4.10.9 3DNLG-9: Large elastic deflection of a pinched hemispherical shell

Product: ABAQUS/Standard

Elements tested

S3 S3R S4 S4R S4R5 S8R S8R5 S9R5

STRI3 STRI65

SC6R SC8R

Problem description

Material:

Young's modulus = 6.825 × 10⁷, Poisson's ratio = 0.3.

Boundary conditions:

Symmetry on plane y = 0 (, symmetry on plane x = 0 (, at A to prevent rigid body motion.

Loading:

Inward and outward diametrical point loads (P = 100).

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 3DNLG-9 from NAFEMS Publication R0024 “A Review of Benchmark Problems for Geometric Non-linear Behaviour of 3-D Beams and Shells (SUMMARY).”

The published results of this problem were obtained with ABAQUS. Thus, a comparison of ABAQUS and NAFEMS results is not an independent verification of ABAQUS. The NAFEMS study includes results from other sources for comparison that may provide a basis for verification of this problem.

Results and discussion

The following table shows the radial displacements at points A and B at three load levels. All the meshes have the same nodal spacing.

Element	P = 40		P = 60		P = 100

S4R	–3.26	2.32	–4.34	2.82	–5.90	3.41
S4	–3.21	2.30	–4.26	2.79	–5.80	3.37
S4R5	–3.28	2.33	–4.36	2.83	–5.95	3.43
S8R	–3.16	2.23	–4.13	2.67	–5.47	3.19
S8R5	–3.23	2.32	–4.30	2.81	–5.83	3.40
S9R5	–3.23	2.32	–4.30	2.81	–5.83	3.40
S3/S3R	–3.12	2.27	–4.15	2.76	–5.67	3.34
STRI3	–3.31	2.31	–4.36	2.80	–5.91	3.38
STRI65	–3.13	2.27	–4.16	2.75	–5.63	3.31
SC6R	–3.06	2.21	–4.03	2.66	–5.42	3.21
SC8R	–3.25	2.30	–4.29	2.78	–5.80	3.35

Response predicted by ABAQUS

Similar load-displacement curves are obtained for all test cases. The response predicted using S4R elements is shown below. The curve for node A is for the negative displacement and load values.