# Second moment of area

## Mathematical construct in engineering / From Wikipedia, the free encyclopedia

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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 $I$ (for an axis that lies in the plane of the area) or with a $J$ (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, m^{4}, or inches to the fourth power, in^{4}, 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 ${\textstyle I_{x}=\iint _{R}y^{2}\,dA}$ or ${\textstyle I_{y}=\iint _{R}x^{2}\,dA,}$ with respect to some reference plane), or the **polar** second moment of area (${\textstyle I=\iint _{R}r^{2}\,dA}$, 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: ${\textstyle I=\int _{Q}r^{2}dm}$, 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.[1]