TOTAL ORTHOGONALIZATION OF REDUNDANTS IN THE FORCE METHOD

If the force method is fully optimized by introducing two appropriate co-ordinate transformations on the internal generalized force unknowns, the self-stress states (or redundant force systems) are determined so as to be an orthonormal set which is also orthogonal to the particular solution. The magnitudes of the redundant forces then become zero for any loading. This eliminates all operations with the redundants. In other words, the particular solution which is found is the solution to the problem. The result is a large decrease in the amount of computation necessary in the force method by eliminating the need to form up and solve the matrices associated with the compatibility equations. Although a different approach, this optimized form of the force method is shown to result in the same numerical procedures as those for the natural factor formulation of the displacement method. The same transformations developed to orthogonalize completely the self-stress states may also be applied directly to the compatibility and equilibrium equations as an alternative procedure. This approach to the solution of the structural equations is designated as the force transformation method.