Engineering Mechanics

(^350) „„„„„ A Textbook of Engineering Mechanics
From the geometry of the figure, we find that when the virtual upward displacement of the
beam at B is y, then the virtual upward displacement of the beam at C and D is 0.2 y and 0.4 y
respectively as shown in Fig. 16.10.
Fig. 16.10.
∴ Total virtual work done by the two reactions RA and RB
= + [(RA × 0) + (RB × y)] = + RB × y ...(i)
...(Plus sign due to reactions acting upwards)
and total virtual work done by the point load at C and uniformly distributed load between D and B.
0.4
–(5 0.2) 2 3
2
yy
y
⎡ ⎛⎞+ ⎤
=× +⎢⎥⎜⎟×
⎣⎦⎝⎠
= – 5.2 y ...(ii)
...(Minus sign due to loads acting downwards)
We know that from the principle of virtual work, that algebraic sum of the total virtual
works done is zero. Therefore
RB × y – 5.2 y = 0
or RB = 5.2 kN Ans.
and RA = 5 + (2 × 3) – 5.2 = 5.8 kN Ans.
Example 16.5. An overhanging beam ABC of span 3 m is loaded as shown in Fig. 16.11.
Fig. 16.11.
Using the principle of virtual work, find the reactions at A and B.
Solution. Given : Span AB = 2 m and span BC = 1 m
Let RA = Reaction at A,
RB = Reaction at B, and
y = Virtual upward displacement of beam at B.