ProductsAbaqus/StandardAbaqus/ExplicitAbaqus/CAE ## Linear section behaviorLinear section response is calculated as follows. At each point in the cross-section the axial stress, $\sigma $, and the shear stress, $\tau $, are given by $$\sigma =E\left(\overline{\theta},{f}_{\beta}\right)\left(\epsilon -{\epsilon}^{th}\right)\text{and}\tau =G\left(\overline{\theta},{f}_{\beta}\right)\gamma ,$$ where - $E\left(\overline{\theta},{f}_{\beta}\right)$
is Young's modulus (which may depend on the temperature, $\overline{\theta}$, and field variables, ${f}_{\beta}$, at the beam axis); - $G\left(\overline{\theta},{f}_{\beta}\right)$
is the shear modulus (which may also depend on the temperature and field variables at the beam axis); - $\epsilon $
is the axial strain; - $\gamma $
is the shear caused by twist; and - ${\epsilon}^{th}$
is the thermal expansion strain.
The thermal expansion strain is given by $${\epsilon}^{th}=\alpha \left(\overline{\theta},{f}_{\beta}\right)\left(\theta -{\theta}^{0}\right)-\alpha \left({\overline{\theta}}^{I},{f}_{\beta}^{I}\right)\left({\theta}^{I}-{\theta}^{0}\right),$$ where - $\alpha \left(\overline{\theta},{f}_{\beta}\right)$
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), and - ${f}_{\beta}^{I}$
are the initial values of the field variables at this point (see Defining initial values of predefined field variables).
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 $\theta $ varies linearly over the section, ${\epsilon}^{th}$ 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 ${x}_{1}$- and ${x}_{2}$-axes: $$\theta =\overline{\theta}+\frac{\partial \theta}{\partial {x}_{1}}{x}_{1}+\frac{\partial \theta}{\partial {x}_{2}}{x}_{2}.$$ The axial force, N; 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 $\sigma $ and the shear stress $\tau $ (see Beam element formulation). These terms are $$\begin{array}{cc}\hfill N=& E\left(A\left({\epsilon}_{c}-{\epsilon}_{c}^{th}\right)+{\mathrm{\Gamma}}_{0}\chi \right),\hfill \\ \hfill {M}_{1}=& E\left({I}_{11}\left({\kappa}_{1}-\frac{\partial {\epsilon}^{th}}{\partial {x}_{2}}\right)-{I}_{12}\left({\kappa}_{2}+\frac{\partial {\epsilon}^{th}}{\partial {x}_{1}}\right)\right),\hfill \\ \hfill {M}_{2}=& E\left(-{I}_{12}\left({\kappa}_{1}-\frac{\partial {\epsilon}^{th}}{\partial {x}_{2}}\right)+{I}_{22}\left({\kappa}_{2}+\frac{\partial {\epsilon}^{th}}{\partial {x}_{1}}\right)\right),\hfill \\ \hfill T=& GJ\varphi +G{I}_{p}{w}_{p},\hfill \\ \hfill W=& E\left({\mathrm{\Gamma}}_{0}\left({\epsilon}_{c}-{\epsilon}_{c}^{th}\right)+{\mathrm{\Gamma}}_{W}\chi \right),\hfill \end{array}$$ where - A
is the area of the section, - ${I}_{11}$
is the moment of inertia for bending about the 1-axis of the section, - ${I}_{12}$
is the moment of inertia for cross-bending, - ${I}_{22}$
is the moment of inertia for bending about the 2-axis of the section, - J
is the torsional constant, - ${\mathrm{\Gamma}}_{0}$
is the sectorial moment of the section, - ${\mathrm{\Gamma}}_{W}$
is the warping constant of the section, - ${\epsilon}_{c}$
is the axial strain measured at the centroid of the section, - ${\epsilon}_{c}^{th}$
is the thermal axial strain, - ${\kappa}_{1}$
is the curvature change about the first beam section local axis, - ${\kappa}_{2}$
is the curvature change about the second beam section local axis, - $\varphi $
is the twist, - $\chi $
is the bicurvature defining the axial strain in the section due to the twist of the beam, and - ${w}_{p}={w}_{f}-w$
is the difference between the unconstrained warping amplitude, ${w}_{f}$, and the actual warping amplitude, w.
${\mathrm{\Gamma}}_{0}$, ${\mathrm{\Gamma}}_{W}$, $\chi $, and ${w}_{p}$ are nonzero only for open-section beam elements. ## Defining linear section behavior for library cross-sections or linear generalized cross-sectionsLinear beam section response is defined geometrically by A, ${I}_{11}$, ${I}_{12}$, ${I}_{22}$, J, and—if necessary—${\mathrm{\Gamma}}_{0}$ and ${\mathrm{\Gamma}}_{W}$. 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); 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 directlyYou can define “generalized” linear section behavior by specifying A, ${I}_{11}$, ${I}_{12}$, ${I}_{22}$, J, and—if necessary—${\mathrm{\Gamma}}_{0}$ and ${\mathrm{\Gamma}}_{W}$ 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. Input File Usage 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, ${\mathrm{\Gamma}}_{0}$, ${\mathrm{\Gamma}}_{W}$ If necessary, use the following option to specify the location of the centroid: CENTROID If necessary, use the following option to specify the location of the shear center: SHEAR CENTER 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 AssignSection: select regions ## Specifying a standard library section and allowing Abaqus to calculate the geometric quantitiesYou can select one of the standard library sections (see Beam cross-section library) 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 AssignSection: select regions Specifying an offset for the section origin is not supported in Abaqus/CAE. ## Defining linear section behavior for meshed cross-sectionsLinear 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 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/StandardIn 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 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 $$\begin{array}{cc}\hfill {A}^{\text{eff}}=& \frac{{A}^{I}+\sqrt{{A}^{I}{A}^{J}}+{A}^{J}}{3},\hfill \\ \hfill {I}_{11}^{\text{eff}}=& \frac{{I}_{11}^{I}+\sqrt[4]{{\left({I}_{11}^{I}\right)}^{3}{I}_{11}^{J}}+\sqrt{{I}_{11}^{I}{I}_{11}^{J}}+\sqrt[4]{{I}_{11}^{I}{\left({I}_{11}^{J}\right)}^{3}}+{I}_{11}^{J}}{5},\hfill \\ \hfill {I}_{22}^{\text{eff}}=& \frac{{I}_{22}^{I}+\sqrt[4]{{\left({I}_{22}^{I}\right)}^{3}{I}_{22}^{J}}+\sqrt{{I}_{22}^{I}{I}_{22}^{J}}+\sqrt[4]{{I}_{22}^{I}{\left({I}_{22}^{J}\right)}^{3}}+{I}_{22}^{J}}{5},\hfill \end{array}$$ where the superscripts $I$ and $J$ 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 $I$ of $\rho {A}^{I}/3$, while node $J$ contributes with $\rho {A}^{J}/3$ and the two off-diagonal contributions equal $\rho \left({A}^{I}+{A}^{J}\right)/12$. Mild tapering is assumed in this formulation, since the total mass of the element totals $\rho \left({A}^{I}+{A}^{J}\right)/2$. 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 cross-sections: 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 AssignSection: select regions ## Nonlinear section behaviorTypically 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), 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 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 responseNonlinear section response is assumed to be defined by $$\begin{array}{cc}\hfill N=& N\left({\epsilon}_{c}-{\epsilon}_{c}^{th},\overline{\theta},{f}_{\beta}\right),\hfill \\ \hfill {M}_{1}=& {M}_{1}\left({\kappa}_{1},\overline{\theta},{f}_{\beta}\right),\hfill \\ \hfill {M}_{2}=& {M}_{2}\left({\kappa}_{2},\overline{\theta},{f}_{\beta}\right),\hfill \\ \hfill T=& T\left(\varphi ,\overline{\theta},{f}_{\beta}\right),\hfill \end{array}$$ where $(\mathit{})$ means a functional dependence on the conjugate variables: $N=N\left(\epsilon \right)$, ${M}_{1}={M}_{1}\left({\kappa}_{1}\right)$, etc. For example, $N\left({\epsilon}_{c}-{\epsilon}_{c}^{th},\overline{\theta},{f}_{\beta}\right)$ means that N is a function of:${\epsilon}_{c}-{\epsilon}_{c}^{th}$; $\overline{\theta}$, 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 behaviorYou can define “generalized” nonlinear section behavior by specifying the area, A; moments of inertia, ${I}_{11}$ for bending about the 1-axis of the section, ${I}_{22}$ for bending about the 2-axis of the section, and ${I}_{12}$ for cross-bending; and torsional constant, J. These values are used only to calculate the transverse shear 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 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 |