446 



ROOF. 



Koof. 



Method of 

 finding the 

 best form 

 of a kirb 

 roof. 

 PLATE 

 CCCCI.XXXI. 

 Fig. 6. 



or the tension produced by the weights n>, rv', tv", are 

 directly as the sines of the angles at thest. joints; and 

 inversely as the products of the sines of the angles which 

 the rafters make with a vertical line, that is, in Fig. 4, 

 Plate LXXX. 



Sin. r C I 



1 ension a c is as ^ TTI E 7 ~J7n 



Sin. r C 4 dxSm dC I 



with these data, we are *now prepared to determine 

 the best form of a kirb roof. 



Let it be required to find the form of a kirb roof 

 ABCDE, whose rafters AB, BC, CD, DE are equal, 

 AE being the width, and CF the height of the roof. As 

 the points A, C, and E, are fixed, this problem resolves 

 itself into finding the position of the point D, in the 

 line DHG, bisecting CE perpendicularly, when the 

 loads at the angles C and D are equal, and consequent- 

 ly in equilibrio. 



From the point G, where DH intersects AE, and with 

 the radius GE describe the circle EKC, passing through 

 C, because CH bisects CE at right angles. Draw HK, 

 parallel to FE, cutting the circle EC in K, and join 

 KC. The point D, where CK cuts GH, produced is 

 the point required, and the lines CD, ED, meeting at 

 this point, show the position of the rafters. 



Produce ED till it cut the vertical bar FC in N, 

 and having given the rafters CB, BD, the same posi- 

 tion as CD, DF, complete the parallelogram BCDP, 

 and draw DB, bisecting CP in R. Join K, F by the 

 lineKF, which is parallel to DP, because CDP=CKF, 

 on account of the parallelism of RD, QK, and the 

 equality of CR, RP and CQ, QF ; make CS equal and 

 parallel to FG, and upon S'with the radius SF, de- 

 scribe the semicircle WKF, which must pass through 

 K, because CG=SF=GE and CQ=QF. Join WK and 

 WS, and produce BC cutting ND in O. Now the an- 

 gle WKF, at the circumference, is equal to WSF at 

 the centre, and is therefore equal to WSC or CGF, and 

 double of CFE, or its alternate angle ECS. But 

 ECS=ECD+DCS, and ECD=1NDC and DCS= 

 DCO, or the alternate angle CDP. Hence WKF = 

 NDC + CDP=NDF, and WK parallel to ND. Conse- 

 quently CF : CW - CP : CN ; and hence CF = CW 

 we have CN = CP. 



Now, in the two triangles CDN, CDP, the sides are 

 to one another as the sines of the opposite angles, as 

 follows. 



CN : CD = Sin. CDN : Sin. CND 



CD : DP = Sin. CPD : Sin. CDP 



DP : CP = Sin. PCD : Sin. CDP 



Hence 



CN : CP = Sin. CDN x Sin. CPD x Sin. PCD : : 

 Sin. CND x Sin. PCD x Sin. CDP, or 

 CN-CP- Sin. CDN ^ 



~Sln7cl^~x~&in<PCD :~Sln7PCD 



But CDN, CDP, are the angles at -the joints 

 and CND, PCD, and .PCD, CPD, are the angles 

 which the rafters make with the plumb lines, conse- 

 quently .CN is to CP, directly as the sines of the 

 angles at the joints, and inversely as the products 

 of the sines of the angles which the rafters make with 

 the vertical; that is, CN : CP as the loads at the joints 

 D and C ; but CN = CP, consequently the loads at 

 the joints are equal, and the rafters being equal, they 

 will be in equilibrio. 



When the rafters CD, DE have any other propor- 

 tion than that of equality, as for example ED', D'C, the 

 point D will be in the circumference of a circle H'D'/*', 

 having its centre in the line CE, and ED' : D'C= 

 CH : HE' = c h' : h'E. 



Roof. 



When a roof requires to be flat on the top, it may 

 be considered as consisting of three rafters A B, B C, W-y-*/ 

 C D, Fig. 7. If B C is horizontal and A B, D C, 

 equally inclined to it, it is obvious that the rafters will 

 be in equilibrio. In order to stiffen this roof, queen Fig. 7. 

 pouts BE, C F, are placed at the angles B, C, and are 

 connected with the tie-beam A D either by mortices 

 or straps. This form of roof though less strong than 

 A G D would have been, of the same scantling, yet it 

 has the advantage of giving more room for garrets. A 

 stronger but less commodious form is shown in Fig. 8. 



In the construction of roofs of all kinds, those parts Fig. 8. 

 which compose it may be divided into two kinds, viz. 

 those which are compressed and require stiffness as 

 well as cohesion, such as rafters, braces, and trusses, 

 and those which are extended only, as tie-beams, king 

 posts and queen posts, and which may be replaced by 

 ropes, chains, or rods of iron. All pieces of timber in 

 a roof, excepting the sarking, ought to perform one or 

 other of these offices, and ought either to be pushed or 

 stretched in the direction of its length. 



As the limits assigned to this article will not permit 

 us to enter into any farther theoretical details on this 

 subject, we shall now communicate for the benefit of 

 the practical mechanic, some useful information on the 

 subject of roofs, for which we are indebted to Mr. Tred- 

 goid's excellent work on the elementary principles of 

 carpentry. 



The general height of roofs varies between one-third Height of 

 and one-sixth of the span. For slates the usual height roofs, 

 is one-fourth, which make the inclination to the hori- 

 zon 26^ degrees. 



The following table, given by Mr. Tredgold, shows 

 the inclination that may be given for other materials. 



The following tables of the scantlings and timbers Tables of 

 for roofs of different spans from 20 to 30 feet ; from 32 the scant- 

 to 46 feet ; from 48 to 60 feet, and from 65 to 90. lings of tim- 



In these tables the pitch of the roof is supposed to ber of roofs 

 be about 27, the covering slate, and the timber good ^ Different 

 Riga or Memel fir. When the timber is soft, spongy, s P ans * 

 or inferior in any way, it will require to be of larger 

 dimensions. One-fourth of an inch hi each dimen- 

 sion will be sufficient to compensate for any difference 

 in quality, unless in the case of knotty timber. 



TABLE I. See PLATE CCCCLXXXI. Fig. Q. 



The trusses are supposed in this table not to be more 

 than 10 feet apart. 



Fig. 9. is drawn with a parapet wall on one side, 

 and with eaves on the other. 



