Second Moments of Area & Bending

The second moments of area are defined as

$$\begin{aligned} I_{yy} &= \int_{A}^{}{z^{2}dA} \\ I_{zz} &= \int_{A}^{}{y^{2}dA} \\ I_{yz} &= \int_{A}^{}{yzdA} \end{aligned}$$

For symmetric sections $I_{xy}$ is zero.

For uniaxial bending or bending about principal axes

$$\begin{aligned} M_{y} &= EI_{yy}\kappa_{y} \\ M_{z} &= EI_{zz}\kappa_{z} \end{aligned}$$

When there is biaxial bending these have to be modified to

$$\left\{\begin{array}{l} M_{y} \\ M_{z} \end{array}\right\}=E\left[\begin{array}{cc} I_{y y} & -I_{y z} \\ -I_{y z} & I_{z z} \end{array}\right]\left\{\begin{array}{l} \kappa_{y} \\ \kappa_{z} \end{array}\right\}$$

As the second moments of area form a tensor these can be rotated to different axes using a rotation matrix

$$\left[\begin{array}{cc} \tilde{I}_{y y} & -\tilde{I}_{y z} \\ -\tilde{I}_{y z} & \tilde{I}_{z z} \end{array}\right]=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{cc} I_{y y} & -I_{y z} \\ -I_{y z} & I_{z z} \end{array}\right]\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$$

Or

$$\begin{aligned} &\tilde{I}_{y y}=I_{y y} \cos ^{2} \theta+I_{z z} \sin ^{2} \theta-2 I_{y z} \sin \theta \cos \theta \\ &\tilde{I}_{z z}=I_{y y} \sin ^{2} \theta+I_{z z} \cos ^{2} \theta+2 I_{y z} \sin \theta \cos \theta \\ &\tilde{I}_{y z}=I_{y y} \sin \theta \cos \theta-I_{z z} \sin \theta \cos \theta+I_{y z}\left(\cos ^{2} \theta-\sin ^{2} \theta\right) \end{aligned}$$

Or in terms of double angles

$$\begin{aligned} &\tilde{I}_{y y}=\frac{I_{y y}+I_{z z}}{2}+\frac{I_{y y}-I_{z z}}{2} \cos 2 \theta-I_{y z} \sin 2 \theta \\ &\tilde{I}_{z z}=\frac{I_{y y}+I_{z z}}{2}-\frac{I_{y y}-I_{z z}}{2} \cos 2 \theta+I_{y z} \sin 2 \theta \\ &\tilde{I}_{y z}=\frac{I_{y y}-I_{z z}}{2} \sin 2 \theta+I_{y z} \cos 2 \theta \end{aligned}$$