<!-- source-page: 321 -->

# By defining the value at the origin and the gradients in the 1- and 2-directions

Temperatures and field variables can be defined by giving the value at the origin of the cross-section and the gradients in the 2- and 1-directions of the cross-section (that is, give $\theta , \partial \theta / \partial X _ { 2 }$ and $\partial \theta / \partial X _ { 1 }$ in the predefined field or initial condition definition). For beams in a plane only and $\partial \theta / \partial X _ { 2 }$ need be given; gradients in the 1-direction are ignored in this case.

Input File Usage: \*BEAM SECTION, TEMPERATURE=GRADIENTS

Abaqus/CAE Usage: Property module: Create Section: select Beam as the section Category and Beam as the section Type: Section integration: During analysis, Linear by gradients

# By defining the values at points through the section

Temperatures and field variables can be defined at a set of points on the section, as indicated for each cross-section in “Beam cross-section library,” Section 29.3.9.

This technique cannot be used for any beam element that is adjacent to a general beam section element, as it can lead to incorrect temperature distributions at the shared cross-section. If you cannot avoid this modeling scenario, you must define the adjacent elements using separate nodes connected by MPCs, as discussed above.

Input File Usage: \*BEAM SECTION, TEMPERATURE=VALUES

Abaqus/CAE Usage: Property module: Create Section: select Beam as the section Category and Beam as the section Type: Section integration: During analysis, Interpolated from temperature points

# Output

Beam section properties such as cross-sectional area, moments of inertia, etc. are printed in the model data output. When a beam section integrated during the analysis is used, section forces, moments, and transverse shear forces and section strains, curvatures, and transverse shear strains can be output for the section (see “Element output” in “Output to the data and results files,” Section 4.1.2, and “Element output” in “Output to the output database,” Section 4.1.3). In addition, stress and strain can be output at each section point. “Beam element library,” Section 29.3.8, lists some of the element output quantities that are available for beam elements.

Axial strains due to warping are included in the stress/strain output from Abaqus/Standard if a beam section integrated during the analysis is used.

Temperature output at the section points can be obtained using the element variable TEMP. If the temperatures are given at specific points through the section, output at the temperature points can be obtained using the nodal variable NTxx. The nodal variable NTxx should not be used for output at the temperature points if the temperatures are specified by defining the value at the origin of the cross-section and specifying the gradients in the local 1- and 2-directions. In this case output variable NT should be requested; NT11 (the reference temperature value) and NT12 and NT13 (the temperature gradients in the local 1- and 2-directions, respectively) will be output automatically.

<!-- source-page: 322 -->

Beam normals are written to the output database automatically for all frames that include field output of nodal displacements. The normal directions can be visualized in the Visualization module of Abaqus/CAE.

<!-- source-page: 323 -->

# 29.3.7 USING A GENERAL BEAM SECTION TO DEFINE THE SECTION BEHAVIOR

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

# References

• “Beam modeling: overview,” Section 29.3.1   
• “Beam section behavior,” Section 29.3.5   
• \*BEAM GENERAL SECTION   
• \*BEAM SECTION OFFSET   
• “Specifying properties for general beam sections” in “Creating beam sections,” Section 12.13.11 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

# Overview

A general beam section:

• is used to define beam section properties that are computed once and held constant for the entire analysis;   
• can be used to define linear or nonlinear section behavior;   
• for linear section behavior can be associated only with linear material behavior   
• enables the use of meshed cross-sections (“Meshed beam cross-sections,” Section 10.6.1); and   
• enables the use of tapered cross-sections (Abaqus/Standard only).

# Linear section behavior

Linear section response is calculated as follows. At each point in the cross-section the axial stress, $\sigma ,$ and the shear stress, , are given by

$$
\sigma = E (\overline {{\theta}}, f _ {\beta}) (\varepsilon - \varepsilon^ {t h}) \quad \text { and } \quad \tau = G (\overline {{\theta}}, f _ {\beta}) \gamma ,
$$

where

$E ( { \overline { { \theta } } } , f _ { \beta } )$ is Young’s modulus (which may depend on the temperature, ${ \overline { { \theta } } } ,$ , and field variables, $f _ { \beta }$ , at the beam axis);

$G ( { \overline { { \theta } } } , f _ { \beta } )$ is the shear modulus (which may also depend on the temperature and field variables at the beam axis);

$\varepsilon$ is the axial strain;   
$\gamma$ is the shear caused by twist; and   
$\varepsilon ^ { t h }$ is the thermal expansion strain.

<!-- source-page: 324 -->

The thermal expansion strain is given by

$$
\varepsilon^ {t h} = \alpha (\overline {{\theta}}, f _ {\beta}) (\theta - \theta^ {0}) - \alpha (\overline {{\theta}} ^ {I}, f _ {\beta} ^ {I}) (\theta^ {I} - \theta^ {0}),
$$

where

$\alpha ( { \overline { { \theta } } } , f _ { \beta } )$ is the thermal expansion coefficient,

$\theta$ is the current temperature at a point in the beam section,

$f _ { \beta }$ are field variables,

$\theta ^ { 0 }$ is the reference temperature for $\alpha ,$

$\theta ^ { I }$ is the initial temperature at this point (see “Defining initial temperatures” in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1), and

$f _ { \beta } ^ { I }$ are the initial values of the field variables at this point (see “Defining initial values of predefined field variables” in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1).

If the thermal expansion coefficient is temperature or field-variable dependent, it is evaluated at the temperature and field variables at the beam axis. Therefore, since we assume that varies linearly over the section, $\varepsilon ^ { t h }$ also varies linearly over the section.

The temperature is defined from the temperature of the beam axis and the gradients of temperature with respect to the local $x _ { 1 } \cdot$ and $x _ { 2 } \mathrm { - a x e s } .$

$$
\theta = \overline {{\theta}} + \frac {\partial \theta}{\partial x _ {1}} x _ {1} + \frac {\partial \theta}{\partial x _ {2}} x _ {2}.
$$

The axial force, $N _ { \ast }$ bending moments, $M _ { 1 }$ and $M _ { 2 }$ about the 1 and 2 beam section local axes; torque, T; and bimoment, W, are defined in terms of the axial stress and the shear stress $\tau$ (see “Beam element formulation,” Section 3.5.2 of the Abaqus Theory Guide). These terms are

$$
N = E \left(A (\varepsilon_ {c} - \varepsilon_ {c} ^ {t h}) + \Gamma_ {0} \chi\right),
$$

$$
M _ {1} = E \left(I _ {1 1} \left(\kappa_ {1} - \frac {\partial \varepsilon^ {t h}}{\partial x _ {2}}\right) - I _ {1 2} \left(\kappa_ {2} + \frac {\partial \varepsilon^ {t h}}{\partial x _ {1}}\right)\right),
$$

$$
M _ {2} = E \left(- I _ {1 2} (\kappa_ {1} - \frac {\partial \varepsilon^ {t h}}{\partial x _ {2}}) + I _ {2 2} (\kappa_ {2} + \frac {\partial \varepsilon^ {t h}}{\partial x _ {1}})\right),
$$

$$
T = G J \phi + G I _ {p} w _ {p},
$$

$$
W = E \left(\Gamma_ {0} (\varepsilon_ {c} - \varepsilon_ {c} ^ {t h}) + \Gamma_ {W} \chi\right),
$$

$A$ is the area of the section,

$I _ { 1 1 }$ is the moment of inertia for bending about the 1-axis of the section,

$I _ { 1 2 }$ is the moment of inertia for cross-bending,

122 $I _ { 2 2 }$ is the moment of inertia for bending about the 2-axis of the section,

<!-- source-page: 325 -->

<table><tr><td>J</td><td>is the torsional constant,</td></tr><tr><td> $\Gamma_0$ </td><td>is the sectorial moment of the section,</td></tr><tr><td> $\Gamma_W$ </td><td>is the warping constant of the section,</td></tr><tr><td> $\varepsilon_c$ </td><td>is the axial strain measured at the centroid of the section,</td></tr><tr><td> $\varepsilon_c^{th}$ </td><td>is the thermal axial strain,</td></tr><tr><td> $\kappa_1$ </td><td>is the curvature change about the first beam section local axis,</td></tr><tr><td> $\kappa_2$ </td><td>is the curvature change about the second beam section local axis,</td></tr><tr><td> $\phi$ </td><td>is the twist,</td></tr><tr><td> $\chi$ </td><td>is the bicurvature defining the axial strain in the section due to the twist of the beam, and</td></tr><tr><td> $w_p = w_f - w$ </td><td>is the difference between the unconstrained warping amplitude,  $w_f$ , and the actual warping amplitude,  $w$ .</td></tr></table>

$\Gamma _ { 0 } , \Gamma _ { W } , \chi _ { \scriptscriptstyle 3 }$ , and $w _ { p }$ are nonzero only for open-section beam elements.

# Defining linear section behavior for library cross-sections or linear generalized cross-sections

Linear beam section response is defined geometrically by $A , I _ { 1 1 } , I _ { 1 2 } , I _ { 2 2 } , J ,$ and—if necessary— $- \Gamma _ { 0 }$ and $\Gamma _ { W }$ .

You can input these geometric quantities directly or specify a standard library section and Abaqus will calculate these quantities. In either case define the orientation of the beam section (see “Beam element cross-section orientation,” Section 29.3.4); give Young’s modulus, the torsional shear modulus, and the coefficient of thermal expansion, as functions of temperature; and associate the section properties with a region of your model.

If the thermal expansion coefficient is temperature dependent, the reference temperature for thermal expansion must also be defined as described later in this section.

# Specifying the geometric quantities directly

You can define “generalized” linear section behavior by specifying $A , \ I _ { 1 1 } , \ I _ { 1 2 } , \ I _ { 2 2 } , \ J ,$ and—if necessary $- \Gamma _ { 0 }$ and $\Gamma _ { W }$ directly. In this case you can specify the location of the centroid, thus allowing the bending axis of the beam to be offset from the line of its nodes. In addition, you can specify the location of the shear center.

<table><tr><td>Input File Usage:</td><td>Use the following option to define generalized linear beam section properties:*BEAM GENERAL SECTION, SECTION=GENERAL, ELSET=name $A, I_{11}, I_{12}, I_{22}, J, \Gamma_{0}, \Gamma_{W}$ If necessary, use the following option to specify the location of the centroid:*CENTROIDIf necessary, use the following option to specify the location of the shear center:*SHEAR CENTER</td></tr></table>

<!-- source-page: 326 -->

Abaqus/CAE Usage: Property module: Create Profile: Name: generalized\_section, Generalized Create Section: select Beam as the section Category and Beam as the section Type: Section integration: Before analysis, Profile name: generalized\_section: Centroid and Shear Center Assign→Section: select regions

Specifying a standard library section and allowing Abaqus to calculate the geometric quantities

You can select one of the standard library sections (see “Beam cross-section library,” Section 29.3.9) and specify the geometric input data needed to define the shape of the cross-section. Abaqus will then calculate the geometric quantities needed to define the section behavior automatically. In addition, you can specify an offset for the section origin.

Input File Usage: \*BEAM GENERAL SECTION, SECTION=library\_section, ELSET=name If necessary, use the following option to specify an offset for the section origin: \*BEAM SECTION OFFSET

Abaqus/CAE Usage: Property module: Create Profile: Name: library\_section Create Section: select Beam as the section Category and Beam as the section Type: Section integration: Before analysis, Profile name: library\_section Assign→Section: select regions Specifying an offset for the section origin is not supported in Abaqus/CAE.

# Defining linear section behavior for meshed cross-sections

Linear beam section response for a meshed section profile is obtained by numerical integration from the two-dimensional model. The numerical integration is performed once, determining the beam stiffness and inertia quantities, as well as the coordinates of the centroid and shear center, for the duration of the analysis. These beam section properties are calculated during the beam section generation and are written to the text file jobname.bsp. This text file can be included in the beam model. See “Meshed beam cross-sections,” Section 10.6.1, for a detailed description of the properties defining the linear beam section response for a meshed section, as well as for how a typical meshed section is analyzed.

Input File Usage: Use the following options: \*BEAM GENERAL SECTION, SECTION=MESHED, ELSET=name \*INCLUDE, INPUT=jobname.bsp

Abaqus/CAE Usage: Meshed cross-sections are not supported in Abaqus/CAE.

# Defining linear section behavior for tapered cross-sections in Abaqus/Standard

In Abaqus/Standard you can define Timoshenko beams with linearly tapered cross-sections. General beam sections with linear response and standard library sections are supported, with the exception of

<!-- source-page: 327 -->

arbitrary sections. The section parameters are defined at the two end nodes of each beam element. The effective beam area and moment of inertia for bending about the 1- and 2-axis of the section used in the calculation of the beam stiffness matrix, section forces, and stresses are

$$
\begin{array}{l} A ^ {\mathrm{eff}} = \frac {A ^ {I} + \sqrt {A ^ {I} A ^ {J}} + A ^ {J}}{3}, \\ I _ {1 1} ^ {\mathrm{eff}} = \frac {I _ {1 1} ^ {I} + \sqrt [ 4 ]{(I _ {1 1} ^ {I}) ^ {3} I _ {1 1} ^ {J}} + \sqrt {I _ {1 1} ^ {I} I _ {1 1} ^ {J}} + \sqrt [ 4 ]{I _ {1 1} ^ {I} (I _ {1 1} ^ {J}) ^ {3}} + I _ {1 1} ^ {J}}{5}, \\ I _ {2 2} ^ {\mathrm{eff}} = \frac {I _ {2 2} ^ {I} + \sqrt [ 4 ]{(I _ {2 2} ^ {I}) ^ {3} I _ {2 2} ^ {J}} + \sqrt {I _ {2 2} ^ {I} I _ {2 2} ^ {J}} + \sqrt [ 4 ]{I _ {2 2} ^ {I} (I _ {2 2} ^ {J}) ^ {3}} + I _ {2 2} ^ {J}}{5}, \\ \end{array}
$$

where the superscripts and refer to the two end nodes of the beam. The remaining effective geometric quantities are calculated as the average between the values at the two end nodes. This approximation suffices for mild tapering along each element, but it can lead to large errors if the tapering is not gradual. Abaqus/Standard issues a warning message during input file preprocessing if the area or inertia ratio is larger than 2.0 and an error message if the ratio is larger than 10.0.

The effective area and inertia are not used in the computation of the mass matrix. Instead, terms on the diagonal quadrants use the properties from the respective nodes, while off-diagonal quadrants use averaged quantities. For example, the axial inertia a linear element would have the diagonal term coming from node of $\rho A ^ { I } / 3$ , while node contributes with $\rho A ^ { J } / 3$ and the two off-diagonal contributions equal $\rho ( A ^ { I } + A ^ { J } ) / 1 2$ . Mild tapering is assumed in this formulation, since the total mass of the element totals $\rho ( A ^ { I } + A ^ { J } ) / 2$ .

Note: When you apply a tapered beam section to geometry in Abaqus/CAE, the full tapering is applied to each element along the beam’s length. For beams that include multiple elements, this modeling style can create a “sawtooth” pattern along the length of the beam. If you want to model gradual tapering along the entire length of the beam in Abaqus/CAE, you must calculate the size and shape of the beam profiles at the intermediate nodes, then apply different tapered beam sections to each beam element along the length.

Input File Usage: Use the following option to define linear section behavior of tapered crosssections:

\*BEAM GENERAL SECTION, TAPER, ELSET=name

Abaqus/CAE Usage: Property module:

Create Profile: Name: library\_section

Create Section: select Beam as the section Category and Beam as the section Type: Section integration: Before analysis,

Beam shape along length: Tapered: Beam start and Beam end options: Profile name: library\_section

Assign→Section: select regions

<!-- source-page: 328 -->

# Nonlinear section behavior

Typically nonlinear section behavior is used to include the experimentally measured nonlinear response of a beam-like component whose section distorts in its plane. When the section behaves according to beam theory (that is, the section does not distort in its plane) but the material has nonlinear response, it is usually better to use a beam section integrated during the analysis to define the section geometrically (see “Using a beam section integrated during the analysis to define the section behavior,” Section 29.3.6), in association with a material definition.

Nonlinear section behavior can also be used to model beam section collapse in an approximate sense: “Nonlinear dynamic analysis of a structure with local inelastic collapse,” Section 2.1.1 of the Abaqus Example Problems Guide, illustrates this for the case of a pipe section that may suffer inelastic collapse due to the application of a large bending moment. In following this approach you should recognize that such unstable section collapse, like any unstable behavior, typically involves localization of the deformation: results will, therefore, be strongly mesh sensitive.

# Calculation of nonlinear section response

Nonlinear section response is assumed to be defined by

$$
\begin{array}{l} N = N (\varepsilon_ {c} - \varepsilon_ {c} ^ {t h}, \overline {{\theta}}, f _ {\beta}), \\ M _ {1} = M _ {1} (\kappa_ {1}, \overline {{\theta}}, f _ {\beta}), \\ M _ {2} = M _ {2} (\kappa_ {2}, \overline {{\theta}}, f _ {\beta}), \\ T = T (\phi , \overline {{\theta}}, f _ {\beta}), \\ \end{array}
$$

where means a functional dependence on the conjugate variables: $N = N ( \varepsilon ) , M _ { 1 } = M _ { 1 } ( \kappa _ { 1 } )$ , etc. For example, ${ \cal N } ( \varepsilon _ { c } - \varepsilon _ { c } ^ { t h } , \overline { { { \theta } } } , f _ { \beta } )$ means that N is a function of: $\varepsilon _ { c } - \varepsilon _ { c } ^ { t h } ; \overline { { \theta } } _ { \mathrm { { ; } } }$ ; , the temperature of the beam axis; and of $f _ { \beta } { } _ { ; }$ , any predefined field variables at the beam axis. When the section behavior is defined in this way, only the temperature and field variables of the beam axis are used: any temperature or field-variable gradients given across the beam section are ignored.

These nonlinear responses may be purely elastic (that is, fully reversible—the loading and unloading responses are the same, even though the behavior is nonlinear) or may be elastic-plastic and, therefore, irreversible.

The assumption that these nonlinear responses are uncoupled is restrictive; in general, there is some interaction between these four behaviors, and the responses are coupled. You must determine if this approximation is reasonable for a particular case. The approach works well if the response is dominated by one behavior, such as bending about one axis. However, it may introduce additional errors if the response involves combined loadings.

# Defining nonlinear section behavior

You can define “generalized” nonlinear section behavior by specifying the area, $A ;$ moments of inertia, $I _ { 1 1 }$ for bending about the 1-axis of the section, $I _ { 2 2 }$ for bending about the 2-axis of the section, and $I _ { 1 2 }$ for cross-bending; and torsional constant, J. These values are used only to calculate the transverse shear

<!-- source-page: 329 -->

stiffness; and, if needed, A is used to compute the mass density of the element. In addition, you can define the orientation and the axial, bending, and torsional behavior of the beam section $( N , M _ { 1 } , M _ { 2 } , T )$ , as well as the thermal expansion coefficient. If the thermal expansion coefficient is temperature dependent, the reference temperature for thermal expansion must also be defined as described below.

Nonlinear generalized beam section behavior cannot be used with beam elements with warping degrees of freedom.

The axial, bending, and torsional behavior of the beam section and the thermal expansion coefficient are defined by tables. See “Material data definition,” Section 21.1.2, for a detailed discussion of the tabular input conventions. In particular, you must ensure that the range of values given for the variables is sufficient for the application since Abaqus assumes a constant value of the dependent variable outside this range.

Input File Usage: Use the following options to define generalized nonlinear beam section properties:
*BEAM GENERAL SECTION, SECTION=NONLINEAR GENERAL, ELSET=name
A, $I_{11}$ , $I_{12}$ , $I_{22}$ , J
*AXIAL for N
*M1 for $M_{1}$ *M2 for $M_{2}$ *TORQUE for T
*THERMAL EXPANSION for the thermal expansion coefficient

Abaqus/CAE Usage: Nonlinear generalized cross-sections are not supported in Abaqus/CAE.

Defining linear response for $N , M _ { I } , M _ { 2 }$ , and T

If the particular behavior is linear, $N , M _ { 1 } , M _ { 2 }$ , and T should be specified as functions of the temperature and predefined field variables, if appropriate.

As an example of axial behavior, if

$$
N = (A E) (\varepsilon_ {c} - \varepsilon_ {c} ^ {t h}),
$$

where is constant for a given temperature, the value of is entered. can still be varied as a function of temperature and field variables.

Input File Usage: Use the following options to define linear axial, bending, and torsional behavior:

*AXIAL, LINEAR
*M1, LINEAR
*M2, LINEAR
*TORQUE, LINEAR

Abaqus/CAE Usage: Nonlinear generalized cross-sections are not supported in Abaqus/CAE.

<!-- source-page: 330 -->

Defining nonlinear elastic response for $N , M _ { I } , M _ { 2 }$ , and T

If the particular behavior is nonlinear but elastic, the data should be given from the most negative value of the kinematic variable to the most positive value, always giving a point at the origin. See Figure 29.3.7–1 for an example.

Input File Usage: Use the following options to define nonlinear elastic axial, bending, and torsional behavior:

\*AXIAL, ELASTIC   
\*M1, ELASTIC   
\*M2, ELASTIC   
\*TORQUE, ELASTIC

Abaqus/CAE Usage: Nonlinear generalized cross-sections are not supported in Abaqus/CAE.

![](images/page-330_1d7844ab059935f75a8794b009b2dc991f990e99e70c31ab9d268d5754283334.jpg)

<details>
<summary>line</summary>

| Curvature, K | Bending moment, M |
| ------------ | ----------------- |
| K₁           | M₁                |
| K₂           | M₂                |
| K₃           | M₃                |
| K₄           | M₄                |
| K₅           | M₅                |
| K₆           | M₆                |
</details>

Figure 29.3.7–1 Example of elastic nonlinear beam section behavior definition.

Defining elastic-plastic response for $N , M _ { I } , M _ { 2 }$ , and T

By default, elastic-plastic response is assumed for $N , M _ { 1 } , M _ { 2 }$ , and T.

The inelastic model is based on assuming linear elasticity and isotropic hardening (or softening) plasticity. The data in this case must begin with the point and proceed to give positive values of the kinematic variable at increasing positive values of the conjugate force or moment. Strain softening is
