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378

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a sufficient number of fixed points, either from the roof itself,or from other immoveable places. There are three points for' this purpose ; if the rafters have a tie-beam below, that is, atthe vertex, and at the two extremities of the rafters, the tri-angle being immoveable at the angles, every force appliedj there tends either to compress or extend the sides of the| frame, without a transverse strain.

j Examples. —Let it be required to divide each of the raftersj into three equal lengths, in order to support two purlins onj each side ; this may be done, as in Plate III. Figure 1, byj pieces, c e, c d, a g, a f, reaching from the two lower angles,

; C and A, and to the opposite sides of the rafters, A b and b C,

' intersecting each other at h and i, and halved upon eachother at these intersections : this mode prevents the raftersfrom sagging, but does not afford any support to the tie-beam. The meeting of so many braces at the same point,too, gives little opportunity of making the ends entirelysecure, even though assisted by iron straps.

Another mode may be by introducing a king-post, b k,Figure 2, to which the struts, d h, e i, f m, g l, may be firmlyjoggled at h, i, m, l, and mortised to the rafters at d, e, f, g :this method keeps up the middle of the beam ; but when theroof is low, and the span great, the struts, d h and f m, requirethemselves to be supported, and are much too oblique toprevent a change of figure.

Another method may be, as in Figure 3, with the king-post in the middle, as before, two queen-posts under therafters at e and f, two struts, h e and i f, joggled to thebottom of the king-post at h and i, and to the top of thequeen-posts at e and f ; and in order to secure the points nand p, two other braces, nq,ph, are joggled to the bottomof the queen-posts at Q and r, and mortised into the raftersat the upper end. This construction supports the tie-beamin three different points, and each of the rafters in two. Thetimbers are much shorter than those of the preceding ; butso many joggles are certainly an objection to this method, asthe shrinking of the timber must be very considerable in threebreadths, which would allow the roof to descend.

When the span is great, and the points to be supportedmany, an excellent method may be as in Figure 4, wherethere are two arches of cast-iron, or good English oak, intro-duced, which abut on the king-post, and at the other extre-mity at the ends of the beam. The rafters and the beam, bythis mode, may be supported by as many equidistant pointsas we please.

When the tie-beam is removed from the bottom, as inFigure 5, and no fixed points are to be found from below, alongitudinal truss may be constructed, the end of which isshown at a b, and the manner of framing it in Figure 6, thetwo ends being supposed to be firmly fixed into the gables ;but where the length is great, the form of Figure 7, with aparabolic arch, would be much better : by this method, therafters will be kept nearly in the same plane, and all lateralpressure from the walls will be removed ; for it is evidentthat if the ridge-tree is supported, there can be no motiondownwards in the direction of the rafters, the whole roofbeing hung to this longitudinal frame.

Proposition YI.— Figure 8. —If a roof be constructedwith two equal rafters, a m, c m, with a tie extending fromthe bottom of each to an intermediate point in the oppositerafter, and the ties halved together at their intersection, b, soas to form with the rafters a quadrilateral, M D b e, at thevertex, and two triangles, adb and c e b ; then, if m d beequal to m e, and C p represent the direction and quantity offorce on the wall at C, the force tending to break the raftersat d and e : a s - fCK X s.pme X lk X dm

_ j! >’

For, completing the parallelogram pci.Ki A ji, tbeequal to C l ; and n o parallel,and mo perpendicular j e vei' 3triangles C be and A b Dmay be looked upon as so qv^(at least with regard to forces applied to the angles,) c tround B. Then the force cp will communicate 1i ^ up 0 ®to the rafter, and c L is the power acting obliquely perp eI>the rafter a m : then, because n o is parallel, and o t i,e ra‘f e ,dicular to A m, o m is the force tending to break ^ ^ ji.

!gl0

Jl®

at D, and ON that pushing it towards a: ^} ere ^ t \\e ^

be considered radius, and O M will be the sine j e jtD M N, or d M e ; for, produce A M to Q, and the a D t j ier ef° rwill be the supplement of the vertical angle D M 0) is ^the sine of n m q, equal to the sine of the angle &same with the sine of n m a ; then, by trigonome }lk:lc::s.lck:s. c k l or s. f c K

n m or l c : m o : : r : s. d m e = s. m n o ;

therefore, l k : m o :: s. i c K X R : s. p c k X s - p

TT S. P C K X S. D M E X L K , force ® c

Hence ho= --— tne

s. l c k x R ,. „ to br *.

perpendicular to a m at m ; but the force tendi g ^ fare®'the rafter at d, is as the lever d m multiplied intoS.PCK x S.DME X LK X DM

that is =-—-- ' -;<rld

S.LCK X R is a

Corollary 1. — Hence, if the angle D sl E ^ p wnangle, the force tending to break the rafters. p c k X l k X d M

s. l e k jjjg con'

Corollary 2.—Hence the rafters of every r0 °^th e aCt .*gystruction must sag in a greater or less degree,, is, 111 jfof the rafters against each other at the point ^.’^ges; b®^will be bent into curves concave on the a PP e1 ^ aD da diagonal connect the two vertical pointschange of figure will be prevented. sgUf e of

Proposition VII.— To remove the ^ atera ^^nt.roof without any intermediate beam, brace, or ^ ra fte r *'

Plate IY. Figure 1. No. 2.—Let a b, bC, b ® E F g, ^ i9 ’and let there be constructed a strong wall-pl ate ’.p { | 10 r°° „firmly bolted together at the angles ; to a coS °„ood

to be gable-ended, after having fixed the ra of 8^1

ridge-tree, let two curves be made he'raft<* s >®‘byEnglish oak, of a parabolic form, and let into to tb 0 (

on the upper or under surface, and firmly se 8 ®°bolts or nails, and at their lower extremities the 1 fnUthe wall-plates, the vertex of each curve me j je ° c on''® t ii0piece on each side of it, or nearly so, as may f 0 rm .° p ed'One half of the plan, No. 1, exhibits f or a ^o^aexecution of a gable-ended roof, and the o „ £0j are s p qroof. The two sides, laid in piano for eac p j, a® ,j, e rin No. 3 and No. 4 ; at hikl and E F

represent the same wall-plate; d g , N °> ^ side

_..5„ _J_5.:_Hw, rid ire on eui- t ,; no6 d,

wall-plate, IK and mp meeting the ridge liipP eu ’ rfi

but it must be observed when the roof is n ~" rethe ridge-tree must be very strong, as thbe very great, the liip-rafters acting Hk e P°. _ wall'P ^ th®

• . Hence, it is evident tha .„ocs

its extremities. Hence, it is evident i"“- - c urv eS , “" e qui*as the tie-beams of a common roof, and 1 i, r idg erafters ; or, more naturally, like an ar0 e q ua ^ ^r» b °^ C

librio. It has already been shown 0 { a P 111 ^

acting in equidistant lines, require an ar jt J ® ^

fnrm to th*»m in P.nuilibriO. ^

Porm ° to keen

considered thif In e <luilibrio. m -- rf 0

upwards, they are in 16 arcbes are P Iaoed witb , mly be S otout very convoy- a , state of compression, and m Jw ere inverted fl ‘ ent ^ in several lengths; but if a tcb

must then be L ‘ ey would be in a state of tension;

°ne piece, as the ridge would be eo V

s. L c K x R