Second moment of area

This article is about the geometrical property of

an area, termed the second moment of area. y
For the moment of inertia dealing with the ro-
tation of an object with mass, see mass moment
of inertia.

For a list, see list of area moments of inertia.

y dA
The second moment of area, also known as moment of
inertia of plane area, area moment of inertia, or sec-
ond area moment, is a geometrical property of an area
which reflects how its points are distributed with regard ρ
to an arbitrary axis. The second moment of area is typi-
cally denoted with either an I for an axis that lies in the
plane or with a J for an axis perpendicular to the plane.
Its unit of dimension is length to fourth power, L4 . x x
In the field of structural engineering, the second moment
of area of the cross-section of a beam is an important A schematic showing how the polar moment of inertia is calcu-
property used in the calculation of the beam’s deflection lated for an arbitrary shape with respect to the z axis. ρ is the
and the calculation of stress caused by a moment applied radial distance to the element dA, with projections x and y on the
to the beam. axes.

Note: Different disciplines use “Moment of In-

ertia” (MOI) to refer to either or both∫of the pla- ∫
nar second moment of area, I = A x2 dA , JBB = ρ2 dA
A
where x is the distance to some reference plane,
∫or the2
polar second moment of area, I = dA = Differential area of the arbitrary shape
A
r dA , where r is the distance to some refer-
ence axis. In each case the integral is over all the ρ = Distance from the axis BB to dA[2]
infinitesimal elements of area, dA, in some two-
dimensional cross-section. In math and physics, For example, when the desired reference axis is the x-axis
“Moment of Inertia” is strictly the second mo- the second moment of area, Ixx (often denoted as Ix ) can
ment of mass be computed in Cartesian coordinates as
∫ with respect to distance from an
axis: I = m r2 dm , where r is the distance
to some potential rotation axis, and the integral ∫∫
is over all the infinitesimal elements of mass, Ix = y 2 dx dy
dm, in a three-dimensional space occupied by A
an object. The MOI, in this sense, is the analog The second moment of the area is crucial in Euler–
of mass for rotational problems. In engineer- Bernoulli theory of slender beams.
ing (especially mechanical and civil), “Moment
of Inertia” commonly refers to the second mo-
ment of the area.[1] 1.1 Product moment of area
More generally, the product moment of area is defined
1 Definition as

∫∫
The second moment of area for an arbitrary shape with
respect to an arbitrary axis BB is defined as Ixy = xy dx dy
A

1
2 5 EXAMPLES

2 Parallel axis theorem 5 Examples

Main article: Parallel axis theorem See list of area moments of inertia for other shapes.

It is often easier to derive the second moment of area with

5.1 Rectangle with centroid at the origin
respect to its centroidal axis, x′ . However, it may be
necessary to calculate the second moment of area with
Consider a rectangle with base b and height h whose
respect to a different, parallel axis, say the x axis. The
centroid is located at the origin. Ix represents the second
parallel axis theorem states
moment of area with respect to the x-axis; Iy represents
the second moment of area with respect to the y-axis; Jz
represents the polar moment of inertia with respect to the
Ix = Ix′ + Ad2y z-axis.

where

A = Area of the shape

dy = Perpendicular distance between the x′ and
x axes[3]

A similar statement can be made about the y axis and the

parallel centroidal y ′ axis. Or, in general, any centroidal
B ′ axis and a parallel B axis.

3 Perpendicular axis theorem ∫ ∫ b/2 ∫ h/2 ∫ b/2

1 h3 bh3
Ix = y 2 dA = y 2 dy dx = dx =
A −b/2 −h/2 −b/2 3 4 12
Main article: Perpendicular axis theorem
∫ ∫ b/2 ∫ h/2 ∫ b/2
b3 h
Iy = x2 dA = x2 dy dx = hx2 dx =
For the simplicity of calculation, it is often desired to A −b/2 −h/2 −b/2 12
define the polar moment of area (with respect to a per-
pendicular axis) in terms of two area moments of inertia Jz = Ix + Iy = bh hb 3 3
bh 2
+ h2 )
12 + 12 = 12 (b
(both with respect to in-plane axes). The simplest case (see Perpendicular axis theorem)
relates Jz to Ix and Iy .

∫ ∫ ∫ ∫ 5.2 Annulus centered at origin

Jz = ρ2 dA = (x2 +y 2 ) dA = x2 dA+ y 2 dA = Ix +Iy
A A A A

This relationship relies on the Pythagorean theorem

which relates x and y to ρ and on the linearity of inte-
gration.

4 Composite shapes
For more complex areas, it is often easier to divide the
area into a series of “simpler” shapes. The second mo-
ment 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.
 3

Consider an annulus whose center is at the origin, outside gon all values will be negative with same absolute value)
radius is ro , and inside radius is ri . Because of the sym-
metry of the annulus, the centroid also lies at the origin.
We can determine the polar moment of inertia, Jz , about 1 ∑ 2
i=N

the z axis by the method of composite shapes. This po- Ix = 2

(y + yi yi+1 + yi+1 )(xi yi+1 − xi+1 yi )
12 i=1 i
lar moment of inertia is equivalent to the polar moment
of inertia of a circle with radius ro minus the polar mo-
1 ∑ 2
i=N
ment of inertia of a circle with radius ri , both centered at Iy = (x + xi xi+1 + x2i+1 )(xi yi+1 − xi+1 yi )
the origin. First, let us derive the polar moment of iner- 12 i=1 i
tia of a circle with radius r with respect to the origin. In
1 ∑
i=N
this case, it is easier to directly calculate Jz as we already
Ixy = (xi yi+1 +2xi yi +2xi+1 yi+1 +xi+1 yi )(xi yi+1 −xi+1 yi )
have r2 , which has both an x and y component. Instead 24 i=1
of obtaining the second moment of area from Cartesian
coordinates as done in the previous section, we shall cal- where xi , yi (with xn+1 = x1 , yn+1 = y1 ) are the co-
culate Ix and Jz directly using Polar Coordinates. ordinates of any polygon vertex.[4]
∫∫ 2 ∫∫ 2
Ix,circle = y dA = (r sin θ) dA =
∫ 2π ∫ r ∫ 2π ∫ r
(r sin θ)2 (r dr dθ) = 0 0 r3 sin2 θ dr dθ = 6 See also
∫02π r04 sin2 θ
0 4 dθ = π4 r4
∫∫ 2 ∫ 2π ∫ r • List of area moments of inertia
Jz,circle = r dA = 0 0 r2 (r dr dθ) =
∫ 2π ∫ r 3 ∫ 2π 4
0 0
r dr dθ = 0 r4 dθ = π2 r4 • List of moments of inertia
Now, the polar moment of inertia about the z axis for an • Moment of inertia
annulus is simply, as stated above, the difference of the
second moments of area of a circle with radius ro and a • Parallel axis theorem
circle with radius ri .
• Perpendicular axis theorem
Jz = Jz,ro − Jz,ri = π 4
2 ro − π2 ri4 = π
2 (ro
4
− ri 4 )
• Radius of gyration
Alternatively, we could change the limits on the dr inte-
gral the first time around to reflect the fact that there is a
hole. This would be done like this.
∫∫ 2 ∫ 2π ∫ ro 2 7 References
Jz = r dA = r (r dr dθ) =
∫ 2π ∫ ro 3 [
∫ 2π ro4
0 ] ri
( )
ri4
0 ri
r dr dθ = 0 4 − 4 dθ = π2 ro4 − ri4 [1] Beer, Ferdinand (2009). Vector Mechanics for Engineers:
Statics. McGraw-Hill. ISBN 978-0-07-352940-0.

[2] Pilkey, Walter D. (2002). Analysis and Design of Elastic

5.3 Any polygon Beams. John Wiley & Sons, Inc. ISBN 0-471-38152-7.

[3] Hibbeler, R. C. (2004). Statics and Mechanics of Mate-

rials (Second ed.). Pearson Prentice Hall. ISBN 0-13-
028127-1.

[4] “Polygons, Meshes”.

8 External links
• Calculator for Second Moment of Area

Any Polygon

The second moment of area for any simple polygon on the

XY-plane can be computed in general by summing con-
tributions from each segment of the polygon. A polygon
is assumed to be counter clock wise (for clockwise poly-
4 9 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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