General Systems
For a general system, the unit dummy force method also comes directly from the virtual work principle. Fig.(a) shows a system with known actual deformations . These deformations, supposedly consistent, give rise to displacements throughout the system. For example, a point A has moved to A', and we want to compute the displacement r of A in the direction shown. For this particular purpose, we choose the virtual force system in Fig.(b) which shows:
- The unit force R* is at A and in the direction of r so that the external virtual work done by R* is, noting that the work done by the virtual reactions in (b) is zero because their displacements in (a) are zero:
- : The desired displacement
- The internal virtual work done by the virtual stresses is
- where the virtual stresses must satisfy equilibrium everywhere.
Equating the two work expressions gives the desired displacement:
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