Discrete Systems
Consider a discrete system such as trusses, beams or frames having members interconnected at the nodes. Let the consistent set of members' deformations be given by, which can be computed using the member flexibility relation. These member deformations give rise to the nodal displacements, which we want to determine.
We start by applying N virtual nodal forces, one for each wanted r, and find the virtual member forces that are in equilibrium with :
In the case of a statically indeterminate system, matrix B is not unique because the set of that satisfies nodal equilibrium is infinite. It can be computed as the inverse of the nodal equilibrium matrix of any primary system derived from the original system.
Imagine that internal and external virtual forces undergo, respectively, the real deformations and displacements; the virtual work done can be expressed as:
- External virtual work:
- Internal virtual work:
According to the virtual work principle, the two work expressions are equal:
Substitution of (1) gives
Since 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. On the other hand, it is important to note that Eq.(2) is for computing displacements or rotations of the nodes only. This is not a restriction because we can make any point into a node when desired.
Finally, the name unit load arises from the interepretation that the coefficients in matrix B are the member forces in equilibrium with the unit nodal force, by virtue of Eq.(1).
Read more about this topic: Unit Dummy Force Method
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