Ill-Conditioned Stiffness Matrices

In practical applications of the stiffness matrix method to structural analysis problems, erroneous solutions are frequently obtained due to ill-conditioning of the stiffness matrix. In this study, a mathematical formulation is obtained to relate the modal distribution of truncation errors to the conditioning of the matrix and to the type of loading on the structure. Truncation errors involved in the solution by Gaussian elimination can be interpreted as modifying the structure by introduction of additional elastic restraints and modification of the stiffnesses of the internal elastic couplings. It is found also that when the deformed configuration of the structure corresponds substantially to its low natural mode of vibration, the error in the calculated stresses is likely to be more significant than the error in the calculated deflections. The converse is true if the deformation corresponds to a very high mode of vibration.