Fig. 57.

If we now draw the lines y x, y t, yz, etc., where the points x, t, and z are the projections of the points x' in the plan, these lines will indicate the position of the tower rafters in the elevation, and where these lines intersect the main roof slope at s, will be points in the curved line of intersection, between the two roofs.

Projecting the points s on each rafter to the plan of each rafter at A, we get the points s', which are all in the line of intersection between the two roofs, as shown at adc, in Fig. 56, and through which the line of intersection a's' c' can be drawn, in Fig. 57. Horizontal lines drawn from the points of intersection s to the outside of the rafter, or slope line yd, will give the length y g of each of the rafters y s.

The plumb-cuts at y and the foot-cuts at d have already been described in connection with Fig. 55, but the foot-cuts of the rafters, where they rest on the surface of the roof vp, are found as follows: With y as a center and a radius y d, draw the arc dh e, on which space off the feet of the rafters, as at a", x", x", etc., and to these points draw radial lines from y. Make yj equal to y w the length of the shortest rafter, and drawy je and jd; then the angle which je or jd makes with the foot of each rafter will be the cheek-cut of that rafter on the roof, and a line g v drawn from the foot of each rafter y g, will give the angle yg v, which is the bevel of the foot-cut of each rafter as it rests upon the roof vp.

The line of intersection between the two roofs, as shown in the elevation at vw, is seen in the plan A at a's' c', and being at an inclination, does not exhibit its exact curvature as it would appear if seen in a horizontal position. To find the points in the line of the true curve, proceed as follows: With v as a center with radii vs, describe the arcs sr, st, so, etc., and where these arcs intersect the line db erect the perpendiculars rr', tt', oo', etc. From the points s', in the plan A, draw the horizontal lines s'r', s't', s'o', etc., and where these intersect the perpendiculars just described, will be the points in the curve required, through which points the true curve a' r' t' o' z' c' may be described.

The advantage of finding this curve is that from it a paper templet maybe made with the points a' r' t' o', etc. located on it, and the curve may thereby be marked out on the boards of the roof vp, and the rafters of the conical roof footed upon the points r' t' o', etc. with great exactness.

Where the tower roof dy b is of large size, and is liable to throw considerable weight upon the main roof vp, short pieces of timber should be framed in between each of a pair of rafters where one of the tower rafters is to set its foot. On small towers the boarding of the roof is usually of sufficient strength in itself, and requires no timbers inserted beneath it.

146. A hexagonal roof and the system of framing it is shown in plan in Fig. 58, where the six hip rafters radiate from the center o to the corners of the hexagon hfbg, etc. The two pairs of rafters c o j and g o h should be set in place first, as in the conical tower, Fig. 55, and the other hip rafters are then fitted against them. To find the plumb-cut and foot-cut for these rafters, lay off o a equal to the height of the roof, and join a and b; then the angle oba is the bevel of the foot-cut, and the angle o a b, the bevel of the plumb-cut, while the distance a b is the length of the hip rafter.

The middle rafters oj, oa, oc, etc. are found by laying off the height of the roof at o d and joining d and c; then the angle ocd is the foot-cut, and the angle odc is the plumb-cut on the rafters oj, oa, oc, etc., and their length is equal to the distance dc.

Fig. 58.

If from b we now lay off b e equal to b a the length of the hip rafters, and draw the lines e b and ef, on each of which, as a center line, the thickness of the rafter is laid off, we can draw in the jack-rafters kx and Ii; then the angle fxk will be the bevel of the cheek-cut, and the angle ode will be the angle of their plumb-cut, which is similar to that of the rafter o c, to which the jack-rafters are parallel.

147. Sometimes a tower, or other small structure, is covered with a roof whose outline is composed of curved lines instead of the usual flat slope. This curved outline is usually compound in form; that is, one end is convex and the other is concave. When such a sloped roof is put upon a circular tower, it is called a bell roof, on account of its resemblance to the shape of a bell; but when applied to a square, or polygonal tower, we call it an ogee roof, from the fact of its outlines being of an ogee, or double curved, shape.

148. In Fig. 59 we have the plan (a) and sectional elevation (b) of the ogee roof of a square tower. The ribs, or rafters a t and t b, shown in the plan (a), are either sawed or bent to the required shape, fitted squarely together at the top, and fastened in place. The ribs at f' t and g' t are next fitted, but are each shorter in length than the former pair by half the thickness of at or tb. The hip ribs lt, q t, mt, and r t are now put in place, and finally the jack-rafters o h.

To determine the length and curve of the hip ribs, or rafters lt, q t, etc., draw tx at right angles to lt and equal in length to the height of the roof c d; divide the rib fd into any number of parts, by making points as at v, and from these points erect perpendicular lines vs parallel to dc; where these lines intersect lt at s, draw the lines s x at right angles to lt, making sx in each case equal to the corresponding r v in the elevation (b); through the points x draw the curve xxx, which will be the profile of the angle rib required.

The length of the jack-rafters o h is found by dropping the perpendiculars o k and ij from each side of their cheek-cuts parallel to cd, and intersecting dg at k and j; then will jg be the extreme length and proper curve of the jack-rafters. while their plumb-cut will be on the line je. The cheek-cut of these jack-rafters is found by drawing a line from j on the outside of the rafter, to k on the inside, which is shown in the plan (a) by the line io. The hip rafters lt, mt, etc. must have their upper edges beveled on each side of the center line, as shown in the plan at m n. This is done in order that the boarding of the roof may have an even bearing on the angle rib, and the edges of the boarding must meet accurately on the center line of the rib m t. The amount of bevel required on the angle rib is found by drawing a line with a scratch gauge parallel to the upper edge, and at a distance m n below this edge; the wood between this line and the center of the top of the rafter is then cut away.