Beam axes

Symmetry about the axis

In most cases beams have symmetry about one axis or both axes (e.g., a rectangular, circular, I-beam or tee section). In these cases the principal axes of the beam correspond with the local axes. This means that the bending behaviour is governed by: $I_{yy}$ and $I_{zz}$ and the $I_{yz}$ term is zero.

No symmetry about the axis

When a section lacks symmetry about both axes, the cross term $I_{yz}$ is no longer zero. This means that a load in the y direction will produce a displacement in y and z directions.

The angle of the principal second moments of area can be calculated from:

$$\theta = tan^{-1} \left( \frac{ 2\times I_{yz}}{I_{zz} - I_{yy}} \right)$$

The second moments of area in the u and v axes are:

$$I_{uu} = I_{yy} \cos^2\theta + I_{zz}\sin^2\theta - I_{yz}\sin(2\theta)$$

$$I_{vv}=I_{yy}\sin^2\theta + I_{zz}\cos^2\theta + I{yz}\sin(2\theta)$$

These relationships can be visualised using a Mohr's circle approach similar to stress.

Note: GSA gives the option to use either principal second moments of area, or local second moments of area (ignoring the $I_{yz}$ term). Principal moments most often apply to asymmetric or angle sections; by using this option it is possible to capture sideways deflection. Therefore, the latter option is not recommended.