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:

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:

ux=0 along edge AB, uy=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 (σyy) 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.