Second moment of area

The second moment of area for an arbitrary shape R with respect to an arbitrary axis ${\displaystyle BB'}$  (${\displaystyle BB'}$  axis is not drawn in the adjacent image; is an axis coplanar with x and y axes and is perpendicular to the line segment ${\displaystyle \rho }$ ) is defined as $${\displaystyle J_{BB'}=\iint _{R}{\rho }^{2}\,dA}$$  where

• ${\displaystyle dA}$  is the infinitesimal area element, and
• ${\displaystyle \rho }$  is the distance from the ${\displaystyle BB'}$  axis.^[2]

For example, when the desired reference axis is the x-axis, the second moment of area ${\displaystyle I_{xx}}$  (often denoted as ${\displaystyle I_{x}}$ ) can be computed in Cartesian coordinates as

$${\displaystyle I_{x}=\iint _{R}y^{2}\,dx\,dy}$$ 

The second moment of the area is crucial in Euler–Bernoulli theory of slender beams.

More generally, the product moment of area is defined as^[3] $${\displaystyle I_{xy}=\iint _{R}yx\,dx\,dy}$$ 

It is sometimes necessary to calculate the second moment of area of a shape with respect to an ${\displaystyle x'}$  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, ${\displaystyle x}$ , and use the parallel axis theorem to derive the second moment of area with respect to the ${\displaystyle x'}$  axis. The parallel axis theorem states $${\displaystyle I_{x'}=I_{x}+Ad^{2}}$$  where

• ${\displaystyle A}$  is the area of the shape, and
• ${\displaystyle d}$  is the perpendicular distance between the ${\displaystyle x}$  and ${\displaystyle x'}$  axes.^[4][5]

A similar statement can be made about a ${\displaystyle y'}$  axis and the parallel centroidal ${\displaystyle y}$  axis. Or, in general, any centroidal ${\displaystyle B}$  axis and a parallel ${\displaystyle B'}$  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 ${\displaystyle J_{z}}$  to ${\displaystyle I_{x}}$  and ${\displaystyle I_{y}}$ .

$${\displaystyle J_{z}=\iint _{R}\rho ^{2}\,dA=\iint _{R}\left(x^{2}+y^{2}\right)dA=\iint _{R}x^{2}\,dA+\iint _{R}y^{2}\,dA=I_{x}+I_{y}}$$ 

This relationship relies on the Pythagorean theorem which relates ${\displaystyle x}$  and ${\displaystyle y}$  to ${\displaystyle \rho }$  and on the linearity of integration.

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 second moments of area for other shapes.

Consider a rectangle with base ${\displaystyle b}$  and height ${\displaystyle h}$  whose centroid is located at the origin. ${\displaystyle I_{x}}$  represents the second moment of area with respect to the x-axis; ${\displaystyle I_{y}}$  represents the second moment of area with respect to the y-axis; ${\displaystyle J_{z}}$  represents the polar moment of inertia with respect to the z-axis.

$${\displaystyle {\begin{aligned}I_{x}&=\iint _{R}y^{2}\,dA=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}\int _{-{\frac {h}{2}}}^{\frac {h}{2}}y^{2}\,dy\,dx=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}{\frac {1}{3}}{\frac {h^{3}}{4}}\,dx={\frac {bh^{3}}{12}}\\I_{y}&=\iint _{R}x^{2}\,dA=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}\int _{-{\frac {h}{2}}}^{\frac {h}{2}}x^{2}\,dy\,dx=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}hx^{2}\,dx={\frac {b^{3}h}{12}}\end{aligned}}}$$ 

Using the perpendicular axis theorem we get the value of ${\displaystyle J_{z}}$ .

$${\displaystyle J_{z}=I_{x}+I_{y}={\frac {bh^{3}}{12}}+{\frac {hb^{3}}{12}}={\frac {bh}{12}}\left(b^{2}+h^{2}\right)}$$ 

Consider an annulus whose center is at the origin, outside radius is ${\displaystyle r_{2}}$ , and inside radius is ${\displaystyle r_{1}}$ . Because of the symmetry of the annulus, the centroid also lies at the origin. We can determine the polar moment of inertia, ${\displaystyle J_{z}}$ , about the ${\displaystyle z}$  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 ${\displaystyle r_{2}}$  minus the polar moment of inertia of a circle with radius ${\displaystyle r_{1}}$ , both centered at the origin. First, let us derive the polar moment of inertia of a circle with radius ${\displaystyle r}$  with respect to the origin. In this case, it is easier to directly calculate ${\displaystyle J_{z}}$  as we already have ${\displaystyle r^{2}}$ , which has both an ${\displaystyle x}$  and ${\displaystyle y}$  component. Instead of obtaining the second moment of area from Cartesian coordinates as done in the previous section, we shall calculate ${\displaystyle I_{x}}$  and ${\displaystyle J_{z}}$  directly using polar coordinates.

$${\displaystyle {\begin{aligned}I_{x,{\text{circle}}}&=\iint _{R}y^{2}\,dA=\iint _{R}\left(r\sin {\theta }\right)^{2}\,dA=\int _{0}^{2\pi }\int _{0}^{r}\left(r\sin {\theta }\right)^{2}\left(r\,dr\,d\theta \right)\\&=\int _{0}^{2\pi }\int _{0}^{r}r^{3}\sin ^{2}{\theta }\,dr\,d\theta =\int _{0}^{2\pi }{\frac {r^{4}\sin ^{2}{\theta }}{4}}\,d\theta ={\frac {\pi }{4}}r^{4}\\J_{z,{\text{circle}}}&=\iint _{R}r^{2}\,dA=\int _{0}^{2\pi }\int _{0}^{r}r^{2}\left(r\,dr\,d\theta \right)=\int _{0}^{2\pi }\int _{0}^{r}r^{3}\,dr\,d\theta \\&=\int _{0}^{2\pi }{\frac {r^{4}}{4}}\,d\theta ={\frac {\pi }{2}}r^{4}\end{aligned}}}$$ 

Now, the polar moment of inertia about the ${\displaystyle z}$  axis for an annulus is simply, as stated above, the difference of the second moments of area of a circle with radius ${\displaystyle r_{2}}$  and a circle with radius ${\displaystyle r_{1}}$ .

$${\displaystyle J_{z}=J_{z,r_{2}}-J_{z,r_{1}}={\frac {\pi }{2}}r_{2}^{4}-{\frac {\pi }{2}}r_{1}^{4}={\frac {\pi }{2}}\left(r_{2}^{4}-r_{1}^{4}\right)}$$ 

Alternatively, we could change the limits on the ${\displaystyle dr}$  integral the first time around to reflect the fact that there is a hole. This would be done like this.

$${\displaystyle {\begin{aligned}J_{z}&=\iint _{R}r^{2}\,dA=\int _{0}^{2\pi }\int _{r_{1}}^{r_{2}}r^{2}\left(r\,dr\,d\theta \right)=\int _{0}^{2\pi }\int _{r_{1}}^{r_{2}}r^{3}\,dr\,d\theta \\&=\int _{0}^{2\pi }\left[{\frac {r_{2}^{4}}{4}}-{\frac {r_{1}^{4}}{4}}\right]\,d\theta ={\frac {\pi }{2}}\left(r_{2}^{4}-r_{1}^{4}\right)\end{aligned}}}$$ 

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 ${\displaystyle n}$  vertices, numbered in counter-clockwise fashion. If polygon vertices are numbered clockwise, returned values will be negative, but absolute values will be correct.

$${\displaystyle {\begin{aligned}I_{y}&={\frac {1}{12}}\sum _{i=1}^{n}\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\left(x_{i}^{2}+x_{i}x_{i+1}+x_{i+1}^{2}\right)\\I_{x}&={\frac {1}{12}}\sum _{i=1}^{n}\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\left(y_{i}^{2}+y_{i}y_{i+1}+y_{i+1}^{2}\right)\\I_{xy}&={\frac {1}{24}}\sum _{i=1}^{n}\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\left(x_{i}y_{i+1}+2x_{i}y_{i}+2x_{i+1}y_{i+1}+x_{i+1}y_{i}\right)\end{aligned}}}$$ 

where ${\displaystyle x_{i},y_{i}}$  are the coordinates of the ${\displaystyle i}$ -th polygon vertex, for ${\displaystyle 1\leq i\leq n}$ . Also, ${\displaystyle x_{n+1},y_{n+1}}$  are assumed to be equal to the coordinates of the first vertex, i.e., ${\displaystyle x_{n+1}=x_{1}}$  and ${\displaystyle y_{n+1}=y_{1}}$ .^{[6][7] [8] [9]}

• List of second moments of area
• List of moments of inertia
• Moment of inertia
• Parallel axis theorem
• Perpendicular axis theorem
• Radius of gyration