EF 202, Module 4, Lecture 2

Second Moment of Area

EF 202 - Week 14

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EF 202, Module 4, Lecture 2

Misnomer•Most people do not use the name

“second area moment.”

•Instead, most people (including our text’s author) use “______________________.”

•Next week, we will study the real ____________________________.

•Confused yet?

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EF 202, Module 4, Lecture 2

Generalized Moment

•The second moment of “anything” about a point O is the product of two things:

•the square of the distance from O to the “anything” and

•the “anything” itself.

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EF 202, Module 4, Lecture 2

Definitions

€

Ix = y 2 + z2( )dA

A

∫

Second moment of area A about the x axis:

Second moment of area A about the y axis:

Second moment of area A about the z axis

€

Iy = x 2 + z2( )dA

A

∫

€

Iz = x 2 + y 2( )dA

A

∫

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EF 202, Module 4, Lecture 2

RestrictionIn this class, all the areas we will consider are in the xy plane (z ). This simplifies two of the previous definitions to

€

Ix = y 2dAA

∫

Iy = x 2dAA

∫

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EF 202, Module 4, Lecture 2

Polar Second Moment

•The second moment Iz is sometimes called the “polar” second moment or “polar” moment of inertia.

•Given the restriction on the previous slide,

• Our text uses the notation

€

Iz = Ix + Iy

€

JO = Iz

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EF 202, Module 4, Lecture 2

Transfer Theorem - 1

•We can “transfer” the second moment (moment of inertia) of an area from one axis to another, provided that the two axes are parallel.

•In other words, if we know the second moment about one axis, we can compute it about any other axis parallel to the first axis.

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EF 202, Module 4, Lecture 2

Transfer Theorem - 2

•If the second moment of an area A about an axis x’ through the centroid is ICx’, and the distance from the x’ axis to the (parallel) axis x is dy, then the second moment of the area about the x axis is

€

Ix = ICx' + Ady2

transfer term{

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EF 202, Module 4, Lecture 2

Transfer Theorem - 3

•The second moment to which the transfer term is added is always the one for an axis through the centroid.

•The second moment about an axis through the centroid is smaller than the second moment about any other parallel axis.

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EF 202, Module 4, Lecture 2

Transfer Theorem - 4

•We can transfer from any axis to a parallel axis through the centroid by subtracting the transfer term.

€

ICx' = Ix − Ady2

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EF 202, Module 4, Lecture 2

Radius of Gyration•By definition, the radius of gyration

of an area A about the x axis is

•Given the area and the radius of gyration,

€

kx =Ix

A

€

Ix = Akx2

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EF 202, Module 4, Lecture 2

Composite Areas

•Since the second moment is an integral, and since the integral over a sum of several areas equals the sum of the integrals over the individual areas, we can find the second moment of a composite area by adding the second moments of its parts.

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EF 202, Module 4, Lecture 2

For the triangular area shown, find the following.

Ix, Iy, kx, ky, JO, ICx’, ICy’

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EF 202, Module 4, Lecture 2

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EF 202, Module 4, Lecture 2

For the T-shaped cross section shown, find the second moments about the x’ and y’ axes.(Problem 10-40)

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EF 202, Module 4, Lecture 2