68 
Proceedings of the Royal Society of Edinburgh. [Sess. 
VIII. — On a Graphical Method of determining Shear Influence 
Lines and Diagrams of Maximum Shearing Force for a 
Beam subjected to a Series of Concentrated Rolling Loads. 
By Alex. R. Horne, B.Sc. (Lond.), Professor of Engineering, Robert 
Gordon’s Technical College, Aberdeen. 
(MS. received November 30, 1920. Read March 7, 1921.) 
The shear influence line is a line the ordinates of which give the values 
of the shearing forces at any one point in a beam or bridge as a load, or a 
series of loads, pass over it. There is thus, for any one beam, an influence 
line for every point in it. 
These influence lines are of great value in the design of structures, such 
as bridges and arches, where it is necessary to determine the greatest maxi- 
mum and minimum shearing forces which occur at every point in them. 
The methods generally used to obtain these lines prove laborious in 
practice, especially when there are, as is often the case, several loads, such 
as the wheel loads of a locomotive. The ordinates of each influence line 
are generally determined by calculation, when it becomes necessary to 
estimate the shearing forces for many positions of the loading. Alterna- 
tively, a graphical method, which requires the construction of funicular 
polygons, and which affords only approximate results, is resorted to. This 
latter method is inconvenient when the load length exceeds the span, as is 
often the case in practice. 
In this paper, a simple graphical method of constructing an influence 
line is explained ; and the system is extended to provide a ready means of 
drawing the influence lines for as many points in the beam as may be 
desired. From these a diagram of maximum positive and negative shears 
can be constructed. No calculation whatever is required, and the method 
is an exact one. Moreover, the system is not limited to the case where the 
total length of the load does not exceed the span of the beam. 
Let a series of loads, W 2 , W 2 , W 3 (fig. 1) cross a beam AB, of span L, 
moving towards the right. When the leading load, W l5 is over the right 
abutment B, the bending moments at A, due to W v W 2 , W 3 , are M 1? M 2 , M 3 
respectively. Let the total bending moment at A, due to these loads, be 
represented by cd to a scale of 1 " = m units. 
If the beam is freely supported at A, the resultant bending moment 
there is zero. It follows that, if R is the reaction of the support at B, 
RL = Mj + M 2 + M 3 = cd (in inches) x m, 
