Patch test for axisymmetric elements with twist

This problem contains basic test cases for one or more Abaqus elements and features.

The following topics are discussed:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/Standard

Elements tested

CGAX3

CGAX3H

CGAX3HT

CGAX3T

CGAX4

CGAX4H

CGAX4HT

CGAX4R

CGAX4RH

CGAX4T

CGAX6

CGAX6H

CGAX6M

CGAX6MH

CGAX8

CGAX8H

CGAX8HT

CGAX8R

CGAX8RH

CGAX8RHT

CGAX8RT

CGAX8T

Problem description

Material:

Linear elastic, Young's modulus = 1.0 × 10⁶, Poisson's ratio = 0.25, conductivity = 4.85 × 10⁻⁴.

Loading for Step 1

A twist of 0.01 per unit length applied to face CD.

$ϕ = $ 1.0 × 10⁻².

Loading for Step 2

Displacement boundary conditions applied to all exterior nodes: $u_{r} = $ 10⁻³r, $u_{z} = $ 10⁻³ $(r + z)$ , $ϕ = $ 0.

Nonuniform body force: To maintain a constant shear stress $σ_{r z} = $ 400 and preserve equilibrium, an equilibrating body force, BZNU, is defined in user subroutine DLOAD as BZNU = $- σ_{r z} / r = $ −400 $ / r$ , where r is the radius of the integration point.

Loading for Step 3

Displacement boundary conditions applied to all exterior nodes: $u_{r} = $ 10⁻²r, $u_{z} = $ 10⁻²z, $ϕ = $ 0.

Loading for Step 4

Displacement boundary conditions applied to the deformed geometry of Step 2 at all exterior nodes: $u_{r} = $ 10⁻³r, $u_{z} = $ 10⁻³ $(r + z)$ , $ϕ = $ 0.

Nonuniform body force (as described for Step 2): BZNU = $- σ_{r z} / r = $ −400 $ / r$ .

Loading for Step 5

The displacement boundary conditions are the same as those applied in Step 3.

Temperatures are prescribed at every node along the boundary of the mesh. $T = C_{1} r + C_{2} z$ , where T is the temperature, $C_{1}$ and $C_{2}$ are arbitrary constants, and r, z denote spatial location.

Nonuniform distributed flux: To maintain a uniform heat flux, q, a distributed heat flux, BFNU, is defined in user subroutine DFLUX as BFNU = $- C_{1} k / r$ , where r is the radius of the integration point and k is the conductivity.

Reference solution

The analytical results for each step are presented below.

Step 1: perturbation

Shear stress, $σ_{23} = G ϕ r$ , where r is the radial distance from the axis of symmetry and G is the shear modulus.

Resultant moment, $M = G ϕ π a^{4} /$ 2 = 6283.2.

Step 2: perturbation

$σ_{r} = σ_{z} = σ_{θ} = $ 2000.
$σ_{r z} = $ 400.
$ϵ_{r} = ϵ_{z} = ϵ_{θ} = $ 10⁻³.
$γ_{r z} = $ 10⁻³.

Step 3: geometrically nonlinear

$σ_{r} = σ_{z} = σ_{θ} = $ 19900.
$σ_{r z} = $ 0.
$ϵ_{r} = ϵ_{z} = ϵ_{θ} = $ 9.95 × 10⁻³.
$γ_{r z} = $ 0.

Step 4: perturbation

$σ_{r} = σ_{z} = σ_{θ} = $ 2000.
$σ_{r z} = $ 400.
$ϵ_{r} = ϵ_{z} = ϵ_{θ} = $ 1 × 10⁻³.
$γ_{r z} = $ 1 × 10⁻³.

Step 5: fully coupled thermal-stress

This step is applied only in tests of coupled temperature-displacement elements (CGAXxT).

Stresses and strains are the same as in Step 3. $q_{r} = - C_{1} k$ ; $q_{z} = - C_{2} k$ .

Results and discussion

The results agree well with the analytical solution for all elements.

Section output requests to the results (.fil) file and to the data (.dat) file are used in the input files with CGAX8RH elements to output accumulated quantities in different sections through the model.

Input files

eca3gfp5.inp: CGAX3 elements.
eca3gfp5.f: User subroutine DLOAD used in eca3gfp5.inp.
eca3ghp5.inp: CGAX3H elements.
eca3ghp5.f: User subroutine DLOAD used in eca3ghp5.inp.
eca3hhp5.inp: CGAX3HT elements.
eca3hhp5.f: User subroutines DLOAD and DFLUX used in eca3hhp5.inp.
eca3hfp5.inp: CGAX3T elements.
eca3hfp5.f: User subroutines DLOAD and DFLUX used in eca3hfp5.inp.
eca4gfp5.inp: CGAX4 elements.
eca4gfp5.f: User subroutine DLOAD used in eca4gfp5.inp.
eca4ghp5.inp: CGAX4H elements.
eca4ghp5.f: User subroutine DLOAD used in eca4ghp5.inp.
eca4hhp5.inp: CGAX4HT elements.
eca4hhp5.f: User subroutines DLOAD and DFLUX used in eca4hhp5.inp.
eca4grp5.inp: CGAX4R elements.
eca4grp5.f: User subroutine DLOAD used in eca4grp5.inp.
eca4gyp5.inp: CGAX4RH elements.
eca4gyp5.f: User subroutine DLOAD used in eca4gyp5.inp.
eca4hfp5.inp: CGAX4T elements.
eca4hfp5.f: User subroutines DLOAD and DFLUX used in eca4hfp5.inp.
eca6gfp5.inp: CGAX6 elements.
eca6gfp5.f: User subroutine DLOAD used in eca6gfp5.inp.
eca6ghp5.inp: CGAX6H elements.
eca6ghp5.f: User subroutine DLOAD used in eca6ghp5.inp.
eca6gkp5.inp: CGAX6M elements.
eca6gkp5.f: User subroutine DLOAD used in eca6gkp5.inp.
eca6glp5.inp: CGAX6MH elements.
eca6glp5.f: User subroutine DLOAD used in eca6glp5.inp.
eca8gfp5.inp: CGAX8 elements.
eca8gfp5.f: User subroutine DLOAD used in eca8gfp5.inp.
eca8ghp5.inp: CGAX8H elements.
eca8ghp5.f: User subroutine DLOAD used in eca8ghp5.inp.
eca8hhp5.inp: CGAX8HT elements.
eca8hhp5.f: User subroutines DLOAD and DFLUX used in eca8hhp5.inp.
eca8grp5.inp: CGAX8R elements.
eca8grp5.f: User subroutine DLOAD used in eca8grp5.inp.
eca8gyp5.inp: CGAX8RH elements.
eca8gyp5.f: User subroutine DLOAD used in eca8gyp5.inp.
eca8hfp5.inp: CGAX8T elements.
eca8hfp5.f: User subroutines DLOAD and DFLUX used in eca8hfp5.inp.