need to add material to bring the faces of the leg out to meet the plan of the prism:

Adding material would be an option, but of course that would require glue and preparing some thin wedges of wood to fit against the leg. That is not something I would be signing up for. A better solution is to start with a leg which is somewhat oversize, and then remove material to get it into the right shape:

That removal of material just mentioned is also called backing the leg in this case. The drawing above shows all faces of the leg being backed, which is what you would need to do if having horizontal pieces meeting the leg along both x and y axes. Once the backing cuts are taken on all four faces like that, the leg will of course no longer be square in section, but more like a diamond. You make the leg into a diamond shape essentially in reverse image to the diamond shape it would form on the floor if it were un-backed. I hope that makes sense - if not, have another beer.

As for the Japanese carpentry methods for determining the backing cut angles, along with the angles the horizontal pieces need to have cut on their ends so they meet the leg cleanly on all four of their faces, I will be detailing those techniques in one of the forthcoming Carpentry essay volumes to be released later this year. One important note though- a pre-requisite for this coming material is the completion/digestion of the material already in print, namely filling in a table, building a small hopper, and the successful completion of a fiendishly complex, dare I say, nightmarish exam. A few of the other carpentry essays I intend to release will not have pre-requisites however, as they cover stuff which falls outside the logical progression of the layout study.

Thanks for dropping by the Carpentry Way today.


  1. Very detailed explanation as usual Chris, but the square-based pyramid at the beginning is not a tetrahedron, a tetrahedron is a triangular-based pyramid.

  2. Michael,

    hey, thanks for noticing that. You're quite right! The pyramidal roof could be called 'half an octohedron', though the term 'pyramid' is just fine, lol. I'll revise the text accordingly.


  3. Another beer and I do think I'll have it. Thanks.


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