ROOF. 



Truncated roofs are chiefly employed in order to diminifh 

 the height, fo as not to predominate over that of the 

 walls. 



When all the four fides of the roof are formed by inclined 

 planes, the roof is faid to be hiped, and is, therefore, called 

 a hiped roof: the inclined ridges, which fpring from the 

 angles of the walls, are called the hips. Roofs of this de- 

 fcription are frequently truncated ; and when the plan of 

 the walls is in the form of a trapezium, the truncation of 

 the roof becomes neceflary. 



Roofs which Hand upon circular bafes, and which have 

 all their horizontal fections circular, and the centres of the 

 circles in :i ftraight line drawn from the centre of the bafe 

 perpendicular to the horizon, are called revolved roofs. 



When the plan of the roof is a regular polygon, or a 

 circle, or an elhpfe, and the horizontal fedtions are all 

 fimilar to the bafe, and a vertical fe&ion a portion of any 

 curve convex on the outfide, the roof is called a dome. 



In order to fave the expence of lead in redtangular roofs, 

 inftead of the flat, a valley is iometimes ufed, which makes 

 the vertical fection in the form of the letter M, or rather an 

 inverted W ; and thus it is that this form of roof has ob- 

 tained the name of an M roof. 



Before we proceed to the confirmation of roofing, it will 

 be neceflary to fhew upon what principles a body or piece 

 of timber may be fupported in various pofitions. 



Theory and Pracl'ue of Roofs. 



Phop. I. 



If a heavy body A BCD (Plate XlAl. Jig. 3. Archi- 

 tedure,) be fulpended by any two inclined firings, D E and 

 C F, in a vertical plane, a right line drawn through the in- 

 terfection, perpendicular to the horizon, will pafs through 

 • he centre of gravity of the body. 



It is (hewn by the writers on mechanics, that if any three 

 forces <l& upon a point, or a body, their directions will tend 

 to the fame point, or lie parallel to each other. It is well 

 known that every body acts with its full force in one point 

 only, viz. in its centre of gravity, and in a direction per- 

 pendicular to the horizon : therefore, if a body is fullained 

 at E and F, it will revolve round thefe points, until the 

 line G H, palling through the interfedtion, H, of the two 

 firings, D £ and C F, and the centre of gravity G, become 

 perpendicular to the horizon. 



Cor. 1. — Hence if any body be fupportcd by two [brings, 

 it may alio be Iupported by two planes perpendicular to 

 thefe firings, provided that the two points of the body fup- 

 portcd are in the direction of the firings ; for every body, 

 acting upon a plane, acts in a line perpendicular to that 



Cor. 2. — Hence, alfo, a body may be fupported by two 



props in any two directions that may be fupported by 



itrings, provided that the furface of the body, at the points 



ntaft, or the ends of the props, be planes at right angles 



to the Itrings. 



Cor. 3. — Hence all the properties that have been demon- 

 flratcd of three forces acting upon a body, fuppofed void of 

 weight, will equally flow from a heavy body Iupported by 

 two Itrings, by fubflituting the weight of the body for the 

 middle force ; and hence, if the direction of any force fup- 

 porting a heavy body be given, the other may cafily be 

 found. 



Pfuor. II. 



Given the pofltion in which a body fliould be placed, 

 .*nd the pofltion uf a plane fupporting the body at one end, 



to find the pofltion of another plane to fupport it at anothei 

 given point, and to find the prcflure on the planes, the 

 weight of the body being given. 



Through the centre of gravity of the body draw a ver- 

 tical line, and through any point on which the body refts 

 on the given plane draw a line perpendicular to that plane, 

 meeting the vertical line ; from the interfeftion draw a line 

 to the other point which is to be fupported ; from that 

 point draw a plane at right angles to this line, which will 

 be the direction of the plane required. And to find the in- 

 tenfity of the forces, take any diltance on the vertical, line 

 to reprefent the weight of the beam from the interfeftion ; 

 then on that line, as a diagonal, complete a parallelogram, 

 whole fides are in the directions of the lines, perpendicular 

 to the fupporting planes ; and the fide of the parallelogram, 

 perpendicular to either plane, will reprefent the force on 

 that plane. 



Example 1. Plate XLU.Jg. 2. 



Let the body A B C D lie upon the top of the wall 

 KC, at C, fo as to touch the lower edge, B C, of the 

 body, at that point C ; it is required to find the direction 

 of a plane that will fupport the lower end at B, and to find 

 the preflure of the body on the wall and on the plane. 



Through the centre of gravity, G, of the body draw 

 the vortical line G F ; draw C F perpendicular to C B, 

 join F B, and draw B I perpendicular to F B, and B I is 

 the direction of the plane required. On the vertical line 

 G F, make F M to reprefent the weight of the body, and 

 complete the parallelogram LMNF; then F N reprefent* 

 the force on the wall-head, in the direction F C ; and F L 

 the force adting perpendicular to the plane, or in the direc- 

 tion B F. But if the vertical and horizontal tlrrults on the 

 wall at C are required, draw N P perpendicular to F G, 

 meeting it in P ; then the force F N is refolved into two 

 forces, F P and P N. P N reprcfents the horizontal part 

 of the force, viz. that which pufhes the wall in a direction 

 parallel to the horizon ; and F P the other part, which 

 tends to prefs it downwards in a direction perpendicular to 

 the horizon. 



Example 2. Fig. 1. 



Let the (loping body, A B C D, be fupported by a wall 

 at its lower end, D, which coincides with the furface of the 

 body, and let G be the centre of gravity ; it is required to 

 cut a notch out of the body, at the upper end C, fo that it 

 may reft upon the top of a wall, which is made to fit the 

 notch, and to find the prcflure on the walls. 



Draw the vertical line G E, from D draw D E per- 

 pendicular to D C, join E C, and make C F at right 

 angles to it ; then the notch, H C F, being cut, the body, 

 A BCD, will be at rell. Then to find the prcflure on 

 the walls, complete the parallelogram EIKL, having a 

 given angle DEC, and its diagonal on the given line E G. 

 Then if KE reprefent the weight of the body, I E will 

 reprefent the preflure in the direction 1) E, upon the wall 

 at D, and LD the preflure in the dim lion C E. The 

 horizontal and perpendicular preflun up ch wall mat 

 be found, as in the preceding example) by refolring . 

 of the forces, IE and L E, into two ; one "f wii' 

 perpendicular to the horizon, and the . ther parallel to it. 



Scholium. — It mult be obferved in tl mple, that the 



notch, which is cut out at C, "ill remove the centre of 

 gravity nearer to the lower end D, and confequently altei 

 the dope C 1" ; but as this can dy In- in a very fmall de- 

 gree, the equilibrium will hardly be affected by it, when 

 the notch is very fmjll. 



Exampk 



