Coming to uniform loads on cantilevers, it is usual to take the collected load, or total weight, as located in the centre of the loaded area, as Fig. 933, from which it will be seen that it is in this case practically halfway along the cantilever, and the formula resolves itself into (Wl)//(2d), twice the depth being taken to give the "half" result as compared with a concentrated load; for we know that the latter causes double the strain of a uniform load. Therefore -
(wl)/(2d) = (4tons x 3feet)/(2x 1foot)= 12/2= 6 tons strain.
Graphically this is ascertained by a similar process to Fig. 950, the diagonal being raised from a point halfway along the bottom flange, as Fig. 953, while the graduation of the strains at intermediate points takes tue form of a concave parabolic curve, as shown on Fig. 954, which should need no further explanation.
The theoretical form of the cantilever would be either as Fig. 955 or Fig. 956.
If two distinct loads are placed on a cantilever their strains should be calculated separately, and the results added together;. while graphically they should be treated as Fig. 957, of which the small diagram is constructed, to give the lines of inclination to the Fig.ure. A perpendicular is dropped from any point as Z, and on it are marked off consecutively from the point the weights or reactions of the loads, Z Y representing to scale 1 ton, and Y X 2 tons. The depth, which always plays such a conspicuous part, is marked off at right angles to Z X from Z, as W; and then W Y and W X are joined. Reverting to the main Fig.ure, from A we draw A B parallel to W Y until it meets a perpendicular raised from the other load; after which A B is continued to C, but parallel to WX. Thus the diagram is formed, and the strains can be scaled off as required. If the cantilever were uniformly loaded, it would be treated similarly to Fig. 954 in principle.
The "shearing" strain on a cantilever, with a concentrated load, is equal to the amount of the reaction all along the cantilever from the point of support to that of application of the load, as Fig. 958; while, on uniformly loaded cantilevers, it gradually diminishes to nothing at the point of application from the amount of the reaction at the support, as Fig. 959, one part of the load assisting the other, as it were.
A cantilever differently loaded at two or more points would have its "shearing" strain shown, as Fig. 960; and if a uniform load is fixed at a distance and clear space from the support, it will be as Fig. 961, taking the form of a concentrated load from A to B.
Girders are treated in a similar manner to cantilevers, their strains being ascertained by the following formula: -
(wl)/(4d) for concentrated loads,
(wl)/(8d) for uniform loads.
So that a girder of 8 feet span and 9 inches deep, supporting a uniform load of 2 tons, would have its strain calculated thus: -
(wl)/(8d)= (2 tonsx 8feet)/(8x 9 inches)= 16 feet/6 feet= 2 2/3 tons strain, which would be doubled if concentrated.
Graphically, a girder, with concentrated load centrally, would be dealt with as Fig. 962, and as Fig. 963 if intermediately loaded; while for uniform loads it would be as Figs. 964 and 965 respectively; one half of the load (or either of the reactions) being set up as the height, when the reactions are equal; and, when they vary, one or the other will answer the purpose, so long as care is taken that the same side of the load is utilised for the diagonal and parallel line to ascertain the strain, as particularly illustrated on Fig. 963. If correct, Fig.ures will always complete themselves.
The theoretical outlines for beams or girders are as Figs. 966 and 967 for concentrated oads, and Figs. 968 and 969 for uniform loads.