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SM1 Bending moment force per unit width about local 2-axis.

SM2 Bending moment force per unit width about local 1-axis.

SM3 Twisting moment force per unit width in local 1–2 plane.

The section force and moment resultants per unit length in the normal basis directions in a given shell section of thickness h can be defined on this basis as

$$
(S F 1, S F 2, S F 3, S F 4, S F 5) = \int_ {- h / 2 - z _ {0}} ^ {h / 2 - z _ {0}} \left(\sigma_ {1 1}, \sigma_ {2 2}, \sigma_ {1 2}, \sigma_ {1 3}, \sigma_ {2 3}\right) d z,
$$

$$
(S M 1, S M 2, S M 3) = \int_ {- h / 2 - z _ {0}} ^ {h / 2 - z _ {0}} (\sigma_ {1 1}, \sigma_ {2 2}, \sigma_ {1 2}) z d z,
$$

where $z _ { 0 }$ is the offset of the reference surface from the midsurface.

The section force SF6, which is the integral of $\sigma _ { 3 3 }$ through the shell thickness, is reported only for finitestrain shell elements and is zero because of the plane stress constitutive assumption. The total number of attributes written to the results file for finite-strain shell elements is 9; SF6 is the sixth attribute.

# Average section stresses

Available for elements with displacement degrees of freedom.

SSAVG1 Average membrane stress in local 1-direction.

SSAVG2 Average membrane stress in local 2-direction.

SSAVG3 Average membrane stress in local 1–2 plane.

SSAVG4 Average transverse shear stress in local 1-direction.

SSAVG5 Average transverse shear stress in local 2-direction.

The average section stresses are defined as

$$
(S S A V G 1, S S A V G 2, S S A V G 3, S S A V G 4, S S A V G 5) = (S F 1, S F 2, S F 3, S F 4, S F 5) / h,
$$

where h is the current section thickness.

# Section strains, curvatures, and transverse shear strains

Available for elements with displacement degrees of freedom.

SE1 Direct membrane strain in local 1-direction.

SE2 Direct membrane strain in local 2-direction.

SE3 Shear membrane strain in local 1–2 plane.

SE4 Transverse shear strain in the local 1-direction (available only for S3/S3R, S3RS, S4, S4R, S4RS, S4RSW, S8R, and S8RT).

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<table><tr><td>SE5</td><td>Transverse shear strain in the local 2-direction (available only for S3/S3R, S3RS, S4, S4R, S4RS, S4RSW, S8R, and S8RT).</td></tr><tr><td>SE6</td><td>Strain in the thickness direction (available only for S3/S3R, S3RS, S4, S4R, S4RS, and S4RSW).</td></tr><tr><td>SK1</td><td>Curvature change about local 2-axis.</td></tr><tr><td>SK2</td><td>Curvature change about local 1-axis.</td></tr><tr><td>SK3</td><td>Surface twist in local 1–2 plane.</td></tr></table>

The local directions are defined in “Shell elements: overview,” Section 29.6.1.

# Shell thickness

STH Shell thickness, which is the current section thickness for S3/S3R, S3RS, S4, S4R, S4RS, and S4RSW elements.

# Transverse shear stress estimates

Available for S3/S3R, S3RS, S4, S4R, S4RS, S4RSW, S8R, and S8RT elements.

<table><tr><td>TSHR13</td><td>13-component of transverse shear stress.</td></tr><tr><td>TSHR23</td><td>23-component of transverse shear stress.</td></tr></table>

Estimates of the transverse shear stresses are available at section integration points as output variables TSHR13 or TSHR23 for both Simpson’s rule and Gauss quadrature. For Simpson’s rule output of variables TSHR13 or TSHR23 should be requested at nondefault section points, since the default output is at section point 1 of the shell section where the transverse shear stresses vanish. For the smallstrain elements in Abaqus/Explicit, transverse shear stress distributions are assumed constant for noncomposite sections and piecewise constant for composite sections; therefore, transverse shear stresses at integration points should be interpreted accordingly.

For element type S4 the transverse shear calculation is performed at the center of the element and assumed constant over the element. Hence, transverse shear strain, force, and stress will not vary over the area of the element.

For numerically integrated shell sections (with the exception of small-strain shells in Abaqus/Explicit), estimates of the interlaminar shear stresses in composite sections—i.e., the transverse shear stresses at the interface between two composite layers—can be obtained only by using Simpson’s rule. With Gauss quadrature no section integration point exists at the interface between composite layers.

Unlike the S11, S22, and S12 in-plane stress components, transverse shear stress components TSHR13 and TSHR23 are not calculated from the constitutive behavior at points through the shell section. They are estimated by matching the elastic strain energy associated with shear deformation of the shell section with that based on piecewise quadratic variation of the transverse shear stress across the section, under conditions of bending about one axis (see “Transverse shear stiffness in composite shells and offsets from the midsurface,” Section 3.6.8 of the Abaqus Theory Guide). Therefore, interlaminar shear stress

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calculation is supported only when the elastic material model is used for each layer of the shell section. If you specify the transverse shear stiffness values, interlaminar shear stress output is not available.

# Heat flux components

Available for elements with temperature degrees of freedom.

HFL1 Heat flux in local 1-direction.

HFL2 Heat flux in local 2-direction.

HFL3 Heat flux in local 3-direction.

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![](images/page-484_36b2794888b748eb1a264ce499f227aab2974aec96fb3c48525ff7209884559b.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["1"] -->|face 1| B["2"]
    B -->|face 2| C["3"]
    C -->|face 3| A
```
</details>

3-node element

![](images/page-484_82f2f7c4ec2659a438fc6462b888d9a2d061d19a8e96a5a8ac73dcc57cbf879d.jpg)

<details>
<summary>text_image</summary>

face 3
face 4 4 3
face 2
1 face 1 2
</details>

4-node element

![](images/page-484_7b77b376b3ef7ba630888b3452e1e84be53f08f2cd4d7d8d631acb53c8a13fac.jpg)

<details>
<summary>flowchart</summary>

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

face 1   
6-node element

![](images/page-484_2611705c5ea1189737ae886d2f60bb56a077d43cf65986172f7aefe72353122d.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["1"] --> B["2"]
    B --> C["3"]
    C --> D["4"]
    D --> E["5"]
    E --> F["6"]
    F --> G["7"]
    G --> H["8"]
    H --> A
    style A fill:#f9f,stroke:#333
    style B fill:#f9f,stroke:#333
    style C fill:#f9f,stroke:#333
    style D fill:#f9f,stroke:#333
    style E fill:#f9f,stroke:#333
    style F fill:#f9f,stroke:#333
    style G fill:#f9f,stroke:#333
    style H fill:#f9f,stroke:#333
```
</details>

face 1

8-node element   
![](images/page-484_e313cfd6a30328a42d692c5bbb3de77c6de504f2ca43cef00dc62d91010ed15f.jpg)

<details>
<summary>flowchart</summary>

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

face 1   
9-node element

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# Stress/displacement analysis

![](images/page-485_f7da6b431b54d171ef9411fa70c830c646697cffc860ff7fca5f0157b14894ac.jpg)

<details>
<summary>text_image</summary>

3
×₁
1
2
</details>

S3R element

![](images/page-485_9eacaed561a8fd7c55e527af3245b4b73012ec984eeb0e97c74c0ea07365a03c.jpg)

<details>
<summary>text_image</summary>

4
×1
1
3
2
</details>

4-nodereduced integrationelement

![](images/page-485_10f62f4fae89b44906a6c3fdf666d6a183cdaf95acb759244177e62c7ef65929.jpg)

<details>
<summary>text_image</summary>

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

9-nodereduced integrationelement

![](images/page-485_bd6996223334845e1df74f9958b6dbd04de1eae2da49211e37c430e81af38e7c.jpg)

<details>
<summary>text_image</summary>

3
×3
×1 2 ×
1 2
</details>

STRI3element

![](images/page-485_e4ff8af7277a4fddb10e2e31118aeaeb908687f7c1bf60d3b08652edb5a59075.jpg)

<details>
<summary>text_image</summary>

4 ×3 4×
×1 2×
1 2 3
</details>

4-node full integrationelement

![](images/page-485_17a75a07293d05c41a694c35c5bf4685a29ef091898822750148a29087c6694a.jpg)

<details>
<summary>text_image</summary>

3
6 ×3
×1
1 4 2×
2
</details>

6-nodeelement

![](images/page-485_a961a88d9767753eab5b204b3a017c5c0912c9c7be8e17bee4bf29fc4dd4d054.jpg)

<details>
<summary>flowchart</summary>

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

8-nodereduced integrationelement

<!-- source-page: 486 -->

# Heat transfer analysis

![](images/page-486_44d365833e99d972e4e11e0079f6e97110c44d22611274ae5bbca1b246741f8f.jpg)

<details>
<summary>text_image</summary>

3
3×
1× 2×
1 2
</details>

DS3

![](images/page-486_c3c22faa9882c64d208905d66d696bd4a6069e382626456f3264c5689ebbe07f.jpg)

<details>
<summary>text_image</summary>

3
3 ×
6 × 5 ×
1 × 4 × 2 ×
1 4 2
</details>

DS6

![](images/page-486_cef4582c775c77f7186ae227e0a795d4ccea5802f597264e5d1518f78229a61c.jpg)

<details>
<summary>text_image</summary>

4
3
3× 4×
1× 2×
1 2
</details>

DS4

![](images/page-486_a88a6a2e29c8bc0659fab1f005ad9ef86fd18de95d41a590f17b98c7d6afc11f.jpg)

<details>
<summary>text_image</summary>

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

<!-- source-page: 487 -->

# 29.6.8 CONTINUUM 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   
• \*SHELL GENERAL SECTION   
• \*SHELL SECTION

# Overview

This section provides a reference to the continuum shell elements available in Abaqus/Standard and Abaqus/Explicit.

# Element types

Stress/displacement elements 

<table><tr><td>SC6R</td><td>6-node triangular in-plane continuum shell wedge, general-purpose, finite membrane strains</td></tr><tr><td>SC8R</td><td>8-node hexahedron, general-purpose, finite membrane strains</td></tr></table>

Active degrees of freedom

1, 2, 3

Additional solution variables

None.

Coupled temperature-displacement elements 

<table><tr><td>SC6RT</td><td>6-node linear displacement and temperature, triangular in-plane continuum shell wedge, general-purpose, finite membrane strains</td></tr><tr><td>SC8RT</td><td>8-node linear displacement and temperature, hexahedron, general-purpose, finite membrane strains</td></tr></table>

Active degrees of freedom

1, 2, 3, 11

Additional solution variables

None.

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Nodal coordinates required

X,Y, Z

Element property definition 

<table><tr><td>Input File Usage:</td><td>Use either of the following options:*SHELL SECTION*SHELL GENERAL SECTION</td></tr><tr><td>Abaqus/CAE Usage:</td><td>Property module: Create Section: select Shell as the section Category and Homogeneous or Composite as the section Type</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>BX</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force (give magnitude as force per unit volume) in the global X-direction.</td></tr><tr><td>BY</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force (give magnitude as force per unit volume) in the global Y-direction.</td></tr><tr><td>BZ</td><td>Body force</td><td> $FL^{-3}$ </td><td>Body force (give magnitude as force per unit volume) in the global Z-direction.</td></tr><tr><td>BXNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force (give magnitude as force per unit volume) in the global X-direction, with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit.</td></tr><tr><td>BYNU</td><td>Body force</td><td> $FL^{-3}$ </td><td>Nonuniform body force (give magnitude as force per unit volume) in the global Y-direction, with magnitude supplied via</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></td><td></td><td></td><td>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 (give magnitude as force per unit volume) in the global Z-direction, with 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 defined as  $\rho\omega^{2}$ , where  $\rho$  is the mass density and  $\omega$  is the angular speed).</td></tr><tr><td> $CENTRIF^{(S)}$ </td><td>Rotational body force</td><td> $T^{-2}$ </td><td>Centrifugal load (magnitude is input as  $\omega^{2}$ , where  $\omega$  is the angular speed).</td></tr><tr><td> $CORIO^{(S)}$ </td><td>Coriolis force</td><td> $FL^{-4}T$  $(ML^{-3}T^{-1})$ </td><td>Coriolis force (magnitude input  $\rho\omega$ , where  $\rho$  is the mass density and  $\omega$  is the angular speed). The load stiffness due to Coriolis loading is not accounted for in direct steady-state dynamics analysis.</td></tr><tr><td>GRAV</td><td>Gravity</td><td> $LT^{-2}$ </td><td>Gravity loading in a specified direction (magnitude is input as acceleration).</td></tr><tr><td> $HPn^{(S)}$ </td><td>Not supported</td><td> $FL^{-2}$ </td><td>Hydrostatic pressure on face n, linear in global Z. A positive pressure is directed into the element.</td></tr><tr><td>Pn</td><td>Pressure</td><td> $FL^{-2}$ </td><td>Pressure on face n. A positive pressure is directed into the element.</td></tr><tr><td>PnNU</td><td>Not supported</td><td> $FL^{-2}$ </td><td>Nonuniform pressure on face n with magnitude supplied via user subroutine DLOAD in Abaqus/Standard and VDLOAD in Abaqus/Explicit. A positive pressure is directed into the element.</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> $\text{ROTA}^{(S)}$ </td><td>Rotational body force</td><td> $T^{-2}$ </td><td>Rotary acceleration load (magnitude is input as  $\alpha$ , where  $\alpha$  is the rotary acceleration).</td></tr><tr><td> $\text{ROTDYNF}^{(S)}$ </td><td>Not supported</td><td> $T^{-1}$ </td><td>Rotordynamic load (magnitude is input as  $\omega$ , where  $\omega$  is the angular velocity).</td></tr><tr><td> $\text{SBF}^{(E)}$ </td><td>Not supported</td><td> $FL^{-5}T^{2}$ </td><td>Stagnation body force in global  $X$ -,  $Y$ -, and  $Z$ -directions.</td></tr><tr><td> $\text{SPn}^{(E)}$ </td><td>Not supported</td><td> $FL^{-4}T^{2}$ </td><td>Stagnation pressure on face  $n$ .</td></tr><tr><td> $\text{TRSHRn}$ </td><td>Surface traction</td><td> $FL^{-2}$ </td><td>Shear traction on face  $n$ .</td></tr><tr><td> $\text{TRSHRnNU}^{(S)}$ </td><td>Not supported</td><td> $FL^{-2}$ </td><td>Nonuniform shear traction on face  $n$  with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td> $\text{TRVECn}$ </td><td>Surface traction</td><td> $FL^{-2}$ </td><td>General traction on face  $n$ .</td></tr><tr><td> $\text{TRVECnNU}^{(S)}$ </td><td>Not supported</td><td> $FL^{-2}$ </td><td>Nonuniform general traction on face  $n$  with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td> $\text{VBF}^{(E)}$ </td><td>Not supported</td><td> $FL^{-4}T$ </td><td>Viscous body force in global  $X$ -,  $Y$ -, and  $Z$ -directions.</td></tr><tr><td> $\text{VPn}^{(E)}$ </td><td>Not supported</td><td> $FL^{3}T$ </td><td>Viscous pressure on face  $n$ , applying a pressure proportional to the velocity normal to the face and opposing the motion.</td></tr></table>

# Foundations

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

<table><tr><td>Load ID(*FOUNDATION)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td> $Fn^{(S)}$ </td><td>Elastic foundation</td><td> $FL^{-3}$ </td><td>Elastic foundation on face n. A positive pressure is directed into the element.</td></tr></table>
