Virtual work

Virtual work is the mathematical product of unrelated force and displacement or moment and rotation.

When a force displaces or a moment rotates, it does work. If both the force, or moment, and the displacement, or rotation, are actual or real, then the work performed is real work. When one or both quantities are not real, i.e. virtual, the work done is termed virtual work.

D'Alembert's principle is a early form of the virtual work principle applied to the dynamics of a particle. It is a statement of the fundamental classical laws of motion. It is equivalent to Newton's second law. It is named after its discoverer, the French physicist Jean le Rond d'Alembert.

The principle states that the sum of the differences between the generalized forces acting on a system and the time derivative of the generalized momenta of the system itself along an infinitesimal displacement compatible with the constraints of the system, is zero. That is:

<math> \sum_{i}\left({ {\mathbf F}_{i} – \dot {\mathbf p}_{i} }\right) \cdot \delta{\mathbf r}_{i} = 0. <math>

The motivation for introducing virtual work can be appreciated by the following simple example from statics of particles. Suppose a particle is in equilibrium under a set of forces <math>Fx_i<math>, <math>Fy_i<math>, <math>Fz_i<math> <math>i = 1,2,...n<math>:

<math>\sum_{i=1}^n Fx_i = 0 <math>

<math>\sum_{i=1}^n Fy_i = 0 <math>

<math>\sum_{i=1}^n Fz_i = 0 \qquad \mathrm{(a)}<math>

Multiplying the three equations with the respective arbitrary constants <math>D_x<math>, <math>D_y<math>, <math>D_z<math>:

<math>D_x\sum_{i=1}^n Fx_i = 0 <math>

<math>D_y\sum_{i=1}^n Fy_i = 0 <math>

<math>D_z\sum_{i=1}^n Fz_i = 0 \qquad\mathrm{(b)}<math>

When the arbitrary constants <math>D_x<math>, <math>D_y<math>, <math>D_z<math> are thought of as virtual displacements of the particle, then the left-hand-sides of (b) represent the virtual work. The total virtual work is:

<math>D_x\sum_{i=1}^n Fx_i + D_y\sum_{i=1}^n Fy_i + D_z\sum_{i=1}^n Fz_i = 0 \qquad\mathrm{(c)}<math>

Since the preceding equality is valid for arbitrary virtual displacements, it leads back to the equilibrium equations in (a). The equation (c) is called the principle of virtual work for a particle. Its use is equivalent to the use of many equilibrium equations.

Applying to a deformable body in equilibrium that undergoes compatible displacements and deformations, we can find the total virtual work by including both internal and external forces acting on the particles. If the material particles experience compatible displacements and deformations, the work done by internal stresses cancel out, and the net virtual work done reduces to the work done by the applied external forces. The total virtual work in the body may also be found by the volume integral of the product of stresses <math>\boldsymbol{\sigma}<math> and virtual strains <math>\delta\boldsymbol{\epsilon}<math>:

<math>\int_{V} \boldsymbol{\sigma}\delta\boldsymbol{\epsilon} \, dV<math>

Thus, the principle of virtual work for a deformable body is:

<math>\mbox{External virtual work} = \int_{V}\boldsymbol{\sigma} \delta\boldsymbol{\epsilon} \, dV<math>

This relation is equivalent to the set of equilibrium equations written for the particles in the deformable body. It is valid irrespective of material behaviour, and hence leads to powerful applications in structural analysis and finite element analysis.