ROOF. 



beams of a common roof, and the curves as the rafters, or 

 more naturally like an arch of a bridge in equilibrium. It 

 has already been /hewn that equal weights, acting in cqui- 

 dillant lines, require an arch of a parabolic form to keep 

 ihsra in equilibrio. In this it is to be confidcrcd, that as the 

 arches are placed with their crown upwards, they are in a 

 ftate of compreflion, and may be got out very conveniently 

 in feveral lengths ; but if the arches were inverted they would 

 be in a Hate of tenfion : each arch mull then be in one 

 piece ; the ridge would be comprefted by the tenfion of the 

 two curves. This inverted difpofition of equilibration is not 

 fo fecure as when the crown of the arches either meets the 

 ridge or lies towards it. Though the above conllruftion 

 will prevent lateral preflure, it will not hinder the rafters 

 from fagging ; but the addition of a collar-beam will effectu- 

 ally anlwer this purpofe in all moderate fpans. 



Prop. XIII. 



Given the conflruftion of a roof, of which not more 

 than three timbers meet at the fame junction, and a force in 

 the direftion of any one of the timbers ; to find the forces 

 communicated to the other timbers, fo that the roof (hall 

 be in equilibrio. 



Begin with the given force, and take a part of the line of 

 its direftion from the junftion to reprefent it ; then with 

 the other two direftions complete a parallelogram, and 

 apply them from the next junctions on the fame flraight 

 line from which they were taken, and complete paral- 

 lelograms as before. Proceed in this manner from one 

 junftion to another, until parallelograms have been made 

 at every junftion. Then the parts of thefe parallelograms 

 in the direftions of the timbers are the forces in thefe 

 direftions ; then to know the Hate of tenfion or com- 

 preflion of any timber, obferve that when two of the 

 angles formed by three direftions are lefs than two right 

 angles, the middle force afts always contrary to the two 

 extreme ones, as has already been explained ; and that when 

 any two of the angles of direftion are greater than two 

 right angles, then the forces will aft towards or from the 

 fame point. 



Example I Let ABC D A (fg. 2.) be a roof, confiding 



of two rafters, A B, B C, two beams, C D, D A, and a king- 

 poll, D B, fupported by the walls A O and C E. Let C E 

 reprefent hall the weight of the roof, or the re-aftion of the 

 wall C E ; complete the parallelogram C E F G ; make D L 

 equal to F C, and complete the parallelogram L M N D, then 

 C F or D L is the force in the direftion of the beam C D 

 or A 1), and D M the force in the direftion of the poll 

 D B ; then becaufe the angles E C F and F C G are lefs 

 than two right angles, and becaufe the point C is p relied by 

 t'he re-aftion of the wall E C, it will alfo be prcfied by the 

 force G C, and drawn by the force C F ; therefore the 

 beam C D is in a Hate of tenfion, and the rafter B C in a 

 ftate of compreflion. Again, becaufe C D B and B D A are 

 greater than the two rigl t angles, and becaufe C D is in 

 a Hate of tenfion, B D and 1) A are alfo in a itate of 

 tenfion. 



If B H be made equal to G C, and the parallelogram 

 BHIK completed, and if B P be made equal to DM, 

 then will P I be equal to twice C E, the prefTure on the 

 walls. 



Example 2— Let A B C D E A {Jig. 3.) be a roof fup- 

 ported by walls in the direftion I' A and Q C, and let there 

 be two pieces of timber, B D and B E, connefting the an- 

 points 1) and E to the ridge at B. 



Tafc C F to reprefent half the weight of the roof, or 

 the re-aftion of the wall Q C : complete the parallelogram 



CFGH, produce C D to K, make D K equal to GC, 

 and complete the parallelogram D I K L ; then G C or 

 D K is the force in the direftion of the timber C I) or A E, 

 and is in a ftate of tenfion, becaufe the angles F C G and 

 G C H are lefs than two right angles, and becaufe C F is 

 in a ftate of compreflion ; C H, the force in a direftion of 

 the rafter B C, is alfo in a ftate of compreflion ; and 

 becaufe any two of the three angles G D B, G D E, 

 E D B, are greater than two right angles, and D C is in a 

 ftate of tenfion, the two pieces, D B and D E, are alfo 

 in a ftate of tenfion : that is, E A, E B, ED, 1) B, DC, 

 are all ties. The force in D B or E B is D L, that in 

 DE is D I. 



If B R and B S be made equal to C H, and the paral- 

 lelogram BRWS completed ; and if B T and BU be 

 made equal to L D, and the parallelogram BTVU com- 

 pleted, then will V W be equal to twice C F, that is, by 

 reducing the force in the direction of the pieces B E and 

 B D to an equivalent one. 



Prop. XIV. 



Given the lengths A B, B C, CD, D E, (PIateXL\l. 

 Jig. I . ) of the rafters of a roof and their angles of pofition, 

 to find thofe angles that require ties, and thole which require 

 ftrutts. 



Let A B be to B C as 3 is to 4, that is, as 6 to 8, the 

 proportion of the weight of the rafters ; then if 8 be taken 



for the weight of each of the upper rafters, 



8+8 



= 8 



is the weight on the vertical angle C, and = 7, will 



be the weight on each of the vertical angles B and D, fo 

 that the weight on the vertical angle is to the weight on 

 each of the lower angles, as 8 is to 7. Draw the vertical 

 line B G F, and draw AG, A F, parallel to the rafters 

 B C, CD; then if F G be to G B as 8 to 7, the rafters 

 are in equilibrio, and require no ties. But fuppofe it lliould 

 be found that F G is to G B as I to 2 ; now as that will 

 keep it in equilibrio, it would then require a very confi- 

 derable addition laid on the angle B to keep it from fpring. 

 ing outwards, fo that if two braces, F G and K L, N° 2, 

 were fixed to the rafters A B, B C, CD, D E, thefe braces 

 would be in a ftate of compreflion, and if the brace H I 

 were fixed at the top it would be in a ftate of tenfion : F G 

 and K L only require firm butments, but HI to be well 

 bolted. It may here be obferved, that if the vertical angle 

 only be braced and fecured to the two rafters, the whole 

 frame will then be immoveable. 



Prop. XV. 



To difcover the effect of bracing the angles of a roof 

 fiat on the top, fupported by puncheons at the bottom of 

 the rafters, to accommodate a femicircular ceiling within. 



Let A B C D EFl/^. 2. N° 1.) be the trufs, duelled 

 of its braces, the bottoms of the punched I irmly on 



the walls at A and F, and the , ! I, E, to be 



quite moveable, like rule joints. Now, as tin-, difpofitionof 

 timbers would fall, and in falling, 1 I aflun the form oi 

 N" 2, the angles at C and D woul I more and more 



obtufe, while thofe at 15 and E would become more and more 

 acute ; thi rould ' raining-pii 



and?' 1 ties: the {training-pieces mull have good 



abutments, and the tie. be Wl II bolted a', their extremities. 

 Let N" 3 be the trufs, we ! '1 in the UrM I 



angles: this difpofition will bend the rafters B C, DE, and 



the puncheons B A, E 1'", convex towards the outfide, winch 



3 R 2 is 



