ch10 - Chapter10 MomentsofInertia :Statics...

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Engineering Mechanics: Statics Chapter 10:  Moments of Inertia
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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.
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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
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Chapter Outline Moments of Inertia for an Area about  Inclined Axes Mohr’s Circle for Moments of Inertia Mass Moment of Inertia 
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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
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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
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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 d F  about the y axis dM = dF = kz 2  dA Resulting moment of the entire stress distribution  = applied moment  M Integral represent the moment of inertia of area  about the y axis = dA z M 2
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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 = = = = A y A x y x dA x I dA y I dA x dI dA y dI 2 2 2 2
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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,  y x A O O I I dA r J dA r dJ + = = = 2 2
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10.1 Moments of Inertia Moment of Inertia  Relationship between J O , I x  and I y  is  possible since r 2  = x 2  + y 2 J O , I x  and I y  will always be positive since 
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