Next, we need to create compatibility equations in order to find . The compatibility equations return the required continuity to the section cuts by setting the displacements relative to the redundant X to zero. which is, using the False Force Unit Method"):.

where.

Equation (7b) can be solved for X, and the member forces are then found from (5) while the nodal displacements can be found by.

where.

The movement of supports taking place at redundant locations can be included in the right side of equation (7), while the movement of supports to other locations must be included in and as well.

Although the choice of redundants in (4) appears to be arbitrary and difficult for automatic calculations, 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 process above that the stiffness matrix method is easy to understand and implement for automatic calculations. It is also easy to extend for advanced applications such as nonlinear analysis, stability, vibrations, etc. For these reasons, the stiffness matrix method is the method of choice for use in general-purpose structural analysis software packages. On the other hand, for linear systems with a low degree of static indeterminacy, the flexibility method has the advantage of being computationally less intensive. This advantage, however, is a moot point given that personal computers are widely available and more powerful. The main redeeming factor in learning this method today is its educational value in imparting the concepts of balance and compatibility in addition to its historical value. In contrast, the direct stiffness method process is so mechanical that it is in danger of being used without a great understanding of structural behavior.

The arguments presented above were valid until the early 1990s. However, recent advances in numerical calculations have shown a reversal of the force method, especially in the case of nonlinear systems. New frameworks have been developed that allow respectively "exact" formulations of the type or nature of the nonlinearity of the system. The main advantages of the flexibility method are that the resulting error is independent of the discretization of the model and that this is actually a very fast method. At the moment, the Elastic-plastic solution of a continuous beam using the force method requires only 4 beam elements while a commercial "stiffness-based (FEM)" requires 500 elements in order to give results with the same precision. To conclude, one can say that in the case where the solution of the problem requires recursive evaluations of the force field as in the case of structural optimization or system identification, the efficiency of the flexibility method is indisputable.

For Statically Indeterminate systems, M > N, and therefore, we can augment (3) with I = M-N equations of the form:.

The vector X is what is also called the Redundancy vector of forces and I is the degree of static indeterminacy of the system. We usually choose j, k, ... , , and such that is a reaction at the support or an internal force at one end of the member. With adjustable choices of redundant forces, the system of equations (3) augmented by (4) can now be solved to obtain:

Substituting into (2) gives:.

Equations (5) and (6) are the solution for the primary system which is the original system that has been made statically determined by cuts that expose the redundant forces. Equation (5) effectively reduces the set of unknown forces to .

Next, we need to create compatibility equations in order to find . The compatibility equations return the required continuity to the section cuts by setting the displacements relative to the redundant X to zero. which is, using the False Force Unit Method"):.

where.

Equation (7b) can be solved for X, and the member forces are then found from (5) while the nodal displacements can be found by.

where.

The movement of supports taking place at redundant locations can be included in the right side of equation (7), while the movement of supports to other locations must be included in and as well.

Although the choice of redundants in (4) appears to be arbitrary and difficult for automatic calculations, 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 process above that the stiffness matrix method is easy to understand and implement for automatic calculations. It is also easy to extend for advanced applications such as nonlinear analysis, stability, vibrations, etc. For these reasons, the stiffness matrix method is the method of choice for use in general-purpose structural analysis software packages. On the other hand, for linear systems with a low degree of static indeterminacy, the flexibility method has the advantage of being computationally less intensive. This advantage, however, is a moot point given that personal computers are widely available and more powerful. The main redeeming factor in learning this method today is its educational value in imparting the concepts of balance and compatibility in addition to its historical value. In contrast, the direct stiffness method process is so mechanical that it is in danger of being used without a great understanding of structural behavior.

The arguments presented above were valid until the early 1990s. However, recent advances in numerical calculations have shown a reversal of the force method, especially in the case of nonlinear systems. New frameworks have been developed that allow respectively "exact" formulations of the type or nature of the nonlinearity of the system. The main advantages of the flexibility method are that the resulting error is independent of the discretization of the model and that this is actually a very fast method. At the moment, the Elastic-plastic solution of a continuous beam using the force method requires only 4 beam elements while a commercial "stiffness-based (FEM)" requires 500 elements in order to give results with the same precision. To conclude, one can say that in the case where the solution of the problem requires recursive evaluations of the force field as in the case of structural optimization or system identification, the efficiency of the flexibility method is indisputable.