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# Temperature and field variable input at specific points for beam sections integrated during the analysis

Give the value at each of the points shown below.

![](images/page-361_4396f06dcb02462159d93a844be2c9ea5ca93aac075f42df935d28e48fbe0088.jpg)

<details>
<summary>text_image</summary>

2
3
2
2
1
1
</details>

Beam in a plane

![](images/page-361_f6c0efd8a4e49ee29d52cf031e31c3299a0e35d5916565f48fafb8d7ed181dc2.jpg)

<details>
<summary>text_image</summary>

2
3
4
2
1
1
</details>

Beam in space

# Rectangular section

Input File Usage: Use one of the following options:

\*BEAM SECTION, SECTION=RECT   
\*BEAM GENERAL SECTION, SECTION=RECT   
\*FRAME SECTION, SECTION=RECT

Abaqus/CAE Usage: Property module: Create Profile: Rectangular

![](images/page-361_b504d1fe9430896563997443c44d86b35b8c05cbca6b69456a53bb6038331ec8.jpg)

<details>
<summary>text_image</summary>

b
a
1
2
3
4
5
</details>

Default integration, beam in a plane

![](images/page-361_fbcafd2bfd4d92109ea779f073dd906c179e4c2943e01bd6470a1cab731f83bf.jpg)

<details>
<summary>scatter</summary>

| Position | Value |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | 10 |
| 11 | 11 |
| 12 | 12 |
| 13 | 13 |
| 14 | 14 |
| 15 | 15 |
| 16 | 16 |
| 17 | 17 |
| 18 | 18 |
| 19 | 19 |
| 20 | 20 |
| 21 | 21 |
| 22 | 22 |
| 23 | 23 |
| 24 | 24 |
| 25 | 25 |
| 26 | 26 |
| 27 | 27 |
| 28 | 28 |
| 29 | 29 |
| 30 | 30 |
| 31 | 31 |
| 32 | 32 |
| 33 | 33 |
| 34 | 34 |
| 35 | 35 |
| 36 | 36 |
| 37 | 37 |
| 38 | 38 |
| 39 | 39 |
| 40 | 40 |
| 41 | 41 |
| 42 | 42 |
| 43 | 43 |
| 44 | 44 |
| 45 | 45 |
| 46 | 46 |
| 47 | 47 |
| 48 | 48 |
| 49 | 49 |
| 50 | 50 |
| 51 | 51 |
| 52 | 52 |
| 53 | 53 |
| 54 | 54 |
| 55 | 55 |
| 56 | 56 |
| 57 | 57 |
| 58 | 58 |
| 59 | 59 |
| 60 | 60 |
| 61 | 61 |
| 62 | 62 |
| 63 | 63 |
| 64 | 64 |
| 65 | 65 |
| 66 | 66 |
| 67 | 67 |
| 68 | 68 |
| 69 | 69 |
| 70 | 70 |
| 71 | 71 |
| 72 | 72 |
| 73 | 73 |
| 74 | 74 |
| 75 | 75 |
| 76 | 76 |
| 77 | 77 |
| 78 | 78 |
| 79 | 79 |
| 80 | 80 |
| 81 | 81 |
| 82 | 82 |
| 83 | 83 |
| 84 | 84 |
| 85 | 85 |
| 86 | 86 |
| 87 | 87 |
| 88 | 88 |
| 89 | 89 |
| 90 | 90 |
| 91 | 91 |
| 92 | 92 |
| 93 | 93 |
| 94 | 94 |
| 95 | 95 |
| 96 | 96 |
| 97 | 97 |
| 98 | 98 |
| 99 | 99 |
| 100 | 100 |
</details>

Default integration, beam in space

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# Geometric input data

a, b

# Default integration (Simpson)

Beam in a plane: 5 points

Beam in space: 5 × 5 (25 total)

# Nondefault integration input for a beam section integrated during the analysis

Beam in a plane: Give the number of points in the second beam section axis direction. This number must be odd and greater than or equal to five.

Beam in space: Give the number of points in the first beam section axis direction, then the number of points in the second beam section axis direction. These numbers must be odd and greater than or equal to five.

# Default stress output points if a beam section integrated during the analysis is used

Beam in a plane: Bottom and top (points 1 and 5 above for default integration).

Beam in space: Corners (points 1, 5, 21, and 25 above for default integration).

# Temperature and field variable input at specific points for beam sections integrated during the analysis

Give the value at each of the points shown below.

![](images/page-362_4856c4ac62e33e3a4fb88a2ab39145332d2566f6f17382169690aa9e4d42ae6a.jpg)

<details>
<summary>text_image</summary>

2
3
2
1
1
</details>

Beam in a plane

![](images/page-362_25debccf163124016f2a9170e54e82e129251fd76e47e543e9be3a3dd3e4eec7.jpg)

<details>
<summary>text_image</summary>

4
2
3
1
1
2
</details>

Beam in space

# Temperature input for a frame section

Constant temperature throughout the element cross-section is assumed; therefore, only one temperature value per node is required.

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# Trapezoidal section

Input File Usage:

Use either of the following options:

\*BEAM SECTION, SECTION=TRAPEZOID   
\*BEAM GENERAL SECTION, SECTION=TRAPEZOID

Abaqus/CAE Usage:

Property module: Create Profile: Trapezoidal

![](images/page-363_d301c08ff9ddd6fd5665724f03957548947239667f1a0c9ef262f7b347d3174a.jpg)

<details>
<summary>text_image</summary>

c
2
5
4
3
2
1
b
d
a
1
</details>

Default integration, beam in a plane

![](images/page-363_84d83ce35d6e7e61e0811ed4cb1d9846853bf47d6a239d40889e145a16cf9bb5.jpg)

<details>
<summary>geo</summary>

| Position | Value |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | 10 |
| 11 | 11 |
| 12 | 12 |
| 13 | 13 |
| 14 | 14 |
| 15 | 15 |
| 16 | 16 |
| 17 | 17 |
| 18 | 18 |
| 19 | 19 |
| 20 | 20 |
| 21 | 21 |
| 22 | 22 |
| 23 | 23 |
| 24 | 24 |
| 25 | 25 |
| c | 2 |
| b | 1 |
| a | 1 |
</details>

Default integration, beam in space

# Geometric input data

$$
a, b, c, d
$$

By allowing you to specify d, the origin of the local cross-section axes can be placed anywhere on the symmetry line (the local 2-axis). In the above figures a negative value of d implies that the origin of the local cross-section axis is below the lower edge of the section. This may be needed when constraining a beam stiffener to a shell.

# Default integration (Simpson)

Beam in a plane: 5 points

Beam in space: 5 × 5 (25 total)

# Nondefault integration input for a beam section integrated during the analysis

Beam in a plane: Give the number of points in the second beam section axis direction. This number must be odd and greater than or equal to five.

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Beam in space: Give the number of points in the first beam section axis direction, then the number of points in the second beam section axis direction. These numbers must be odd and greater than or equal to five.

# Default stress output points if a beam section integrated during the analysis is used

Beam in a plane: Bottom and top (points 1 and 5 above for default integration).

Beam in space: Corners (points 1, 5, 21, and 25 above for default integration).

# Temperature and field variable input at specific points for beam sections integrated during the analysis

Give the value at each of the points shown below.

![](images/page-364_87c3d0fdb472e23caa62055d7a8e18e92e8aae1e3aae8fa2c2957d024f05685e.jpg)

<details>
<summary>text_image</summary>

2
3
b/2
2
b/2
1
</details>

Beam in a plane

![](images/page-364_a362ed66b71d90492b91a85b279c16ead575d13112a9a1f329b135dcb744ad5d.jpg)

<details>
<summary>text_image</summary>

2
4
3
1
1
2
1
</details>

Beam in space

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# 29.4 Frame elements

• “Frame elements,” Section 29.4.1   
• “Frame section behavior,” Section 29.4.2   
• “Frame element library,” Section 29.4.3

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# 29.4.1 FRAME ELEMENTS

# Product: Abaqus/Standard

# References

• “Beam modeling: overview,” Section 29.3.1   
• “Frame section behavior,” Section 29.4.2   
• “Frame element library,” Section 29.4.3   
• \*FRAME SECTION

# Overview

# Frame elements:

• are 2-node, initially straight, slender beam elements intended for use in the elastic or elastic-plastic analysis of frame-like structures;   
• are available in two or three dimensions;   
• have elastic response that follows Euler-Bernoulli beam theory with fourth-order interpolation for the transverse displacements;   
• have plastic response that is concentrated at the element ends (plastic hinges) and is modeled with a lumped plasticity model that includes nonlinear kinematic hardening;   
• are implemented for small or large displacements (large rotations with small strains);   
• output forces and moments at the element ends and midpoint;   
• output elastic axial strain and curvatures at the element ends and midpoint and plastic displacements and rotations at the element ends only;   
• admit, optionally, a uniaxial “buckling strut” response where the axial response of the element is governed by a damaged elasticity model in compression and an isotropic hardening plasticity model in tension and where all transverse forces and moments are zero;   
• can switch to buckling strut response during the analysis (for pipe sections only); and   
• can be used in static, implicit dynamic, and eigenfrequency extraction analyses only.

# Typical applications

Frame elements are designed to be used for small-strain elastic or elastic-plastic analysis of frame-like structures composed of slender, initially straight beams. Typically, a single frame element will represent the entire structural member connecting two joints. A frame element’s elastic response is governed by Euler-Bernoulli beam theory with fourth-order interpolations for the transverse displacement field; hence, the element’s kinematics include the exact (Euler-Bernoulli) solution to concentrated end forces and moments and constant distributed loads. The elements can be used to solve a wide variety of civil engineering design applications, such as truss structures, bridges, internal frame structures of

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buildings, off-shore platforms, and jackets, etc. A frame element’s plastic response is modeled with a lumped plasticity model at the element ends that simulates the formation of plastic hinges. The lumped plasticity model includes nonlinear kinematic hardening. The elements can, thus, be used for collapse load prediction based on the formation of plastic hinges.

Slender, frame-like members loaded in compression often buckle in such a way that only axial force is supported by the member; all other forces and moments are negligibly small. Frame elements offer optional buckling strut response whereby the element only carries axial force, which is calculated based on a damaged elasticity model in compression and an isotropic hardening plasticity model in tension. This model provides a simple phenomenological approximation to the highly nonlinear geometric and material response that takes place during buckling and postbuckling deformation of slender members loaded in compression.

For pipe sections only, frame elements allow switching to optional uniaxial buckling strut response during the analysis. The criterion for switching is the “ISO” equation together with the “strength” equation (see “Buckling strut response for frame elements,” Section 3.9.3 of the Abaqus Theory Guide). When the ISO and strength equations are satisfied, the elastic or elastic-plastic frame element undergoes a one-time-only switch in behavior to buckling strut response.

# Element cross-sectional axis system

The orientation of the frame element’s cross-section is defined in Abaqus/Standard in terms of a local, right-handed $( \mathbf { t } , \mathbf { n } _ { 1 } , \mathbf { n } _ { 2 } )$ axis system, where is the tangent to the axis of the element, positive in the direction from the first to the second node of the element, and ${ \bf n } _ { 1 }$ and $\mathbf { n } _ { 2 }$ are basis vectors that define the local 1- and 2-directions of the cross-section. ${ \bf n } _ { 1 }$ is referred to as the first axis direction, and $\mathbf { n } _ { 2 }$ is referred to as the normal to the element. Since these elements are initially straight and assume small strains, the cross-section directions are constant along each element and possibly discontinuous between elements.

# Defining the $\boldsymbol { \mathsf { n } } _ { 1 }$ -direction at the nodes

For frame elements in a plane the $\mathbf { n } _ { 1 }$ -direction is always $( 0 . 0 , 0 . 0 , - 1 . 0 )$ ; that is, normal to the plane in which the motion occurs. Therefore, planar frame elements can bend only about the first axis direction.

For frame elements in space the approximate direction of $\mathbf { n } _ { 1 }$ must be defined directly as part of the element section definition or by specifying an additional node off the element’s axis. This additional node is included in the element’s connectivity list (see “Element definition,” Section 2.2.1).

• If an additional node is specified, the approximate direction of ${ \bf n } _ { 1 }$ is defined by the vector extending from the first node of the element to the additional node.   
• If both input methods are used, the direction calculated by using the additional node will take precedence.   
• If the approximate direction is not defined by either of the above methods, the default value is (0.0, $0 . 0 , - 1 . 0 )$ .

The $\mathbf { n } _ { 1 }$ -direction is then the normal to the element’s axis that lies in the plane defined by the element’s axis and this approximate $\mathbf { n } _ { 1 } { \mathrm { - d i r e c t i o n } }$ . The $\mathbf { n } _ { 2 }$ -direction is defined as $\mathbf { t } \times \mathbf { n } _ { 1 }$ .

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# Large-displacement assumptions

The frame element’s formulation includes the effect of large rigid body motions (displacements and rotations) when geometrically nonlinear analysis is selected (see “General and linear perturbation procedures,” Section 6.1.3). Strains in these elements are assumed to remain small.

# Material response (section properties) of frame elements

For frame elements the geometric and material properties are specified together as part of the frame section definition. No separate material definition is required. You can choose one of the section shapes that is valid for frame elements from the beam cross-section library (see “Beam cross-section library,” Section 29.3.9). The valid section shapes depend upon whether elastic or elastic-plastic material response is specified or whether buckling strut response is included. See “Frame section behavior,” Section 29.4.2, for a complete discussion of specifying the geometric and material section properties.

Input File Usage: \*FRAME SECTION, SECTION=section\_type

# Mechanical response and mass formulation

The mechanical response of a frame element includes elastic and plastic behavior. Optionally, uniaxial buckling strut response is available.

# Elastic response

The elastic response of a frame element is governed by Euler-Bernoulli beam theory. The displacement interpolations for the deflections transverse to the frame element’s axis (the local 1- and 2-directions in three dimensions; the local 2-direction in two dimensions) are fourth-order polynomials, allowing quadratic variation of the curvature along the element’s axis. Thus, each single frame element exactly models the static, elastic solution to force and moment loading at its ends and constant distributed loading along its axis (such as gravity loading). The displacement interpolation along an element’s axis is a second-order polynomial, allowing linear variation of the axial strain. In three dimensions the twist rotation interpolation along an element’s axis is linear, allowing constant twist strain. The elastic stiffness matrix is integrated numerically and used to calculate 15 nodal forces and moments in three dimensions: an axial force, two shear forces, two bending moments, and a twist moment at each end node, and an axial force and two shear forces at the midpoint node. In two dimensions 8 nodal forces and moments exist: an axial force, a shear force, and a moment at each end, and an axial force and a shear force at the midpoint. The forces and moments are illustrated in Figure 29.4.1–1.

# Elastic-plastic response

The plastic response of the element is treated with a “lumped” plasticity model such that plastic deformations can develop only at the element’s ends through plastic rotations (hinges) and plastic axial displacement. The growth of the plastic zone through the element’s cross-section from initial yield to a fully yielded plastic hinge is modeled with nonlinear kinematic hardening. It is assumed that the plastic deformation at an end node is influenced by the moments and axial force at that node only. Hence, the

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![](images/page-370_90781c2210d32355e9fe2a9864c01f1962fd0a84df4d582a9d18d2dc847571ba.jpg)

<details>
<summary>text_image</summary>

N₂
N₁
N₂
N₁
N₂
N₁
T
M₁
1
N
3
N
2
N
T
M₁
M₂
M₂
L/2
L/2
t
n₂
n₁
</details>

Figure 29.4.1–1 Forces and moments on a frame element in space.

yield function at each node, also called the plastic interaction surface, is assumed to be a function of that node’s axial force and three moment components only. No length is associated with the plastic hinge. In reality, the plastic hinge will have a finite size determined by the element’s length and the specific loading that causes yielding; the hinge size will influence the hardening rate but not the ultimate load. Hence, if the rate of hardening and, thus, the plastic deformation for a given load are important, the lumped plasticity model should be calibrated with the element’s length and the loading situation taken into account. For details on the elastic-plastic element formulation, see “Frame elements with lumped plasticity,” Section 3.9.2 of the Abaqus Theory Guide.

# Uniaxial linear elastic and buckling strut response with tensile yield

You can obtain a frame element’s response to uniaxial force only, based on linear elasticity, buckling strut response, and tensile yield. In that case all transverse forces and moments in the element are zero. For linear elastic response the element behaves like an axial spring with constant stiffness. For buckling strut response if the tensile axial force in the element does not exceed the yield force, the axial force in the element is constrained to remain inside a buckling envelope. See “Frame section behavior,” Section 29.4.2, for a description of this envelope. Inside the envelope the force is related to strain by a damaged elastic modulus. The cyclic, hysteretic response of this model is phenomenological and approximates the response of thin-walled, pipe-like members. When the element is loaded in tension beyond the yield force, the force response is governed by isotropic hardening plasticity. In reverse loading the response is governed by the buckling envelope translated along the strain axis by an amount equal to the axial plastic strain. For details of the buckling strut formulation, see “Buckling strut response for frame elements,” Section 3.9.3 of the Abaqus Theory Guide.

# Mass formulation

The frame element uses a lumped mass formulation for both dynamic analysis and gravity loading. The mass matrix for the translational degrees of freedom is derived from a quadratic interpolation of the axial
