Default Orientation Rules for Frame Elements

Default Orientation Rules for Frame Elements

Frame element local coordinate system is defined in two steps. First, it is formed according to default orientation rules and then rotated about local 1-axis according to the rotation angle (if it is specified).

Default Orientation of Frame Element Local Coordinate System

  • Local axis 1 of the element is along the element's axial axis with the origin being at the start node.
  • Then local axis 2 is oriented as follows:
    • Local axis 2 is in the plane formed by local axis 1 and global Z-axis unless the element (local axis 1) is parallel to the global Z axis.
    • Local axis 2 is in the direction of positive X-axis if the local axis 1 is parallel to the global Z-axis in positive direction.
    • Local axis 2 is in the direction of negative X-axis if the local axis 1 is parallel to the global Z-axis in negative direction.
  • Finally, local axis 3 is determined by the cross product of local 1-axis and local 2-axis.

The mathematical procedure to form the frame element local axes according to the default orientation rule is summarized below:

  1. Let V1, V2 and V3 be the unit vectors that represent the local 1,2 and 3 axes respectively and we also have an auxiliary unit vector, Vn the definition of which is explained in the following procedure.
  2. V1 is the vector from start node to end node.
  3. Then the vector Vn is
    • [0, 0, 1] if local axis 1 is not parallel to global Z-axis.
    • [1, 0, 0] if local axis 1 is in the direction of positive Z-axis of the global coordinate system.
    • [-1, 0, 0] if local axis 1 is in the direction of negative Z-axis of the global coordinate system.
  4. Obtain V3 as V3 = V1 x Vn
  5. Obtain V2 as V2 = V3 x V1

Rotation Angle

The orientations of frame elements that are different from the default orientation are defined using the element rotation angle. It is the angle by which the local 2 and 3 axes are rotated about local 1-axis.
Notes
The direction of rotation follows the right hand rule. i.e. when you curl your hands about local 1-axis while your thumb pointing to the positive direction of local 1-axis, the direction of rotation of your hand indicates the rotation direction.
Following figures show some frame elements with different rotation angles and different orientations according to the default orientation rule.