What is the principle of virtual work?

For a rigid body in equilibrium, the total virtual work done by all external forces through any compatible virtual displacement is zero. This gives one equation per independent virtual displacement, from which unknown reactions can be found.

What is a virtual displacement?

A virtual displacement is a small, imaginary, kinematically admissible displacement that satisfies the constraints of the structure. It does not actually occur; it is used as a mathematical device to extract equilibrium equations.

When is virtual work positive or negative?

Virtual work = Force × virtual displacement. It is positive when the force and displacement are in the same direction, negative when opposite. The sum of all virtual works equals zero at equilibrium.

Can this tool handle distributed loads?

Yes — a uniformly distributed load (UDL) can be entered as its resultant force acting at the centroid of its distribution. Enter the total load magnitude and the virtual displacement at that centroid point.

Why does the unknown virtual displacement have to be non-zero?

If the virtual displacement at the unknown force location is zero, that force does no virtual work and drops out of the equation — you cannot solve for it. Choose a virtual displacement pattern that gives a non-zero value at the unknown.

Virtual Work Calculator

Name: Virtual Work Calculator
Availability: InStock
Author: Nham Vu

Define applied forces and their virtual displacements to solve for an unknown reaction using the principle of virtual work.

Force System

Enter known forces with their virtual displacements, then define the unknown reaction.

Known Forces

Force (N) Virtual disp. (m)

Unknown Reaction

Label

Virtual disp. (m)

Positive = same direction as force; negative = opposite.

Quick Presets

Add forces and click Calculate to see results.

Sign convention

Virtual work W_i = F_i × δ_i. Positive when force and virtual displacement are in the same direction. Equilibrium requires ΣW_i = 0, so: unknown = −Σ(known virtual work) / (virtual displacement at unknown).

Summary

Define applied forces and their virtual displacements to solve for an unknown reaction using the principle of virtual work.

How it works

Select the structure type: simply supported beam or custom force system.
Enter each known force and its corresponding virtual displacement.
Specify the virtual displacement assigned to the unknown reaction.
Click Calculate — the tool sums virtual work and solves for the unknown.
Review the step-by-step solution showing each force contribution.

Use cases

Find reactions at beam supports without writing multiple equilibrium equations.
Verify beam reaction results from free-body diagram analysis.
Solve pin-jointed frame reactions using generalized virtual displacements.
Demonstrate the principle of virtual work in mechanics coursework.
Check results from the direct stiffness or moment distribution methods.
Analyze indeterminate structures after introducing a release.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu