The second moment of area, or second area moment, or quadratic moment of area 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 I {\displaystyle I} (for an axis that lies in the plane of the area) or with a J {\displaystyle J} (for an axis perpendicular to the plane). In both cases, it is calculated with a multiple integral over the object in question. Its dimension is L (length) 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 or the US customary system. 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. Different disciplines use the term moment of inertia (MOI) to refer to different moments. It may refer to either of the planar second moments of area (often I x = ∬ R y 2 d A {\textstyle I_{x}=\iint _{R}y^{2}\,dA} or I y = ∬ R x 2 d A , {\textstyle I_{y}=\iint _{R}x^{2}\,dA,} with respect to some reference plane), or the polar second moment of area ( I = ∬ R r 2 d A {\textstyle I=\iint _{R}r^{2}\,dA} , where r is the distance to some reference axis). In each case the integral is over all the infinitesimal elements of area, dA, in some two-dimensional cross-section. In physics, moment of inertia is strictly the second moment of mass with respect to distance from an axis: I = ∫ Q r 2 d m {\textstyle I=\int _{Q}r^{2}dm} , where r is the distance to some potential rotation axis, and the integral is over all the infinitesimal elements of mass, dm, in a three-dimensional space occupied by an object Q. The MOI, in this sense, is the analog of mass for rotational problems. In engineering (especially mechanical and civil), moment of inertia commonly refers to the second moment of the area.
Definition
The second moment of area for an arbitrary shape R with respect to an arbitrary axis B B ′ {\displaystyle BB'} ( B B ′ {\displaystyle BB'} axis is not drawn in the adjacent image; is an axis coplanar with x and y axes and is perpendicular to the line segment ρ {\displaystyle \rho } ) is defined as
J B B ′ = ∬ R ρ 2 d A {\displaystyle J_{BB'}=\iint _{R}{\rho }^{2}\,dA}
where
d A {\displaystyle dA} is the infinitesimal area element, and
ρ {\displaystyle \rho } is the distance from the B B ′ {\displaystyle BB'} axis. For example, when the desired reference axis is the x-axis, the second moment of area I x x {\displaystyle I_{xx}} (often denoted as I x {\displaystyle I_{x}} ) can be computed in Cartesian coordinates as
I x = ∬ R y 2 d x d y {\displaystyle I_{x}=\iint _{R}y^{2}\,dx\,dy}
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
I x y = ∬ R y x d x d y {\displaystyle I_{xy}=\iint _{R}yx\,dx\,dy}
Parallel axis theorem
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