Strength of materials - Geometric properties of an area

( ) 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