--- title: "Virtual work and its principle" subject: "Engineering Mechanics" chapter: 1 key: "semester_3/engineering_mechanics:virtual-work-principle-definition" source: https://www.bctengineer.com/q/semester_3/engineering_mechanics/virtual-work-principle-definition site: BCT Engineering --- # Virtual work and its principle Engineering Mechanics · Chapter 1: Virtual Work - Asked in: year not given, 4 marks - Video: [Mechanical Engineering: Ch 13: Virtual Work Applications (1 of 39) What is Virtual Work? 1](https://www.youtube.com/watch?v=BBaU3qIJwIg) — Michel van Biezen - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/engineering_mechanics/virtual-work-principle-definition - Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_mechanics/chapter-1 ## Question Define virtual displacement and virtual work. State the principle of virtual work for a particle and for a rigid body, and list two uses of the principle. ## Short answer Virtual displacement is an imaginary infinitesimal displacement consistent with constraints. Virtual work is work done by forces during that displacement. For a particle, equilibrium requires total virtual work zero for any virtual displacement; for a rigid body, total virtual work of external forces and couples is zero. Uses: finding unknown forces in linkages without dismembering, and determining equilibrium positions. ## Answer A **virtual displacement** $\delta r$ is an imaginary, infinitesimal displacement of a point, consistent with the constraints of the system, taken at a fixed instant of time with no change in the applied forces. **Virtual work** $\delta U$ is the work done by the forces during a virtual displacement. #### Principle for a particle A particle is in equilibrium if and only if the total virtual work of all forces acting on it is zero for every virtual displacement: $$ \delta U = \sum \mathbf{F}_i \cdot \delta \mathbf{r} = 0 $$ This is equivalent to $\Sigma F = 0$, because $\delta \mathbf{r}$ is arbitrary. #### Principle for a rigid body A rigid body is in equilibrium if the total virtual work of all external forces and couples is zero for every virtual displacement (translation and rotation) consistent with the constraints: $$ \delta U = \sum \mathbf{F}_i \cdot \delta \mathbf{r}_i + \sum M_j\, \delta\theta_j = 0 $$ A couple $M$ does virtual work $M\,\delta\theta$ for a virtual rotation $\delta\theta$. For a system of connected rigid bodies the equation is written in terms of one independent coordinate $\theta$ (or $x$), so that $\delta U = Q\,\delta\theta = 0$. Forces that do no virtual work are left out: reactions at smooth pins and smooth surfaces, the internal forces of a rigid body, and forces at fixed supports. #### Uses 1. Finding unknown forces or couples in machines and linkages (jacks, toggles, scissors) without dismembering them, since the unknown reactions at smooth supports never appear. 2. Finding the equilibrium position of a system under given loads, for example the angle at which a ladder or a spring-loaded linkage rests. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which description fits a virtual displacement? A. A small real displacement that occurs in time dt under the applied forces B. An imaginary infinitesimal displacement consistent with the constraints, at a fixed instant with forces unchanged C. Any large displacement the body may take, ignoring constraints D. The displacement of the body from its initial to its final position Answer: B. A virtual displacement is imagined, infinitesimal, compatible with constraints and taken at a frozen instant with the forces unchanged. (Key point: Virtual displacement) 2. For a particle, the principle of virtual work says equilibrium exists if the virtual work is zero for: A. One chosen virtual displacement only B. Every virtual displacement C. Only displacements along the resultant force D. Only the actual displacement Answer: B. Because dr is arbitrary, requiring sum F.dr = 0 for every dr is equivalent to sum F = 0. (Key point: Principle for a particle) 3. In the virtual work equation for a rigid body, how is the work of a couple M written? A. M x dr B. M dtheta C. M / dtheta D. M dr Answer: B. A couple does virtual work M dtheta for a virtual rotation dtheta. (Key point: Rigid body principle) 4. Which forces are omitted from the virtual work equation? A. Weights of the members B. Applied external loads C. Reactions at smooth pins and smooth surfaces D. Spring forces Answer: C. Reactions at smooth supports, internal forces of a rigid body and forces at fixed supports do no virtual work, so they are left out. (Key point: Forces left out) ## Related in Engineering Mechanics - [Kinetic energy and work of rigid body](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Akinetic-energy-rigid-body-work-energy) — Engineering Mechanics, chapter 7 - [Ladder equilibrium by virtual work](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Aladder-virtual-work-friction) — Engineering Mechanics, chapter 1 - [Work-energy principle, power and efficiency](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Awork-energy-principle-power-efficiency) — Engineering Mechanics, chapter 3 ## Related in other subjects - [Concept of particle, rigid and deformable body](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aconcept-particle-rigid-deformable) — Applied Mechanics, chapter 1 - [Truss horizontal displacement at C by virtual work](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-virtual-work-horizontal-c) — Theory of Structures I, chapter 3 - [Derive virtual work principle](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Avirtual-work-principle-derive) — Theory of Structures I, chapter 3 - [Two balls impact e = 0.67; work-energy principle](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Atwo-balls-e067-work-energy) — Applied Mechanics (Dynamics), chapter 2