Comment on "Effect of Higher Order Terms in Certain Nonlinear Finite Element Models"

Abstract

O et al. conclude that convergence will be slowed if terms which characterize nonlinear behavior are approximated by a simpler polynomial than that used for the conventional stiffness matrix. The present Comment offers numerical evidence pertinent to this theoretical conclusion. The problem chosen is of the type used in Ref. 1. A cantilever beam is modeled by ten elements, and at its free end carries an axial compressive load P and a transverse load Q = 0.001. Properties are chosen such that the Euler buckling load is Pcr = 1.000. The solution method is based on convected coordinates, as clearly detailed for plate problems by Murray and Wilson. Thus the technique is pure Newton-Raphson iteration; under-relaxation, convergence-acceleration schemes, etc., were not used. Iterative cycling at a given load level was terminated by a convergence test on displacements. The next load level was then applied, and iteration begun again, starting with the displacements obtained at the previous load level. The conventional stiffness matrix of each element (in its convected coordinate system) was based on a cubic polynomial. Two forms of 'geometric' or 'initial stress' stiffness matrix were considered. The first, [kGC], was based on the cubic polynomial, and the second, [kGL], was based on a linear polynomial. In other words, with reference to Eqs. (3) of Ref. 1, [kcc] was based upon the lateral displacement v(x) = (x) and [kGL] upon v(x) = ^O)v. Let d/L be the ratio of lateral tip displacement to beam length. With P = 0.99, d/L converged to 0.0606 in 9 cycles when [kGL] was used, and to 0.0701 in 15 cycles when [kGC] was used. With the geometric stiffness matrix omitted altogether, d/L reached 0.0257 at 40 cycles but had not converged. The correct result is d/L = 0.0811, which is some 100 times the value produced by lateral load Q acting alone. Table 1 lists tip rotations obtained by progressive increases in load level, with a milder convergence tolerance than used in the preceding paragraph.