Lecture Strength of Materials I: Chapter 5 - PhD. Tran Minh Tu

Chapter 5 - Geometric properties of an area. The following will be discussed in this chapter: First moment of area, moment of inertia for an area, moment of inertia for some simple areas, parallel - axis theorem.
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Lecture Strength of Materials I: Chapter 5 - PhD. Tran Minh Tu STRENGTH OF MATERIALS1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, VietnamCHAPTER 5 Geometric Properties of an Area 1/10/2013 Contents 5.1. Introduction 5.2. First moment of area 5.3. Moment of inertia for an area 5.4. Moment of inertia for some simple areas 5.5. Parallel - axis theorem 5.6. Examples1/10/2013 35.1. Introduction Dimension, shape?1/10/2013 45.2. First Moment of Area 5.2.1. Definition• The first moment of a plane A aboutthe x- and y-axes are defined as Sx   ( A) ydA Sy   xdA ( A)• Value: positive, negative or zero• Dimension: [L3]; Unit: m3, cm3,...• Centroidal axes: are axes, which first moment of a plane A about themis zero5.2.2. The centroid of an area• The centroid C of the area is defined as the point in the xy-plane thathas the coordinates1/10/2013 55.2. First Moment of Area Sy Sx xC  yC  A A yC C• If the origin of the xy-coordinate system xCis the centroid of the area then Sx=Sy=0• Whenever the area has an axis ofsymmetry, the centroid of the area will lieon that axis • If the area can be subdivided in to simple geometric shapes (rectangles, circles, etc., then n n Sx   S i x S y   S yi i 1 i 11/10/2013 65.2. First Moment of Area y5.2.3. The centroid of composite area n yC1 C1 Sy x Ci Ai C2 xC   i 1 n A A i i 1 C3 xC1 x n Sx y Ci Ai yC   i 1 n A A i i 11/10/2013 75.3. Moment of Inertia for an Area5.3.1. Moment of inertia Ix   ( A) y 2dA Iy   ( A) x 2dA5.3.2. Polar moment of inertia Ip  ( A)   2 dA  I x  I y 5.3.3. Product of inertia • The value of moment of inertia and polar moment of inertia always positive, but the I xy   xydA ( A) product of inertia can be positive, negative, or zero • Dimension: [L4]; Unit: m4, cm4,...1/10/2013 85.3. Moment of Inertia for an Area- The product of inertia Ixy for an area will be zero if either the x or the yaxis is an axis of symmetry for the area- The area with hole, then the hole’sarea is given by minus sign.- The composite areas: n Sx   S n S y   S yi i x i 1 i 1 n n Ix   I i x I y   I yi i 1 i 11/10/2013 95.4. Moment of Inertia for some simple areas • Rectangular y bh3 hb3 Ix  Iy  12 12 y • Circle h ...