The pinched cylinder problem

The results for the regular Abaqus/Standard mesh are shown in Table 1. The second-order elements (types S8R5 and S9R5) provide the most accurate solutions, whereas element type S8R (also a second-order element, but designed primarily for thick shell applications) provides a rather less accurate solution. Element type STRI3 provides the most accuracy among the first-order elements. None of the first-order elements provides acceptable solutions with the coarsest meshes used.

Element type STRI65 appears to converge rather slowly compared to the other element types. This result may appear counterintuitive, especially when compared to the STRI3 results, which demonstrate better convergence in this problem. Compared to STRI3, which is a flat facet element, element type STRI65 is preferable for modeling bending of thin shells and has complete quadratic representation of membrane strains; therefore, STRI65 is expected to perform better than STRI3 provided the number of elements in the two meshes is the same. In the present convergence study we have instead retained an equal number of nodes, which results in the relatively poor performance of the STRI65 element.

The results for the regular Abaqus/Explicit mesh are shown in Table 2. The results suggest that element types S3R and S4R are initially stiff but then converge to the correct solution. In addition, an energy plot is provided in Figure 7, which shows that by the end of the analysis a steady, static solution is obtained.

The second type of irregular mesh has more distorted element shapes than the first type of irregular mesh. The results for the two irregular Abaqus/Standard meshes are given in Table 3; and, as discussed in The barrel vault roof problem, they show less accurate results than the regular mesh problems.

Element types S8R5 and S9R5 again provide reasonably accurate results with fine meshes, although the coarse mesh results with these elements demonstrate poor accuracy. Interestingly, in this case all the first-order quadrilateral elements provide quite accurate values even with coarse meshing. This result may be fortuitous and should not be taken as a general indication of the quality of the elements in distorted meshes. For element type S4R both stiffness-based and enhanced hourglass controls are used to study the effect of mesh refinement and skew sensitivity. As expected, the coarse mesh results for enhanced hourglass control show poor accuracy compared with the fine mesh results.

The results for the irregular Abaqus/Explicit meshes are given in Table 4. These irregular meshes are more accurate despite the increased distortion because mesh refinement is concentrated in the area of highest solution gradients.

Results from the submodeled Abaqus/Standard analyses for the shell-to-shell cases are given in Table 5. Clearly, the submodeling technique provides a more accurate solution in the vicinity of the point load than the coarser global analyses. When S4R elements are used on the global level, the radial displacement at the point of load application is within 40% of Lindberg's solution for the coarse mesh and 13% for the finer mesh. The submodeling technique significantly improves these results, giving radial displacements in the shell submodels within 11% and 2% for all four combinations of meshes.

When S8R elements are used to mesh the quarter cylinder, solution accuracy improves from within 6% on the global level to within 0.7% on the submodel level. Displacement contours for the shell submodels are shown in Figure 8 for a representative analysis in which a 5 × 5 mesh of S8R elements is used on the global level and a 10 × 10 mesh of S8R elements is used on the submodel level.

Submodel analyses are tested with output from input files pinchcyl_s4r_reg55.inp, pinchcyl_s4r_reg1010.inp, and pinchcyl_s8r_reg55.inp. If five degree of freedom shells (S4R5, S8R5, etc.) are used at the global level, only the displacement degrees of freedom on the submodel boundary are driven since the rotations are not written to the results file for these elements.

A shell-to-solid submodel is also available for this problem, with a 10 × 10 C3D8I element mesh and four elements across the shell thickness. The submodel is driven from a 12 × 12 S4R element global model. The results are in good agreement with the shell-to-shell submodel results. Since the submodel in this case is made of solid elements, no comparison to the shell analytical solution is offered. The use of the shell-to-solid submodeling capability would be more justified in the case of concentrated loading applied on a finite area instead of the point load.

Six shell-to-solid coupling cases are analyzed in Abaqus/Standard, as listed in Table 6. In all six cases a 12 × 12 shell element mesh is used. As is clearly seen, the shell-to-solid coupling analyses provide accurate solutions in the vicinity of the point load. The radial displacement at the point of load application is within 4.1% of Lindberg's solution for all six cases. As mentioned for submodeling, the use of the shell-to-solid coupling capability would be more justified in the case of concentrated loading applied on a finite area instead of the point load.

The results for the Abaqus/Explicit shell-to-solid coupling analysis are given in Table 7. The radial displacement at the point of load application is within 32% of Lindberg's solution.