THE FORCE METHOD

The traditional method of cutting a statically indeterminate structure, applying redundant forces, and solving for the redundant forces by setting the relative displacements at the cuts to zero has been the most popular method of structural analysis, if hand calculations are used. At this point in time, there appears to be no compelling reason for using the force method within a computer program for solving large structural systems. In fact, programs based on the displacement approach are simple to program and, in general, require less computer time to execute. Another significant advantage of a displacement approach is that the method is easily extended to the dynamic response of structures.

To develop the stiffness of one-dimensional elements, however, the force method should be used because the internal forces can be expressed exactly in terms of the forces at the two ends of the element. Therefore, the force method will be presented here for a single-element system.

Neglecting thermal strains, the energy function can be written as:

The internal forces can be expressed in terms of the node forces using the following equation:

For linear material   and the energy function can be written as:

Where the element flexibility matrix is:

We can now minimize the complementary energy function by requiring that:

The node displacements can now be expressed in terms of node forces by:

The element stiffness can now be numerically evaluated from:

The element stiffness can be used in the direct stiffness approach where the basic unknowns are the node displacements. One can also derive the element flexibility by applying the virtual force method.