VIRTUAL AND REAL WORK
The principles of virtual work are very simple and are clear statements of conservation of energy. The principles apply to structures that are in equilibrium in a real displaced position when subjected to loading . The corresponding real internal deformations and internal forces are and respectively.
The principle of virtual forces states if a set of infinitesimal external forces, , in equilibrium with a set of infinitesimal internal forces that exist before the application of the real loads and displacements, the external virtual work is equal to the internal virtual work.
If only one joint displacement is to be calculated, only one external virtual load exists, . For this case, the equation is the same as the unit load method.
It is apparent for nonlinear analysis that the principle of virtual forces cannot be used, because the linear relationship between and may not hold after the application of the real loads and displacements.
The principle of virtual displacements states if a set of infinitesimal external displacements, , consistent with a set of internal virtual displacements, , and boundary conditions are applied after the application of the real loads and displacements, the external virtual work is equal to the internal virtual work.
It is important to note that the principle of virtual displacements does apply to the solution of nonlinear systems because the virtual displacements are applied to real forces in the deformed structure.
In the case of finite element analysis of continuous solids, the virtual work principles are applied at the level of stresses and strains; therefore, integration over the volume of the element is required to calculate the virtual work terms.
For linear analysis, it is apparent that the real external work, or energy, is given by:
The real internal work, or strain energy, is given by:
Reference
Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson