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

By Mohit Uniyal|Updated : August 3rd, 2022

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.

## 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.

Any infinitesimal virtual displacement in configuration space that is consistent with the constraints doesn't require any work, according to the virtual work principle. In simple words, the total virtual work of forces operating on a particle when it is in equilibrium is zero for any virtual displacement. Mathematically, it can be written as;

U = F.δx = M.δθ = 0

## Advantages of Principle of Virtual Work Method

1. It is not essential to disassemble the systems to develop relationships between the active forces.
2. Without considering the reactive forces, relationships between the active forces can be found directly.
3. This method is particularly helpful in identifying the position of equilibrium of a system under known loads.
4. One significant advantage of the virtual work principle is that it just requires active forces when the system goes through a virtual displacement to identify the system's mechanics.
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## FAQs on Principle of Virtual Work

• The idea is that, for a set of infinitesimally small virtual displacements from equilibrium, the total virtual work performed by all forces operating on a system is zero. The virtual displacements are so-called because they do not necessarily need to be obtained by a displacement that actually takes place in the system. The work performed by virtual displacements, which can be arbitrary as long as they are consistent with the system's limitations, is called the principle of virtual work.

• The principle of virtual work is a concept in mechanics that wasn't frequently utilised to calculate work since when it was applied to objects in equilibrium, the work was immediately equal to zero. The virtual work approach allows for determining a system's equilibrium conditions without introducing any constraint factors, greatly simplifying the solution and broadening the range of issues that can be resolved.

• By first approximating the structure's unknown displacement field with a shape or form that has a limited number of unknown characteristics, the principle of virtual work is applied. Virtual work is one of the most useful and extensively used methods for estimating deflections in beams and frames.

• Any displacement that satisfies the boundary criteria at the supports and is consistent with the limitations of the structure is referred to as a virtual displacement. Virtual displacement is not experienced in reality but is assumed to exist for the analysis of the system.

• A real force operating through a virtual displacement or a virtual force acting through a real displacement is said to have performed "virtual work." The forces are taken to be constant during the virtual displacement in the equation for virtual work. This holds true even when the forces have a small but significant displacement.

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