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In some structural applications the beam element may be a one-dimensional approximation of a structure with complex cross-sectional geometry and mass distribution. In such a cross-section there may be inertia contributions that represent heavy machinery, cargo loaded in a ship compartment, fluid-filled ballast tanks, or any other mass distributed along the length of the beam that is not part of the beam’s structural stiffness. In such cases you can define additional mass and rotary inertia associated with the beam section properties. Multiple masses per unit length (with location other than the origin of the beam cross-section) and rotary inertias per unit length can be specified. Mass proportional damping (alpha or composite damping) associated with this additional inertia can also be specified. Abaqus will use the mass weighted average (based on the material damping and the added inertial damping) for the element mass proportional damping. See “Material damping,” Section 26.1.1, for details.
Additional inertia due to immersion in fluid
When a beam is fully or partially submerged, the effect of the surrounding fluid can be modeled as an additional distributed inertia on the beam. See “Additional inertia due to immersion in fluid” in “Beam section behavior,” Section 29.3.5, for details.
Warping (open-section) beams
When modeling beams in space, a further consideration arises from the possible warping of the beam’s cross-section under torsional loading. For all but circular sections the beam’s cross-section will deform out of its original plane when subject to torsion. This warping deformation will modify the shear strain distribution throughout the section.
Open sections will typically twist very easily if warping is not prevented, especially if the walls that form the beam section are thin. Constraint of this warping at certain points along the beam (such as where the beam is built into some other member, Figure 29.3.3–1, or into a wall) is then a major determinant of the beam’s overall torsional response.
Element types B31OS, B32OS (and their “hybrid” equivalents) have the warping magnitude, w, as a degree of freedom at each node; they are available only in Abaqus/Standard. In these elements Abaqus/Standard assumes that the warping of the cross-section follows a certain pattern as a function of position in the cross-section (Abaqus will calculate this warping pattern if you have specified a standard library section or an “arbitrary” section): only the warping magnitude varies with position along the beam’s axis. These elements are meant for the analysis of thin-walled open sections in which warping constraints play a role and the axial strains due to warping cannot be neglected. Examples of such open sections that may warp in this fashion are the I-section and any open arbitrary section. In the other beam element types warping is considered unconstrained and any axial stress due to warping is neglected; torsional behavior will not be represented adequately when these element types are used with thin-walled, open sections.
In general, the warping magnitude can be continuous only when the beam axis is continuous through a node and the beam cross-section is the same on both sides of the node. Thus, if open-section members intersect at a node (such as the cross-member of a vehicle chassis abutting a longitudinal member, Figure 29.3.3–1), separate nodes may have to be used for the intersecting members with different axial directions and appropriate constraints must be chosen for the warping amplitudes in each member at this point. The choice of these constraints is a matter of detail of the local construction. For
natural_image
Isometric line drawing of a structural beam with dashed internal planes (no text or symbols)
Figure 29.3.3–1 Intersection of open section beams.
example, if the joint is reinforced, warping may be prevented; therefore, degree of freedom 7 should be fully constrained with a boundary condition on the appropriate members at the joint.
“Pipe” elements
The pipe elements in Abaqus assume a hollow circular section. The internal stress caused by internal or external pressure loading in the pipe is included in these elements so that on the pipe cross-section a point under tension will have different yield than a point under compression (Figure 29.3.3–2), thus causing an asymmetry in the section’s response to inelastic bending. Two formulations are available for pipe elements in Abaqus. The thin-walled pipe formulation assumes constant hoop stress across the crosssection and neglects the radial stress, whereas thick-walled pipes (available only in Abaqus/Standard) allow the hoop and radial stress components to vary across the cross-section.
The hoop stress in thin-walled pipe elements is computed as the average stress in equilibrium with the internal and external pressure loading on the pipe section. For the thin-walled formulation, an integration rule with one point through the thickness suffices to obtain an accurate solution.
For thick-walled pipes, the hoop stress and radial stress variation under applied internal and/or external pressure are calculated using Lamé’s equations. The constitutive calculations at each material point take into account the imposed hoop and radial stress values to determine the structural response. A two-dimensional integration rule is used for thick-walled pipes to capture the effect of stress variation across the section accurately.
“Hybrid” beams
Hybrid beam element types (B21H, B33H, etc.) are provided in Abaqus/Standard for use in cases where it is numerically difficult to compute the axial and shear forces in the beam by the usual finite element
text_image
σ hoop hoop stress caused by pressurization σ axial asymmetric stress limits in tension and compression Mises yield surface
Figure 29.3.3–2 Yield behavior in thin-walled PIPE elements.
displacement method. This problem arises most commonly in geometrically nonlinear analysis when the beam undergoes large rotations and is very rigid in axial and transverse shear deformation, such as a link in a vehicle’s suspension system or a flexing long pipe or cable. The problem in such cases is that slight differences in nodal positions can cause very large forces, which, in turn, cause large motions in other directions. The hybrid elements overcome this difficulty by using a more general formulation in which the axial and transverse shear forces in the elements are included, along with the nodal displacements and rotations, as primary variables. Although this formulation makes these elements more expensive, they generally converge much faster when the beam’s rotations are large and, therefore, are more efficient overall in such cases.
Additional references
• Archer, J. S., “Consistent Matrix Formulations for Structural Analysis using Finite-Element Techniques,” American Institute of Aeronautics and Astronautics Journal, vol. 3, pp. 1910–1918, 1965.
• Cowper, R. G., “The Shear Coefficient in Timoshenko’s Beam Theory,” Journal of Applied Mechanics, vol. 33, pp. 335–340, 1966.
29.3.4 BEAM ELEMENT CROSS-SECTION ORIENTATION
Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
References
• “Beam modeling: overview,” Section 29.3.1
• “Beam cross-section library,” Section 29.3.9
• “Beam section behavior,” Section 29.3.5
• “Assigning a beam orientation,” Section 12.15.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
The orientation of a beam cross-section:
• is defined in terms of a local, right-handed axis system; and
• can be user-defined or calculated by Abaqus.
Beam cross-sectional axis system
The orientation of a beam cross-section is defined in Abaqus 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. \mathbf { n } _ { 1 } is referred to as the first beam section axis, and \mathbf { n } _ { 2 } is referred to as the normal to the beam. This beam cross-sectional axis system is illustrated in Figure 29.3.4–1.
Defining the { \boldsymbol { \mathsf { n } } } _ { 1 } -direction
For beams in a plane the { \bf n } _ { 1 } -direction is always (0.0, 0.0, −1.0); that is, normal to the plane in which the motion occurs. Therefore, planar beams can bend only about the first beam-section axis.
For beams in space the approximate direction of \mathbf { n } _ { 1 } must be defined directly as part of the beam section definition or by specifying an additional node off the beam axis as part of the element definition (see “Element definition,” Section 2.2.1). This additional node is included in the element’s connectivity list.
• If an additional node is specified, the approximate direction of \mathbf { n } _ { 1 } is defined by the vector extending from the first node of the element to the additional node.
• If { \bf n } _ { 1 } is defined directly for the section and an additional node is specified, 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).
text_image
z y x n₂ n₁ t 1 2
Figure 29.3.4–1 Local axis definition for beam-type elements.
This approximate { \bf n } _ { 1 } -direction may be used to determine the \mathbf { n } _ { 2 } -direction (discussed below). Once the \mathbf { n } _ { 2 } -direction has been defined or calculated, the actual { \bf n } _ { 1 } -direction will be calculated as \mathbf { n } _ { 2 } \times \mathbf { t } . , possibly resulting in a direction that is different from the specified direction.
Input File Usage: Use the following option to specify the { \bf n } _ { 1 } -direction directly for a beam section integrated during the analysis:
*BEAM SECTION
-direction (the data line number depends on the value of the SECTION parameter)
Use the following option to specify the { \bf n } _ { 1 } -direction directly for a general beam section:
*BEAM GENERAL SECTION
-direction (the data line number depends on the value of the SECTION parameter)
Use the following option to specify an additional node off the beam axis to define the \mathbf { n } _ { 1 } -direction:
*ELEMENT
Abaqus/CAE Usage: Property module: Assign→Beam Section Orientation: select region and enter the { \bf n } _ { 1 } -direction
Specifying an additional node off the beam axis is not supported in Abaqus/CAE.
For beams in space you can define the nodal normal ( -direction) by giving its direction cosines as the fourth, fifth, and sixth coordinates of each node definition or by giving them in a user-specified normal definition; see “Normal definitions at nodes,” Section 2.1.4, for details. Otherwise, the nodal normal will be calculated by Abaqus, as described below.
If the nodal normal is defined as part of the node definition, this normal is used for all of the structural elements attached to the node except those for which a user-specified normal is defined. If a user-specified normal is defined at a node for a particular element, this normal definition takes precedence over the normal defined as part of the node definition. If the specified normal subtends an angle that is greater than 20° with the plane perpendicular to the element axis, a warning message is issued in the data (.dat) file. If the angle between the normal defined as part of the node definition or the user-specified normal and \mathbf { t } \times \mathbf { n } _ { 1 } is greater than 90°, the reverse of the specified normal is used.
Input File Usage: Use the following option to specify the -direction as part of the node definition:
*NODE
node number, nodal coordinates, nodal normal coordinates
Use the following option to define a user-specified normal:
*NORMAL
Abaqus/CAE Usage: Defining the nodal normal is not supported in Abaqus/CAE; the nodal normal calculated by Abaqus is always used.
Calculation of the average nodal normals by Abaqus
If the nodal normal is not defined as part of the node definition, element normal directions at the node are calculated for all shell and beam elements for which a user-specified normal is not defined (the “remaining” elements). For shell elements the normal direction is orthogonal to the shell midsurface, as described in “Shell elements: overview,” Section 29.6.1. For beam elements the normal direction is the second cross-section direction, as described in “Beam element cross-section orientation,” Section 29.3.4. The following algorithm is then used to obtain an average normal (or multiple averaged normals) for the remaining elements that need a normal defined:
- If a node is connected to more than 30 remaining elements, no averaging occurs and each element is assigned its own normal at the node. The first nodal normal is stored as the normal defined as part of the node definition. Each subsequent normal is stored as a user-specified normal.
- If a node is shared by 30 or fewer remaining elements, the normals for all the elements connected to the node are computed. Abaqus takes one of these elements and puts it in a set with all the other elements that have normals within 20° of it. Then:
a. Each element whose normal is within 20° of the added elements is also added to this set (if it is not yet included).
b. This process is repeated until the set contains for each element in the set all the other elements whose normals are within 20°.
c. If all the normals in the final set are within 2 0 ^ { \circ } of each other, an average normal is computed for all the elements in the set. If any of the normals in the set are more than 2 0 ^ { \circ } out of line from even a single other normal in the set, no averaging occurs for elements in the set and a separate normal is stored for each element.
d. This process is repeated until all the elements connected to the node have had normals computed for them.
e. The first nodal normal is stored as the normal defined as part of the node definition. Each subsequently generated nodal normal is stored as a user-specified normal.
This algorithm ensures that the nodal averaging scheme has no element order dependence. A simple example illustrating this process is included below.
Example: beam normal averaging
Consider the three beam element model in Figure 29.3.4–2. Elements 1, 2, and 3 share a common node 10, with no user-specified normal defined.
flowchart
graph TD
A["10"] -->|1| B["20"]
A -->|2| C["30"]
A -->|3| D["40"]
B -->|2| E["3"]
C -->|1| F["1"]
Figure 29.3.4–2 Three-element example for nodal averaging algorithm.
In the first scenario, suppose that at node 10 the normal for element 2 is within 2 0 ^ { \circ } of both elements 1 and 3, but the normals for elements 1 and 3 are not within 20° of each other. In this case, each element is assigned its own normal: one is stored as part of the node definition and two are stored as user-specified normals.
In the second scenario, suppose that at node 10 the normal for element 2 is within 2 0 ^ { \circ } of both elements 1 and 3 and the normals for elements 1 and 3 are within 2 0 ^ { \circ } of each other. In this case, a single average normal for elements 1, 2, and 3 would be computed and stored as part of the node definition.
In the last scenario, suppose that at node 10 the normal for element 2 is within 2 0 ^ { \circ } of element 1 but the normal of element 3 is not within 2 0 ^ { \circ } of either element 1 or 2. In this case, an average normal is computed and stored for elements 1, and 2 and the normal for element 3 is stored by itself: one is stored as part of the node definition and the other is stored as a user-specified normal.
Appropriate beam normals
To ensure proper application of loads that act normal to the beam cross-section, it is important to have beam normals that correctly define the plane of the cross-section. When linear beams are used to model a curved geometry, appropriate beam normals are the normals that are averaged at the nodes. For such cases it is preferable to define the cross-sectional axis system such that beam normals lie in the plane of curvature and are properly averaged at the nodes.
Initial curvature and initial twist
In Abaqus/Standard normal direction definitions can result in a beam element having an initial curvature or an initial twist, which will affect the behavior of some elements.
When the normal to an element is not perpendicular to the beam axis (obtained by interpolation using the nodes of the element), the beam element is curved. Initial curvature can result when you define the normal directly (as part of the node definition or as a user-specified normal) or can result when beams intersect at a node and the normals to the beams are averaged as described above. The effect of this initial curvature is considered in cubic beam elements. Initial curvature resulting from normal definitions is not considered in quadratic beam elements; however, these elements do properly account for any initial curvature represented by the node positions.
• Similarly, nodal-normal directions that are in different orientations about the beam axis at different nodes imply a twist. The effect of an initial twist, which could result from normal averaging or user-defined normal definitions, is considered in quadratic beam elements.
Since the behavior of initially curved or initially twisted beams is quite different from straight beams, the changes caused by averaging the normals may result in changes in the deformation of some beam elements. You should always check the model to ensure that the changes caused by averaging the normals are intended. If the normal directions at successive nodes subtend an angle that is greater than 2 0 ^ { \circ } , a warning message is issued in the data (.dat) file. In addition, a warning message will be issued during input file preprocessing if the average curvature computed for a beam differs by more than 0.1 degrees per unit length or if the approximate integrated curvature for the entire beam differs by more than 5 degrees as compared to the curvature computed without nodal averaging and without user-defined normals.
In Abaqus/Explicit initial curvature of the beam is not taken into account: all beam elements are assumed to be initially straight. The element’s cross-section orientation is calculated by averaging the - and \mathbf { n } _ { 2 } -directions associated with its nodes. These two vectors are then projected onto the plane that is perpendicular to the beam element’s axis. These projected directions \mathbf { n } _ { 1 } and \mathbf { n } _ { 2 } are made orthogonal to each other by rotating in this plane by an equal and opposite angle.