660 
Transactions of the Royal Society of South Aftica. 
from a fixed point on the beam ; let Gr be the bending moment at P (con- 
sidered positive when the concavity is upwards), and w the weight per 
foot run, then 
Gr = K + La? — I wx"- 
where K and L are constants, which are different for each span. 
The term — ^ wx'^ is the same for all spans, and so the various parts of 
the bending moment graph are all parabolas of the same latus rectum with 
their axes perpendicular to the length of the beam. 
Suppose we have lines drawn through the points of support perpendicular 
to the beam, and lengths set off on them to represent on a proper scale the 
bending moments at these points. To get the bending moment graph 
between these points we have to draw through the ends of the perpendicular 
lines a parabola with the proper latus rectum and with its axis perpendicular 
to the beam. If we have a parabola of suitable latus rectum cut out of 
cardboard, the work can be done by adjusting this cardboard parabola, 
keeping its axis perpendicular to the beam, till it passes through both 
points. If pins are put through the points the adjustment is quickly 
made. 
The shear may lie easily got from the same figure. If H is the shear 
is proportional to the slope of the parabola, which again is . propor- 
tional to the distance from the vertex measured parallel to the tangent at 
the vertex. Thus in drawing and cutting out the parobola, care should be 
taken that for any point on the parabola the abscissa of the point can be 
easily read off. If the parabola is properly adjusted in place, its boundary 
is the bending moment graph, and the abscissa readings represent the 
shear on some scale. When the shear is known at two adjacent points of 
support, the shear diagram is completed by drawing a straight line. When 
the shears on two sides of one point of support are known, their difference 
gives their pressure on that support. 
§ 3, Consider the c[uestion of suitable scale for the parabola. 
Let X be the distance in feet along a uniform beam from a point where 
the shear is zero, and the bending moment Ctq, then 
Gr = Gr^) — i WX"^, H = WX. 
We propose to use a parabola »3 = drawn on squared paper to repre- 
sent the values of G and H corresponding to any value of a; ; | is to repre- 
sent length along the beam and also shear, while n is to represent Gtq — Gr. 
Suppose the scale of the figure is such that one unit of | represents n 
feet; then on some scale | represents shear and n bending moment. 
Suppose one unit of | represents x units of shear (lb. weight) 
and ,, ») „ V bending moment (foot-lbs,). 
