Common examples in wooden construction and carpentry that require calculation of difficult angles are diagonal wall-bracing, hip rafters, and common rafters. There are others, but these are some of the more common ones. Today, we’ll cover how to cut out diagonal wall-bracing. Generally speaking, there are 2 ways to brace a wooden house in shear: diagonal wall-bracing or shear walls.
A shear wall or shear panel is a wall that is constructed with a sheet of structural plywood nailed to its outer face, with a designated pitch — often as small as 75 millimeters — to resist lateral forces. As an example, the 2-by-4 housebuilding method uses this type of structural wall extensively. Structures with diagonal wall-bracing on the other hand, may use plywood on its outer face, but fastener-pitch is much less stringent as it is not what provides the bulk of the shear resistance.
Here in Japan, diagonal wall-bracing is shaped and fit snug between 2 structural columns (柱、はしら), each typically measuring 105 millimeters square (3 sun, 5 bu), and fastened to the adjacent column with special steel plates. The distance between 2 structural wall columns that would have diagonal bracing between them is typically either 910 or 1820 millimeters, center to center (3 or 6 shaku, respectively).
Now we measure or calculate the inside distance between the 2 columns, as well as from the foundation beam (土台、どだい) to the second floor beam (梁、はり) stretching above. If the boneframe is good, a calculation should be more or less accurate. Simply subtract 105 millimeters from 910, and you get 805, which is the inside measure between the columns. The height is roughly 2730 millimeters (9 shaku).
We now have the lengths of the rise and run of an imaginary right triangle between the 2 columns. Calculating the hypotenuse of this right triangle can be done via the Pythagorean Theorem (a2+b2=c2, where abc are side lengths of a right triangle, c being the hypotenuse), which amounts to roughly 2846 millimeters. Let’s assume that the center of the wall-brace will meet the internal corners on both ends.
By scaling the length of the 2 legs down by a factor of 10 each, we get 80.5 and 273. We can also do the same thing to the hypotenuse and get 284.6 millimeters to serve as a confirming third measurement. Now mark the center of the wall-brace which will make the “point” that will meet the internal corner where the column meets the horizontal beam, and snap an ink line down the length of the board.
Take your sashigane and line both the 80.5 millimeter mark on the short leg and the 273 millimeter mark on the long leg with the center ink line — as shown in the image above — which would be A and C respectively. As a confirmation, the distance from A to C should be 284.6 millimeters, as per the calculation above. The distance D to E equals the non-scaled distance of A to C, which is 2846 millimeters.
The next and final step is cutting the remaining short leg of the point at C, completing the brace. Going purely by calculation, cutting a perpendicular line from point C at both ends should result in the correct length. However, because real-life situations are rarely that accurate, I would suggest cutting one end but leaving the other end until you can match it up to the actual spot it will be used in.
As long as one end is cut, you can simply place the finished end pointing down, and butt the unfinished end against the beam above and make a pencil line where it needs to be cut. Using the image as reference, the finished end would be C pointing down, and the unfinished end A would protrude above past the beam where you would then make your pencil line.
This method results in a snug fit that can compensate for small variations in measure. Note also that a calculation based on real measurements will result in a brace that can sometimes “straighten” a leaning structure. In other words, if the columns are leaning slightly, and the brace is cut out so as to only fit between 2 plumb columns, it can be forced or hammered in, consequently straightening out the columns.
Once the wall-brace is cut out and in position between the wall columns, it needs to be bracketed to the wall columns on both the top and the bottom ends with the certified brace plates and fasteners. There are several types, but the image shows one example.
Wall-brace plates (sujikai plates) come with a specific number of certified fasteners, all of which must be used for proper adherence to code. Note also that there will at times be longer fasteners and shorter fasteners with the same brace plates, depending on type. The longer ones always go into the column, and the shorter ones into the brace.
Additional reading (Japanese): How to choose your column cross-section – 3.5 or 4 sun columns?