T1: Plane stress elements—membrane with hot-spot

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/Standard

Elements tested

CPS3

CPS4

CPS4I

CPS4R

CPS6

CPS6M

CPS8

CPS8R

Problem description

Material:

Linear elastic, Young's modulus = 100 GPa, Poisson's ratio = 0.3.

Boundary conditions:

A quarter-section is modeled with the symmetry conditions $u_{y} =$ 0 along $y =$ 0 and $u_{x} =$ 0 along $x =$ 0.

Loading:

Within the hot-spot, thermal strain ( $α T$ ) = 1.0 × 10⁻³. Outside the hot-spot, thermal strain ( $α T$ ) = 0.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test T1 from NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks,” October 1990.

Target solution: $σ_{y y}$ at point D (outside the hot-spot) = 50.0 MPa.

Results and discussion

The results are shown in the following table. The values enclosed in parentheses are percentage differences with respect to the reference solution.

Element	$σ_{y y}$ at D
CPS3	54.25 MPa (8.4%)
CPS4	50.94 MPa (1.9%)
CPS4I	46.29 MPa (−7.4%)
CPS4R	23.65 MPa (−52.7%)*
CPS6	54.46 MPa (8.9%)
CPS6M	54.25 MPa (8.5%)
CPS8	51.53 MPa (3.1%)
CPS8R	44.06 MPa (−11.9%)*

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

Input files

nt1xxf3x.inp: CPS3 elements.
nt1xxf4x.inp: CPS4 elements.
nt1xxf6x.inp: CPS6 elements.
nt1xxf8x.inp: CPS8 elements.
nt1xxi4x.inp: CPS4I elements.
nt1xxm6x.inp: CPS6M elements.
nt1xxr4x.inp: CPS4R elements.
nt1xxr8x.inp: CPS8R elements.