528 



CARPENTRY, 



Carpentry. 



PLATE 

 CXXII. 



Pig. 1 



Construe- PROB. VII. The figure ABCD of the wall plate 

 tive of a hipped span roof, and the height of the roof be- 

 ing given, to find the backing of the hips, the angles 

 made upon the sides of the purlins by their longitu- 

 dinal arrises, and the angles made upon the sides of 

 the jack rafters, the roof being equally inclined to 

 the different sides of the building, Fig. 1. 



Let the two sides, AB, AD, and DC of the wall 

 plate be at right angles to each other, and the end 

 CB at oblique angles to AB and CD ; draw the seat 

 EF of the ridge, in the middle of the breadth, pa- 

 rallel to AB and DC ; make AG and DH equal to 

 half the breadth of the building ; join GH, which 

 will be the seat of the common rafters adjoining the 

 hips. Make E f equal to the height of the roof, and 

 draw IG and IH, which are the length of the com- 

 mon rafters. Draw ED and EA, the seats of the 

 hips; make EK equal to El, and draw KA, which 

 gives the length of each hip. Through any point L, 

 in the seat of the hip AK, draw MN perpendicular to 

 AE, cutting the adjacent sides of the wall plate at 

 M and N ; take the nearest distance from L to the 

 rafter AK, and make LO equal to it, and draw OM 

 and ON, and MON is the backing of the hip, repre- 

 sented by their seats AE and DE. This operation 

 is the same as having the two legs of a right angled 

 solid angle to find the angle opposite to one of the 

 legs, the angle MON being exactly double of the 

 angle so found ; for the hipped angle of the roof con- 

 sists of two equal solid angles. 



Suppose that the bevel end at CB is inclined at a 

 different angle to the other sides and let FC and FB 

 be the seats of the hips, draw FQ perpendicular to 

 FC, and FP perpendicular to FB, and draw QC and 

 PB, which are the lengths of the hip rafters. The 

 backings, SUT and VWX, are found in the same 

 manner as above, and may be described in the same 

 words. 



From A, with the distance AK, describe an arc, 

 cutting GH at J, and join AJ, GJ A will be the side 

 bevel which the jack- rafters make with the hipa ; and 

 if a right angle be added to the same angle G.TA, 

 the obtuse angle thus formed will be the angle which 

 the upper arris of the side of the purlin, placed in 

 the inclined side of the roof, makes with the hip 

 rafter. 



Let a be the position of a purlin in the rafter 

 HI ; in GH take any point b, and draw be parallel 

 to the inward direction of the purlin a ; from b t 

 with any distance b c, describe an arc c d, cutting 

 GH at d ; draw be, c/, and dg, parallel to EF, the 

 former two cutting ED at e andf; draw^/g parallel 

 to GH, and join eg; produce be to h, and he g, or 

 beg will be the angle required, according to which 

 side it is applied. This will be found to be the same 

 thing as one of the legs, and the adjacent angle of a 

 right-angled solid angle being given to find the hy- 

 pothenuse. In the same manner, if neither side of the 

 purlin should be parallel to the inclined side of the 

 roof, as at k in the rafter GI, the bevel or angle up- 

 on each side will be found. 



Fig. 2. shows half the angle of the backing of the 

 hips ; the length of the common and hip rafters ; the 

 bevel of the jack-rafters on their upper sides in an 

 equal inclined roof, without laying down or drawing 



fig. 2 



any more than the necessary seats ; and this is all that Construe^ 



is necessary when each side of the roof is alike ; AB tive 



being the wall plate between the hip and the rafter 



which joins the top of the hip, AC the seat of the 



rafter which joins the top of the hip, BC that of the CXXir. 



hip, AF the length of the rafter which joins the Fig. 2. 



hip, BE the length of the hips, CHG half the back- 



ing, ADB the angle which the jack-rafters form 



with the hips on their upper sides, and consequently, 



with the addition of a right-angle, the side bevel of 



the purlin. 



Fig. 3. shows the same bevels, excepting that the Fig. 3. 

 side joint of the purlin is found by a different pro- 

 cess, thus : From B, with tht distance BA, describe 

 an arc at D ; from G, with the distance AC, de- 

 scribe another arc, cutting the former at D ; join 

 BD, and the angle CBD will be the angle in the 

 plane of the roof, made by the lower arris of the 

 purlin, and the joint against the hip rafter. 



Fig. 4. is a diagram, showing the lengths of the Fig. 4. 

 parts and angles concerned in the roof, in the same 

 manner as above ; but the plan of the building, or 

 form of the wall plate, is a quadrilateral, which has 

 neither pair of its opposite sides parallel. The me- 

 thod of executing the roof in this case, is to form a 

 level on the top, from the top of the hips at the 

 narrow end to the other end, as otherwise the roof 

 must either wind, or be brought to a ridge, which 

 will form a line inclined to the horizon : Either of 

 the two last casts is very unsightly. 



Besides the angles already mentioned, (see Fig. 2.) 

 AFC shows the angle formed by the upper side of 

 the rafter, and the ridge piece ; and the angle BEC, 

 the angle which the top side of the hips makes with 

 a vertical line ; also, the angle FAC, shows the form 

 of the heel of the common rafters ; and EBC, that 

 of the hips. 



It may be here noticed, once for all, that the bevels, 

 or angles, concerned in hip roofing, are no more than 

 the sides or angles of right-angled solid angles, con- 

 sisting of three plane angles. The quantities in de- 

 grees may be found as in spherical trigonometry, by 

 considering the plane angles, which constitute the 

 solid angles as sides, and the inclinations of the planes 

 as angles. 



Circular, Elliptical, and Polygonal Roofs. 



A circular roof may be executed with timbers dis- circular, 

 posed in vertical planes, whether the ribs or rafters &c. roofs, 

 are convex, concave, or straight, without any tie be- 

 tween the rafters or ribs, even though the wall were 

 ever so thin ; provided that it be only sufficient to 

 sustain the weight of the roof, by joining the wall 

 plate, so as to form a chain, a ring, or endless plate, 

 and by strutting the rafters in one or more horizon- 

 tal courses, without any danger of lateral pressure, 

 or of the timbers themselves being bent by the weight 

 of the covering ; but the same cannot be done with 

 the roof of a rectangular building, for single parallel 

 rafters would not only obtain a concave curvature, 

 but would thrust the walls outwards. Hence, the 

 means of executing circular roofs with safety are sim- 

 ple ; but those for straight sided buildings are com- 

 plex, and require much more skill in contriving, ac- 





