Can an arrangement be found which will satisfy these diverse requirements? I decided to explore different polygon shapes to see if one might offer any advantages over another. First, I started with the square plan:
Next I looked at a pentagonal plan:
Here the holy grail would appear to be realized: the reciprocal rafters cross one another under the hip rafters,just a bit inboard of the middle of the span, and the beams are able to cantilever out to the mid-span of the wall plates. The angle of the cantilever relative to the common rafter line is only modestly askew, and yet the line of the cantilever is parallel to the line of the wall plate. This was looking good.
How about a hexagonal plan?:
Here the arrangement is not too bad, however the crossing point for the reciprocal beams is now a little outboard of the mid-span zone. The reciprocal beams are radically more askew in relation to the line of the common rafters, which is less desirable. A little less ideal all around than a pentagonal plan I would say.
Next I considered a heptagonal plan, one of my favorite polygons if one can have such a thing, and the result was even less satisfactory:
Just in case things might get better, I checked out an octagonal plan:
Worse again - decidedly. I could see that the greater the number of sides, the further outboard the crossing point would get, and the more askew the reciprocal beam centerline would be from the common rafter line.
It was obvious to me that the pentagonal plan offered the best configuration for my requirements. Some may be wondering whether having so many requirements might unnecessarily constrain the design somehow - I guess I'd ask you to hold off that assessment until you've seen the full development of this plan.
Since the reciprocal beam would project beyond the wall plate, and would have its end cut to be in plane with the wall, if left as a rectilinear section the beam would appear to look like a parallelogram section. I therefore decided to make the reciprocal beams parallelogram in section in the opposite direction so that once in place, the appearance on the end of the beam would be of a rectilinear section. Also, making the beam into a parallelogram section, which is akin to taking the backing, makes for some more straightforward joinery - for example, the connection of the reciprocal beam atop the wall plate, and placement of stub posts on the reciprocal beam.
In all of my layout texts, in 4 different languages, had no resources to draw upon for determining the geometry of parallelogram-shaped reciprocal beams - I don't know whether it has ever been done before. So, I worked it out on my own. I've been studying the topic of descriptive geometry for carpentry sufficiently long now that I can usually solve most layout problems - not always with immediate elegance and ease, but I get there in the end. It is a field of ongoing, life-long study.
I'm not going to spend time here discussing in detail the geometrical solution, as that would involve several posts and veer seriously off topic. However an overview of the drawing I will happily share:
A little closer view shows the reciprocal beam, inside and outside faces:
Once the beam shape was sorted, I placed the pieces atop the wall plate ring:
Click on any of these drawings to see an enlarged view.
Next up were the lower exposed hip rafters - these have to tie into the lap-crossing point of the reciprocal beams, and be of a slope which relates to the slack pitch of the decorative eave rafters, or keshō-daruki. I tried a variety of solutions. In the end, I decided to place the upper end of the hip so that it tenoned into the neutral axis of the lower reciprocal beam. In order to meet cleanly without overhanging bits, the lower hip section was turned into an irregular hexagonal. A complex solution, perhaps, but the result is clean I think:
Here's a closer look at the tenoned connection joining the upper end of the hip to the lower reciprocal beam:
Next I placed the ring of the dashi-geta:
The dashi geta purlin ring sits a bit less than half-way out along the eave's total projection distance.
A view from underneath reveals how things are shaping up aesthetically:
Just getting started with this roof of course, and many of the joinery details are not in the drawing yet. In the next post in this series I'll continue a look at the build-up of this roof and share more of the design process.
Thanks for coming by the Carpentry Way. Comments always appreciated. On to part 7