Second moment of area

From Wikipedia, the free encyclopedia

This article is about the moment of inertia as related to the bending of a plane. For the moment of inertia dealing with rotation of an object, see Moment of inertia.

The second moment of area, also known as the area moment of inertia and less precisely as the moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. It is analogous to the polar moment of inertia, which characterizes an object's ability to resist torsion.

The second moment of area is not the same thing as the moment of inertia, which is used to calculate angular acceleration. Many engineers refer to the second moment of area as the moment of inertia and use the same symbol $I$ for both, which may be confusing. Which inertia is meant (accelerational or bending) is usually clear from the context and obvious from the units.

[edit] Definition

$I_x = \int y^2\, dA$

I_x = the moment of inertia about the axis x
dA = an elemental area
y = the perpendicular distance to the element dA from the axis x

[edit] Unit

The SI unit for second moment of area is metre to the fourth power (m⁴)

[edit] Second moment of area - rectangular cross section

Rectangle: $I_{x}=\frac{bh^3}{12}$

b = width (x-dimension),
h = height (y-dimension)

See List of area moments of inertia for other shapes.

[edit] Second moment of area - composite cross section

In composite cross sections the second moment of area is given as

$I_{xx}= \sum y^{2}A +I_\mathrm{local}$

The above can only be used on its own, when sections are symmetrical about the x-axis. When this is not the case, the second moment of area about the xx-, yy-, and xy-axes are required.

$I_{yy}= \sum x^{2}A +I_\mathrm{local}$

$I_{xy}= \sum yxA$

y = distance from x-axis
x = distance from y-axis
A = surface area of part

Note that $I l o c a l$ is the second moment of area for that part of the composite.

I x y

is also known as Product Moment of Area

[edit] Parallel axis theorem

$I_z = I_{CG}+Ad^2\,$

I_z = the second moment of area with respect to the z-axis
I_CG = the second moment of area with respect to an axis parallel to z and passing through the centroid of the shape (coincides with the neutral axis)
A = area of the shape
d = the distance between the z-axis and the centroidal axis