Beam Element Cross-Section Orientation

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 $\mathbf{t}$ is the tangent to the axis of the element, positive in the direction from the first to the second node of the element, and ${\mathbf{n}}_1$ 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 1.

Local axis definition for beam-type elements.

For beams 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 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). 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 ${\mathbf{n}}_1$ 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).

This approximate ${\mathbf{n}}_1$-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 ${\mathbf{n}}_1$-direction will be calculated as ${\mathbf{n}}_2\times \mathbf{t}$, possibly resulting in a direction that is different from the specified direction.

For beams in space you can define the nodal normal (${\mathbf{n}}_2$-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 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.

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 20°, 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 ${\mathbf{n}}_1$- 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.