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Copyright © 2010 Pearson Education South Asia Pte Ltd Chapter Objectives Method for determining the moment of inertia for an area Introduce product of inertia and show determine the maximum and minimum moments of inertia for an area Discuss the mass moment of inertia Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd 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 Composite Areas Product of Inertia for an Area Moments of Inertia for an Area about Inclined Axes Mohr’s Circle for Moments of Inertia Mass Moment of Inertia Copyright © 2010 Pearson Education South Asia Pte Ltd

10.1 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 Copyright © 2010 Pearson Education South Asia Pte Ltd

10.1 Definition of Moments of Inertia for Areas 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 Copyright © 2010 Pearson Education South Asia Pte Ltd

10.1 Definition of Moments of Inertia for Areas 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, Copyright © 2010 Pearson Education South Asia Pte Ltd

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 Copyright © 2010 Pearson Education South Asia Pte Ltd

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 Copyright © 2010 Pearson Education South Asia Pte Ltd

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) Copyright © 2010 Pearson Education South Asia Pte Ltd

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 For radii of gyration Similar to finding moment of inertia of a differential area about an axis Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd 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. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Part (a) Differential element chosen, distance y’ from x’ axis. Since dA = b dy’, Part (b) By applying parallel axis theorem, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Part (c) For polar moment of inertia about point C, Copyright © 2010 Pearson Education South Asia Pte Ltd

10.4 Moments of Inertia for Composite Areas Composite area consist of a series of connected simpler parts or shapes Moment of inertia of the composite area = algebraic sum of the moments of inertia of all its parts Procedure for Analysis Composite Parts Divide area into its composite parts and indicate the centroid of each part to the reference axis Parallel Axis Theorem Moment of inertia of each part is determined about its centroidal axis Copyright © 2010 Pearson Education South Asia Pte Ltd

10.4 Moments of Inertia for Composite Areas Procedure for Analysis Parallel Axis Theorem When centroidal axis does not coincide with the reference axis, the parallel axis theorem is used Summation Moment of inertia of the entire area about the reference axis is determined by summing the results of its composite parts Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Example 10.4 Compute the moment of inertia of the composite area about the x axis. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Composite Parts Composite area obtained by subtracting the circle form the rectangle. Centroid of each area is located in the figure below. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Parallel Axis Theorem Circle Rectangle Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Summation For moment of inertia for the composite area, Copyright © 2010 Pearson Education South Asia Pte Ltd

10.5 Product of Inertia for an Area Moment of inertia for an area is different for every axis about which it is computed First, compute the product of the inertia for the area as well as its moments of inertia for given x, y axes Product of inertia for an element of area dA located at a point (x, y) is defined as dIxy = xydA Thus for product of inertia, Copyright © 2010 Pearson Education South Asia Pte Ltd

10.5 Product of Inertia for an Area Parallel Axis Theorem For the product of inertia of dA with respect to the x and y axes For the entire area, Forth integral represent the total area A, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Example 10.6 Determine the product Ixy of the triangle. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Differential element has thickness dx and area dA = y dx Using parallel axis theorem, locates centroid of the element or origin of x’, y’ axes Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Due to symmetry, Integrating we have Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Differential element has thickness dy and area dA = (b - x) dy. For centroid, For product of inertia of element Copyright © 2010 Pearson Education South Asia Pte Ltd

10.6 Moments of Inertia for an Area about Inclined Axes In structural and mechanical design, necessary to calculate Iu, Iv and Iuv for an area with respect to a set of inclined u and v axes when the values of θ, Ix, Iy and Ixy are known Use transformation equations which relate the x, y and u, v coordinates Copyright © 2010 Pearson Education South Asia Pte Ltd

10.6 Moments of Inertia for an Area about Inclined Axes Integrating, Simplifying using trigonometric identities, Copyright © 2010 Pearson Education South Asia Pte Ltd

10.6 Moments of Inertia for an Area about Inclined Axes We can simplify to Polar moment of inertia about the z axis passing through point O is, Copyright © 2010 Pearson Education South Asia Pte Ltd

10.6 Moments of Inertia for an Area about Inclined Axes Principal Moments of Inertia Iu, Iv and Iuv depend on the angle of inclination θ of the u, v axes The angle θ = θp defines the orientation of the principal axes for the area Copyright © 2010 Pearson Education South Asia Pte Ltd

10.6 Moments of Inertia for an Area about Inclined Axes Principal Moments of Inertia Substituting each of the sine and cosine ratios, we have Result can gives the max or min moment of inertia for the area It can be shown that Iuv = 0, that is, the product of inertia with respect to the principal axes is zero Any symmetric axis represent a principal axis of inertia for the area Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Example 10.8 Determine the principal moments of inertia for the beam’s cross-sectional area with respect to an axis passing through the centroid. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Moment and product of inertia of the cross-sectional area, Using the angles of inclination of principal axes u and v, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution For principal of inertia with respect to the u and v axes Copyright © 2010 Pearson Education South Asia Pte Ltd

10.7 Mohr’s Circle for Moments of Inertia It is found that In a given problem, Iu and Iv are variables and Ix, Iy and Ixy are known constants When this equation is plotted on a set of axes that represent the respective moment of inertia and the product of inertia, the resulting graph represents a circle Copyright © 2010 Pearson Education South Asia Pte Ltd

10.7 Mohr’s Circle for Moments of Inertia The circle constructed is known as a Mohr’s circle with radius and center at (a, 0) where Copyright © 2010 Pearson Education South Asia Pte Ltd

10.7 Mohr’s Circle for Moments of Inertia Procedure for Analysis Determine Ix, Iy and Ixy Establish the x, y axes for the area, with the origin located at point P of interest and determine Ix, Iy and Ixy Construct the Circle Construct a rectangular coordinate system such that the abscissa represents the moment of inertia I and the ordinate represent the product of inertia Ixy Copyright © 2010 Pearson Education South Asia Pte Ltd

10.7 Mohr’s Circle for Moments of Inertia Construct the Circle Determine center of the circle O, which is located at a distance (Ix + Iy)/2 from the origin, and plot the reference point a having coordinates (Ix, Ixy) By definition, Ix is always positive, whereas Ixy will either be positive or negative Connect the reference point A with the center of the circle and determine distance OA (radius of the circle) by trigonometry Draw the circle Copyright © 2010 Pearson Education South Asia Pte Ltd

10.7 Mohr’s Circle for Moments of Inertia Principal of Moments of Inertia Points where the circle intersects the abscissa give the values of the principle moments of inertia Imin and Imax Product of inertia will be zero at these points Principle Axes This angle represent twice the angle from the x axis to the area in question to the axis of maximum moment of inertia Imax The axis for the minimum moment of inertia Imin is perpendicular to the axis for Imax Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Example 10.9 Using Mohr’s circle, determine the principle moments of the beam’s cross-sectional area with respect to an axis passing through the centroid. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Determine Ix, Iy and Ixy Moments of inertia and the product of inertia have been determined in previous examples Construct the Circle Center of circle, O, lies from the origin, at a distance Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Construct the Circle With reference point A (2.90, -3.00) connected to point O, radius OA is determined using Pythagorean theorem Principal Moments of Inertia Circle intersects I axis at points (7.54, 0) and (0.960, 0) Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Principal Axes Angle 2θp1 is determined from the circle by measuring CCW from OA to the direction of the positive I axis The principal axis for Imax = 7.54(109) mm4 is therefore orientated at an angle θp1 = 57.1°, measured CCW from the positive x axisto the positive u axis. v axis is perpendicular to this axis. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd 10.8 Mass Moment of Inertia Mass moment of inertia is defined as the integral of the second moment about an axis of all the elements of mass dm which compose the body For body’s moment of inertia about the z axis, The axis that is generally chosen for analysis, passes through the body’s mass center G Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd 10.8 Mass Moment of Inertia If the body consists of material having a variable density ρ = ρ(x, y, z), the element mass dm of the body may be expressed as dm = ρ dV Using volume element for integration, When ρ being a constant, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd 10.8 Mass Moment of Inertia Procedure for Analysis Shell Element For a shell element having height z, radius y and thickness dy, volume dV = (2πy)(z)dy Disk Element For disk element having radius y, thickness dz, volume dV = (πy2) dz Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Example 10.10 Determine the mass moment of inertia of the cylinder about the z axis. The density of the material is constant. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Shell Element For volume of the element, For mass, Since the entire element lies at the same distance r from the z axis, for the moment of inertia of the element, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Integrating over entire region of the cylinder, For the mass of the cylinder So that Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd 10.8 Mass Moment of Inertia Parallel Axis Theorem If the moment of inertia of the body about an axis passing through the body’s mass center is known, the moment of inertia about any other parallel axis may be determined by using parallel axis theorem Using Pythagorean theorem, r 2 = (d + x’)2 + y’2 For moment of inertia of body about the z axis, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd 10.8 Mass Moment of Inertia Parallel Axis Theorem For moment of inertia about the z axis, I = IG + md2 Radius of Gyration For moment of inertia expressed using k, radius of gyration, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Example 10.12 If the plate has a density of 8000kg/m3 and a thickness of 10mm, determine its mass moment of inertia about an axis perpendicular to the page and passing through point O. Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution The plate consists of 2 composite parts, the 250mm radius disk minus the 125mm radius disk. Disk For moment of inertia of a disk, Mass center of the disk is located 0.25m from point O Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd Solution Hole For moment of inertia of plate about point O, Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 1. The definition of the Moment of Inertia for an area involves an integral of the form A)  x dA. B)  x2 dA. C)  x2 dm. D)  m dA. 2. Select the SI units for the Moment of Inertia for an area. A) m3 B) m4 C) kg·m2 D) kg·m3 Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 3. A pipe is subjected to a bending moment as shown. Which property of the pipe will result in lower stress (assuming a constant cross-sectional area)? A) Smaller Ix B) Smaller Iy C) Larger Ix D) Larger Iy 4. In the figure to the right, what is the differential moment of inertia of the element with respect to the y-axis (dIy)? A) x2 ydx B) (1/12)x3dy C) y2 x dy D) (1/3)ydy M y Pipe section x x y=x3 x,y y Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 5. The parallel-axis theorem for an area is applied between A) An axis passing through its centroid and any corresponding parallel axis. B) Any two parallel axis. C) Two horizontal axes only. D) Two vertical axes only. 6. The moment of inertia of a composite area equals the ____ of the MoI of all of its parts. A) Vector sum B) Algebraic sum (addition or subtraction) C) Addition D) Product Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 7. For the area A, we know the centroid’s (C) location, area, distances between the four parallel axes, and the MoI about axis 1. We can determine the MoI about axis 2 by applying the parallel axis theorem ___ . A) Directly between the axes 1 and 2. B) Between axes 1 and 3 and then between the axes 3 and 2. C) Between axes 1 and 4 and then axes 4 and 2. D) None of the above. d3 d2 d1 4 3 2 1 A • C Axis Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 8. For the same case, consider the MoI about each of the four axes. About which axis will the MoI be the smallest number? A) Axis 1 B) Axis 2 C) Axis 3 D) Axis 4 E) Can not tell. d3 d2 d1 4 3 2 1 A • C Axis Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ A=10 cm2 • C d2 d1 3 2 1 d1 = d2 = 2 cm 9. For the given area, the moment of inertia about axis 1 is 200 cm4 . What is the MoI about axis 3? A) 90 cm4 B) 110 cm4 C) 60 cm4 D) 40 cm4 10. The moment of inertia of the rectangle about the x-axis equals A) 8 cm4. B) 56 cm4 . C) 24 cm4 . D) 26 cm4 . 2cm 3cm x Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 11. The formula definition of the mass moment of inertia about an axis is ___________ . A)  r dm B)  r2 dm C)  m dr D)  m dr 12. The parallel-axis theorem can be applied to determine ________ . A) Only the MoI B) Only the MMI C) Both the MoI and MMI D) None of the above. Note: MoI is the moment of inertia of an area and MMI is the mass moment inertia of a body Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 13. Consider a particle of mass 1 kg located at point P, whose coordinates are given in meters. Determine the MMIof that particle about the z axis. A) 9 kg·m2 B) 16 kg·m2 C) 25 kg·m2 D) 36 kg·m2 14. Consider a rectangular frame made of 4 slender bars with four axes perpendicular to the screen and passing through P, Q, R, and S respectively. About which of the four axes will the MMI of the frame be the largest? A) zP B) zQ C) zR D) zS E) Not possible to determine z x y ·P(3,4,6) P • S • Q • R Copyright © 2010 Pearson Education South Asia Pte Ltd

Copyright © 2010 Pearson Education South Asia Pte Ltd QUIZ 15. A particle of mass 2 kg is located 1 m down the y-axis. What are the MMI of the particle about the x, y, and z axes, respectively? A) (2, 0, 2) B) (0, 2, 2) C) (0, 2, 2) D) (2, 2, 0) • 1 m x y z Copyright © 2010 Pearson Education South Asia Pte Ltd