Orientation

An orientation is required for 3D beam elements to indicate how the section is oriented. The orientation object also handles transformation between global and local coordinate systems of beam/truss elements.

Similar configurations are also required in other FEM software, see, for example, Section 26.3.4 Beam element cross-section orientation of the Abaqus Analysis User's Manual.

The local $x$-axis is along the cord of the beam element. The local $z$-axis is temporarily taken as the vector assigned by user. The local $y$-axis is defined to be the cross product of the local $z$-axis and the local $x$-axis. After obtaining the local $y$-axis, the local $z$-axis is redefined to be the cross product of the local $x$-axis and the local $y$-axis.

Depending on the different literature and textbooks, different conventions may be used. Thus, the term local $z$-axis may not be the same axis you may expect. Please validate the orientation first before using it.

global and local coordinate systems

For 2D beam elements, the local $z$-axis is defined to be the global $z$-axis.

Syntax

orientation (1) (2) (3) (4) (5)
# (1) int, unique orientation tag
# (2) string, orientation type
# (3) double, x component of local z axis
# (4) double, y component of local z axis
# (5) double, z component of local z axis

Remarks

1. For type (2), the following options are available.

   1. B3DL (stands for linear 3D beam transformation) and B3DC (corotational formulation). The nodal size is 6 and the local quantity size is 6.

   2. B3DOSL (linear for beams of open sections such as EB31OS) and B3DOSC (corotational). The nodal size is 7 (including an extra warping DoF) and the local quantity size is 9.

2. The B3DC/B3DOSC formulation is based on Crisfield's work with modifications. See de Souza's thesis.

3. For brevity, the mass distribution does not follow the corotational formulation, implying that the mass matrix is always a constant matrix. This is often sufficient for applications in earthquake engineering. However, it is not recommended for other applications. An energy-reserving corotational formulation is required, see Crisfield's work.

Correct orientations are required for different element types. For example, B31/F31 require B3DL or B3DC. EB31OS requires B3DOSL or B3DOSC.

One can use the following model to check how the orientation works.

The following model defines a number of beam elements with the WT12X125 section that is placed at the centre of web.

The elements are subject to axial forces, but since loads are applied at the centre of web, the elements are also subject to bending.

Different orientations of local reference frame are defined. The nodal forces are different for different orientations.

node 1 0 0 0
node 2 1 0 0
node 3 1 0 0
node 4 1 0 0
node 5 1 0 0
orientation B3DL 1 0. 0. 1.
orientation B3DL 2 0. 1. 0.
material Elastic1D 1 1
section US3D WT12X125 1 1
element B31 1 1 2 1 1
element B31 2 1 3 1 2
element B31 3 4 1 1 1
element B31 4 5 1 1 2
fix2 1 E 1
displacement 1 0 1 1 2
displacement 2 0 1 1 3
displacement 3 0 1 1 4
displacement 4 0 1 1 5
step static 1
analyze
peek node 2 3 4 5
exit