Second moment of area

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.

The second moment of area is not the same thing as the moment of inertia that is used to calculate angular acceleration, although many engineers refer to the second moment of area as the moment of inertia and use the same symbol $I$ for both.

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

Unit

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

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 second moments of area for other shapes.

Second moment of area - Composite cross section

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

I_{xx}= \Sigma\ y^{2}A +I_{local}

The above can only be used on its own, when sections are symetrical about the X axis When this is not the case, The second moment of area about the XX,YY and XY axis' are required.

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

$I_{xy}= \Sigma\ yxA$

y -distance from X axis
x -distance from Y axis
A -Surface area of part

Note that

I_{local}

is the second moment of area for that part of the composite.

Parallel axis rule

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

I_z - the second moment of area with respect to axis z
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

Stress in a beam

The classic bending formula for a beam is:

{\sigma}= \frac{M y}{I_x}

${\sigma}$ is the bending stress
M - the moment at the neutral axis
y - the perpendicular distance to the neutral axis
I_x - the second moment of inertia about the neutral axis x

External links

Illustrations and examples (de.wikipedia.org)

Continuum mechanics | Structural engineering