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The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The second moment of area is typically denoted with either an for an axis that lies in the plane or with a for an axis perpendicular to the plane. Its unit of dimension is length to fourth power, L4.
The second moment of area for an arbitrary shape with respect to an arbitrary axis is defined as
For example, when the desired reference axis is the x-axis the second moment of area, (often denoted as ) can be computed in Cartesian coordinates as
It is often easier to derive the second moment of area with respect to its centroidal axis, . However, it may be necessary to calculate the second moment of area with respect to a different, parallel axis, say the axis. The parallel axis theorem states
A similar statement can be made about the axis and the parallel centroidal axis. Or, in general, any centroidal axis and a parallel axis.
For the simplicity of calculation, it is often desired to define the polar moment of area(with respect to a perpendicular axis) in terms of two area moments of inertia (both with respect to in-plane axes). The simplest case relates to and .
For more complex areas, it is often easier to divide the area into a series of "simpler" shapes. The second moment of area for the entire shape is the sum of the second moment of areas of all of its parts about a common axis. This can include shapes that are "missing" (i.e. holes, hollow shapes, etc.), in which case the second moment of area of the "missing" areas are subtracted, rather than added. In other words, the second moment of area of "missing" parts are considered negative for the method of composite shapes.
See list of area moments of inertia for other shapes.
Consider a rectangle with base and height whose centroid is located at the origin. represents the second moment of area with respect to the x-axis; represents the second moment of area with respect to the y-axis; represents the polar moment of inertia with respect to the z-axis.
Consider an annulus whose center is at the origin, outside radius is , and inside radius is . Because of the symmetry of the annulus, the centroid also lies at the origin. We can determine the polar moment of inertia, , about the axis by the method of composite shapes. This polar moment of inertia is equivalent to the polar moment of inertia of a circle with radius minus the polar moment of inertia of a circle with radius , both centered at the origin. First, let us derive the polar moment of inertia of a circle with radius with respect to the origin. In this case, it is easier to directly calculate as we already have , which has both an and component. Instead of calculating and as done in the previous section, we shall calculate directly using Polar Coordinates.
Now, the polar moment of inertia about the axis for an annulus is simply, as stated above, the difference of the second moments of area of a circle with radius and a circle with radius .
Alternatively, we could change the limits on the integral the first time around to reflect the fact that there is a hole. This would be done like this.
The second moment of area for any simple convex polygon on the XY-plane can be computed in general by summing contributions from each segment of the polygon (method of composite shapes)(order of iteration along the polygon vertex gives sign to the value calculated).
where (with ) are the coordinates of any polygon vertex.