# Strength of materials - Geometric properties of an area

STRENGTH OF MATERIALS TRAN MINH TU - University of Civil Engineering, Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam 1/10/2013 1 5 CHAPTER 1/10/2013 Geometric Properties of an Area 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 3 5.1. Introduction 1/10/2013 4 Dimension, shape? 1/10/2013 5 5.2. First Moment of Area 5.2.1. Definition 21 trang | Chia sẻ: huong20 | Ngày: 21/01/2022 | Lượt xem: 17 | Lượt tải: 0
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( ) x A S ydA  ( ) y A S xdA  • Centroidal axes: are axes, which first moment of a plane A about them is zero • The first moment of a plane A about the x- and y-axes are defined as • Value: positive, negative or zero • Dimension: [L3]; Unit: m3, cm3,... 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 6 y C S x A  xC S y A  xC yC C 5.2. First Moment of Area • If the origin of the xy-coordinate system 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 1 n i x x i S S   1 n i y y i S S   • If the area can be subdivided in to simple geometric shapes (rectangles, circles, etc., then 1/10/2013 7 5.2.3. The centroid of composite area 5.2. First Moment of Area 1 1 n Ci i y i C n i i x A S x A A       1 1 n Ci i x i C n i i y A S y A A       x y C1 C2 C3 xC1 yC1 1/10/2013 8 5.3. Moment of Inertia for an Area 2 ( ) x A I y dA  2 ( ) y A I x dA  5.3.1. Moment of inertia 5.3.2. Polar moment of inertia 2 ( ) p x y A I dA I I   5.3.3. Product of inertia ( ) xy A I xydA  • 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 9 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: 1 n i x x i I I   1 n i y y i I I   1 n i x x i S S   1 n i y y i S S   1/10/2013 10 5.4. Moment of Inertia for some simple areas • Rectangular • Circle • Triangular 3 12 x bh I  3 12 y hb I  4 4 40,1 2 32 p R D I D      4 4 40,05 4 64 x y R D I I D       3 12 x bh I  h b x y D x y b h x 1/10/2013 11 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: • Geometric properties of an area in the coordinates O'uv are: u x b  v y a  .u xS S a A  .v yS S b A  22u x xI I aS a A   22v y yI I bS b A   uv xy y xI I aS bS abA    1/10/2013 12 5.5. Paralell-axis Theorem If O go through centroid C, then: C C 2 u xI I a A  2 v yI I b A  uv xyI I abA  . Radius of gyration The radius of gyration of an area about the x and y axes, and the point O are defined as ; yx x y II r r A A   1/10/2013 13 5.5. Paralell-axis Theorem 1/10/2013 14 5.5. Paralell-axis Theorem 1/10/2013 15 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 2 simple areas 1 2 1 2 x0 y0 1. Determine the centroid: - xC=0 (y0 – axis of symmetry) Example 5.1 1/10/2013 16 1 2 x 0 y0- Draw the principal coordinates Cxy - The Principal moment of inertia for an area: Example 5.1 1/10/2013 17 Problem 5.2. Example 5.2 1/10/2013 18 Example 5.2 1/10/2013 19 Example 5.3 1/10/2013 20 Example 5.3 THANK YOU FOR ATTENTION ! 1/10/2013 21

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