Discussion of “Nonlinear Truss Analysis by One Matrix Inversion” by A. Fafitis

Abstract

The author presents a method of analysis for truss structures with material and geometric nonlinearity. The method consists of a single factorization of the stiffness matrix, followed by its successive application to corrective force vectors in order to find structural equilibrium when nonlinearities arise due to strainhardening/softening, buckling, breaking, and stiffness degradation. The author points out that the stiffness matrix used for the iteration does not necessarily have to be the exact stiffness matrix of the structure, and the proposed analysis method treats proportional, nonproportional, and cyclic loadings uniformly. Although the developments are clear and the applications are practical, this method of analysis is not significantly different from the modified Newton–Raphson algorithm, where the stiffness matrix is held constant over the course of a load increment. The modified Newton–Raphson algorithm is discussed extensively in a reference Crisfield 1991 cited by the author; however, the author neither acknowledges this material nor cites any of the numerous references available in the mathematical literature on the convergence properties of the modified Newton– Raphson algorithm, such as Shamanskii 1967, Stoer and Bulirsch 1993, and Kelley 1995 to name but a few. Due to its slow convergence rate, the excessive number of iterations required to achieve equilibrium with the modified Newton–Raphson algorithm can far outweigh the computational savings afforded by a single matrix factorization for structures with a small to moderate number of degrees of freedom. For such structures, the computational cost of a matrix factorization is cheap relative to the cost of evaluating the corrective force vector, making the full Newton– Raphson algorithm more efficient. Furthermore, the use of equation solvers Mackay et al. 1991; Demmel et al. 1999; Davis 2003 that exploit the sparse matrix topology generated by a finite-element analysis can mitigate the computational expense of full Newton–Raphson for large structural systems. The discussers acknowledge that the full Newton–Raphson algorithm requires an exact stiffness matrix be computed at every iteration, which may not be possible when using complex numerical models of constitutive behavior. This is not the case, however, for closed-form scalar expressions, such as the bilinear stress-strain law and the Euler buckling formula Eq. 5 the author uses in the numerical examples. The author proposes holding an inexact stiffness matrix, Kinexact, constant during the equilibrium iteration. This approach can increase the number of iterations required to reach equilibrium. Consistent with the numerical properties of the modified Newton–Raphson algorithm, if the spectral radius of the matrix