as the primary unknowns, the number of nodal equilibrium equations is insufficient for solution, in general—unless the system is statically determinate.
To resolve this difficulty, first we make use of the nodal equilibrium equations in order to reduce the number of independent unknown member forces.
The compatibility equations restore the required continuity at the cut sections by setting the relative displacements
That is, using the unit dummy force method: where Equation (7b) can be solved for X, and the member forces are next found from (5) while the nodal displacements can be found by where Supports' movements taking place at the redundants can be included in the right-hand-side of equation (7), while supports' movements at other places must be included in
While the choice of redundant forces in (4) appears to be arbitrary and troublesome for automatic computation, this objection can be overcome by proceeding from (3) directly to (5) using a modified Gauss–Jordan elimination process.
This is a robust procedure that automatically selects a good set of redundant forces to ensure numerical stability.
It is apparent from the above process that the matrix stiffness method is easier to comprehend and to implement for automatic computation.
On the other hand, for linear systems with a low degree of statical indeterminacy, the flexibility method has the advantage of being computationally less intensive.
The main redeeming factor in learning this method nowadays is its educational value in imparting the concepts of equilibrium and compatibility in addition to its historical value.
In contrast, the procedure of the direct stiffness method is so mechanical that it risks being used without much understanding of the structural behaviors.
However, recent advances in numerical computing have shown a comeback of the force method, especially in the case of nonlinear systems.
New frameworks have been developed that allow "exact" formulations irrespectively of the type or nature of the system nonlinearities.