In Fig. 198 the rafters C C are valley rafters and, although the bevels for these rafters are not the same as the common rafter in either roof surface, yet the bevels depend upon the relation between the common rafters and the valley rafters.

It is best to consider the common rafter as the hypotenuse of a right triangle or as the diagonal of a rectangle whose length is the run of the rafter and whose width is the rise of the rafter. In studying the valley rafter it is evident that there are three dimensions to be considered. Rafter C extends to the right to the ridge of the main roof besides rising. It may, therefore, be considered as the diagonal of a rectangular solid. For instance, if the run of the common rafter is 12 feet, the rise 10 feet, and the distance M R is 8 feet, the valley rafter will form the diagonal of a rectangular solid 12 inches X 10 inches X 8 inches, and its length and bevels can be found as shown in Figs. 199 and 200. In Fig. 198 we find the run which is the hypotenuse of the triangle C R M. That is, the run of the valley rafter is taken from the distance between the 12 and 8 on the square. It is 14 5/12 inches, showing that the run of the valley is 14 feet 5 inches. Now the rise is the same as the rise of the common rafter C R. That is, it is 10 feet and the bevel at the foot of the rafter is cut along the blade of the square when the figures read 14 5/12 inches on the blade and 10 inches on the tongue.

Fig. 199. Method of Finding Bevels for Various Runs of Rafters.

Fig. 200. Cutting Bevels on Common Rafter.

The plumb cut at the top of the rafter is made by holding the square in the same position and cutting along the tongue.

The Length of the rafter is determined either by measuring the distance from the 14 5/12 and 10 on the square or by finding the square root of the sums of the squares of the three dimensions. The latter method gives feet 6 inches (approx.).

The layout of a hip rafter is the same in principle as the layout of a valley rafter. To find the run of a hip rafter, find the diagonal of a square whose sides are equal to the run of the common rafter. That is, if the run of the common rafter is 10 feet, the run of the hip rafter is the hypotenuse of a right triangle whose sides are 10 feet and this distance is 14.14 feet or 14 feet 1 1/2 inches. The rise of the hip is the same as the rise of the common rafter. If, then, the rise is 8 feet, use 14 1/8 inches on the blade and 8 inches on the tongue to lay off the horizontal and the plumb cuts. The length of the hip is the hypotenuse of the triangle between the 14 1/8-inch mark on the blade and the 8-inch mark on the tongue. To compute this mathematically we have feet, or 16 feet 3 inches.

Fig. 201. Plan for Valley Rafter Connecting Two Roofs of Unequal Pitch and Width.

When a valley rafter serves to connect two roofs of unequal pitch and width, the problem is more complex. In Fig. 201 a 10X12 foot roof covers the main building and an 8X12 foot roof covers the ell on the left. The rise of rafter A C is 13 feet 4 inches, the rise of rafter C D is 7 feet 6 inches, and the ridge of the main roof is nearly 6 feet above the ridge of the ell.

One of the valley rafters C F runs to the ridge of the main roof, its rise being 13 feet 4 inches. In extending to F the valley runs 16 feet toward the main ridge, and the distance A F is found by proportion or by drawing the plan to scale and measuring.

In using proportion, take the run of the common rafters A C and CD. If the ridge of the ell roof coincides with the ridge of the main roof, the common rafter C M would be in proportion with C D, thus:

Rise of CD: rise of C M:: run of CD: run of C M Substituting 7' - 6": 13' - 4" :: 10' : run of C M

90:160 :: 120: run of CM Run of CM=213 1/3" = 17'-9 1/3"

Now find the diagonal distance C F by mathematics or the use of the square. On the square use 17f inches on the blade and 16 inches on the tongue.

The distance is 23 feet 11 inches. The rise is 13 feet 4 inches. Hence, use 23 11/12 inches on the blade and 13 1/3 inches on the tongue to give the horizontal and plumb bevels and length of the valley.

To cut the side bevel at the top, use the distances C M and A C, cutting along the C M side. In order, however, that this cut car be made accurately, the rafter must be backed and the square laid on the backed surface. Few carpenters, if any, ever back a Vahey rafter and consequently a roundabout method is used to get this bevel. The common result is, that the bevel very rarely fits snugly against the ridge. Where the rafter is not more than 2 inches thick, the misfit is not so noticeable, but in 4-inch material the open joint must be "doctored" by gauging and resawing after it has been tried.

When the rafter is cut properly and set in place it will be found that the plane of its top surface does not lie in either of the two roof surfaces. The surface of the top lies at an equal angle to each roof surface and one edge extends up above the other rafter in both roofs.

Fig. 202A shows how the edges of a hip rafter extend over the plate at the bottom. To overcome this the rafter can be backed or cut shorter as shown in Figs. 202B and 202C. To back the rafter, lay the square on the bevel at the bottom end in the position the plate will occupy. Now mark the points C D and draw the lines D F and C G from these points parallel to the edge of the rafter and cut away the triangular part A B D F H and A E C G H This is an expensive means of making the rafter conform to the roof surface and most carpenters merely shorten the rafter until the outside corners conform to the surfaces, as shown in Fig. 203.

Fig. 202. Cutting Rafter to Prevent Ends from Projecting Over Plate.

If the rafter is not to be backed, the effect of backing can be easily obtained by nailing a thin board on the top of the rafter and giving this board the proper bevel. The method is illustrated in Fig. 204.

The clapboard A is nailed at the edges and the one side wedged up to the angle the backing would take, care being taken to allow the square to touch the edge B B of the rafter at C C. The square used on the side of the rafter gives the plumb cut C D, and C M over the clapboard gives the side bevel. In cutting, the saw is held at an angle to coincide with both C D and C M.

In the rafter B E, Fig. 201, the horizontal cut at E is obtained by using 23 feet 11 inches and 13 feet 4 inches and is the same as the cut at C. The length of the rafter can be found by using proportion or by finding the length of B D.

Fig. 203. Simpler Method of Avoiding Projecting Hip Rafter.

In using proportion, it is evident that B E, the run of the short valley, is to C F as 7 feet 6 inches, or 90 inches, is to 13 feet 4 inches, or 160 inches.

B E: 287:: 90:160

B E= 287 x 90 / 160 = 161.5 inches= 13 feet 5 1/2 inches

The rise is 7 feet 6 inches and the run is 13 feet 5 1/2 inches.

Another way in which the problem may be solved, is to find where the ridge of the ell intersects the main roof surface. The intersection is at a height of 7 feet 6 inches which is 90/160 of the run of AC, or 9 of 16 feet and the distance B D is just 9 feet. Hence, the run of B E is feet 5 1/2 inches, and this is 9/16 of the run of the rafter C F. Hence, the run of C F is 13.45 x 16//9 = 23 feet 11 inches.

Fig. 204. Method of Using Clapboard to Cut Bevel on Rafter.

The end cut at B on B E, that is, the bevel that fits against C F, to be cut accurately, must be handled like the side bevel at F. First cut the bevel at the plate and get the backing line that makes B E lie in the main roof surface. Now, at B, either back the rafter a short distance, or use a clapboard as in Fig. 204.