  # Principle of Virtual Work: What is Virtual Work? Explain it

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Updated on: September 25th, 2023 Since the beginning of statics research, the concept of the Principle of Virtual Work has been applied to some degree. It was known as “the law of lever” and was employed by the Greeks, medieval Arabs, and Latins. The principle of virtual work and the associated calculus of variations were initially created to examine systems of rigid structures. Still, they were later expanded to include the study of the mechanics of deformable bodies.

The study of forces and movement in a mechanical system leads to virtual work when the concept of least action is applied. Because the Principle of virtual work could be taken alone as the basis for all of the mechanics, the virtual work concept was thought to be more basic than the present understanding that least action does not account for non-conservative forces.

Table of content ## What is the Principle of Virtual Work?

According to the principle of virtual work, total virtual work done by all the forces acting on a system in static equilibrium is zero for a set of infinitesimal virtual displacements from equilibrium.” John Bernoulli devised this technique in 1717. We must be extremely clear about the definitions of work, virtual work, and virtual displacement to understand the “principle of virtual work.” When an object is moved over a distance by an external force, at least a portion of that force must be applied in the direction of the displacement. This is known as “work” in physics. If the force is constant, work is calculated by multiplying the length of the path by the component of the force acting along the path. Mathematically, the work can be given as; W= f.d

Any system of forces that is in equilibrium is a virtual force. A virtual displacement of a system can be defined as “an infinitesimal displacement of any mechanical system point that is compatible with the system’s constraints that may or may not happen.” so any displacement that is possible without violating any constraints is referred to as a virtual displacement. In continuation of this, virtual work is defined as, ”The work performed by a real force operating through a virtual displacement or a virtual force acting through a real displacement is known as virtual work”.

## Mathematical Relation of Principle of Virtual Work

A special type of problem in mechanics involves problems of systems in equilibrium. In these systems, the problem is finding the system’s configuration in equilibrium subjected to any type of constraint.

According to the virtual work principle, any infinitesimal virtual displacement in configuration space consistent with the constraints doesn’t require any work. Simply put, the total virtual work of forces operating on a particle in equilibrium is zero for any virtual displacement. Mathematically, it can be written as;

U = F.δx = M.δθ = 0 GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 help@byjusexamprep.com