LE5: Z-section cantilever

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
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

S3
S3R
S4R
S4R5
S4RS
S4RSW
S8R
S8R5
S9R5
STRI3
STRI65

B31OS
B32OS

Problem description

Model:

Z-section cantilever under torsional loading.

Material:

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

Boundary conditions:

All displacements are zero along the edge at $x = $ 0.

Loading:

Torque of 1.2 MN-m applied at $x = $ 10. The torque is applied by two uniformly distributed edge shears of 0.6 MN at each flange when shell elements are used. In the explicit dynamic analysis the loading rate 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 LE5 from NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks,” October 1990.

Target solution: Axial stress, $σ_{x x} = $ −108 MPa at midsurface, point A.

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. Slow convergence toward the target solution is seen as the mesh is refined.

Table 1. Abaqus/Standard analysis.
Element	$σ_{x x}$ , Coarse Mesh	$σ_{x x}$ , Refined Mesh
S3/S3R	−24.266 MPa (−78%)	−92.166 MPa (−15%)
S4	−110.36 MPa (2.2%)	−110.38 MPa (2.2%)
S4R	−50.480 MPa (−53%)	−96.732 MPa (−10%)
S4R5	−50.116 MPa (−54%)	−96.378 MPa (−11%)
S8R	−109.85 MPa (1.7%)	—
S8R5	−109.72 MPa (1.6%)	—
S9R5	−109.72 MPa (1.6%)	—
STRI3	−30.389 MPa (−72%)	−94.532 MPa (−12%)
STRI65	−107.32 MPa (−0.63%)	—
B31OS	−108.09 MPa (0.08%)	—
B32OS	−107.34 MPa (−0.61%)	—

Table 2. Abaqus/Explicit analysis.
Element	$σ_{x x}$ , Coarse Mesh	$σ_{x x}$ , Refined Mesh
S4R	−49.5 MPa (−54%)	−100.3 MPa (−7.1%)
S4RS	−87.5 MPa (−19%)	−100.3 MPa (−7.1%)
S4RSW	−87.7 MPa (−19%)	−100.3 MPa (−7.1%)

Input files

Abaqus/Standard input files

Coarse mesh tests:

nle5xf3c.inp: S3/S3R elements.
nle5xe4c.inp: S4 elements.
nle5xf4c.inp: S4R elements.
nle5x54c.inp: S4R5 elements.
nle5x68c.inp: S8R elements.
nle5x58c.inp: S8R5 elements.
nle5x59c.inp: S9R5 elements.
nle5x63c.inp: STRI3 elements.
nle5x56c.inp: STRI65 elements.
nle5xb2c.inp: B31OS elements.
nle5xb3c.inp: B32OS elements.

Fine mesh tests:

nle5xf3f.inp: S3/S3R elements.
nle5xe4f.inp: S4 elements.
nle5xf4f.inp: S4R elements.
nle5x54f.inp: S4R5 elements.
nle5x63f.inp: STRI3 elements.

Abaqus/Explicit input files

Coarse mesh tests:

le5_c.inp: S4R elements.
le5_c_s4rs.inp: S4RS elements.
le5_c_s4rsw.inp: S4RSW elements.

Fine mesh tests:

le5_f.inp: S4R elements.
le5_f_s4rs.inp: S4RS elements.
le5_f_s4rs_subcyc.inp: S4RS elements and subcycling.
le5_f_s4rsw.inp: S4RSW elements.