### Principle of Virtual Work

• Apply principal of virtual work in statics by understanding virtual displacements and virtual work.

“If a system in equilibrium under the action of a set of forces is given a virtual displacement, the virtual work done by the forces will be zero.”

The terms used in this statement are defined as follows:

• A virtual displacement $\delta$s is an imaginary infinitesimal variation of the coordinate given instantaneously. The virtual displacement must be compatible with the constraints of the system.

• Virtual work $\delta$W is the work done by all the active forces in a virtual displacement. Since there is no significant change geometry associated with the virtual displacement, the force acting on the system are assumed to remain unchanged for the calculation of $\delta$W.

### Solved Example:

#### 24-6-01

Using principle of virtual work, determine the force P which will keep the weightless bar AC in equilibrium. Take length AB as 2m and length AC as 8m. The bar makes an angle of 30$^\circ$ with horizontal. All the surfaces in contact are smooth. Refer figure.

Solution:

By using the principle of virtual work, \begin{align*} P\dfrac {d}{d\theta }\left( 8\cos \theta \right) +800\left( \dfrac {d}{d\theta }6\sin \theta \right) &=0\\ P\left( -8\sin\theta \right) +800\left( 6\cos \theta \right) &=0\\ 8P\sin \theta &=4800\cos \theta \\ P&=600\cot \theta \\ &=600 \cot 30\\ &=1038\ N \end{align*}

### Solved Example:

#### 24-6-02

Virtual work refers to: (TNTRB ME 2017)

### Solved Example:

#### 24-6-03

The principle of virtual work states that, for a body to be in equilibrium, the virtual work should be: (CIL MT Civil 2020)