Second moment of area explained

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

(for an axis that lies in the plane of the area) or with a

(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 $I_x = \iint_ y^2\, dA$ or $I_y = \iint_ x^2\, dA,$ with respect to some reference plane), or the polar second moment of area ( $I = \iint_ r^2\, dA$ , 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: $I = \int_ r^2 dm$ , 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 . 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]

Definition

The second moment of area for an arbitrary shape with respect to an arbitrary axis

BB'

(

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

\rho

) is defined as

J_ = \iint_ ^2 \, dA

where

is the infinitesimal area element, and

\rho

is the distance from the

BB'

axis.^[2]

For example, when the desired reference axis is the x-axis, the second moment of area

I_xx

(often denoted as

I_x

) can be computed in Cartesian coordinates as

$I_ = \iint_ y^2\, dx\, dy$

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^[3] $I_ = \iint_ yx\, dx\, dy$

Parallel axis theorem

See main article: article and 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

I_ = I_x + A d^2

where

is the area of the shape, and

is the perpendicular distance between the

and

axes.^[4] ^[5]

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

See main article: article and Perpendicular axis theorem. 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

J_z

I_x

and

I_y

$J_z = \iint_ \rho^2\, dA= \iint_ \left(x^2 + y^2\right) dA= \iint_ x^2 \, dA + \iint_ y^2 \, dA= I_x + I_y$

This relationship relies on the Pythagorean theorem which relates

and

\rho

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" (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.

Examples

See list of second moments of area for other shapes.

Rectangle with centroid at the origin

Consider a rectangle with base

and height

whose centroid is located at the origin.

I_x

represents the second moment of area with respect to the x-axis;

I_y

represents the second moment of area with respect to the y-axis;

J_z

represents the polar moment of inertia with respect to the z-axis.

$\begin I_x &= \iint_ y^2\, dA = \int^\frac_ \int^\frac_ y^2 \,dy \,dx = \int^\frac_ \frac\frac\,dx = \frac \\ I_y &= \iint_ x^2\, dA = \int^\frac_ \int^\frac_ x^2 \,dy \,dx = \int^\frac_ h x^2\, dx = \frac\end$

Using the perpendicular axis theorem we get the value of

J_z

$J_z = I_x + I_y = \frac + \frac = \frac\left(b^2 + h^2\right)$

Annulus centered at origin

Consider an annulus whose center is at the origin, outside radius is

r₂

, and inside radius is

r₁

. Because of the symmetry of the annulus, the centroid also lies at the origin. We can determine the polar moment of inertia,

J_z

, 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

r₂

minus the polar moment of inertia of a circle with radius

r₁

, 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

J_z

as we already have

r²

, 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

I_x

and

J_z

directly using polar coordinates.

$\begin I_&= \iint_ y^2\,dA = \iint_ \left(r\sin\right)^2\, dA = \int_0^\int_0^r \left(r\sin\right)^2\left(r \, dr \, d\theta\right) \\&= \int_0^\int_0^r r^3\sin^2\,dr \, d\theta= \int_0^ \frac\,d\theta= \fracr^4 \\
J_&= \iint_ r^2\, dA = \int_0^\int_0^r r^2\left(r\,dr\,d\theta\right) = \int_0^\int_0^r r^3\,dr\,d\theta \\&= \int_0^ \frac\,d\theta = \fracr^4\end$

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

r₂

and a circle with radius

r₁

$J_z = J_ - J_ = \fracr_2^4 - \fracr_1^4 = \frac\left(r_2^4 - r_1^4\right)$

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.

$\begin J_&= \iint_ r^2 \, dA = \int_0^\int_^ r^2\left(r\, dr\, d\theta\right) = \int_0^\int_^ r^3\, dr\, d\theta \\&= \int_0^\left[\frac{r_2^4}{4} - \frac{r_1^4}{4}\right]\, d\theta = \frac\left(r_2^4 - r_1^4\right)\end$

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.

$\begin I_y &= \frac\sum_^ \left(x_i y_ - x_ y_i\right)\left(x_i^2 + x_i x_ + x_^2 \right) \\ I_x &= \frac\sum_^ \left(x_i y_ - x_ y_i\right)\left(y_i^2 + y_i y_ + y_^2 \right) \\ I_ &= \frac\sum_^ \left(x_i y_ - x_ y_i\right) \left(x_i y_ + 2 x_i y_i + 2 x_ y_ + x_ y_i \right)\end$

where

x_i,y_i

are the coordinates of the

-th polygon vertex, for

1\lei\len

. Also,

x_n+1,y_n+1

are assumed to be equal to the coordinates of the first vertex, i.e.,

x_n+1=x₁

and

y_n+1=y₁

.^[6] ^[7] ^[8] ^[9]

Notes and References

Book: Beer. Ferdinand P.. Vector Mechanics for Engineers. 2013. New York. McGraw-Hill. 978-0-07-339813-6. 10th. 471. The term second moment is more proper than the term moment of inertia, since, logically, the latter should be used only to denote integrals of mass (see Sec. 9.11). In engineering practice, however, moment of inertia is used in connection with areas as well as masses..
Book: Pilkey. Walter D. . Analysis and Design of Elastic Beams. limited. 15. 2002. John Wiley & Sons, Inc. 978-0-471-38152-5.
Book: Beer. Ferdinand P.. Vector Mechanics for Engineers. 2013. New York. McGraw-Hill. 978-0-07-339813-6. 10th. 495. Chapter 9.8: Product of inertia.
Hibbeler, R. C. (2004). Statics and Mechanics of Materials (Second ed.). Pearson Prentice Hall. .
Book: Beer. Ferdinand P.. Vector Mechanics for Engineers. 2013. New York. McGraw-Hill. 978-0-07-339813-6. 10th. 481. Chapter 9.6: Parallel-axis theorem.
David. Hally . Calculation of the Moments of Polygons . Technical Memorandum 87/209 . Canadian National Defense . 1987 . https://web.archive.org/web/20200323095244/https://apps.dtic.mil/dtic/tr/fulltext/u2/a183444.pdf. live. March 23, 2020.
Book: Obregon, Joaquin . Mechanical Simmetry . 978-1-4772-3372-6 . 2012 . AuthorHouse .
Web site: Steger, Carsten . On the Calculation of Arbitrary Moments of Polygons . https://web.archive.org/web/20181003014106/https://pdfs.semanticscholar.org/fd18/036ba9b78174a2a161b184148028a43881a8.pdf . dead . 2018-10-03 . 1996 . 17506973.
Web site: Soerjadi, Ir. R. . On the Computation of the Moments of a Polygon, with some Applications.

Second moment of area explained

Definition

Product moment of area

Parallel axis theorem

Perpendicular axis theorem

Composite shapes

Examples

Rectangle with centroid at the origin

Annulus centered at origin

Any polygon

See also

Notes and References