<!-- source-page: 261 -->

<table><tr><td>Load ID(*DLOAD)</td><td>Units</td><td>Description</td></tr><tr><td>TRSHR</td><td> $FL^{-2}$ </td><td>Shear traction on the element reference surface.</td></tr><tr><td> $TRSHRNU^{(S)}$ </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> $FL^{-2}$ </td><td>General traction on the element reference surface.</td></tr><tr><td> $TRVECNU^{(S)}$ </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></table>

# Foundations

Foundations are specified as described in “Element foundations,” Section 2.2.2.

<table><tr><td>Load ID(*FOUNDATION)</td><td>Units</td><td>Description</td></tr><tr><td>F</td><td> $FL^{-3}$ </td><td>Elastic foundation.</td></tr></table>

# Surface-based loading

# Distributed loads

Surface-based distributed loads are specified as described in “Distributed loads,” Section 34.4.3.

<table><tr><td>Load ID(*DSLOAD)</td><td>Units</td><td>Description</td></tr><tr><td>HP</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 to the surface normal.</td></tr><tr><td>P</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> $FL^{-2}$ </td><td>Nonuniform pressure on the element reference surface with magnitude supplied</td></tr></table>

<!-- source-page: 262 -->

<table><tr><td>Load ID(*DSLOAD)</td><td>Units</td><td>Description</td></tr><tr><td></td><td></td><td>via user subroutine DLOAD. The pressure is positive in the direction opposite to the surface normal.</td></tr><tr><td>TRSHR</td><td> $FL^{-2}$ </td><td>Shear traction on the element reference surface.</td></tr><tr><td> $TRSHRNU^{(S)}$ </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> $FL^{-2}$ </td><td>General traction on the element reference surface.</td></tr><tr><td> $TRVECNU^{(S)}$ </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></table>

# Element output

If a local orientation (“Orientations,” Section 2.2.5) is not used with the element, the stress/strain components are expressed in the default directions on the surface defined by the convention given in “Conventions,” Section 1.2.2. If a local orientation is used with the element, the stress/strain components are in the surface directions defined by the orientation. In large-displacement problems the local directions defined in the reference configuration are rotated into the current configuration by the average material rotation. See “State storage,” Section 1.5.4 of the Abaqus Theory Guide, for details.

# 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>Local 11 direct stress.</td></tr><tr><td>S22</td><td>Local 22 direct stress.</td></tr><tr><td>S12</td><td>Local 12 shear stress.</td></tr></table>

# Section thickness

STH Current thickness.

<!-- source-page: 263 -->

# Node ordering and face numbering on elements

![](images/page-263_ced4d4cca6efe7ec4112a1786e9074c05cd8a952f6ef85e46812d0705514f0f1.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    1 --> 2
    2 --> 3
    3 --> 4
    4 --> 6
    6 --> 1
```
</details>

6-node element

![](images/page-263_78157a8dfa5741a452862da48de9c0e476438ff2616c92fc90cbf73f90778ca5.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    1 --> 2
    2 --> 3
    3 --> 4
    4 --> 7
    5 --> 6
    6 --> 7
    7 --> 8
    8 --> 9
    9 --> 2
```
</details>

9-node element

# Numbering of integration points for output

![](images/page-263_ee9c657b2bae27328b2152c76c5f07e80fe60ac12f7b92b6f5265cfb74643965.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    3 --> 5
    5 --> 2
    2 --> 1
    1 --> 6
    6 --> 4
    4 --> 3
    3 --> 2
    2 --> 1
    1 --> 2
    2 --> 3
    3 --> 4
    4 --> 6
    6 --> 1
    3 -->|× 3| 2
    2 -->|× 2| 3
    1 -->|× 1| 4
```
</details>

6-node element

![](images/page-263_5290cadccbe1277c8a8e1e774510c583abd788a566f7dca33786fa8decb5fcc2.jpg)

<details>
<summary>text_image</summary>

3
7 ×6
×5
4
6
×4
9 ×2
×3
8
1
2
5
</details>

9-node element

<!-- source-page: 264 -->

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# 29.1.4 AXISYMMETRIC MEMBRANE ELEMENT LIBRARY

Products: Abaqus/Standard Abaqus/CAE

# References

• “Membrane elements,” Section 29.1.1   
• \*MEMBRANE SECTION   
• \*NODAL THICKNESS

# Overview

This section provides a reference to the axisymmetric membrane elements available in Abaqus/Standard.

# Conventions

Coordinate 1 is $\boldsymbol { r } ,$ coordinate 2 is z. At , the r-direction corresponds to the global X-direction and the z-direction corresponds to the global Y-direction. This is important when data are required in global directions. Coordinate 1 should be greater than or equal to zero.

Degree of freedom 1 is $u _ { r }$ , degree of freedom 2 is $u _ { z }$ . Generalized axisymmetric elements with twist have an additional degree of freedom, 5, corresponding to the twist angle $\phi$ (in radians).

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

Point loads and moments should be given as the value integrated around the circumference; that is, the total value on the ring.

# Element types

# Regular axisymmetric membranes

<table><tr><td>MAX1</td><td>2-node linear, without twist</td></tr><tr><td>MAX2</td><td>3-node quadratic, without twist</td></tr></table>

Active degrees of freedom

1, 2

Additional solution variables

None.

# Generalized axisymmetric membranes

<table><tr><td>MGAX1</td><td>2-node linear, with twist</td></tr><tr><td>MGAX2</td><td>3-node quadratic, with twist</td></tr></table>

<!-- source-page: 266 -->

Active degrees of freedom

1, 2, 5

Additional solution variables

Nodal coordinates required

R, Z

Element property definition 

<table><tr><td>Input File Usage:</td><td>*MEMBRANE SECTIONIn addition, use the following option for variable thickness membranes:*NODAL THICKNESS</td></tr><tr><td>Abaqus/CAE Usage:</td><td>Property module:Create Section:select Shell as the sectionCategory and Membrane as the section TypeYou cannot define variable thickness membranes in Abaqus/CAE.</td></tr></table>

Element-based loading   
Distributed loads

Distributed loads are specified as described in “Distributed loads,” Section 34.4.3.

<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 in the radial (1 or  $r$ ) direction.</td></tr><tr><td>BZ</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force in the axial (2 or  $z$ ) direction.</td></tr><tr><td>BRNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force in the radial direction with magnitude supplied via user subroutine DLOAD.</td></tr><tr><td>BZNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force in the axial direction with magnitude supplied via user subroutine DLOAD.</td></tr><tr><td>CENT</td><td>Not supported</td><td> $FL^{-4}$  $(ML^{-3}T^{-2})$ </td><td>Centrifugal load (magnitude is input as  $\rho\omega^{2}$ , where  $\rho$  is the mass density per unit volume,  $\omega$  is the angular</td></tr></table>

<!-- source-page: 267 -->

<table><tr><td>Load ID (*DLOAD)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td></td><td></td><td></td><td>velocity). Since only axisymmetric deformation is allowed, the spin axis must be the z-axis.</td></tr><tr><td>CENTRIF</td><td>Rotational body force</td><td> $T^{-2}$ </td><td>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</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. The pressure is positive in the direction of the positive element normal.</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 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</td></tr></table>

<!-- source-page: 268 -->

Load ID (\*DLOAD)

Abaqus/CAE Load/Interaction

Units

Description

magnitude and direction supplied via user subroutine UTRACLOAD.

# Foundations

Foundations are specified as described in “Element foundations,” Section 2.2.2.

Load ID(\*FOUNDATION)

Abaqus/CAE Load/Interaction

Units

Description

F

Elastic foundation

FL−3

Elastic foundation. For MGAX elements the elastic foundations are applied to degrees of freedom $u _ { r }$ and $u _ { z }$ only.

# Surface-based loading

# Distributed loads

Surface-based distributed loads are specified as described in “Distributed loads,” Section 34.4.3.

<table><tr><td>Load ID(*DSLOAD)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>HP</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 to 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. The pressure is positive in the direction opposite of the surface normal.</td></tr><tr><td>TRSHR</td><td>Surface traction</td><td> $FL^{-2}$ </td><td>Shear traction on the element reference surface.</td></tr></table>

<!-- source-page: 269 -->

<table><tr><td>Load ID(*DSLOAD)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</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></table>

# Incident wave loading

Surface-based incident wave loads are available. They are specified as described in “Acoustic and shock loads,” Section 34.4.6. If the incident wave field includes a reflection off a plane outside the boundaries of the mesh, this effect can be included.

# Element output

The default local material directions are such that local material direction 1 lies along the line of the element and local material direction 2 is the hoop direction.

# 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>Local 11 direct stress.</td></tr><tr><td>S22</td><td>Local 22 direct stress.</td></tr><tr><td>S12</td><td>Local 12 shear stress. Only available for generalized axisymmetric membrane elements.</td></tr></table>

# Section thickness

STH Current thickness.

<!-- source-page: 270 -->

# Node ordering on elements

![](images/page-270_4c0a39df0ee692b27ef030daceb600dbed3ee1820208a15e4707d3bf16d24cdb.jpg)

<details>
<summary>natural_image</summary>

Simple diagonal line with two labeled points (1 and 2) at endpoints (no additional text or symbols)
</details>

2 - node element

![](images/page-270_e74070dab192c4515d0e6d7fec786e957f014318faf77e81d68b5598fe893dde.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    1 --> 2
    2 --> 3
```
</details>

3 - node element

# Numbering of integration points for output

![](images/page-270_ea385ec2245199f3803cdadd5d670e842c405c2291580691a864b8d8d6324b10.jpg)

<details>
<summary>text_image</summary>

1
+
1
2
</details>

2 - node element

![](images/page-270_9a207245b9eff370bc1f8cd1c648d904c7cfaf30eb6fdf8cefb764f0909c5869.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    1["1"] -->|×| 1
    1 -->|×| 2["2"]
    2 -->|×| 2
    2 -->|×| 3["3"]
```
</details>

3 - node element
