Chapter 7-2.pdf. .

Chapter 7: Moment of inertia
- Introduction about moment of inertia
- Transfer of axes (parallel axes theorem)
- Radius of Gyration
- Polar moment of inertia
- Moment of inertia of composite areas
- Moment of inertia of curved areas

In physics and engineering mechanics, moment is the product
of a quantity and the distance from that quantity to a given
point or axis. For example, in Statics, a force acting on a wrench
handle produces a torque, or moment, about the axis of a bolt:
M=P×L . This is the moment of a force.
7.1 Introduction about moment of Inertia
We can also describe moments of areas. Consider a beam with a
rectangular cross-section. The horizontal centroidal axis of this
beam is the x-x axis in the drawing. Take a small area a within
the cross-section at a distance y from the x-x centroidal axis of
the beam. The first moment of this area is a x y

The second moment of this area is I x=(a×y)× y= . In Strength of
Materials, “second moment of area” is usually abbreviated “moment of
inertia”. If we divide the total area into many little areas, then the
moment of inertia of the entire cross-section is the sum of the
moments of inertia of all of the little areas. We can also calculate
the moment of inertia about the vertical y-y centroidal axis: Iy=(a×x)
×x= .The x and y in Ix and Iy refer to the centroidal axis.

We can estimate the moment of inertia for the entire area as the
sum of the moments of inertia of the segments, written as
where n = the total number of segments
As the segment size drops, the estimates converge on a
solution. Split the beam into an infinite number of
infinitely-small segments to get the actual solution, derived
from calculus:
where b is the width and h is the depth.

The moment of inertia of an area a non-centroidal axis may be
easily expressed in terms of the moment of inertia about a parallel
centroidal axis.
For that purpose, the following formula can be use.
7.2 Transfer of axes (parallel axes theorem)

Columns are tall, thin structures loaded in compression which fail
at stresses below the expected yield strength of the material.
Column analysis uses the radius of gyration to calculate failure
loads. The radius of gyration is defined as
The greater the radius of gyration, the more resistant the column is to
buckling failure.
7.3 Radius of Gyration

7.4 Polar Moment of Inertia
The polar moment of inertia of an area A with respect to the pole
O is defined as
The distance from O to the
element of area dA is r.
Observing that
we established the relation

7.5 Moment of inertia of composite areas

Example 7.1:
A 5 cm*16 cm rectangular plank is glued to an 8cm×3cm as
shown, so they share the same neutral x-x axis. What is the
moment of inertia about the centroidal axis?
Solution
7.5 Moment of inertia of composite areas

Example 7.3:
A standard 2" steel pipe has an outside diameter do=2.375 in. and
an inside diameter di=2.067 in. What is the moment of inertia about
the x-x centroidal axis? Report the answer in in4
.
Solution: For solid circle
Subtract the moment of inertia of the outside dimensions from the
moment of inertia of the hollow space.

Example 7.4:
Find the centroidal moment of inertia for a T-shaped area. 1) First,
locate the centroid of each rectangular area relative to a common
base axis, then... 2) determine the location of the centroid of the
composite. 3) Find centroidal moment of inertia about the x-axis.
4) Find the centroidal moment of inertia about the y-axis.
5) Determine the polar moment of inertia.

The polar moment of inertia J= Ixc+Iyc=115.4 +14.6=130

Determination of the Moment of Inertia of an Area by
Integration

Example 7.7: Determine the moments of inertia of the
crosshatched area about the x and y axes.

Chapter 7-2.pdf. .

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