The principle of equilibrium demands, for the sake of stability, that the strains which these stresses impose upon structures and their different members shall be met by a resistance in the nature and strength of the materials capable of withstanding those strains. Their actions must be met by corresponding reactions, which are located, as it were, at the points of support.
For instance, a girder, weighing 1 ton, and carrying a load of 11 tons, is supported by two columns, which, it is obvious, must be of sufficient strength to be able to keep up or resist between them the 12 tons of pressure or strain which the stress or action of the load imposes upon them; or, reverting to the "tug of war," the competing teams must be evenly balanced to keep in equilibrium, thus making it clear that actions must be met by reactions of equal or greater strength in order that the latter may succeed or hold their own.
When a girder is uniformly loaded, or carries its load in the centre, it is obvious that each support will carry an equal proportion; therefore the two reactions required to meet the strain on the supports must together be equivalent to the load, and of equal (or greater) power.
Thus, a girder, weighing 1 ton, and carrying 11 tons, either at its centre, as Fig. 932, or uniformly distributed over its whole length, as Fig. 933 (the total load, or external force, being 12 tons), brings about an action of 6 tons at each support, and each of these has to be met by a reaction of an upward tendency, equivalent (at least) to 6 tons, to prevent the columns being crushed down by those loads.
When these loads, however, are placed in different positions at unequal distances from the supports, it is plain that the stress from the loads acts unequally on the supports; that nearest the load, of course, taking the lion's share.
In such cases the principle employed in ascertaining the amount of strain imposed on each support is that of "proportion," as regulated by the different lengths into which the girder is divided by the load. Thus a girder 20 feet long, with a load of 10 tons 5 feet from one of the supports and 15 feet from the other, would transmit its load to the supports in the proportion of 5 and 15, the one nearest to the load, of course, taking the 15 parts - i.e., three-quarters (as Fig. 934). The rule for calculating the reactions required is therefore: "Multiply the load by its distance from the other support, and divide by the span," to give the reaction. Thus: -
Reaction at A =
50 =2 l/2 tons 20
= 10 tons.
Reaction at B =
150 = 7 1/2 "
The sum of the two reactions, as calculated, should be exactly the same as the load.
When two or more loads are placed on one girder, each load must be dealt with separately, as shown below in reference to Fig. 935, which illustrates a girder 20 feet long, loaded with 6 tons at a point 5 feet from A, 4 tons 4 feet from B, and 8 tons in the centre, making, in all, a load of 18 tons to be carried by the two supports in the proportions ascertained.
Necessary for lead of 6 tons =
4 1/2 tons
9 3/10 tons.
" 8 " =
" 4 " =
Necessary to complete load of 6 tons =
1 1/2 "
8 7/10 tons.
8 " =
4 " =
3 1/2 "
6 tons + 4 tons + 8 tons = total of l8 tons
With regard to cantilevers, which have only one support, the reaction required is the same as the load to be carried by the cantilever, together with its own weight.
It will have been noticed that the principle of "leverage" plays a conspicuous part in the proportioning of the loads to the supports and their necessary reactions; and it is this same principle, in a different form, which gives us the undoubted and undisputed fact that concentrated loads cause double the stress and strain of the same load uniformly distributed; each part of the latter load, nearer the support, assisting to balance its outer neighbour, as it were; just as a man would carry a given quantity of water in two buckets, one on each side, more easily than he would carry it if it were all in one large bucket, and carried on one side.
For method and convenience of calculation, these distributed loads are grouped together, as it were, and reckoned as one load at its centre - i.e., the centre of the distribution, as Fig. 936.