Engineering Mechanics

(^348) „„„„„ A Textbook of Engineering Mechanics
Let y = Virtual upward displacement of the beam at C.
From the geometry of the figure, we find that when virtual upward displacement of the beam at
C is y, then the virtual upward displacement of beam at E is
6
1.2
5
y
= y as shown in Fig. 16.7 (a)
Therefore total virtual work done by the two reactions RA and RC
= + [(RA × 0) + (RC × y)] = + RC × y ...(i)
...(Plus sign due to reaction at C acting upwards)
and virtual work done by the point load at E
= – (500 × 1.2y) = 600y ...(ii)
...(Minus sign due to load acting downwards)
We know that from principle of virtual work, that algebraic sum of the total virtual work
done is zero. Therefore
∴ RC × y – 600 y=0
or RC= 600 N Ans.
Now consider the beam BD with loads at C and F as shown in Fig. 16.7 (b)
Let x = Virtual upward displacement of the beam at B.
From the geometry of the figure, we find that when virtual upward displacement of the beam at
B is x, then the virtual upward displacement of the beam at C and F is^6 0.75
8
x
= x and
3
0.375
8
x
= x
respectively as shown in Fig. 16.7 (b).
Therefore total virtual work done by the two reactions RB and RD
= + (RB × x) + (RD × 0) = + RB × x ... (iii)
...(Plus sign due to reactions acting upwards)
and virtual work done by the point loads at C and F
= – [(600 × 0.75 x) + (1000 × 0.375 x)] = – 825 x
...(Minus sign due to loads acting downwards)
We know that from principle of virtual work, that algebraic sum of the total virtual works done
is zero. Therefore
RB × x – 825x=0
or RB= 825 N Ans.
Note. In this case, we have assumed the virtual upward displacement of the beam at B, because
it is hinged at D. However, if we assume the virtual upward displacement at D, it is not wrong. In this
case, we shall obtain the value of reaction at D.
16.7.APPLICATION OF THE PRINCIPLE OF VIRTUAL WORK FOR BEAMS
CARRYING UNIFORMLY DISTRIBUTED LOAD
Consider a beam AB of length l simply supported at its both ends, and carrying a uniformly
distributed load of w per unit length for the whole span from A to B as shown in Fig. 16.8 (a).
Let RA = Reaction at A, and
RB = Reaction at B.