84CARPENTRY AND JOINERY.
that may be safely carried on the joist is a certain fraction of the breaking weight — that is, of the load that would break the joist. This fraction varies, for the different purposes for which the scantling is to be used, from one-fifth to one-tenth. In the case of floor timbers, where the joist has to sustain a live load, it should not exceed one- seventh or one-eighth the breaking weight. In the example given above, the joist has to carry a load of 18 cwt. Hence the breaking weight is equal to 18 x 8 = 114 cwt. (3) The breadth or thickness of the joist must bear a certain proportion to the depth so as to be satisfactory as regards strength and economy. Let this proportion for a bridging joist be decided by the formula b = '3 d, where b = the breadth and D the depth — all in inches. It is evident that the joist in such a case must be considered as strutted. The preliminary calculations as regards the joist having been made, a formula applicable to every case for calculating the strength of timber, no matter where or for what purpose the scantling may be required, must be decided on. A piece of wood of the same kind as that used for the joist, and 1 ft. long by 1 in. square, loaded at the centre till it breaks, will be the constant for all purposes of calculation when dealing with the same material. It will be found that the strength varies directly as the breadth, directly as the square of the depth, and inversely as the length ; this may be proved by increasing the breadth, length, and depth, and carefully noting the difference in the loads required to break the beam in each case. Briefly, the formula may be stated thus : BW = cbd/L that is, for a central load. But a floor- joist carries a distributed load, and this load will be found to be equal to twice the load it will carry when centrally loaded. Then the formula will be : — BW = 2cbd^2/L
114 = (2 x 4 x 3d x d^2 )/18 and d^3 = 144 x 18 )/2 x 4 x .3 = 1080.
d = 3V1080 = 10 in. nearly, and b = .3 x d = .3 x 10 = 3 in.
Let c be the constant = 4 cwt. ; b the breadth in inches ; d the depth in inches ; L the length in feet ; B.W. the breaking weight = 114 cwt. Therefore a joist 10 in. by 3 in. would be suitable for a span of 18 ft., and would carry a load of 1 cwt. per ft. super. The following rule is given by Tredgold for fir joist : — D = 3VL^2/B x 2-2 In this case a breadth must be assumed, which is, in most cases, a difficult and very uncertain proceeding ; however, assuming for the present example the breadth to be 3 in., Then -7». .0 b = ;/**i! x o.o. D = / 08 o.o -5 x 2-2 = 9 9 in. The result is very much the same as in the previous example, but the advantage of the first method will be obvious when dealing with further calculations, as it is applicable to other beams than floor timber. Determining Size of Binder. Say it is required to determine the size of a binder 10 ft. long and fixed 6 ft. apart, capable of carrying a floor weighing 1 cwt. pecxr ft. super. Make, as before, the necessary preliminary calculation. (1) Total load carried by the binder =10 x 6 = 60 cwt. (2) Breaking weight (say) seven times safe load = 60 x 7 = 420 cwt, (3) Let the ratio of the breadth and the depth be as 6 is to 10, that is *6 d, which is a very suitable ratio for all purposes where stiffness is required. (4) Let c the constant = 4 cwt. Then, using the same formula as before, breaking weight = 2cbd^2 / L
=-
x 4 x -§d x d 2 10
x 4 x ?3
.
x 10
d° — ~ ~ A a = 8 ?5
x 4 x "o
