Virtual work and its principle

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.