This section is from the book "A Treatise On Architecture And Building Construction Vol2: Masonry. Carpentry. Joinery", by The Colliery Engineer Co. Also available from Amazon: A Treatise On Architecture And Building Construction.
149. When two gables of different heights and spans intersect, the framing of the valley rafters is somewhat different from the case where both gables are the same height. Fig. 60 is the plan of a portion of a gable roof where a smaller gable at (b) intersects its slope. The inclination of the main gable (a) is laid off at gm, and the elevation of the end of the smaller gable (b) is drawn at pbu. The height a b of the smaller gable is laid off at c d and the line de is drawn parallel to cm; ei is then drawn parallel to the ridge kc; fe, therefore, represents the height of the smaller gable in comparison to the large one, and e is the point of the slope where the ridge gable (b) will intersect; therefore, by drawing the ridge line a i of the gable, we intersect the line drawn from e parallel to the small main ridge at i, and the lines ip and in will, therefore, be the lines of the valley rafters of the small gable, as seen in plan. But, in framing our roof, we will carry one of these valley rafters in past the point of intersection i to the main ridge at k, in order that it may have a firm support at its upper end. The rafter n k, therefore, will have its plumb-cheek and foot-cuts the same as though the intersecting gables were of the same height.
The rafter ip, however, though cut at the foot with the same bevel as is n k, has a different cut at i. With m as a center and a radius mg, we strike the arcgr, and with the radius me we strike the arc ej; drawing rl and jh, Irm will be the plan of the roof on the flat surface, as though there were no pitch at all, the plan then shows the rafters at their true lengths and bevels. The rafter n k will, in this revolved plan, occupy the position n l, and lt will be its cheek-cut, while the rafter p i will now appear at p h, and u h will be the bevel of its cheek-cut. The plumb-cut of the rafter n k will be on the angle formed at w by making k w equal to eg and perpendicular to nk, and drawing wn; the bevel of the foot-cut will be the angle w n k.
The lengths and bevels of the jack-rafters in the main roof are shown in their true proportions in the revolved plan l r m, and can be scaled from that, while the necessary measurements for the jack-rafters in the small roof (b) can be obtained by revolving the pitch n b on n as a center until it lies on the level plane n o. The triangle osn will then be the actual shape and proportion of one slope of the small gable roof, and the length and bevels of the jack-rafters can be scaled therefrom.
If the line p n from which the small gable springs should be, instead of at the eaves, as in this case, at some point on the slope of the roof above the eave line p m, the valley rafters h n and hp would be carried down to the plate, as at u' and v' in Fig. 72, in order to secure them a good footing, and a piece of timber t' would be framed, from which the short rafters q' would extend to the eaves. Sometimes, when the main gable is very large and the smaller one is much less in size, the valley rafters are not carried down to the plate, as above described, but are simply framed into the nearest common rafter on each side and a rod is put through along the side of the piece which is shown framed at t; this rod, when its nuts are screwed up tight on each end, prevents the common rafters from spreading under the thrust of the valley rafters framed against them.
150. It is sometimes desirable, in framing a roof, to make the rafters of very heavy material and space them much farther apart than in ordinary cases, as is always done when trusses are used in the roof.
In order to properly fasten the roofing material to these widely spaced rafters, it is necessary to introduce a series of small roof beams at right angles to, and with their ends resting on, the rafters. These roof beams are called purlins, and though they possess neither plumb-cut nor foot-cut, they have a peculiar miter-cut, where they meet on hips or in valleys, illustrated in Fig. 61, in which abcd is the plan of a roof with two slopes meeting on the hip db. From d we lay off de perpendicular to db and df parallel to a b, and each equal to the height of the roof; then eb will be the slope of the roof over the hip db, and fa will be the slope of the roof over the common rafter da; ghj is a purlin on the slope of the common rafters, and the dotted lines j I, hm, and gn show the position of the purlin in plan, and lmn the direction of its miter on the hip rafter db. The line Iso is now drawn square across the plan of the purlin.
Fig. 62 is a perspective sketch of the purlin, where the line v zx is drawn square across each side of the timber, and z r and xy are laid off equal, respectively, to s m and o n, Fig. 61. The line vry is, then, the bevel on which to cut the miter of the purlin.
151. When rooms are divided in the attic, or under the roof of a building, and the partitions run neither parallel to the rafters nor to the ridge, the studs have to be cut at their upper ends to fit an inclined plane which passes over their diagonally opposite corners.
Fig. 63 shows at a b cf the plan of a stud, the sides of which are parallel neither to the rafter line ml, nor to the eave line li. The line kl shows the pitch of the roof over m /, to the plane of the under side of which the stud is to be fitted. The lines fj, c h, and b d, therefore, are the corners fc b of the stud in elevation, and at jh d are the points where these corners intersect this roof pitch. By drawing dn at right angles to db, we find that the corner of the stud c is longer than the corner b by the length of the line eh, and the corner f is longer than the corner b by the length of the line nj. Therefore, if we draw a line around a stud, square with each edge, as top, in Fig. 64, and lay off t s and o r equal to nj and e h, in Fig. 63, we have, on connecting the points s rp, the bevel line of the end of the stud, as required.