Virtual Work

by Dr. Abhijit Chaudhuri
Dept. of Applied Mechanics
IIT Madras, Chennai, India

VIRTUAL WORK

F
F

1. Work on a particle:

U   r  F

r

The particle can move along a specified a path

2. Work on a rigid body:

F
i) Due to translation

F
U   r  F   r W
r
M

ii) Due to rotation

M



 U    M

by Dr. Abhijit Chaudhuri
Dept. of Applied Mechanics
IIT Madras, Chennai, India

The equilibrium position of the particle under the action of different forces, needs to be determined

 r is an imaginary and arbitrarily small
1

displacement away from its natural position.

r
2

4

3

It is called virtual displacement if it is consistent with the system constraints.
FBD of the particle at its natural position:

F1

F2

F4
F3

The virtual work ( U ) due to virtual displacement is,

 U   r  F1   r  F2   r  F3   r  F4   r   F1  F2  F3  F4    r  R
If the particle is in equilibrium i.e. R

 0, the virtual work,  U  0

So for any equilibrium system, the virtual work is always zero.

by Dr. Abhijit Chaudhuri
Dept. of Applied Mechanics
IIT Madras, Chennai, India

In case of rigid body under equilibrium condition:


As all particles inside the rigid body are in equilibrium, the virtual work on all of them will be zero



Thus the virtual work done on the entire rigid body is zero.



So the virtual work done by the external forces is zero since all internal forces occur in pairs of equal, opposite and collinear forces.

Y
C

P

Find out the reaction at B by method of virtual work

 C,x

P

h
X
A



B

l

 C, x  h ,  B, y  l





 U  P  C , x  RB, y   B, y  0  RB, y 

P  C ,x

 B, y



Ph l  B, y

RB,y

Degrees of freedom (DoF)

by Dr. Abhijit Chaudhuri
Dept. of Applied Mechanics
IIT Madras, Chennai, India

Number of independent coordinates needed to completely specify the configuration of the system.

1 DoF system

1 DoF system

2 DoFs system

l1

A

h

by Dr. Abhijit Chaudhuri
Dept. of Applied Mechanics