1/11/2013 STATICS: CE201 Chapter 10 Moments of Inertia Notes are prepared based on: Engineering Mechanics, Statics by R. C. Hibbeler, 12E Pearson Dr M. Touahmia & Dr M. Boukendakdji Civil Engineering Department, University of Hail (2012/2013) 10. Moments of Inertia ________________________________________________________________________________________________________________________________________________ Chapter Objective: 1. 2. Define the moments of inertia (MoI) for an area. Determine the MoI for an area by integration. Contents: 10.1 10.2 10.3 10.4 Definition of Moment of Inertia for Areas Parallel-Axis Theorem for an Area Radius of Gyration of an Area Moments of Inertia for Composite Areas 1 1/11/2013 10.1 Definition of Moments of Inertia for Areas Many structural members like beams and columns have cross sectional shapes like I, L, C, etc. Some others are made of tubes rather than solid squares or rounds. Why do they usually not have solid rectangular, square, or circular cross sectional areas? What primary property of these members influences design decisions? How can we calculate this property? 10.1 Definition of Moments of Inertia for Areas Consider three different possible cross sectional shapes and areas for the beam RS. All have the same total area and, assuming they are made of same material, they will have the same mass per unit length. For the given vertical loading F on the beam, which shape will develop less internal stress and deflection? Why? The answer depends on a property called Moment of Inertia (MoI) of the beam about the x-axis. It turns out that Section (A) has the highest MoI because most of the area is farthest from the x axis. Hence, it has the least stress and deflection: (σ = M.y/I); as I (MoI) increases, σ (stress) decreases and deflection decreases also. 2 1/11/2013 10.1 Definition of Moments of Inertia for Areas The Area Moment of Inertia of a beams cross-sectional area measures the beams ability to resist bending. The MoI is a geometrical property of a beam and depends on a reference axis. It is frequently used in formulas related to the strength and stability of structural members. The larger the MoI the less the beam will bend. The smallest MoI about any axis passes through the centroid. 10.1 Definition of Moments of Inertia for Areas The area moment of Inertia represents the second moment of the area about an axis. The moments of inertia of a differential area dA about the x and y are: dI x y 2 dA dI y x 2 dA The moment of inertia of dA about the “pole” O or z axis is then: dJ O r 2 dA For the entire area A, the moments of inertia are determined by integration: I x y 2 dA A I y x 2 dA A 2 The polar moment of inertia is: J O Ar dA I x I y 3 1/11/2013 The step-by-step procedure for analysis is: 1. Choose the element dA: There are two choices: a vertical strip or a horizontal strip. Some considerations about this choice are: a) The element parallel to the axis about which the MoI is to be determined usually results in an easier solution. For example, we typically choose a horizontal strip for determining Ix and a vertical strip for determining Iy. b) If y is easily expressed in terms of x (e.g., y = x2 + 1), then choosing a vertical strip with a differential element dx wide may be advantageous. 2. Integrate to find the MoI. Area Moment of Inertia of Common Shapes: 4 1/11/2013 10.2 Parallel-Axis Theorem for an Area The parallel-axis theorem can be used to find the moment of inertia of an area about any axis that is parallel to an axis passing through the Centroid and about which the moment of inertia is known ( I x I J ). y C The MoI for an area about any axis parallel to the Centroid axis is equal to its MoI about an axis passing through the area’s Centroid plus the product of the area and the square of the perpendicular distance between the axes. I x I x Ad y2 I y I y Ad x2 J O J C Ad 2 10.3 Radius of Gyration of an area For a given area A and its MoI, Ix , imagine that the entire area is located at distance kx from the x axis: I x k x2 A kx Ix A This kx is called the radius of gyration of the area about the x axis. Similarly: y A ky Iy A kx kO JO A x 5 1/11/2013 Example 1 Determine the moment of inertia for the rectangle area shown in the figure with respect to: (a) The Centroid x’ axis. (b) The axis xb passing through the base of the rectangle. (c) The pole or z’ axis perpendicular to the x’- y’ plan and passing through the Centroid C. Solution 1 (a) MoI about x’ axis: Integration from y’=- h/2 to y’= h/2 I x y 2 dA A h2 2 y bdy b h 2 h2 y 2 dy h 2 1 3 bh 12 (b) MoI about xb: I xb 1 1 h I x Ad bh 3 bh bh 3 12 3 2 (c) Polar MoI about point C: 2 2 y I y 1 hb 3 12 J C I x I y 1 bh h 2 b 2 12 6 1/11/2013 Example 2 Determine the moment of inertia for the shaded area shown in the figure about the x axis. Solution 2 A differential element dA parallel to the x axis is chosen for integration: dA 100 x dy Integration with respect to y, from y = 0 to y = 200 mm, yields: I x y 2 dA A 2 y 100 x dy 0 200mm 200mm 0 200m 0 y2 dy y 2 100 400 y 100y 2 dy 107.10 6 mm 4 400 4 7 1/11/2013 10.4 Moments of Inertia for Composite Areas A composite area is made by adding or subtracting a series of “simple” shaped areas like rectangles, triangles, and circles. For example, the area on the figure can be made from a rectangle minus a triangle and circle. The MoI of these “simpler” shaped areas about their Centroidal axes are found in most engineering handbooks. Using these data and the parallel-axis theorem, the MoI for a composite area can easily be calculated. The MoI of the composite area is equal to the algebraic sum of the moments of inertia of each of its parts. Example 3 Determine the moment of inertia of the area shown in the figure about the x axis. 8 1/11/2013 Solution 3 Composite Parts: 1. Divide the given area into its simpler shaped parts. 2. Locate the Centroid of each part and indicate the perpendicular distance from each Centroid to the desired reference axis. = Solution 3 Parallel-Axis Theorem: 1 (a) Circle: I I Ad 25 25 75 11.410 mm 4 4 2 x x 2 2 6 4 y (b) Rectangle: I x I x Ad y2 1 1001503 100150752 112.510 6 mm 4 12 Algebraic Summation: The MoI for the entire area is the algebraic summation of the individual MoI: I 112.510 11.410 101.110 mm 6 6 6 4 x 9