The Unit dummy force method provides a convenient means for computing displacements in structural systems.
It is applicable for both linear and non-linear material behaviours as well as for systems subject to environmental effects, and hence more general than Castigliano's second theorem.
Consider a discrete system such as trusses, beams or frames having members interconnected at the nodes.
These member deformations give rise to the nodal displacements
We start by applying N virtual nodal forces
, one for each wanted r, and find the virtual member forces
: In the case of a statically indeterminate system, matrix B is not unique because the set of
contains arbitrary virtual forces, the above equation gives It is remarkable that the computation in (2) does not involve any integration regardless of the complexity of the systems, and that the result is unique irrespective of the choice of primary system for B.
It is thus far more convenient and general than the classical form of the dummy unit load method, which varies with the type of system as well as with the imposed external effects.
This is not a restriction because we can make any point into a node when desired.
Finally, the name unit load arises from the interpretation that the coefficients
For a general system, the unit dummy force method also comes directly from the virtual work principle.
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: Equating the two work expressions gives the desired displacement: