engineering mechanics: statics course outcomesportal.unimap.edu.my/portal/page/portal30/lecture...

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Engineering Mechanics: StaticsEngineering Mechanics: Statics

Lect04: Moments of Inertia

Lect04: Moments of Inertia

Dr. Wan Mohd Sabki Wan Omar

Structure & Construction Eng.

Email: [email protected]/

[email protected]

Phone: 013-9335477

Dr. Wan Mohd Sabki Wan Omar

Structure & Construction Eng.

Email: [email protected]/

[email protected]

Phone: 013-9335477

WMSWO

Course OutcomesCourse Outcomes

� Able to develop a method for determining the moment of inertia for an area.

� Able to introduce the product of inertia and determine the maximum and minimum moments of inertia for an area.

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OutlineOutline

� Definitions of Moments of Inertia for Areas

� Parallel-Axis Theorem for an Area

� Moments of Inertia for an Area by Integration

� Moments of Inertia for Composite Areas

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4.1 Moments of Inertia4.1 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

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� 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



� 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

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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


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,

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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

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4.2 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

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� 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



� 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)

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� 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

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4.4 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

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Procedure 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

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Procedure 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

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Procedure 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

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Example 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.

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Solution

Part (a)

� Differential element chosen, distance y’ from x’ axis

� Since dA = b dy’



Solution

Part (b)

� Moment of inertia about an axis passing through the base of the rectangle obtained by applying parallel axis theorem



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Solution

Part (c)

� For polar moment of inertia about point C



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Example 10.2

Determine the moment of

inertia of the shaded area

about the x axis

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Solution

� 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



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Solution

� 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



Solution

� 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,

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Solution

� For centroid of the element from the x axis

� Moment of inertia of the element

� Integrating



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4.5 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

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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

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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

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Procedure for Analysis

Summation

� 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

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Example 10.5

Compute the moment of

inertia of the composite

area about the x axis.

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Solution

Composite Parts

� Composite area obtained by subtracting the circle form the rectangle

� Centroid of each area is located in the figure



Solution

Parallel Axis Theorem

� Circle

� Rectangle

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Solution

Summation

� For moment of inertia for the composite area,

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Example 10.6

Determine the moments

of inertia of the beam’s

cross-sectional area

about the x and y

centroidal axes.

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Solution

Composite Parts

� Considered as 3 composite areas A, B, and D

� Centroid of each area is located in the figure



Solution


� Rectangle A

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Solution


� Rectangle B



Solution


� Rectangle D

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Solution

Summation

� For moment of inertia for the entire cross-sectional area,

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Lecture Summary Lecture Summary

Area Moment of Inertia� Represent second moment of area about an axis� Frequently used in equations related to strength

and stability of structural members or mechanical elements

� If the area shape is irregular, a differential element must be selected and integration over the entire area must be performed

� Tabular values of the moment of inertia of common shapes about their centroidal axis are available

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Area Moment of Inertia� To determine moment of inertia of these shapes

about some other axis, parallel axis theorem must be used

� If an area is a composite of these shapes, its moment of inertia = sum of the moments of inertia of each of its parts

Product of Inertia� Determine location of an axis about which the

moment of inertia for the area is a maximum or minimum

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Product of Inertia� If the product of inertia for an area is known

about its x’, y’ axes, then its value can be determined about any x, y axes using the parallel axis theorem for product of inertia

Principal Moments of Inertia� Provided moments of inertia are known, formulas

or Mohr’s circle can be used to determine the maximum or minimum or principal moments of inertia for the area, as well as orientation of the principal axes of inertia

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Mass Moments of Inertia

� Measures resistance to change in its rotation

� Second moment of the mass elements of the body about an axis

� For bodies having axial symmetry, determine using wither disk or shell elements

� Mass moment of inertia of a composite body is determined using tabular values of its composite shapes along with the parallel axis theorem

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Lecture ReviewLecture Review

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engineering mechanics: statics course outcomesportal.unimap.edu.my/portal/page/portal30/lecture...

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