In case a concealed gutter is used and the rafter is set directly over the wall, the line DP coincides with the line M N, Fig. 193, and the rafter has only the horizontal cut at the bottom or a hori zontal and vertical cut, as shown in Figs. 196 and 197.
Valley and Hip Rafters. In Fig. 19S 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 sum-face, 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 arc 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 rectangu lar solid. For instance, if the run of the common rafter is 12 feet, the rise 10 feet, and the distance MR is 8 feet, the valley rafter will form the diagonal of a rectangular solid 12 inches X 10 inches X inches, and its length and bevels can be found as shown in Pigs. 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 dis tance between the 12 and 8 on the square. It is 141-'1- 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 inches on the blade and 10 inches on the tongue.
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 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 1/144+100+64=17.5=17 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 11 inches. The rise of the
hip is the same as the rise of the common rafter. If, then, the rise is 8 feet, use inches on the blade and S 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 141-inch mark on the blade and the 8-inch mark on the tongue. To compute this ically we have 264=16.25 feet, or 16 feet 3 inches. When a valley rafter serves to connect two roofs of unequal pitch and width, the problem is more complex. In Fig. 201 a 10X 12 foot roof covers the main building and an SX 12 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 propor tion or by drawing the plan to scale and measuring.
In using proportion, take the run of the common rafters A C and C D. 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: Now find the diagonal distance C F by mathematics or the use of the square. On the square use 1711 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 23-12 inches on the blade and 131 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 CM and A C, cutting along the CM sick. 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 1, al.ey 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 resolving after it has been tried.