Short Answer:

The principle of virtual work states that if a system is in equilibrium, the total virtual work done by all the forces acting on the system during a virtual displacement consistent with its constraints is zero. In other words, when a body is at rest or in equilibrium, the sum of the work done by all forces for any small imaginary (virtual) displacement is equal to zero.

In simple terms, this principle helps determine equilibrium conditions without calculating reaction forces. It is widely used in the analysis of structures, mechanisms, and machines where the forces and motions follow geometric constraints.

Detailed Explanation :

Principle of Virtual Work

The principle of virtual work is one of the most powerful methods in mechanics for studying the equilibrium of systems of particles and rigid bodies. It is based on the concept of virtual displacement, which is an infinitesimal imaginary change in the configuration of a system consistent with its constraints.

This principle is particularly useful in solving complex equilibrium problems where several forces act on a body, and not all reaction forces are known. It allows engineers to determine unknown forces and reactions using energy concepts instead of directly applying Newton’s laws of motion.

Definition of Principle of Virtual Work

The principle of virtual work can be stated as:

“If a body or system of bodies is in equilibrium, then the total virtual work done by all the forces (including external and internal forces) acting on the body during any virtual displacement consistent with the constraints is zero.”

Mathematically, this can be expressed as:

Where,

•  = total virtual work,
•  = force acting on the particle or point ,
•  = virtual displacement of the point in the direction of the force.

If the algebraic sum of the virtual works done by all forces is zero, the system is said to be in equilibrium.

Concept of Virtual Work

In mechanics, work is said to be done when a force causes a displacement in the direction of the force. Similarly, virtual work is the work done by forces when a body undergoes a small imaginary (virtual) displacement, consistent with its constraints.

The displacement considered here is not an actual motion but a hypothetical one, small enough to assume linear behavior of the system. The virtual displacement must always obey the system’s geometric restrictions — for example, if a particle is constrained to move along a straight line or a circular path, the virtual displacement must also occur along that same path.

Conditions for the Principle of Virtual Work

For the principle of virtual work to hold true, the following conditions must be satisfied:

1. The system must be in equilibrium.
2. The virtual displacements considered must be consistent with the system’s constraints.
3. The virtual displacements are infinitesimal and instantaneous — they do not take place over time.
4. The work done by reaction forces due to constraints (like hinges or supports) is zero, because such forces act perpendicular to the virtual displacement.

Mathematical Formulation for a Particle

Let a particle of mass  be acted upon by several forces . If the particle undergoes a small virtual displacement , then the total virtual work done is:

For equilibrium, the resultant of all forces must be zero, i.e.,

Hence,

This shows that the total virtual work done by all forces on a body in equilibrium is zero.

Example of Principle of Virtual Work

Example: Determining the Reaction at a Support

Consider a simply supported beam  carrying a load  at its midpoint . Let  and  be the reactions at supports  and .

Now, imagine a small virtual displacement  at the midpoint , such that point  moves slightly downward. Because of this:

• The beam rotates about the supports  and .
• The displacements at  and  are zero since they are fixed supports.

The virtual work done by:

• Load  =  (downward)
• Reaction at , , and at ,  =  (no movement at supports)

For equilibrium, total virtual work must be zero:

Since the load  acts at the midpoint, the reactions are equal and each support carries half of the load:

This example shows that the principle of virtual work can be used to find unknown reactions without directly using equations of equilibrium.

Applications of Principle of Virtual Work

The principle of virtual work has wide applications in mechanical and civil engineering, especially in systems where direct force analysis is difficult. Some common applications include:

1. Structural Analysis:
   Used to determine support reactions and internal forces in beams, trusses, and frames.
2. Mechanisms and Machines:
   Applied to study equilibrium and motion in linkages, gears, and robotic systems.
3. Statics and Dynamics:
   Helps in analyzing systems of particles and rigid bodies both at rest and in motion.
4. Energy Methods:
   Forms the foundation of more advanced analytical methods such as Lagrange’s equations and Hamilton’s principle.
5. Engineering Design:
   Used in determining efficiency, load distribution, and deformation in mechanical components.

Advantages of Using Virtual Work Principle

• Simplifies problems with multiple forces and constraints.
• Eliminates the need to consider reaction forces directly.
• Provides an energy-based approach for analyzing equilibrium.
• Can be applied to both static and dynamic systems.

Conclusion

The principle of virtual work states that for a system in equilibrium, the total virtual work done by all forces during any small, imaginary displacement consistent with the system’s constraints is zero. It is a powerful and general method used for analyzing equilibrium in complex systems of particles and rigid bodies. By applying this principle, engineers can easily determine unknown forces and reactions without solving multiple equilibrium equations. Hence, it plays a vital role in structural and mechanical system analysis.