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# 29.6.9 AXISYMMETRIC SHELL ELEMENT LIBRARY

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

# References

• “Shell elements: overview,” Section 29.6.1   
• “Choosing a shell element,” Section 29.6.2   
• \*NODAL THICKNESS   
• \*SHELL GENERAL SECTION   
• \*SHELL SECTION

# Overview

This section provides a reference to the axisymmetric shell elements available in Abaqus/Standard and Abaqus/Explicit. For axisymmetric shell geometries in which nonaxisymmetric behavior is expected, use the SAXA elements available in Abaqus/Standard (see “Axisymmetric shell elements with nonlinear, asymmetric deformation,” Section 29.6.10).

# Conventions

Coordinate 1 is r, coordinate 2 is z. The r-direction corresponds to the global X-direction, and the zdirection corresponds to the global Y-direction. Coordinate 1 should be greater than or equal to zero.

Degree of freedom 1 is $u _ { r }$ , degree of freedom 2 is $u _ { z }$ , and degree of freedom 6 is rotation in the r–z plane.

Abaqus does not automatically apply any boundary conditions to nodes located along the symmetry axis. You should apply radial or symmetry boundary conditions on these nodes if desired.

Point loads and concentrated fluxes should be given as the value integrated around the circumference (that is, the load on the complete ring).

The meridional direction is the direction that is tangent to the element in the r–z plane; that is, the meridional direction is along the line that is rotated about the axis of symmetry to generate the full three-dimensional body.

The circumferential or hoop direction is the direction normal to the r–z plane.

# Element types

# Stress/displacement elements

SAX1 2-node thin or thick linear shell   
SAX2(S) 3-node thin or thick quadratic shell

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Active degrees of freedom

1, 2, 6

Additional solution variables

None.

# Heat transfer elements

DSAX1(S)

2-node shell

DSAX2(S)

3-node shell

Active degrees of freedom

11, 12, 13, etc. (temperatures through the thickness as described in “Choosing a shell element,” Section 29.6.2)

Additional solution variables

None.

# Coupled temperature-displacement element

SAX2T(S)

3-node thin or thick shell, quadratic displacement, linear temperature in the shell surface

Active degrees of freedom

1, 2, 6 at all three nodes

11, 12, 13, etc. (temperatures through the thickness as described in “Choosing a shell element,” Section 29.6.2) at the end nodes

Additional solution variables

None.

# Nodal coordinates required

r, z, and optionally for shells with displacement degrees of freedom, , , the direction cosines of the shell normal at the node.

# Element property definition

Input File Usage:

Use either of the following options for stress/displacement elements:

\*SHELL SECTION

\*SHELL GENERAL SECTION

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Use the following option for heat transfer or coupled temperature-displacement elements:

\*SHELL SECTION

In addition, use the following option for variable thickness shells:

\*NODAL THICKNESS

Abaqus/CAE Usage: Property module: Create Section: select Shell as the section Category and Homogeneous or Composite as the section Type

# Element-based loading

# Distributed loads

Distributed loads are available for elements with displacement degrees of freedom. They are specified as described in “Distributed loads,” Section 34.4.3.

Distributed load magnitudes are per unit area or per unit volume. They do not need to be multiplied by .

Body forces and centrifugal loads must be given as force per unit area if a general shell section is used.

<table><tr><td>Load ID (*DLOAD)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>BR</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force per unit volume in the radial direction.</td></tr><tr><td>BZ</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force per unit volume in the axial direction.</td></tr><tr><td>BRNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force per unit volume in the radial direction, with the magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit.</td></tr><tr><td>BZNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force per unit volume in the global z-direction, with the magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit.</td></tr><tr><td> $CENT^{(S)}$ </td><td>Not supported</td><td> $FL^{-4}$  $(ML^{-3}T^{-2})$ </td><td>Centrifugal load (magnitude given as  $\rho\omega^{2}$ , where  $\rho$  is the mass density and  $\omega$  is the angular velocity). Since only</td></tr></table>

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<table><tr><td>Load ID (*DLOAD)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>CENTRIF(S)</td><td>Rotational body force</td><td> $T^{-2}$ </td><td>axisymmetric deformation is allowed, the spin axis must be the z-axis. Centrifugal load (magnitude is input as  $\omega^2$ , where  $\omega$  is the angular velocity). Since only axisymmetric deformation is allowed, the spin axis must be the z-axis.</td></tr><tr><td>GRAV</td><td>Gravity</td><td> $LT^{-2}$ </td><td>Gravity loading in a specified direction (magnitude input as acceleration).</td></tr><tr><td>HP(S)</td><td>Not supported</td><td> $FL^{-2}$ </td><td>Hydrostatic pressure applied to the element reference surface and linear in global Z. The pressure is positive in the direction of the positive element normal.</td></tr><tr><td>P</td><td>Pressure</td><td> $FL^{-2}$ </td><td>Pressure applied to the element reference surface. The pressure is positive in the direction of the positive element normal.</td></tr><tr><td>PNU</td><td>Not supported</td><td> $FL^{-2}$ </td><td>Nonuniform pressure applied to the element reference surface with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit. The pressure is positive in the direction of the positive element normal.</td></tr><tr><td>SBF(E)</td><td>Not supported</td><td> $FL^{-5}T^2$ </td><td>Stagnation body force in radial and axial directions.</td></tr><tr><td>SP(E)</td><td>Not supported</td><td> $FL^{-4}T^2$ </td><td>Stagnation pressure applied to the element reference surface.</td></tr><tr><td>TRSHR</td><td>Surface traction</td><td> $FL^{-2}$ </td><td>Shear traction on the element reference surface.</td></tr><tr><td>TRSHRNU(S)</td><td>Not supported</td><td> $FL^{-2}$ </td><td>Nonuniform shear traction on the element reference surface with</td></tr></table>

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<table><tr><td>Load ID(*DLOAD)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td></td><td></td><td></td><td>magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td>TRVEC</td><td>Surface traction</td><td> $FL^{-2}$ </td><td>General traction on the element reference surface.</td></tr><tr><td> $TRVECNU^{(S)}$ </td><td>Not supported</td><td> $FL^{-2}$ </td><td>Nonuniform general traction on the element reference surface with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td> $VBF^{(E)}$ </td><td>Not supported</td><td> $FL^{-4}T$ </td><td>Viscous body force in radial and axial directions.</td></tr><tr><td> $VP^{(E)}$ </td><td>Not supported</td><td> $FL^{-3}T$ </td><td>Viscous surface pressure. The viscous pressure is proportional to the velocity normal to the element face and opposing the motion.</td></tr></table>

# Foundations

Foundations are available for Abaqus/Standard elements with displacement degrees of freedom. They are specified as described in “Element foundations,” Section 2.2.2.

<table><tr><td>Load ID(*FOUNDATION)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $F^{(S)}$ </td><td>Elasticfoundation</td><td> $FL^{-3}$ </td><td>Elastic foundation in the direction ofthe shell normal.</td></tr></table>

# Distributed heat fluxes

Distributed heat fluxes are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID(*DFLUX)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $BF^{(S)}$ </td><td>Body heat flux</td><td> $JL^{-3} \ T^{-1}$ </td><td>Body heat flux per unit volume.</td></tr><tr><td> $BFNU^{(S)}$ </td><td>Body heat flux</td><td> $JL^{-3} \ T^{-1}$ </td><td>Nonuniform body heat flux per unit volume with magnitude supplied via user subroutine DFLUX.</td></tr><tr><td> $SNEG^{(S)}$ </td><td>Surface heat flux</td><td> $JL^{-2} \ T^{-1}$ </td><td>Surface heat flux per unit area into the bottom face of the element.</td></tr></table>

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<table><tr><td>Load ID(*DFLUX)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $SPOS^{(S)}$ </td><td>Surface heat flux</td><td> $JL^{-2} \ T^{-1}$ </td><td>Surface heat flux per unit area into the top face of the element.</td></tr><tr><td> $SNEGNU^{(S)}$ </td><td>Not supported</td><td> $JL^{-2} \ T^{-1}$ </td><td>Nonuniform surface heat flux per unit area into the bottom face of the element with magnitude supplied via user subroutine DFLUX.</td></tr><tr><td> $SPOSNU^{(S)}$ </td><td>Not supported</td><td> $JL^{-2} \ T^{-1}$ </td><td>Nonuniform surface heat flux per unit area into the top face of the element with magnitude supplied via user subroutine DFLUX.</td></tr></table>

# Film conditions

Film conditions are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID(*FILM)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $FNEG^{(S)}$ </td><td>Surface film condition</td><td> $JL^{-2} T^{-1} \theta^{-1}$ </td><td>Film coefficient and sink temperature (units of  $\theta$ ) provided on the bottom face of the element.</td></tr><tr><td> $FPOS^{(S)}$ </td><td>Surface film condition</td><td> $JL^{-2} T^{-1} \theta^{-1}$ </td><td>Film coefficient and sink temperature (units of  $\theta$ ) provided on the top face of the element.</td></tr><tr><td> $FNEGNU^{(S)}$ </td><td>Not supported</td><td> $JL^{-2} T^{-1} \theta^{-1}$ </td><td>Nonuniform film coefficient and sink temperature (units of  $\theta$ ) provided on the bottom face of the element with magnitude supplied via user subroutine  $\text{FILM}$ .</td></tr><tr><td> $FPOSNU^{(S)}$ </td><td>Not supported</td><td> $JL^{-2} T^{-1} \theta^{-1}$ </td><td>Nonuniform film coefficient and sink temperature (units of  $\theta$ ) provided on the top face of the element with magnitude supplied via user subroutine  $\text{FILM}$ .</td></tr></table>

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# Radiation types

Radiation conditions are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID(*RADIATE)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>RNEG(S)</td><td>Surface radiation</td><td>Dimensionless</td><td>Emissivity and sink temperature (units of θ) provided for the bottom face of the shell.</td></tr><tr><td>RPOS(S)</td><td>Surface radiation</td><td>Dimensionless</td><td>Emissivity and sink temperature (units of θ) provided for the top face of the shell.</td></tr></table>

# Surface-based loading

# Distributed loads

Surface-based distributed loads are available for elements with displacement degrees of freedom. They are specified as described in “Distributed loads,” Section 34.4.3.

Distributed load magnitudes are per unit area or per unit volume. They do not need to be multiplied by .

<table><tr><td>Load ID(*DSLOAD)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>HP(S)</td><td>Pressure</td><td> $FL^{-2}$ </td><td>Hydrostatic pressure on the element reference surface and linear in global Z. The pressure is positive in the direction opposite the surface normal.</td></tr><tr><td>P</td><td>Pressure</td><td> $FL^{-2}$ </td><td>Pressure on the element reference surface. The pressure is positive in the direction opposite to the surface normal.</td></tr><tr><td>PNU</td><td>Pressure</td><td> $FL^{-2}$ </td><td>Nonuniform pressure on the element reference surface with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit. The pressure is positive in the direction opposite to the surface normal.</td></tr></table>

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<table><tr><td>Load ID(*DSLOAD)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $SP^{(E)}$ </td><td>Pressure</td><td> $FL^{-4}T^{2}$ </td><td>Stagnation pressure applied to the element reference surface.</td></tr><tr><td>TRSHR</td><td>Surface traction</td><td> $FL^{-2}$ </td><td>Shear traction on the element reference surface.</td></tr><tr><td> $TRSHRNU^{(S)}$ </td><td>Surface traction</td><td> $FL^{-2}$ </td><td>Nonuniform shear traction on the element reference surface with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td>TRVEC</td><td>Surface traction</td><td> $FL^{-2}$ </td><td>General traction on the element reference surface.</td></tr><tr><td> $TRVECNU^{(S)}$ </td><td>Surface traction</td><td> $FL^{-2}$ </td><td>Nonuniform general traction on the element reference surface with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td> $VP^{(E)}$ </td><td>Pressure</td><td> $FL^{-3}T$ </td><td>Viscous surface pressure. The viscous pressure is proportional to the velocity normal to the element surface and opposing the motion.</td></tr></table>

# Distributed heat fluxes

Surface-based heat fluxes are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID(*DSFLUX)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $S^{(S)}$ </td><td>Surface heat flux</td><td> $JL^{-2}$   $T^{-1}$ </td><td>Surface heat flux per unit area into the element surface.</td></tr><tr><td> $SNU^{(S)}$ </td><td>Surface heat flux</td><td> $JL^{-2}$   $T^{-1}$ </td><td>Nonuniform surface heat flux per unit area into the element surface with magnitude supplied via user subroutine DFLUX.</td></tr></table>

# Film conditions

Surface-based film conditions are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

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<table><tr><td>Load ID(*SFILM)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $F^{(S)}$ </td><td>Surface film condition</td><td> $JL^{-2} T^{-1} \theta^{-1}$ </td><td>Film coefficient and sink temperature (units of  $\theta$ ) provided on the element surface.</td></tr><tr><td> $FNU^{(S)}$ </td><td>Surface film condition</td><td> $JL^{-2} T^{-1} \theta^{-1}$ </td><td>Nonuniform film coefficient and sink temperature (units of  $\theta$ ) provided on the element surface with magnitude supplied via user subroutineFILM.</td></tr></table>

# Radiation types

Surface-based radiation conditions are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID(*SRADIATE)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $R^{(S)}$ </td><td>Surface radiation</td><td>Dimensionless</td><td>Emissivity and sink temperature (units of  $\theta$ ) provided for the element surface.</td></tr></table>

# Incident wave loading

Surface-based incident wave loads are available. They are specified as described in “Acoustic, shock, and coupled acoustic-structural analysis,” Section 6.10.1. If the incident wave field includes a reflection off a plane outside the boundaries of the mesh, this effect can be included.

# Element output

# Stress, strain, and other tensor components

Stress and other tensors (including strain tensors) are available for elements with displacement degrees of freedom. All tensors have the same components. For example, the stress components are as follows:

<table><tr><td>S11</td><td>Meridional stress.</td></tr><tr><td>S22</td><td>Hoop (circumferential) stress.</td></tr></table>

# Section forces, moments, and transverse shear forces

Available for elements with displacement degrees of freedom.

SF1 Membrane force per unit width in the meridional direction.

SF2 Membrane force per unit width in the hoop direction.

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<table><tr><td>SF3</td><td>Transverse shear force per unit width in the meridional direction (available only from Abaqus/Standard).</td></tr><tr><td>SF4</td><td>Integrated stress in the thickness direction; always zero (available only from Abaqus/Standard).</td></tr><tr><td>SM1</td><td>Bending moment per unit width about the hoop direction.</td></tr><tr><td>SM2</td><td>Bending moment per unit width about the meridional direction.</td></tr></table>

# Section strains, curvature changes, and transverse shear strains

Available for elements with displacement degrees of freedom.

<table><tr><td>SE1</td><td>Membrane strain in the meridional direction.</td></tr><tr><td>SE2</td><td>Membrane strain in the hoop direction.</td></tr><tr><td>SE3</td><td>Transverse shear strain in the meridional direction (available only from Abaqus/Standard).</td></tr><tr><td>SE4</td><td>Strain in the thickness direction (available only from Abaqus/Standard).</td></tr><tr><td>SK1</td><td>Curvature change about the hoop direction.</td></tr><tr><td>SK2</td><td>Curvature change about the meridional direction.</td></tr></table>

# Shell thickness

STH Shell thickness, which is the current thickness for SAX1, SAX2, and SAX2T elements.

# Heat flux components

Available for elements with temperature degrees of freedom.

HFL1 Heat flux in the meridional direction.

HFL2 Heat flux in the thickness direction.

# Node ordering on elements

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Simple line diagram with two labeled points (1 and 2) connected by a straight line (no additional text or symbols)
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2 - node element

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1
2
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3 - node element
