Area Moments of Inertia
TECH 3401
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Introduction
The strength of a member, in terms of bending and buckling, depends on that member’s shape & orientation.
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Area Moment of Inertia
Property of a 2D plane shape.
Area moment is not observable, but you have seen the mass moment of inertia before.
Note: the Mass moment of inertia (what is usually just called “moment of inertia” becomes the area moment as the part becomes very thin.)
The resistance to rotating
of the thin 2D section is
analogous to bending resistance
Also called
“Second Moment of Area”
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IMPORTANT
The moment of inertia is computed with respect to a particular axis.
Usually, we will be using an axis through the centroid.
Notice units go to the FOURTH power. Always positive.
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See Properties of Areas
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Moments of Structural Shapes
(reference only, not for design)
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Example
Calculate the moment of inertia with respect to x-x centroidal axis for the area shown in the figure below.
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Combined Moments - Easy Case
If the the areas share the same centroidal axis, you can simply add the moments. What is the moment of inertia of the x-x axis of this shape (the circles are voids).
Subtract the Ix of the
voids from Ix of the
solid.
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Transfer Formula
The transfer formula is the general case solution for adding moments of inertia. But...moment axes must be parallel.
(AKA “Parallel Axis Theorem”)
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Moment of Inertia of Composite Areas
Do the transfer formula on each area.
It’s this simple if the centroidal axis is known
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A More Realistic Example
Location of Centroid is not known. Need to find it first.
Need to know:
Formula for I of a rectangle.
How to find location of centroid.
Transfer formula.
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Radius of Gyration (for strengths of materials, not for dynamics)
Useful for buckling resistance.
The distance from a reference axis at which the entire area (in the shape of a line) may assume to be located without changing its moment of inertia.
Used to replace the relationship between the moment of area and the cross-sectional area.
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Understanding what r is...
...is harder than calculating it.
Each area (and the red line) have the same moment of inertia with respect to the X-X axis.
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Example
Calculate the radius of gyration with respect to the X-X centroidal axis of the figure below.
Units are inches
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Polar
Moment
The moment of inertia of an area with respect to a perpendicular axis.
By Ansgar Walk - Own work, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=1552224
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Polar Moment of Inertia
This is related to the resistance to twisting along the z axis.
Just as with the other moments, can be seen by analogy as resistance to spinning.
In this case around z axis (but this is the mass, we are looking at area).
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Polar Moment of Inertia
This is the same
(pythagorean thm).
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Example
Calculate the (centroidal) polar moment of inertia for a hollow circular shaft with an outside diameter of 4.0 inches and a wall thickness of .5 inches.
You will be able to look up the formulas in your book.
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NX Example
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