Transformation of principal axis angle

General thin-walled cross-sections often have asymmetrical geometries. The principal axes of such sections are then not parallel to the horizontally and vertically aligned axes Y and Z. When determining the cross-section properties, the angle α between the center-of-gravity axis y and the principal axis u is determined in addition to the principal axis-related moments of inertia.

SHAPE-THIN determines the principal axis angle α from the moments of inertia according to the following formula:

\tan (2 α) = \frac{2 \cdot I_{yz}}{I_{z} - I_{y}}

α	Principal axis angle
I_yz	Biaxial second moment of area
I_z	Moment of inertia about the z-axis
I_y	Moment of inertia about the y-axis

Principal Axis Inclination α

The axes y and z represent the centroidal axes parallel to the global axes Y and Z. In the case of a flat cross-section of which the width is greater than the height, it can happen that the "major" principal axis u points in the direction that actually represents the "minor" principal axis v. In this case, the table shows a moment of inertia I_u that is less than the moment of inertia I_v for the weak axis.

Cross-Section with "Swapped" Moments of Inertia

The "Transform principal axes so that I_u (major axis) is always greater than I_v (minor axis)" function allows you to compare the position of the principal axes with the input.

Setting for Principal Axis Inclination

Thus, the cross-section does not need to be rotated 90° in order to obtain the moments of inertia as usual.

Principal Moments of Inertia

According to the formula given above, only angles between +90° and -90° are possible. If the cross-section is to be rotated and internal forces with reference to the principal axes u and v are already available, you can use an additional function to ensure that the position of the principal axes is retained when the cross-section is subsequently rotated: "Maintain the position of the axes related to the cross-section geometry when rotating the cross-section".