10.4: Principal Moments of Area
In mechanics, the product of inertia and moments of inertia of area help to calculate the stability and performance of various structures and components. The coordinate transformation relations are used to calculate the moments and products of inertia for an area about the inclined axes. Further, the moments and products of inertia with respect to the principal axes can be determined using the moments and products of inertia about the inclined axes.
The principal moment of inertia axes are the set of axes corresponding to the maximum and minimum values of the moments of inertia. The orientation of these axes is determined by differentiating the moment of inertia about the inclined axis with respect to the inclination angle and equating the result to zero.
The roots of the above equation define two angles 90° apart from each other. These angles specify the orientation angle with respect to the principal axes.
Substituting these orientation angles into the product of inertia expression gives a zero value. The product of inertia with respect to the principal axis is always zero. Further, the moments of inertia about the inclined axes can be rewritten by incorporating the sine and cosine terms. The expression is further simplified to obtain the principal moments of inertia. The principal moments of inertia can be either the maximum or the minimum, depending on the sign of the expression.
If an area has an axis of symmetry passing through the origin, this is a principal axis of the area about the origin. However, a principal axis does not necessarily have to be an axis of symmetry. When the origin coincides with the centroid, the principal axes denote the principal centroidal axes.
