Two-Plane Bending

Normal stresses for beams in bending often occur in both the $xy$ and $xz$ planes, such that there are bending moments $M_z$ and $M_y$ respectively.

• The subscript for $M$ is the axis which the moment is around

The bending stress can then be given by:

$${\sigma }_x=-\frac{M_{z}y}{I_z}+\frac{M_{y}z}{I_y}$$

The maximum occurs when the summation gives the greatest positive stresses (tensile) and negative stresses (compressive).

For solid circular cross sections, the maximum is calculated as:

$${\sigma }_{\text{max}}=\frac{Mc}{I}=\frac{32}{\pi d^3}(M_y^2+M_z^2{)}^{1/2}$$