LE1: Plane stress elements—elliptic membrane

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

CPS3

CPS4

CPS4I

CPS4R

CPS6

CPS6M

CPS8

CPS8R

Problem description

Model:

Plane stress problem with shape defined by ABCD. Functions defining the curves BC and AD are given above.

Mesh:

A coarse and a fine mesh are tested for each element. In addition, a very fine mesh is tested for each element in the explicit dynamic analysis.

Material:

Linear elastic, Young's modulus = 210 GPa, Poisson's ratio = 0.3, density = 7800 kg/m³.

Boundary conditions:

$u_{x} = 0$ along edge AB, $u_{y} = 0$ along edge CD.

Loading:

Uniform outward pressure of 10 MPa at outer edge BC. In the explicit dynamic analysis the loading is applied such that a quasi-static solution is obtained.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test LE1 from NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks,” October 1990.

Target solution: Tangential edge stress ( $σ_{y y}$ ) at D is 92.7 MPa.

Results and discussion

The results are shown in Table 1 and Table 2. The values enclosed in parentheses are percentage differences with respect to the reference solution.

Table 1. Abaqus/Standard analysis.
Element	Coarse Mesh	Fine Mesh
CPS3	51.04 MPa (−45%)	71.26 MPa (−23%)
CPS4	66.73 MPa (−28%)	84.54 MPa (−9%)
CPS4I	58.82 MPa (−37%)	78.21 MPa (−16%)
CPS4R*	40.48 MPa (−56%)	56.18 MPa (−39%)
CPS6	89.10 MPa (−4%)	94.01 MPa (1%)
CPS6M	85.88 MPa (−7%)	93.71 MPa (1%)
CPS8	84.54 MPa (−9%)	92.81 MPa (0.12%)
CPS8R	85.80 MPa (−7%)	90.07 MPa (−3%)

*A comparison of the results for reduced-integration and full-integration lower-order elements indicates that the full-integration elements perform significantly better for problems with stress concentrations of this type.

Table 2. Abaqus/Explicit analysis.
Element	Coarse Mesh	Fine Mesh	Very Fine Mesh
CPS3	51.2 MPa (−45%)	71.5 MPa (−23%)	85.7 MPa (−8%)
CPS4R	39.6 MPa (−57%)	55.7 MPa (−40%)	87.3 MPa (−6%)
CPS6M	86.12 MPa (−7%)	92.93 MPa (−0.2%)	—

Input files

Abaqus/Standard input files

Coarse mesh tests:

nle1xf3c.inp: CPS3 elements.
nle1xf4c.inp: CPS4 elements.
nle1xi4c.inp: CPS4I elements.
nle1xr4c.inp: CPS4R elements.
nle1xf6c.inp: CPS6 elements.
nle1xm6c.inp: CPS6M elements.
nle1xf8c.inp: CPS8 elements.
nle1xr8c.inp: CPS8R elements.

Fine mesh tests:

nle1xf3f.inp: CPS3 elements.
nle1xf4f.inp: CPS4 elements.
nle1xi4f.inp: CPS4I elements.
nle1xr4f.inp: CPS4R elements.
nle1xf6f.inp: CPS6 elements.
nle1xm6f.inp: CPS6M elements.
nle1xf8f.inp: CPS8 elements.
nle1xr8f.inp: CPS8R elements.

Abaqus/Explicit input files

Coarse mesh tests:

le1_cps3_c.inp: CPS3 elements.
le1_cps4r_c.inp: CPS4R elements.
le1_cps6m_c.inp: CPS6M elements.

Fine mesh tests:

le1_cps3_f.inp: CPS3 elements.
le1_cps4r_f.inp: CPS4R elements.
le1_cps6m_f.inp: CPS6M elements.

Very fine mesh tests:

le1_cps3_vf.inp: CPS3 elements.
le1_cps4r_vf.inp: CPS4R elements.