R o o r. 



445 



the df&ired-cflVct may be produced by an assemblage 

 of two or more pieces of wood or beams. 



The simple-it form of a roof is when two beams are 

 uied, as in I'l 1*1 ('('('( I. XXXI. Fig. 1, where AC 

 i.i the distance to be crossed ; and AH, BC, the two 

 beam- employed Cor this pnrpo-e, 4- ther to carry a roof 

 or the roadway ot .1 bridge. The first question for our 

 consideration here i-, the pitch or declivity of the roof 

 or the angle ABC. We h.ive already shown under 

 C.viirt.NTiiY, p. .'>()(), that two beams, AH, AD will 

 bear the same load, being both equal to a horizontal 

 beam AE. If a beam Al), therefore, is just able to 

 carry the rooting which it is to bear, the more inclined 

 beam AH, having to support a greater quantity of 

 roofing tiom its greater length, will require to be 

 stron^tr than AD. Hence steeper roofs must always 

 require stronger begins, or the same beams to be placed 

 closer to each other in the proportion of their greater 

 length ; or the depth of the beam AB must be to that 

 of A D as the square root of A B is to t hesquare root of A D. 



In this construction of a roof, it is obvious that the 

 walls or abutments at A and C are supposed capable 

 of sustaining the joists at A and C, pressing them 

 outwards ; but as the ordinary walls of houses are 

 quite unfit to resist any such force, it becomes neces- 

 sary to resirt it by another of sufficient power. This 

 effect is obtained by introducing the beam AC, which 

 is called a te, from its binding together the feet 

 A and C, of the two inclined beams AB, CB. The 

 whole weight of the roof, therefore, in place of pushing 

 out the walls, is exerted in drawing out or stretching 

 the tie-beam AC in the direction of its length. If we 

 consider this tie-beam merely as a part of the roof, its 

 place might be supplied by a chain, or wire, or rope of 

 equal strength; but as it is often used to carry the 

 weight of the ceiling of the room below it, and some- 

 times to carry a flooring, it is generally made strong 

 and connected with the feet A and C of the rafter by a 

 mortice and tenon. 



When the tie-beam AC is long, it has a tendency 

 to bend down or sway at its middle E. It therefore 

 requires to be sustained at that part, and this is ef- 

 fected by suspending it by a short beam BE, from the 

 ridge B. This beam BE, is called the king pout, and 

 performs the part of a string or chain. The various 

 methods of joining the king posts or rafters, &c. -has 

 already been explained under CARPENTRY, Vol. V. 

 p. 537, nnd in Plate CXXVIII. Fig. 4, &c. 



When the rafters AB, BC are long, or considerably 

 loaded, they also have a tendency to bend. In order 

 to prevent this, braces or struts EF, EG are morticed 

 into them at C and F, and also into joggles at the 

 foot of the king post. By this means, the rafters have 

 their relative strength quadrupled, in consequence of 

 being reduced to half their original length. 



Having thus explained the construction of roofs, 

 consisting of two principal rafters, we shall proceed to 

 the consideration of those of a more complicated form, 

 where the rafters are more than two in number; in 

 which case it is generally called a kirb roof. 



We have already demonstrated in our article BRIDGE, 

 Vol. IV. p. 490, that if a string or festoMi of heavy 

 bodies connected together, is suspended from its two 

 extremities, (See Plate LXXX. Fig. 2.) they will ar- 

 range themselves into a Catenarian curve by the force of 

 gravity; and that if this assemblage of bodies is in- 

 verted so as to rest upon the former points of suspen- 

 sion, it will form an arch of equilibration. The same 

 is obviously true of any number of bars of metal or 



beams of wood, connected together by m /veabte joints, ' 

 o M to take the pmjtion of equilibrium, winch the > ' 

 force of gravity acting upon each of the beams mutt 

 necessarily. give them. 



The slightest conkideration is sufficient to convince 

 us that such a position of the beams i- that which they 

 should have when formed into a roof, fn thit posi- 

 tion, all the rafters re in equilibrium with each other, 

 and are acting on each other in the direction of their 

 lengths, and consequently resisting any external 

 and uniformly distributed strains acting in the direc- 

 tion of gravity with the greatest force. 



In Fig. 5, for example, let Afl, BC, CD, DE, be p, g . 

 rafters moving round flexible joint*, and arranging 

 themselves in the curve of equilibrium A<BCI>1 

 is the position which must be given them when fixed 

 into a kirbed roof, with this difference only, that they 

 must be placed in an inverted position. 



If they have any other position different from th.it 

 of equilibrium, such as is shown by the dotted lines in 

 Fig. 5, where A'>, (>c, cd, dE, are the rafters, then the 

 rafter cd must be held in its depressed position by some 

 external force ; and, consequently, when the whole u 

 inverted to form a roof, the rafter cd must have a 

 tendency to assume the position of equilibrium 

 CD, and in consequence of this unbalanced force cd, 

 and all the other beams will not act upon each other 

 in the direction of their lengths, and consequently 

 will not be in their strongest position. When they 

 are placed, on the other hand, in a position of equili- 

 brium, the tie beam, the kingpost, ami the braces, &c. 

 have to perform no other office but that of preserving 

 the rafters in their position of equilibrium. 



If the strain is uniformly diffused over the roof, 85 

 in houses covered with slate or lead, or if unequal loads 

 are symmetrically placed upon it, then the form of the 

 roof, or the curve of equilibrium will be symmetrical, 

 and its two halves will be equal and similar; but if 

 it is loaded more in one place than another, and if that 

 place is not on the ridge, then the form of the rafters 

 must be unsymmetrical. Thus in Fig. 5, if the rafters 

 AB, BC, CD, DE, are made equally heavy, the curve 

 will be as in the figure ; and is symmetrical, the 

 angle ABC being equal to the angle CDE. If BC 

 and CD, made equally heavy, are heavier than AB and 

 DE, which are equally heavy, then BC and CD will 

 fall lower, increasing the angles ABC and CDE, 

 and diminishing the angle BCD ; but still the curve 

 passing through the joints, will be symmetrical. If 

 the part of the roof CD is to be loaded with lead, 

 while all the rest is to carry only slating, then CD be- 

 ing much heavier than any of the other rafters, will 

 sink in the experiment of suspension to cd, raising the 

 joint B to b, and depressing C to c, and D to d. The 

 form of the roof will therefore be that of AAcrfE, 

 which is no longer symmetrical. 



Although it is now easy for the practical mechanic 

 to determine experimentally the position of the rafters 

 of a kirb roof mechanically, by loading the centres of 

 gravity of his experimental beams, in the same man- 

 ner as the corresponding beams in the roof are to be 

 loaded ; yet it is desirable to have a mathematical me- 

 thod of determining the best form of a kirb roof. 



By referring to our article BRIDGE, Vol. IV. p. 490, 

 and to Plate LXXX. Fig. 4, where the rafters are re- 

 presented as cords, it will be found to be demonstrated, 



1. That the tension in any part of the cord is in- 

 versely as the sine of its inclination to the vertical, and 

 1. That the loads on the different joints (C, C', C" ) 



