Standard tests, curved shells

Results.

Problem. The curved shell tests measure whether the element carries membrane and bending action together on curved surfaces. The Scordelis-Lo roof is solved under its own weight, and the pinched cylinder and the hemisphere with a hole under opposite point loads.

Reference document. The reference values are those of MacNeal and Harder (1985) for the Scordelis-Lo roof and the hemisphere, and of Belytschko et al. (1985) for the cylinder.

Numerical model. The curved surfaces are represented piecewise with flat shell elements. The models use symmetry: a quarter of the roof and the hemisphere and an eighth of the cylinder. On the symmetry edges the corresponding translations and rotations are held, and at the ends of the roof and the cylinder the translations in the plane of the section. Each problem is solved on meshes refined from four to thirty-two elements per side.

Scordelis-Lo roof#

Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy

Note. The reference 0.3024 is the value used by MacNeal and Harder. Krysl and Chen (2023) propose 0.302 for shear-flexible (Mindlin) shells, against which the deviation is 0.3%.

Pinched cylinder with end diaphragms#

Published source. Belytschko et al. (1985), standard problems for shell elements

Note. The reference values come from thin shell theory. Under a point load the exact solution of shear-flexible shell theory is not finite, so on very fine meshes the result can keep rising slowly; the comparison is meaningful for the standard meshes.

Hemisphere with an 18° hole#

Published source. MacNeal and Harder (1985), A proposed standard set of problems to test finite element accuracy

Note. The reference values come from thin shell theory. Under a point load the exact solution of shear-flexible shell theory is not finite, so on very fine meshes the result can keep rising slowly; the comparison is meaningful for the standard meshes.