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Second moment of area

The second moment of area, also known as the second moment of inertia and the area moment of inertia, is a property of a shape that is used to predict its resistance to bending.

It is derived with use of the parallel axis theorem. The second moment of area is not the same as thing as the moment of inertia, which is used to calculate the angular acceleration.

Contents

1 Definition

2 Unit

3 Second moment of area - common shapes

4 Parallel axis theorem equation

5 Stress in a beam

6 See also

7 External links

Definition

$I_x = \int y^2 dA$

I_x - the second 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 - common shapes

Rectangle: $I_{xx}=\frac{bd^3}{12}$ , b = width (x-dimension), d = height (y-dimension)

Parallel axis theorem equation

$I_\mathrm{ZZ} = \sum {I_\mathrm{CG}+md^2}$

I_CG - *I_ZZ - the second moment of area for axis one is moving a shape to
m - area of the moved shape
d - the distance between the new axis and the axis of the shape

This equation is useful for calculating the second moment of area for I-beams.

Stress in a beam

The classic bending formula for a beam is:

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

$σ$ 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

When one is learning about the second moment of inertia the above formula is where one usually sees it first.

External links

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10-26-2009 08:16:03
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