Chap. XXII.
TIMBER ROOFS.
1319
Fig. 2107.
Fig. 2108.
Tie-beams, at their introduction, com-pletely altered the principles of framingroofs ; the ends of the rafters were nowheld together at the feet, by framing theminto a horizontal piece of timber whichacted like a rope, and resisted their beingpulled asunder, in the direction of itslength. When a heavy body is supportedby two pieces of timber, A C and B C, itseffects depend upon their position ; the far-ther the ends of the timbers are apart,the greater is the strain upon them : theweight resolves itself into two forces, onein the direction of each beam.
The weight W might be sup-ported upon an upright post in thedirection of Cc, where its verticalforce is equal to the weight: thismay be resolved again into twoforces, acting in the direction of thebeams, that would produce the sameeffect as the vertical force Cc.
If the vertical line Cc be drawnthrough the centre of the weight,and ac be drawn parallel to therafter or beam A C , also be parallelto B C ; then the weight and pres-sures exerted will be found thus: —
as the line Cc, is to the line Cb, so is the weight W, to the pressure in the direction of thebeam Ac .• and as the line Cc, is to the line Ca , so is the weight W, to the pressure in thedirection of the beam C B.
If the position of the beam were that shown by the dotted lines CE the magnitude ofthe strain would be considerably increased on both beams : by drawing lines parallel tothem in this position, expressing the weight by the line Cc, then the pressure on the beam,in the position CE, is expressed by the line Cd, instead of Ca, and the strain on C A bythe line Ce ; we find that considerable strains are often the result of comparatively triflingweights, when a change is made in the position of the supports A, B, and E. By drawingout a variety of diagrams we are enabled to form some idea of the changes which posi-tion can effect, and to judge of the alterations in the duties which the several timbers com-bined into a truss or piece of framing have to perform ; the object of the carpenter oughtalways to be that of not relieving one piece at the expense of another, but of so arrangingthe whole that the burden may be equally distributed.
An easy method of ascertaining how much each sus-tains is, by passing a string over two pulleys, as B and-C, and tying another to any point at A, which has aweight attached to it at W: if the sum of the weightsb and c be greater than the single weight W, it is aposition in which they may be placed, and will allowthe three weights to be at rest, or in equilibrio.
"When the whole is in this state of equilibrium, draw thefigure on paper, and from a scale of equal parts makeA F equal to the number of pounds contained in theweight \Y, and continue the line B A to E, and drawthe line EF parallel to A C : then FE measured onthe scale will express the number of pounds on theweight at c; and the length of A E by the same scalewill show the number of pounds in the weight b .
When the three weights are equal, the three linesA F, F E, and A E will be equal, and all the anglesround the point A, where the perpendicular string istied, will be equal. When these three forces are in thesame plane, and meet in a point, they are always inequilibrio, and the forces are represented by the threesides of a triangle drawn parallel to the directions of theforces. When a weight is kept at rest by three forces, and any two are represented inmagnitude and direction by the two sides of a triangle, the third side, taken in order, willshow the magnitude and direction of the other force.
4 v 4