LE3: Hemispherical shell with point loads

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

The following topics are discussed:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/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:

$u_{x} = u_{y} = u_{z} =$ 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: $u_{x} =$ 185 mm at point A.

Results and discussion

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

Element	$u_{x}$ at A (Coarse)	$u_{x}$ 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 $\cos 2 θ$ 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.

Input files

Abaqus/Standard input files

nle3xf3x.inp: S3/S3R elements.
nle3xe4x.inp: S4 elements.
nle3xf4x.inp: S4R elements.
nle3xf4x_eh.inp: S4R elements with enhanced hourglass control.
nle3x54x.inp: S4R5 elements.
nle3x68x.inp: S8R elements.
nle3x58x.inp: S8R5 elements.
nle3x59x.inp: S9R5 elements.
nle3x63x.inp: STRI3 elements.
nle3x56x.inp: STRI65 elements.
nle3xntx.inp: SAXA12 elements.
nle3xnxx.inp: SAXA22 elements.
nle3_std_sc6r.inp: SC6R elements.
nle3_std_sc8r.inp: SC8R elements.
nle3_std_sc6r_stackdir_sphori.inp: SC6R elements using the STACK DIRECTION=ORIENTATION parameter with a spherical orientation system to define the element thickness direction.
nle3_std_sc8r_stackdir_sphori.inp: SC8R elements using the STACK DIRECTION=ORIENTATION parameter with a spherical orientation system to define the element thickness direction.
nle3_std_sc8r_sgs.inp: SC8R elements using SHELL GENERAL SECTION to define section properties.

Abaqus/Explicit input files

le3_s4r.inp: S4R elements with enhanced hourglass control.
le3_sc8r.inp: SC8R elements with the default “relax stiffness” hourglass control.