tailieunhanh - 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. | Lecture Strength of Materials I: Chapter 5 - PhD. Tran Minh Tu STRENGTH OF MATERIALS 1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam CHAPTER 5 Geometric Properties of an Area 1/10/2013 Contents . Introduction . First moment of area . Moment of inertia for an area . Moment of inertia for some simple areas . Parallel - axis theorem . Examples 1/10/2013 3 . Introduction Dimension, shape? 1/10/2013 4 . First Moment of Area . Definition • The first moment of a plane A about the 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 them is zero . The centroid of an area • The centroid C of the area is defined as the point in the xy-plane that has the coordinates 1/10/2013 5 . First Moment of Area Sy Sx xC yC A A yC C • If the origin of the xy-coordinate system xC is the centroid of the area then Sx=Sy=0 • Whenever the area has an axis of symmetry, the centroid of the area will lie on 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 1 1/10/2013 6 . First Moment of Area y . 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 1 1/10/2013 7 . Moment of Inertia for an Area . Moment of inertia Ix ( A) y 2dA Iy ( A) x 2dA . Polar moment of inertia Ip ( A) 2 dA I x I y . 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: .