Friday, July 23, 2010

Fan of the Fan (II)

In part I of this thread, I looked at one of the geometrical problems inherent with fan rafters, that is the fundamental disparity between placing the rafter in a good alignment with both the view of the underside of the eave and the edge of the eave being taken into consideration. As I mentioned, the circle and the square don't get along so well in that instance.

Today I will explore another angle to the problem. Heh! - that's a pun, and will make more sense when I'm done with this posting. Consider the following picture, which shows a simple (-looking) example of a fanned rafter - two in fact, one to either side of a common rafter:


The above model is the current Level 2 carpentry exam in Japan. I did this model about 5 years back to familiarize myself with the exam material and learn some fun new stuff, and I wrote an article about it.

Here's another view, from the perspective of someone looking more or less directly up at the eave edge:


This model served as the focus for an article, the third in a series on Japanese compound joinery, that I submitted to Timber Framing Journal. This was several years back. Interestingly, after my first two articles, it was the submission of the 3rd one which prompted the editor of that journal to write back to me expressing his belief that the problem above was, in his view, not a compound joinery problem. He thought it was just a simple case of mitering the 'fan' rafters to fit against the common. He didn't get it, in other words, and I suspect that if the editor of a technical journal on Timber Frame carpentry didn't get it, there may well be quite a few others out there who would respond similarly. So, it looks simple enough -what's the big deal with the above model? Today, I'll explain that matter more thoroughly, as here on my blog I can take as much space and time as I like for such things.

What the editor of the Journal was envisioning, I do believe, was a case in which the two rafters at the side of the common are simply mitered and attached to that common. Here's that scenario:


These stick could be 2x4's, for example, and I have placed the three on a flat surface so they are all normal to one another.

In the next picture I have put them together as a group:


I'm sure for more readers the assembled view is exactly what on would expect - no fancy shmancy compound anything going on.

Next, I'll place the assembly at a slope and put in onto a wall beam (plate):


This may not look too much different than the model I showed in the first few pictures. However, when I bring a framing square in and place it plumb to one of those fanned rafters, in orientation to the long axis of the plate, this is what you see:


The rafter is rotated over on its side, unlike the common, which has sides that are plumb.

How about a look at how this arrangement fits against the plate? If we take a glance from below, you can see that I've placed a horizontal line along the intersection of the lower surface of the rafters with the front face of the plate:


The fan rafter's lower surface is in plane with the common rafters lower surface, that should be clear. Similarly, if I place a flat surface on top of the rafters, in simulation of the roof decking, here's what you get:


Thus, the upper and lower surfaces are in plane with the common, but not the sides. Now, unlike the model which has a covering board fitted, if I want to have exposed rafter tips, I would have to bring the ends of the rafters into the same cutting plane with one another. In the next picture I have done that and added some skinny rectangles to the rafter tips so we may more easily compare their alignment:


There is some skewing due to the perspective view in the drawing, but I can assure you that the vertical alignments of those three rafter tips are in complete parallelism. That's great!

However,
if I now consider the view looking up at the rafters from below, the rotation of the side faces of the two fan rafters is very apparent:


So, a simple fanning of regular square-section timbers produces an arrangement with rafters in plane with one another in one orientation but not the other. Also, the notch on the plate to receive the rafters would have sidewalls that are not perpendicular to the long axis of the stick. Less than ideal.

What if we fanned the rafters so that the fanned rafters keep their sides plumb? The would look like this:


The result of that is that our fan rafters no longer meet the common rafter cleanly:


A close up of the interface between the lower surface of the rafter and the view of the front of the plate reveals that though the sides of the rafter are plumb, now the lower surface is rotated out of alignment with the common:


And, since the stick is square in section, what goes for the bottom surface of the stick also goes for the top:


You can see in the above picture that a portion of each fanned rafter sticks out above the plane of the roofing boards, which would obviously make for a lousy fit between those boards and the rafters. The notches on the plate to receive the rafters would have sidewalls perpendicular to the long axis of the plate, but the lower surface of the notch would be askew. Not so great - like the previous example, with the sides of the fan rafters rotated to match the common, you lose something else in the bargain.

Let's look at that Level II model again - here's the orientation of the side walls of the fan rafters to the plate and common:


And now let's look at the top surfaces and their alignment:


How is this possible? The trick my friends is that the fan rafters are not square in section, but are parallelogram-shaped in section. That's the nature of the compound joinery in this situation. Because the fan rafters flare out from the common, and the whole affair is tilted up at a slope, you end up with a compound slope. The behavior of wood in compound slopes, I have found, is counter-intuitive for many folks, if not baffling altogether.

Consider then the behavior of the cluster of rafters that make up an eave with several fanning rafters - each radiates out a different amount away from the common rafter, and thus each has a different slope than the common rafter. As each fan rafter progresses away from the common and towards the hip, the slope of each rafter becomes a bit slacker than the one before it. After all, if you brought a fan rafter all the way over to be in alignment to the hip rafter, it would have the same slope as that hip rafter. Hip rafters, of course, have a lower slope than a common rafter because their run is much longer than the common while their height is the same as the common.

And each fan rafter, having a different effective slope has a different degree of parallelogram-ism (if i might take the liberty of inventing a word) from the next. The common has no parallelogram-ism, the next adjacent fan rafter has a slight amount, the next even more, and so on. The fan rafters are differentially parallelogram. There are not something you cut in a batch.

Wait! there's more! If you also curve the hip rafter itself upwards in fan-raftered roof, the hip will also carry the roof surface upwards as it curves. The fan rafters have to follow that curve, and as they are not attached to the hip, they can't exactly go along for the ride. That means that as the curve increases as you move along the eave edge towards the hip, each fan rafter must curve a little bit more than its preceding neighbor. Therefore, not only are the fan rafters individually shaped in cross-section, but in curvature as well. AND, with all this, one has to solve the issue of how to distribute them around the fan so that have a pleasing appearance at the eave edge and from the view directly underneath. AND, on has to also take into account the view of the rafter tips and how that would change, when viewing the eave's edge from out in the yard, as they rafters fan more and more over to the hip. Each tip needs to be cut differently, in both axes, regardless of whether the tips are exposed to view of hidden behind a covering board.

Yes, there's a lot going on with fan rafters. In the next post I'll take a look at some of the methods for solving the distribution problem, and compare them. I also intend to take a look at the difference between Chinese and Japanese approaches to fan rafters. Stay tuned.

Thanks for coming by and comments always welcome. Let's go to post III

7 comments:

  1. Chris

    FANtastic! Models are so helpful in understanding roofing problems, but those fan rafters make an unequal hip look like child's play.

    Tom

    ReplyDelete
  2. Hmm..the editor or the magazine should have taking his thinking one step further. Using the extreme placements of the fan rafter in relationship to the plate would illustrate the change in angle quite easily.

    If the plane in which the rafter fan is situated is parallel to the top of the plate, the sides of the rafter will be perpendicular to the top of the plate. Easy enough, just as you say, Chris.

    However...

    ...if the plane in which the rafter fan is situated is perpendicular to the top of the plate, the sides of the fanned rafters will be at an angle to the top of the plate equal to the angle of the fanned rafters to the common.

    Any other situation (between parallel and perpendicular planes) will give an angle between top plate top and fanned rafter side somewhere between 90 degrees and the angle of the fanned rafters to the common.

    I use this type of example (extreme placement of planes) to illustrate exactly this type of situation to my students when discussing compound angles. Exaggeration really does help understanding!

    Perhaps the editor hadn't had his second cup of coffee that morning. :)

    Jerome

    ReplyDelete
  3. If it was just a matter of mitering the jacks one would expect to see more fan rafters in roof framing. I'm loving this!!!

    Ian

    ReplyDelete
  4. Thanks for the comments gentlemen - very thoughtful and supportive. Part III to come...

    ~Chris

    ReplyDelete
  5. I may well have a go at recreating this in sketchup and try it out with Sensei. I of course need to make a sashigane in SU first ;P

    ReplyDelete
  6. I have just read this article on fan rafters and I want to congratulate you. That's the best ever explanation on fan rafters anywhere. I too love geometry, and it's only when we apply these theory to the real world that we truly begin to understand and hopefully master. Brilliant.

    Simo

    ReplyDelete
    Replies
    1. Simo,

      glad you liked it. A reread also helped me spot a spelling error.

      ~C

      Delete

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