jr t: *

E O O

379

E O O

'.‘ensio

H

^"'Hbr' 0 ^ tbe tw0 curves - This inverted disposition

ration is not so secure as when the crown of the

6r raeets the ridge or lies towards it.r e, i t 6 . jbe above construction will preventlitj 0n p not hinder tlie rafters from sagging

lateral pres-but the

U JUoH a Co ^ ar “keam will effectually answer this purpose4op„ erate spans.

^ich tiQf lTl ° N VIII.— Given the construction of a roof, of8 fo ni0re than three timbers meet at the same junction,the f CB ln ( he direction of any one of the timbers ; to°°f sfi a /i r ce f communicated to the other timbers, so that thej &egj n . tn equilibria.

(.a i r<:c ti' Vlt ' 1 t * le 8' verl force, and take a part of the line oftvfQ 11 / 1 ^ 0111 tbe junction to represent it ; then, with theJt> f ro directions, complete a parallelogram, and applyp"'clit[j e t|e next junctions on the same straight line fromr °eee(j / w ®re taken, and complete parallelograms, as before.talo» n ^*' a m anner, from one junction to another, untilj Is of S aitls have been made at every one. Then the

th e f ese parallelograms, in the directions of the timbers.to rces ...... . .._. .... ...

Uh

. ten 8 j 0 r ° ea in these directions; and to ascertain the statet ° the ° r cotn P ress i° n °f an y timber, observe, that whenth° ''gltt an ®* es formed by three directions are less than6 t'Vo „ an gies, the middle force acts always contrary toW hetl trerne ones, as has already been explained ; andtwo '-he angles of direction are greater than

l l*at

of

barrio' a . n gles, then the forces will act towards or

.4„‘ P°"U

from

,>ȣl r-

t'Vo t -—Figure 2.—Let

A b c d A be a roof, consisting

•' j. I'after, , , . . .

| ’ »app 0 j" s > ab.bc, two beams, C D, d a, and a king-post,

•th e V the walls AO and ce. Let C E represent

Qq ~ j wans ii aiiu v u ic^icacin

deplete tbe ro °f’ or the reaction of the wall, c e ;

He f„ 0tn Plet

Parallelogram c e f g ; make d l equal to F C,. e the parallelogram lmnd; then c f, or d l. is'n the direction of the beam c d, or A d, and d mln the direction of the post d b ; now, because

the a f ° r ce

’“ v UUCtjtlUll VI U1C JJUOt U U ) HUtVj wt^uuou

h aa Se t j le F 0 an( i F c G are less than two right angles, and0 j also c * s pressed by the reaction of the wall e c,

t a f ’ th ere j e Passed by the force G C, and drawn by the forceatj er 6 c i ° re tbe beam c d is in a state of tension, and the^ , # t). a state of compression. Again, because c d bO ft* 8 a 6 ® reater than the two right angles, and becausejj>i 0n 8tate of tension, B d and D a are also in a state

^PleteJ* 6 tnac f e equal to gC, and the parallelogram bhikfr* to tvyf n< * Bp be made equal to d m, then will p i be

C E ’ the pressure on the walls,tn^alls i n ‘r~-Figure 3.—Let ABCDEAbea roof, supportedJ)^ 68 of ti ffi , ^i re otions pa and qc, and let there be two6, t n r > B D and b e, connecting the angular points.

Vi^c * ri ' ,rr °

p a ^ a oe Q D tae wall q c : complete the parallelogram CFGHmake d k equal to G C, and complete thetj 6( . ct *0n 0 j. , D I K l ; then G C, or D k, is the force in thethe 6 t ‘ mbep C d or A E, and is in a state of tension,fop, 88 ’ and i, an S 1 es fog and gch are less than two right

v F 1 p idge at b.t of t , 0 represent half the weight of the roof, or the

, „ u >e wall O n • r>nm t\laftVio rtnrsillplncrriim f! P ft II 1

atl 'i be,

a dir

:Ca use c f is in a state of compression ; C u, the. ®'°0 ; anf C K° n 0t ' the rafter B C, is also in a state of com-a S [ >kna because any two of the three angles, gdb,8 tat<j* e °f ten**' 6 ® rea * er than two right angles, and D C is in

fbe-°f ten *'° n ’ tbe two P‘ eces > D B an( ^ u E > are a ' so ajf lop ce i n Sl ° n : 'bat is. e a, e b, e d, d b, d c, are all ties,i ^ a k and D B or e b is d e, that in d e is d i.

u Co mrif S * )e m ade equal to c h, and the parallelogramta e p U[ P : and if b t and b u be made equal to l d,a lelog ram b t y U completed, then will vw be

equal to twice C F ; that is, by reducing the force in the direc-tion of the pieces b e and bd to an equivalent one.

Proposition IX.— Given the lengths a b. b c, c r>, d e, ofthe rafters of a roof and their angles of position, to find thoseangles that require ties, and those which require struts.

Plate V. Figure 1.—Let A b be to b c as 3 to 4, that is, as6 to 8, the proportion of the weight of the rafters ; thenif 8 be taken for the weight of each of the upper rafters,

8 -f* 8

—2 -= 8 will be the weight on the vertical angle c, and

a i o

= 7, will be the stress on each of the vertical angles •

2

b and d ; so that the weight on the vertical angle is to thaton each of the lower angles, as 8 to 7. Draw the verticalline B G F, and draw, A G, A F, parallel to the rafters BC,CB;then if F G be to G b as 8 to 7, the rafters will be in equilibrio,and require no ties. But suppose it should be found that f gis to g b as 1 to 2 ; as that will keep it in equilibrio, it wouldrequire a very considerable addition laid on the angle, B, tokeep it from springing outwards, so that if two braces, f g andk l, No. 2, were fixed to the rafters, a b, b c, C d, d e, thesewould be in a state of compression ; and if the brace, li I,were fixed at the top, it would be in a state of tension : f gand k i. only require firm butments, but h i to be well bolted.It may here be observed, that if the vertical angle only bebraced and secured to the two rafters, the whole frame willthen be immoveable.

Proposition X. — To discover the effect of bracing theangles of a roof flat on the top, supported by puncheons atthe bottom oj the rafters, to accommodate a semicircularceiling within.

Figure 2. No. 1.—Let A b C d e f be the truss, divested ofits braces, the bottoms of the puncheons resting firmly on thewalls at a and f, and the joints at B, C, d, e, to be quite move-able, like rule-joints. Now, as this disposition of timberswould fall, and in falling, would assume the form of No. 2,the angles at c and d would become more and more obtuse,while those at A and e would, in the same proportion, becomeacute ; the latter would, therefore, require straining-pieces,and the former ties ; the straining-pieces must have goodabutments, and the ties must be well bolted at theirextremities.

Let No. 3 be the truss, with braces disposed in thelower angles : this disposition will bend the rafters b c,de, and the puncheons, BA, ef, convex towards the out-side, which is entirely occasioned by the braces, G n, N o : thecamber-beam, CD, is no otherwise effected than by its ownweight. Let it now be supposed, that the angles c and d,No. 4, are braced atiK, lh. In this disposition, the pun-cheons, b a, e f, are not affected in respect of transversestrains; the rafters C B, be, and the beam c d, would allbecome concave on the outside; and the points b and e, atthe bottom of the rafters and top of the puncheons, would bepushed out beyond the perpendicular of a and f, at the bottom ;here it is necessary to observe that the effect produced in thiscase on the rafters C b and d e is contrary to the effect pro-duced in No. 3, by the braces being disposed in the lowerangles. Lastly, suppose that all the angles are braced, as inNo. 5, it is evident, since the braces, IIG, N O, produce a con-trary effect to the braces ki, lm, these bending the raftersdownwards, and those upwards, that the rafters C B and dewill become nearly straight, or assume an undulated line ; thepuncheons b a and E F, receiving the force of the braces H gand n o at the points G and o, must still be bent, so long asthe under ends, G and o, of the braces do not coincide withthe under ends, a and e, of the puncheons : in this case, there