118 APPLIED MECHANICS 
The portion AB of the beam may be looked upon as a cantilever 
fixed at B, like that in Fig. 156, the bending moment diagram being 
AHKB, and the deflection P,Q measured from the horizontal B,C will 
be equal to ka,(%,— x), where / is a constant. The total deflection A,C 
is equal to ka,t,. Hence P,P, =A,C—P,Q=ka,%, —ka,(%,—«), and 
P, Py _ Gyy — a(Z, — %) _ AB, + Ay, — 0, (@, —%) _ Asko + He 
ae asks Agits Abs 
Consider the figure AHKS to be a load diagram, Let A’DD’S’ be 
the shearing force diagram, and A,P,B,S, the bending moment diagram 
corresponding to the load diagram AHKS. Then the point where the 
shearing force is zero must be in a vertical line through B,, the lowest 
point in the bending moment diagram, and the reaction R, must equal 
the load represented by the area AHKB, therefore R, = ga,, where g is a 
constant. Then the bending. moment at P is equal to 
R,x — qd,(% — %) = gage — ga,(a — Z) = P,P. 
Also, the bending moment at B is equal to 
Ril — qas(t — &5) = gash — qag(t — %,) = Gast, = AC, 
P,P, _ aye — a,(@ — %) _ (4 + Ay)a * A(t — Fy) _ Ax+ ay%, 6 
and = P= 
A,C re I As%y A,X, 
Hence whether the curve A,P,B,S, be considered as a deflection curve 
or a bending moment curve, the ratio of P,P, to A,C is the same, and 
therefore the bending moment curve will represent the deflection at every 
point. 
127. Beam of Uniform Section Supported at the Ends and Loaded 
at any Intermediate Point.—AB (Fig. 159) is a beam resting on supports 
whose distance apart is L. This 
beam carries a load W at a point C 
at distances a and 6 from A and B 
respectively. The bending moment 
- Wab 
at C is 
nate CD equal to this bending U 
moment, and joining A and B to of Whe 
D, the figure ADB is the bending ns m 
moment diagram for the beam carry- '*—--- © ---~-—> ae 
ing the load W at C. R 
Now consider ADB to be a load ™ ae 
diagram. The resultant P, of the load G 
represented by the triangle ADC is . 
Wad 
2L 
vertical line through the centre of gravity of the triangle ADC. The 
resultant P, of the load represented by the triangle BDC is equal to 
Wal? 
2L 
, and making the ordi- 
Fig. 159. 
equal to , and P, acts in a 
, and P, acts in a vertical line through the centre of gravity of the 
