The 2ndmoment of area, or second area moment and also known as the area moment of inertia, 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 or with a . In both cases, it is calculated with a multiple integral over the object in question. Its dimension is L to the fourth power. Its unit of dimension when working with the International System of Units is meters to the fourth power, m4, or inches to the fourth power, in4, when working in the Imperial System of Units. In structural engineering, the second moment of area of a beam is an important property used in the calculation of the beam's deflection and the calculation of stress caused by a moment applied to the beam. In order to maximize the second moment of area, a large fraction of the cross-sectional area of an I-beam is located at the maximum possible distance from the centroid of the I-beam's cross-section. The planar second moment of area provides insight into a beam's resistance to bending due to an applied moment, force, or distributed load perpendicular to its neutral axis, as a function of its shape. The polar second moment of area provides insight into a beam's resistance to torsional deflection, due to an applied moment parallel to its cross-section, as a function of its shape.
Definition
The second moment of area for an arbitrary shape with respect to an arbitrary axis is defined as where For example, when the desired reference axis is the x-axis, the second moment of area can be computed in Cartesian coordinates as The second moment of the area is crucial in Euler–Bernoulli theory of slender beams.
Product moment of area
More generally, the product moment of area is defined as
Parallel axis theorem
It is sometimes necessary to calculate the second moment of area of a shape with respect to an axis different to the centroidal axis of the shape. However, it is often easier to derive the second moment of area with respect to its centroidal axis,, and use the parallel axis theorem to derive the second moment of area with respect to the axis. The parallel axis theorem states where A similar statement can be made about a axis and the parallel centroidal axis. Or, in general, any centroidal axis and a parallel axis.
Perpendicular axis theorem
For the simplicity of calculation, it is often desired to define the polar moment of area in terms of two area moments of inertia. The simplest case relates to and. This relationship relies on the Pythagorean theorem which relates and to and on the linearity of integration.
Composite shapes
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", 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.
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. Using the perpendicular axis theorem we get the value of.
Annulus centered at origin
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 obtaining the second moment of area from Cartesian coordinates as done in the previous section, we shall calculate and 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.
Any polygon
The second moment of area about the origin for any simple polygon on the XY-plane can be computed in general by summing contributions from each segment of the polygon after dividing the area into a set of triangles. This formula is related to the shoelace formula and can be considered a special case of Green's theorem. A polygon is assumed to have vertices, numbered in counter-clockwise fashion. If polygon vertices are numbered clockwise, returned values will be negative, but absolute values will be correct. where are the coordinates of the -th polygon vertex, for. Also, are assumed to be equal to the coordinates of the first vertex, i.e., and.
