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

This problem provides evidence that Abaqus can reproduce the result from the benchmark defined by NAFEMS and cited as the reference solution.

This page discusses:

Elements tested
Problem description
Reference solution
Results and discussion
Response predicted by Abaqus
Input files

ProductsAbaqus/Standard

Elements tested

CSS8

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 ( $U_{y} = θ_{x} = θ_{z} = 0)$ , symmetry on plane x = 0 ( $U_{x} = θ_{y} = θ_{z} = 0)$ , $U_{z} = 0$ 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 3D 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
	$U_{r} (A)$	$U_{r} (B)$	$U_{r} (A)$	$U_{r} (B)$	$U_{r} (A)$	$U_{r} (B)$
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	−3.16	2.29	−4.20	2.79	−5.76	3.37
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
CSS8	–3.15	2.27	–4.18	2.74	–5.64	3.30

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.

Input files

n3g9xf3x.inp: S3/S3R elements.
n3g9xe4x.inp: S4 elements.
n3g9xf4x.inp: S4R elements.
n3g9x54x.inp: S4R5 elements.
n3g9x68x.inp: S8R elements.
n3g9x58x.inp: S8R5 elements.
n3g9x59x.inp: S9R5 elements.
n3g9x63x.inp: STRI3 elements.
n3g9x56x.inp: STRI65 elements.
nlg9_std_sc6r.inp: SC6R elements.
nlg9_std_sc6r_sgs.inp: SC6R elements using SHELL GENERAL SECTION.
nlg9_std_sc8r.inp: SC8R elements.
nlg9_std_sc8r_stackdir_sphori.inp: SC8R elements with STACK DIRECTION=ORIENTATION and a spherical orientation system.
nlg9_std_css8.inp: CSS8 elements.