What is the principle of virtual work?

Short Answer:

The principle of virtual work states that when a system is in equilibrium, the total virtual work done by all the forces acting on the system during a small virtual displacement is equal to zero. In simple words, if a body is in equilibrium, then the sum of the work done by all applied and reaction forces for any small imaginary movement of the body is zero.

This principle is widely used in mechanics and structural analysis to determine unknown forces and reactions without directly solving the equations of motion. It is a powerful method because it considers equilibrium through work instead of forces.

Detailed Explanation:

Principle of Virtual Work

The principle of virtual work is a fundamental concept in engineering mechanics and structural analysis. It provides an alternate way of expressing equilibrium conditions for a system of forces acting on a body.

Instead of directly applying Newton’s laws of motion, the principle uses the concept of work and virtual displacement to determine whether a body or structure is in equilibrium. The word “virtual” here means imaginary or assumed—that is, the displacement does not actually occur but is considered hypothetically to test equilibrium.

In simple terms, if a body is in equilibrium under the action of various forces, then the total work done by all these forces for a small imaginary displacement compatible with the constraints of the system must be zero.

Definition

The principle of virtual work can be defined as:

“If a system of forces acting on a body or structure is in equilibrium, then the total virtual work done by all the forces during a small virtual displacement consistent with the constraints of the system is equal to zero.”

Mathematically, it is expressed as:

or,

where,

  •  = virtual work done by a force during a virtual displacement,
  •  = magnitude of the ith force,
  •  = corresponding virtual displacement of the point of application of the force in the direction of the force.

Explanation of the Principle

To understand the concept better, consider a body acted upon by several forces. If the body is in equilibrium, then these forces balance each other. Now imagine the body is given a small virtual displacement — an assumed small movement that does not actually happen but helps in the analysis.

During this imaginary movement:

  • Each force does some virtual work equal to the product of the force and its virtual displacement component in its line of action.
  • For equilibrium, the sum of all these small virtual works must be zero because there is no net unbalanced force to cause actual motion.

Thus,

This condition ensures that the system is in equilibrium under all forces acting on it.

Virtual Displacement

A virtual displacement is an imaginary, infinitesimally small displacement given to a body in equilibrium while keeping all the constraints intact.

  • It is not a real displacement but a hypothetical one used for analysis.
  • The direction of the virtual displacement must be consistent with the system’s geometry and constraints.

For example, if a particle is constrained to move along a straight line, its virtual displacement will also be along that line.

Work Done by Forces

If a force  acts on a body and it undergoes a virtual displacement , then the virtual work done is given by:

where,

  •  is the angle between the force and displacement directions.
  • If the force and displacement are in the same direction, the virtual work is positive.
  • If they are in opposite directions, the virtual work is negative.
  • If the displacement is perpendicular to the force, the virtual work is zero.

In equilibrium, the algebraic sum of all such virtual works must be zero.

Mathematical Expression for a System

For a system of forces acting on a body in equilibrium,

This equation is known as the mathematical statement of the principle of virtual work. It is valid for both particle and rigid body systems.

Applications of the Principle of Virtual Work

  1. Determination of Unknown Reactions:
    The principle is used to find reactions in statically determinate and indeterminate structures like beams, trusses, and frames.
  2. Structural Analysis:
    Applied in determining internal forces and deflections in structures under various loads.
  3. Mechanics of Machines:
    Used to analyze linkages, levers, pulleys, and gears to find required forces or torque.
  4. Equilibrium of Particles and Rigid Bodies:
    Helps verify if the system is in equilibrium under the influence of applied forces and constraints.
  5. Elastic Systems and Deformations:
    Used in energy methods to find deflections and stresses in deformable bodies.

Advantages of the Principle

  1. Simplifies complex equilibrium problems without directly resolving forces.
  2. Applicable to both linear and angular systems.
  3. Reduces the number of equations needed compared to using Newton’s laws.
  4. Helpful in analyzing constrained systems where motion is restricted.
  5. Useful for both static and dynamic problems.

Example of Virtual Work Principle

Example:
Consider a beam  supported at two points with an external load  at its center.
If the beam is in equilibrium, the reactions at the supports  and  balance the applied load.

Now imagine the beam undergoes a small virtual downward displacement  at the point of load application.
The virtual works done by:

  • The load :  (positive, since displacement is in the direction of load).
  • The reactions  and :  and  (negative, as they act opposite to displacement).

For equilibrium:

From this equation, the unknown reactions  and  can be determined.

Conditions for the Principle to Hold True

  1. The body must be in equilibrium.
  2. Displacements must be infinitesimally small (virtual).
  3. Virtual displacements must be consistent with the system’s constraints.
  4. All forces considered must be acting at the instant of equilibrium.

Practical Use in Mechanical Engineering

  1. Truss Analysis:
    Used to calculate forces in members without solving multiple equations of equilibrium.
  2. Beam Deflection:
    Helps determine deflections and slopes in loaded beams using virtual work equations.
  3. Mechanism Design:
    Applied to study input-output relationships in levers, linkages, and cams.
  4. Energy Methods:
    Provides a foundation for advanced energy-based methods like Castigliano’s theorem and strain energy principles.
  5. Machine Components:
    Used in designing gears, pulleys, and shafts where torque and force relationships are studied.
Conclusion

The principle of virtual work states that for a body in equilibrium, the algebraic sum of all virtual works done by the forces during a small imaginary displacement is zero. It is a fundamental principle used in analyzing equilibrium problems in mechanics, machines, and structures. The method simplifies complex systems by using work instead of direct force analysis. It forms the basis for many advanced mechanical and structural analysis methods, making it an essential concept in mechanical engineering.