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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.
STRENGTH OF MATERIALS 1/10/2013 TRAN MINH TU - University of Civil Engineering, Giai Phong Str 55, Hai Ba Trung Dist Hanoi, Vietnam CHAPTER 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 Examples 1/10/2013 5.1 Introduction Dimension, shape? 1/10/2013 5.2 First Moment of Area 5.2.1 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 5.2.2 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 5.2 First Moment of Area xC Sy A Sx yC A yC • If the origin of the xy-coordinate system is the centroid of the area then Sx=Sy=0 C xC • 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 Sx S i 1 1/10/2013 n i x S y S yi i 1 5.2 First Moment of Area y 5.2.3 The centroid of composite area yC1 n xC Sy A x i 1 n Ci Ai i y Ci i 1 C3 x xC1 n Ai n A i 1 1/10/2013 C2 A i 1 Sx yC A C1 i 5.3 Moment of Inertia for an Area 5.3.1 Moment of inertia Ix y 2dA Iy x 2dA ( A) ( A) 5.3.2 Polar moment of inertia Ip dA I x I y ( A) 5.3.3 Product of inertia I xy xydA ( A) • The value of moment of inertia and polar moment of inertia always positive, but the product of inertia can be positive, negative, or zero • Dimension: [L4]; Unit: m4, cm4, 1/10/2013 5.3 Moment of Inertia for an Area - The product of inertia Ixy for an area will be zero if either the x or the y axis is an axis of symmetry for the area - The area with hole, then the hole’s area is given by minus sign - The composite areas: n Sx S i 1 i 1 1/10/2013 n S y S yi i 1 n Ix I i x n i x I y I yi i 1 5.4 Moment of Inertia for some simple areas • Rectangular hb3 Iy 12 Ip R Ix I y • y h bh3 Ix 12 • Circle y R4 D 32 x 0,1D D4 64 x b 0,05D D Triangular 1/10/2013 h bh3 Ix 12 x b 10 5.5 Paralell-axis Theorem • In the xy coordinates, an area has geometric properties: Sx, Sy, Ix, Iy, Ixy • In the uv coordinates: O'u//Ox, O'v//Oy và: u xb v ya • Geometric properties of an area in the coordinates O'uv are: Su S x a A Sv S y b A 1/10/2013 Iu I x 2aS x a A I v I y 2bS y b2 A Iuv I xy aS y bS x abA 11 5.5 Paralell-axis Theorem If O go through centroid C, then: Iu I x a A I v I y b2 A Iuv I xy abA C C Radius of gyration The radius of gyration of an area about the x and y axes, and the point O are defined as 1/10/2013 Iy Ix rx ; ry A A 12 5.5 Paralell-axis Theorem 1/10/2013 13 5.5 Paralell-axis Theorem 1/10/2013 14 Example 5.1 Problem 5.6.1 An area with the shape and the dimension as shown in the figure Determine the principal moment of inertia for area Solution Choosing the primary coordinates x0y0 as shows in the figure Divide the composite area to simple areas y0 Determine the centroid: - xC=0 (y0 – axis of symmetry) x0 1/10/2013 15 Example 5.1 - Draw the principal coordinates Cxy y0 - The Principal moment of inertia for an area: x 1/10/2013 16 Example 5.2 Problem 5.2 1/10/2013 17 Example 5.2 1/10/2013 18 Example 5.3 1/10/2013 19 Example 5.3 1/10/2013 20 THANK YOU FOR ATTENTION ! 1/10/2013 21 .. .CHAPTER 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... Parallel - axis theorem 5. 6 Examples 1/10/2013 5. 1 Introduction Dimension, shape? 1/10/2013 5. 2 First Moment of Area 5. 2.1 Definition • The first moment of a plane A about the x- and y-axes are... 1/10/2013 15 Example 5. 1 - Draw the principal coordinates Cxy y0 - The Principal moment of inertia for an area: x 1/10/2013 16 Example 5. 2 Problem 5. 2 1/10/2013 17 Example 5. 2 1/10/2013 18 Example 5. 3