4.2.3 LE3: Hemispherical shell with point loads

Products: ABAQUS/Standard ABAQUS/Explicit

Elements tested

S3 S3R S4 S4R S4R5 S8R S8R5 S9R5 STRI3 STRI65 SC6R SC8R

SAXA12 SAXA22

Problem description

Model:

The model is illustrated in the figure above. In addition, two input files are provided for the continuum shell element model to illustrate the use of the STACK DIRECTION=ORIENTATION parameter to define the element thickness (stacking) direction independent of the nodal connectivity using a spherical system.

Material:

Linear elastic, Young's modulus = 68.25 GPa, Poisson's ratio = 0.3.

Boundary conditions:

0 at point E. Along edge AE, symmetry about the z–x plane. Along edge CE, symmetry about the y–z plane.

Loading:

Concentrated radial loads of 2 kN outward at A, inward at C.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test LE3 from NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks,” October 1990.

Target solution: 185 mm at point A.

Results and discussion

The values enclosed in parentheses are percentage differences with respect to the reference solution.

Element	at A (Coarse)	at A (Fine)
S3/S3R	0.080 (–57%)	0.161 (–13%)
S4	0.083 (–55%)	0.175 (–5%)
S4R	0.180 (–2.7%)	0.180 (–2.7%)
S4R*	0.072 (–61%)	0.170 (–8.1%)
S4R**	0.058 (–68%)	0.168 (–9.1%)
S4R5	0.190 (2.7%)	0.183 (–1.1%)
S8R	0.101 (–45%)	0.178 (–3.8%)
S8R5	0.179 (–3.2%)	0.185 (0.0%)
S9R5	0.179 (–3.2%)	0.185 (0.0%)
STRI3	0.173 (–1.2%)	0.185 (0.0%)
STRI65	0.169 (–8.6%)	0.182 (–1.6%)
SC6R	0.088 (–52.4%)	0.167 (–9.7%)
SC8R	0.210 (13.5%)	0.188 (1.6%)
SC8R***	0.194(4.9%)	0.185(0.0%)
SAXA12****	0.179 (–3.2%)
SAXA22****	0.178 (–3.8%)

* ABAQUS/Explicit finite-strain element with enhanced hourglass control. **ABAQUS/Standard finite-strain element with enhanced hourglass control. *** ABAQUS/Explicit continuum shell element with the default “relax stiffness” hourglass control. **** Due to the loading position, only the Mode 2 and Mode 4 elements can be used. Furthermore, due to the symmetries of the problem, only the Fourier interpolator

contributes to the solution. Thus, the Mode 4 elements produce identical results. Since Mode 4 is the highest-order Fourier term provided, no further circumferential mesh refinement is possible, and only coarse mesh results can be obtained.

The continuum shell element meshes using the STACK DIRECTION=ORIENTATION parameter yield identical results to the continuum shell element meshes in which the thickness direction is defined by the element nodal connectivity.