Beam Element Overview

Local CS

The beam section possesses a local coordinate system that differs from the global coordinate system. The local \(x\)-axis is aligned with beam cord. Strong axis is represented by local \(z\)-axis while weak axis is represented by local \(y\)-axis. The local \(xyz\) axes form a right-handed coordinate system.

Orientation

For 2D beams, it is assumed that local \(z\)-axis coincides with global \(z\)-axis so that there is no need to explicitly define any orientation for 2D beams.

For 3D beams, local \(z\)-axis shall be defined in the associated section orientation.

Local Deformation

The local deformation vector consists of six components, namely,

\[ \mathbf{d}=\begin{bmatrix} u&\theta_{zi}&\theta_{zj}&\theta_{yi}&\theta_{yj}&\theta_{xj}-\theta_{xi} \end{bmatrix}. \]

They correspond to axial elongation, strong axis rotation at first and second node, weak axis rotation at first and second node, and torsion rotation.

For 2D beams, only the first three are present.

Any sections shall take the above deformation vector over element length, that is,

\[ \mathbf{d}/L \]

as input and produce the corresponding force conjugates (section resistance) as the output. Please note the deformation vector is normalised by element length. This offers some convenience so that sections are independent of element properties. The same section can be used for different elements with different element lengths.

Which Element To Be Used?

A few different general purpose beam elements are available.

Displacement Based

The B21 and B31 are classic Bernoulli beam elements, which are probably the first beam elements introduced in any FEM textbooks. They are displacement based elements, meaning that the displacement profile along beam chord always follows the Hermite polynomial. This is fine for elastic analysis but may not be suitable for plastic analysis. The nonlinear response is often over stiff. Accuracy can be improved by mesh refinement, but it appears a bit cumbersome to define multiple elements within the same span/floor with interior nodes do not connect additional frames.

To model hinge connection at one of two ends, one can use B21E so that either the first or the second end has zero moment. To specify the explicit plastic hinge length, one can use B21H. Plasticity is only allowed at ends with a fixed length while the interior remains elastic.

Force Based

The force based beam elements appear to be superior in all cases. This category includes F21 and F31 elements. The moment distribution along beam chord is always linear in absence of distributed load. It is shown that force based elements result in less error with the same number of integration points, and converges faster with increasing number of integration points, see this paper.

Typically, five to seven integration points would be sufficient for each element. Mesh refinement is often not necessary and can be alternatively replaced by adding more integration points to a single element.

Generalised Plasticity Based

Without using sections, it is possible to model nonlinear beams at element level. The NMB21 and NMB31 are two examples of generalised plasticity based beam elements. The efficiency of this type of elements is the best of the three as there are no sections, no integration points. Only local plasticity iterations are performed. The applicability mainly depends on the nonlinear behaviour of NMSection.