This section is from the book "The Building Trades Pocketbook", by International Correspondence Schools. Also available from Amazon: Building Trades Pocketbook: a Handy Manual of reference on Building Construction.
The first step is to lay out a roof plan on a board or sheet of drawing paper, to a scale of, say, 1 1/2 in. to 1 ft. Fig. 7 (a) represents such a plan of a roof of uniform pitch, the wing being the same width as the main building; one end of the roof is hipped, while the other ends are finished with gables, as may be readily understood by reference to the perspective-view (6). In both views, the same letters refer to the same parts.
At the center of a 6 in (a), erect a perpendicular c c' equal to the height of the ridge above the wall plates; join c' and b; then, the angle at b is the foot-cut, and that at c', the plumb-cut of the common rafters. The length may he found by scaling. The angles are transferred by means of a bevel to the rafter, as shown in (c), in which b is the foot-cut, and c the plumb-cut.
On the line that represents the hip on plan - as e g in (a) - erect a perpendicular e e' equal to the height c c' of the ridge; join e' and g; the angles at e' and g are the plumb- and foot-cuts, respectively, and the length may be found by scaling. The lengths and cuts for the valley rafter i h are in this case the same as for the hip rafter, and are found in the same way, as shown at h and i' in (a). Both hip and valley rafters have cheek-cuts, which are the same as those of the jack-rafters.
Erect a perpendicular m e" at the center of the span f g; with f as a center, and e' g, the true length of the hip rafter, as a radius, strike an arc cutting m e" at e"; join f and e"; then the angle at e" is the cheek-cut. The foot- and plumb-cuts will be the same as for the common rafters. The length of any jack-rafter, as op, is found by prolonging it to cut f e", as at p' then op', measured by scale, is the length of op.
On the face of the rafter draw the plumb-cut, as at a 6, Fig. 8; parallel to a b draw d e at a distance c equal to the thickness of the rafter; square over from d to f, and join f and a, thus obtaining the cheek-cut.
Lengths of Hip or Valley Rafters; Wall Plates at Right Angles. Having fixed the rise and run by the square for the common rafter, take 17 in. for the run of the hip or valley rafter. Thus, if the roof is one-third, i.e., 8 in. rise and 12 in. run, the hip or valley rafters will have the same rise, 8 in., and 17 in. run. This rule gives results too great by about 5/16 in in 10 ft.
Having obtained the lengths of the first two adjacent jack-rafters at the toe of the hip rafter, by the graphic or other method, measure the difference between their lengths, and keep adding this difference for the subsequent ones.
Miter Cuts for Purlins. On the squared end of the purlin draw a plumb-line, as at a b, Fig. 9. Draw perpendiculars, as c and d, from the corners of the purlin to this line. On the upper edge of the purlin lay off a distance d' equal to d, and on the lower edge lay off c' equal to c; from e draw lines to the points marked, thus obtaining the lines for the cut. This is for a cut over a hip rafter; where the cut Is over a valley rafter, the bevel will be the same, but d' is laid off on the lower, and c' on the upper, edge.
In Fig. 10 the arrangement of the rafters is shown in plan, eg being a hip rafter, and h l a jack-rafter. In the elevation, b'c' is the span and o e' is the rise. To find the shape of the hip rafter eg, make e"g" equal in length to eg, and lay off e"k", k" h", etc., equal to ek, kh. etc. Erect perpendiculars at e", k", h", etc. At e, k, h, c, etc., drop perpendiculars cutting the curve e' g' at k', h', etc. From k' h', etc. draw horizontals cutting perpendiculars from e", k", h", etc., at the points e"', h'", etc. A curve drawn through these points gives the required outline of the hip rafter. The shape of the jack-rafter h I is the same as the curve h' g'.
A method of determining the pitches for a gambrel roof is shown in Fig. 11. The line af is made equal to the width of the front plus the eave projections. On the center g, with a radius equal to one-half of this measurement, describe a semicircle ahf; divide this into five equal parts, as at b, c, d, and e, and erect g h perpendicular to af. Draw a b and fe, the side slopes, and b h and eh, the upper slopes: then abhef will be the outline of the roof.