Post on 12-Jan-2016
Engineering Mechanics: StaticsEngineering Mechanics: Statics
Chapter 10: Moments of
Inertia
Chapter 10: Moments of
Inertia
Chapter ObjectivesChapter Objectives
To develop a method for determining the moment of inertia for an area.
To introduce the product of inertia and show how to determine the maximum and minimum moments of inertia for an area.
To discuss the mass moment of inertia.
Chapter OutlineChapter Outline
Definitions of Moments of Inertia for Areas
Parallel-Axis Theorem for an AreaRadius of Gyration of an AreaMoments of Inertia for an Area by
IntegrationMoments of Inertia for Composite
AreasProduct of Inertia for an Area
Chapter OutlineChapter Outline
Moments of Inertia for an Area about Inclined Axes
Mohr’s Circle for Moments of InertiaMass Moment of Inertia
10.1 Moments of Inertia10.1 Moments of Inertia
Definition of Moments of Inertia for Areas Centroid for an area is determined by
the first moment of an area about an axis
Second moment of an area is referred as the moment of inertia
Moment of inertia of an area originates whenever one relates the normal stress σ or force per unit area, acting on the transverse cross-section of an elastic beam, to applied external moment M, that causes bending of the beam
10.1 Moments of Inertia10.1 Moments of Inertia
Definition of Moments of Inertia for Areas Stress within the beam varies linearly
with the distance from an axis passing through the centroid C of the beam’s cross-sectional area
σ = kz For magnitude of the force acting
on the area element dA dF = σ dA = kz dA
10.1 Moments of Inertia10.1 Moments of Inertia
Definition of Moments of Inertia for Areas Since this force is located a distance z from
the y axis, the moment of dF about the y axis
dM = dF = kz2 dA Resulting moment of the entire stress
distribution = applied moment M
Integral represent the moment of inertia of area about the y axis
dAzM 2
10.1 Moments of Inertia10.1 Moments of Inertia
Moment of Inertia Consider area A lying in the x-y plane Be definition, moments of inertia of the
differential plane area dA about the x and y axes
For entire area, moments of inertia are given by
Ay
Ax
yx
dAxI
dAyI
dAxdIdAydI
2
2
22
10.1 Moments of Inertia10.1 Moments of Inertia
Moment of Inertia Formulate the second moment of dA
about the pole O or z axis This is known as the polar axis
where r is perpendicular from the pole (z axis) to the element dA
Polar moment of inertia for entire area,
yxAO
O
IIdArJ
dArdJ
2
2
10.1 Moments of Inertia10.1 Moments of Inertia
Moment of Inertia Relationship between JO, Ix and Iy is
possible since r2 = x2 + y2
JO, Ix and Iy will always be positive since they involve the product of the distance squared and area
Units of inertia involve length raised to the fourth power eg m4, mm4
10.2 Parallel Axis Theorem for an Area
10.2 Parallel Axis Theorem for an Area
For moment of inertia of an area known about an axis passing through its centroid, determine the moment of inertia of area about a corresponding parallel axis using the parallel axis theorem
Consider moment of inertia of the shaded area
A differential element dA is located at an arbitrary distance y’ from the centroidal x’ axis
10.2 Parallel Axis Theorem for an Area
10.2 Parallel Axis Theorem for an Area
The fixed distance between the parallel x and x’ axes is defined as dy
For moment of inertia of dA about x axis
For entire area
First integral represent the moment of inertia of the area about the centroidal axis
AyAyA
A yx
yx
dAddAyddAy
dAdyI
dAdydI
22
2
2
'2'
'
'
10.2 Parallel Axis Theorem for an Area
10.2 Parallel Axis Theorem for an Area
Second integral = 0 since x’ passes through the area’s centroid C
Third integral represents the total area A
Similarly
For polar moment of inertia about an axis perpendicular to the x-y plane and passing through pole O (z axis)
2
2
2
0;0'
AdJJ
AdII
AdII
ydAydAy
CO
xyy
yxx
10.2 Parallel Axis Theorem for an Area
10.2 Parallel Axis Theorem for an Area
Moment of inertia of an area about an axis = moment of inertia about a parallel axis passing through the area’s centroid plus the product of the area and the square of the perpendicular distance between the axes
10.3 Radius of Gyration of an Area
10.3 Radius of Gyration of an Area
Radius of gyration of a planar area has units of length and is a quantity used in the design of columns in structural mechanics
Provided moments of inertia are known For radii of gyration
Similar to finding moment of inertia of a differential area about an axisdAydIAkI
AJ
kA
Ik
AI
k
xxx
Oz
yy
xx
22
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration When the boundaries for a planar area
are expressed by mathematical functions, moments of inertia for the area can be determined by the previous method
If the element chosen for integration has a differential size in two directions, a double integration must be performed to evaluate the moment of inertia
Try to choose an element having a differential size or thickness in only one direction for easy integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationProcedure for Analysis If a single integration is performed to
determine the moment of inertia of an area bout an axis, it is necessary to specify differential element dA
This element will be rectangular with a finite length and differential width
Element is located so that it intersects the boundary of the area at arbitrary point (x, y)
2 ways to orientate the element with respect to the axis about which the axis of moment of inertia is determined
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationProcedure for AnalysisCase 1 Length of element orientated parallel to the
axis Occurs when the rectangular element is
used to determine Iy for the area Direct application made since the element
has infinitesimal thickness dx and therefore all parts of element lie at the same moment arm distance x from the y axis
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationProcedure for AnalysisCase 2 Length of element orientated
perpendicular to the axis All parts of the element will not lie at the
same moment arm distance from the axis For Ix of area, first calculate moment of
inertia of element about a horizontal axis passing through the element’s centroid and x axis using the parallel axis theorem
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationExample 10.1Determine the moment of inertia for the rectangular area with respect to (a) the centroidal x’ axis, (b) the axis xb passing through the base of the rectangular, and (c) the pole or z’ axis perpendicular to the x’-y’ plane and passing through the centroid C.
SolutionPart (a) Differential element chosen, distance y’
from x’ axis Since dA = b dy’
3
2/
2/
22
12
1
''
bh
dyydAyIh
hAx
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
SolutionPart (b) Moment of inertia about an axis
passing through the base of the rectangle obtained by applying parallel axis theorem
32
3
2
3
1
212
1bh
hbhbh
AdII xxb
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
SolutionPart (c) For polar moment of inertia about point
C
)(12
1
12
1
22
'
3'
bhbh
IIJ
hbI
yxC
y
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationExample 10.2Determine the moment
of inertia of the shaded
area about the x axis
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationSolution A differential element of area that is
parallel to the x axis is chosen for integration
Since element has thickness dy and intersects the curve at arbitrary point (x, y), the area
dA = (100 – x)dy All parts of the element lie at the same
distance y from the x axis
Solution
46
200
0
200
0
42
2200
0
2
2
2
)10(107
400
1100
400100
)100(
mm
dyydyy
dyy
y
dyxy
dAyI
A
Ax
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationSolution A differential element
parallel to the y axis is chosen for integration
Intersects the curve at arbitrary point (x, y)
All parts of the element do not lie at the same distance from the x axis
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationSolution Parallel axis theorem used to determine
moment of inertia of the element For moment of inertia about its centroidal
axis,
For the differential element shown
Thus, 3
3
121
121
dxyId
yhbxb
bhI
x
x
Solution For centroid of the element from the x axis
Moment of inertia of the element
Integrating
46
100
0
2/33
32
32
10107
4003
1
3
1
3
1
212
1~
2/~
mm
dxxdxydII
dxyy
ydxdxyydAIddI
yy
Axx
xx
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationExample 10.3Determine the moment of inertia with
respect to the x axis of the circular area.
SolutionCase 1 Since dA = 2x dy
4
2
)2(
4
222
2
2
a
dyyay
dyxy
dAyI
a
a
A
Ax
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
SolutionCase 2 Centroid for the element
lies on the x axis Noting
dy = 0 For a rectangle,
3' 12
1bhI x
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
Solution
Integrating with respect to x
4
3
2
3
2
212
1
4
2/322
3
3
a
dxxaI
dxy
ydxdI
a
ax
x
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationExample 10.4Determine the moment of inertia of
the shaded area about the x axis.
SolutionCase 1 Differential element parallel to x axis
chosen Intersects the curve at (x2,y) and (x1, y)
Area, dA = (x1 – x2)dy All elements lie at the same distance y
from the x axis 41
0
42/7
1
0
21
0 2122
0357.04
1
7
2myyI
dyyyydyxxydAyI
x
Ax
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationView Free Body Diagram
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationSolutionCase 2 Differential element
parallel to y axis chosen Intersects the curve at (x,
y2) and (x, y1) All elements do not lie at
the same distance from the x axis
Use parallel axis theorem to find moment of inertia about the x axis
10.4 Moments of Inertia for an Area by
Integration
10.4 Moments of Inertia for an Area by
IntegrationSolution
Integrating
41
0
74
1
0
63
6331
32
2
12112
312
2
3'
0357.021
1
12
13
13
1
3
1
212
1~
12
1
mxx
dxxxI
dxxxdxyy
yyydxyyyydxydAIddI
bhI
x
xx
x
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
A composite area consist of a series of connected simpler parts or shapes such as semicircles, rectangles and triangles
Provided the moment of inertia of each of these parts is known or can be determined about a common axis, moment of inertia of the composite area = algebraic sum of the moments of inertia of all its parts
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
Procedure for AnalysisComposite Parts Using a sketch, divide the area into its
composite parts and indicate the perpendicular distance from the centroid of each part to the reference axis
Parallel Axis Theorem Moment of inertia of each part is
determined about its centroidal axis, which is parallel to the reference axis
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
Procedure for AnalysisParallel Axis Theorem If the centroidal axis does not coincide with
the reference axis, the parallel axis theorem is used to determine the moment of inertia of the part about the reference axis
Summation Moment of inertia of the entire area about
the reference axis is determined by summing the results of its composite parts
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
Procedure for AnalysisSummation If the composite part has a hole, its
moment of inertia is found by subtracting the moment of inertia of the hole from the moment of inertia of the entire part including the hole
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
Example 10.5Compute the moment of inertia of the composite area about the x axis.
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionComposite Parts Composite area
obtained by subtracting the circle form the rectangle
Centroid of each area is located in the figure
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionParallel Axis
Theorem Circle
Rectangle
4623
2'
46224
2'
105.1127515010015010012
1
104.117525254
1
mm
AdII
mm
AdII
yxx
yxx
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionSummation For moment of inertia for the composite
area,
46
66
10101
105.112104.11
mm
I x
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
Example 10.6Determine the moments of inertia of the beam’s cross-sectional area about the x and y centroidal axes.
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionComposite PartsConsidered as 3
composite areas A, B, and D
Centroid of each area is located in the figure
View Free Body Diagram
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionParallel Axis
Theorem Rectangle A
4923
2'
4923
2'
1090.125030010010030012
1
10425.120030010030010012
1
mm
AdII
mm
AdII
xyy
yxx
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionParallel Axis
Theorem Rectangle B
493
2'
493
2'
1080.160010012
1
1005.010060012
1
mm
AdII
mm
AdII
xyy
yxx
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionParallel Axis
Theorem Rectangle D
4923
2'
4923
2'
1090.125030010010030012
1
10425.120030010030010012
1
mm
AdII
mm
AdII
xyy
yxx
10.5 Moments of Inertia for Composite Areas
10.5 Moments of Inertia for Composite Areas
SolutionSummation For moment of inertia for the entire
cross-sectional area,
49
999
49
999
1060.5
1090.11080.11090.1
1090.2
10425.11005.010425.1
mm
I
mm
I
y
x