Popular benchmark problems for geometric nonlinear analysis of shells

Abstract

In most, if not all, of the previous work on finite element formulation and nonlinear solution procedures, results of geometric nonlinear benchmark problems of shells are presented in the form of load-deflection curves. In this paper, eight sets of popularly employed benchmark problems are identified and their detailed reference solutions are obtained and tabulated. It is hoped that these solutions will form a convenient basis for subsequent comparison and that the tedious yet inaccurate task of reconstructing data points by graphical measurement of previously reported load-deflection curves can be avoided. Moreover, the relative convergent difficulty of the problems are revealed by the number of load increments and the total number of iterations required by an automatic load incrementation scheme for attaining the converged solutions under the maximum loads.

Computational shell mechanics by helicoidal modeling, II: Shell element

“…good agreement with the best results reported in [Sze et al 2004] is observed. Some representative load-displacement pairs are listed in Table 5.…”

“…The load-displacement curves for four meshes are plotted in Figure 22 and compared with the results published in [Sze et al 2004]: a good agreement with the reference results is observed. Displacements every 10% of the final force are listed in Table 9.…”

“…The free edge displacements with increasing load are plotted in Figure 4 for three different meshes and compared with the best results reported by [Sze et al 2004]. The results from [Li and Zhan 2000], obtained with a shell element endowed with the drilling degree of freedom and based on Biot strain, are included in Figure 4.…”

“…with those obtained with a refined mesh in [Sze et al 2004]. Loads and displacements at every history step are listed in Table 4.…”

“…The snapping behavior is easily captured by the arc-length procedure. The load-displacement curves, in Figure 16, at the central point A with the 4 × 4 mesh are plotted and compared with the best results reported in [Sze et al 2004]; the results with the 8 × 8 mesh are not included in Figure 16 as they are indistinguishable from those of [Sze et al 2004]. The thin panel curve (Figure 16a) is traced in 24 steps up to exceed the final load of 3000, with an average of 6.3 iterations per step; the thick panel curve (Figure 16b) is traced in 17 steps, with an average of 5.6 iterations per step.…”