Engineering Mechanics: Statics Chapter 10: Moments of Inertia Engineering Mechanics: Statics

Chapter 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 Outline Definitions of Moments of Inertia for Areas Parallel-Axis Theorem for an Area Radius of Gyration of an Area Moments of Inertia for an Area by Integration Moments of Inertia for Composite Areas Product of Inertia for an Area

Chapter Outline Moments of Inertia for an Area about Inclined Axes Mohr’s Circle for Moments of Inertia Mass Moment of Inertia

10.1 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 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 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

10.1 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

10.1 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,

10.1 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 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 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

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)

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 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 axis

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 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

10.4 Moments of Inertia for an Area by Integration Procedure for Analysis Case 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 Procedure for Analysis Case 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 Example 10.1 Determine 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.

10.4 Moments of Inertia for an Area by Integration Solution Part (a) Differential element chosen, distance y’ from x’ axis Since dA = b dy’

10.4 Moments of Inertia for an Area by Integration Solution Part (b) Moment of inertia about an axis passing through the base of the rectangle obtained by applying parallel axis theorem

10.4 Moments of Inertia for an Area by Integration Solution Part (c) For polar moment of inertia about point C

10.4 Moments of Inertia for an Area by Integration Example 10.2 Determine the moment of inertia of the shaded area about the x axis

10.4 Moments of Inertia for an Area by Integration 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

10.4 Moments of Inertia for an Area by Integration Solution

10.4 Moments of Inertia for an Area by Integration 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

10.4 Moments of Inertia for an Area by Integration 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,

10.4 Moments of Inertia for an Area by Integration Solution For centroid of the element from the x axis Moment of inertia of the element Integrating

10.4 Moments of Inertia for an Area by Integration Example 10.3 Determine the moment of inertia with respect to the x axis of the circular area.

10.4 Moments of Inertia for an Area by Integration Solution Case 1 Since dA = 2x dy

10.4 Moments of Inertia for an Area by Integration Solution Case 2 Centroid for the element lies on the x axis Noting dy = 0 For a rectangle,

10.4 Moments of Inertia for an Area by Integration Solution Integrating with respect to x

10.4 Moments of Inertia for an Area by Integration Example 10.4 Determine the moment of inertia of the shaded area about the x axis.

10.4 Moments of Inertia for an Area by Integration View Free Body Diagram Solution Case 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

10.4 Moments of Inertia for an Area by Integration Solution Case 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 Solution Integrating

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 Procedure for Analysis Composite 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 Procedure for Analysis Parallel 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 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

10.5 Moments of Inertia for Composite Areas Example 10.5 Compute the moment of inertia of the composite area about the x axis.

10.5 Moments of Inertia for Composite Areas Solution Composite 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 Solution Parallel Axis Theorem Circle Rectangle

10.5 Moments of Inertia for Composite Areas Solution Summation For moment of inertia for the composite area,

10.5 Moments of Inertia for Composite Areas Example 10.6 Determine 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 View Free Body Diagram Solution Composite Parts Considered as 3 composite areas A, B, and D Centroid of each area is located in the figure

10.5 Moments of Inertia for Composite Areas Solution Parallel Axis Theorem Rectangle A

10.5 Moments of Inertia for Composite Areas Solution Parallel Axis Theorem Rectangle B

10.5 Moments of Inertia for Composite Areas Solution Parallel Axis Theorem Rectangle D

10.5 Moments of Inertia for Composite Areas Solution Summation For moment of inertia for the entire cross-sectional area,