Learning topic
Principal Axes, Principal Moments, and Section Moduli
Learn principal centroidal axes and moments of area, calculate section modulus W = I/ymax, and use these geometric properties in bending-stress calculations.
Principal centroidal axes are mutually perpendicular axes through the centroid for which the product of inertia is zero and the second moments of area take extreme values.
When orthogonal axes are rotated through an angle θ, the second moments and product of inertia transform as:
$$I_{x'}=\frac{I_x+I_y}{2}+\frac{I_x-I_y}{2}\cos2\theta-I_{xy}\sin2\theta,$$
$$I_{y'}=\frac{I_x+I_y}{2}-\frac{I_x-I_y}{2}\cos2\theta+I_{xy}\sin2\theta,$$
$$I_{x'y'}=\frac{I_x-I_y}{2}\sin2\theta+I_{xy}\cos2\theta.$$
Principal centroidal axes satisfy $I_{x'y'}=0$. Their orientation follows from:
$$\tan2\theta_p=-\frac{2I_{xy}}{I_x-I_y},$$
with the quadrant and the adopted positive direction of angle measured consistently.
Principal second moments
For known centroidal $I_x$, $I_y$, and $I_{xy}$, the principal values are:
$$I_{1,2}=\frac{I_x+I_y}{2}\pm\sqrt{\left(\frac{I_x-I_y}{2}\right)^2+I_{xy}^2}.$$
The larger value is commonly denoted $I_1$ and the smaller $I_2$. Their sum equals $I_x+I_y$.
Section modulus
For bending about a selected neutral axis, the geometric section modulus is:
$$W=\frac{I}{y_{\max}}.$$
It has dimensions of length cubed, for example mm³, and enters directly into the maximum normal-stress relation for simple bending: $|\sigma_{\max}|=|M|/W$.
Unsymmetrical section
If the neutral axis does not divide the section depth symmetrically, the distances to the extreme fibers differ. Separate section moduli are then used: $W_+=I/y_+$ and $W_-=I/y_-$.
Standard shapes
| Shape | Area A | Centroidal second moment | Section modulus |
|---|---|---|---|
| Rectangle b × h, x-axis parallel to side b | $bh$ | $I_x=bh^3/12$ | $W_x=bh^2/6$ |
| Rectangle b × h, y-axis parallel to side h | $bh$ | $I_y=hb^3/12$ | $W_y=hb^2/6$ |
| Circle of diameter d | $\pi d^2/4$ | $I_x=I_y=\pi d^4/64$ | $W_x=W_y=\pi d^3/32$ |
| Circle of diameter d, polar property | $\pi d^2/4$ | $J_p=\pi d^4/32$ | $W_p=\pi d^3/16$ |
| Annulus D, d | $\pi(D^2-d^2)/4$ | $I_x=I_y=\pi(D^4-d^4)/64$ | $W_x=2I_x/D$ |
Engineering meaning
A larger second moment of area reduces beam curvature for a given bending moment, while a larger section modulus reduces the maximum normal stress. Efficient beam sections therefore place material appropriately relative to the neutral axis.
About this topic
This topic explains principal centroidal axes and principal second moments of area, their determination for unsymmetrical sections, and their relationship to section modulus. It covers W = I/ymax, separate section moduli for unequal extreme-fiber distances, and the use of these properties in bending calculations.