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THE USE OF A STANDAED PAEABOLA FOE DEAWINd 
DIAOEAMS OF BENDING MOMENT AND OF SHEAE IN 
A BEAM UNIFOEMLY LOADED. 
By Alexander Brown. 
(From the Applied Mathematics Laboratory, South African College.) 
(Eead September 15, 1915.) 
§ 1. The important stresses in a uniform continuous beam are the shear 
and the bending moment ; they are best shown in the form of graphs, where 
length along the beam is taken as abscissa and the required function as 
ordinate. The values of the bending moment at the points of support can 
be obtained by the equation of three moments, and the variation between 
supports follows the parabolic law. The diagram of bending moment may 
therefore be drawn point by point. The pressures may be got by using the 
equation giving pressure in terms of three bending moments, or by taking 
moments about successive points of support. 
The form of bending moment graph generally used by engineers for a con- 
tinuous beam of several spans is derived from the graphs for the separate 
spans considered as discontinuous beams ; the difference between the bending 
moments for the two cases is a linear function of the distance along the 
beam ; the linear function is obtained from knowledge of the bending 
moments at the points of support ; and the bending moment for the con- 
tinuous beam is represented graphically as the difference of the ordinates of 
the graph for discontinuous beams, and of a straight line. This method is 
particularly valuable in irregular loading of the beam. 
For a uniformly loaded beam a simpler method may be used, and in what 
follows it is shown that a parabola of standard latus rectum may be used to 
obtain the graph of bending moment, the shear diagram, and the values 
of the pressures without further calculation than is recjuired to give the 
values of the bending moment at the points of support. The case of a uni- 
form beam loaded at isolated points is included, the isolated points being 
considered as points of support. 
The method holds for beams of variable section provided the bending 
moments at the points of support have been determined, the variation of 
bending moment and of shear between such points depending only on the 
intermediate loading. 
§ 2. Let X be the distance of a point P on a uniform continuous beam 
