Engineering Mechanics

Chapter 16 : Virtual Work „„„„„ 351

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 is 1.5 y as shown in Fig. 16.12.

Fig. 16.12.

∴ Total virtual work done by the two reactions RA and RB

= +[(RA × 0) + (RB × y)] = + RB × y

...(Plus sign due to reactions acting upwards)

and total virtual work done by the point load at C and uniformly distributed load between A and C.

01.5

–(1 1.5) 2 3

2

y

y

⎡ ⎛⎞+ ⎤
=× +⎢⎥⎜⎟×
⎣⎦⎝⎠
= – (1.5y + 4.5 y) = – 6y
...(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 – 6y =0

or RB = 6 kN Ans.

and RA = (2 × 3) + 1 – 6 = 1 kN Ans.

EXERCISE 16.1

1. A simply supported beam AB of span 4 m is subjected to a point load of 10 kN at a
   distance of 1.5 m from A. Using the principle of virtual work, determine the reactions at
   the two supports.
   (Ans. 3.75 kN ; 6.25 kN)


2. Two beams AD and DF of spans 6m and 4m respectively are hinged at C and supported at
   A, D and F. The beams are loaded as shown in Fig. 16.13.

Fig. 16.13.

Using the principle of virtual work, find the reaction at D.(Ans. 22.9 kN)