Virtual work

The principle of virtual work says that if a body is in equilibrium, the total work done by all forces during any small imagined (virtual) displacement consistent with its supports is zero. The displacement is not real; it is a thought experiment. Its power is that unknown reactions that do no work in the chosen displacement simply drop out, so a single equation can give the quantity you want.

For rigid bodies and mechanisms, we give the structure a virtual displacement by releasing the one restraint we are interested in. To find a support reaction of a beam, replace the support with the unknown force and let the beam rotate as a rigid body about the other support. Each load does work equal to force times the virtual displacement of its point of application, in the direction of the force. Setting the sum to zero gives the reaction. To find an internal bending moment, insert a hinge at the section and rotate the two halves.

For deformable structures, the principle splits into two forms. The principle of virtual displacements gives equilibrium equations and is the base of the finite element method. The principle of virtual forces, often called the unit load method, gives deflections: apply a unit load where you want the deflection, compute the resulting internal forces, and integrate their product with the real strains. For trusses this becomes Σ n N L / (A E); for beams, ∫ m M / (E I) dx.

Virtual work also builds influence lines quickly (Müller-Breslau principle): the shape of the deflected structure, when a restraint is released and given a unit displacement, is the influence line for that restraint. In plastic analysis, the same principle equates the work of loads to the work dissipated in plastic hinges to find collapse loads.