10. Moments of Inertia

Transcription

1 //0 STTCS: CE0 Chapter 0 Moments of nertia Notes are prepared based on: Engineering Mechanics, Statics b R. C. Hibbeler, E Pearson Dr M. Touahmia & Dr M. Boukendakdji Civil Engineering Department, Universit of Hail (0/0) 0. Moments of nertia Chapter Objective:. Define the moments of inertia (Mo) for an area.. Determine the Mo for an area b integration. Contents: 0. Definition of Moment of nertia for reas 0. Parallel-is Theorem for an rea 0. Radius of Gration of an rea 0. Moments of nertia for Composite reas

2 //0 0. Definition of Moments of nertia for reas Man structural members like beams and columns have cross sectional shapes like, L, C, etc. Some others are made of tubes rather than solid squares or rounds. Wh do the usuall not have solid rectangular, square, or circular cross sectional areas? What primar propert of these members influences design decisions? How can we calculate this propert? 0. Definition of Moments of nertia for reas Consider three different possible cross sectional shapes and areas for the beam RS. ll have the same total area and, assuming the are made of same material, the 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? Wh? The answer depends on a propert called Moment of nertia (Mo) of the beam about the -ais. t turns out that Section () has the highest Mo because most of the area is farthest from the ais. Hence, it has the least stress and deflection: (σ = M./); as (Mo) increases, σ (stress) decreases and deflection decreases also.

3 //0 0. Definition of Moments of nertia for reas The rea Moment of nertia of a beams cross-sectional area measures the beams abilit to resist bending. The Mo is a geometrical propert of a beam and depends on a reference ais. t is frequentl used in formulas related to the strength and stabilit of structural members. The larger the Mo the less the beam will bend. The smallest Mo about an ais passes through the centroid. 0. Definition of Moments of nertia for reas The area moment of nertia represents the second moment of the area about an ais. The moments of inertia of a differential area d about the and are: d d d d The moment of inertia of d about the pole O or z ais is then: d O r d For the entire area, the moments of inertia are determined b integration: d The polar moment of inertia is: O d r d

4 //0 The step-b-step procedure for analsis is:. Choose the element d: There are two choices: a vertical strip or a horizontal strip. Some considerations about this choice are: a) The element parallel to the ais about which the Mo is to be determined usuall results in an easier solution. For eample, we tpicall choose a horizontal strip for determining and a vertical strip for determining. b) f is easil epressed in terms of (e.g., = + ), then choosing a vertical strip with a differential element d wide ma be advantageous.. ntegrate to find the Mo. rea Moment of nertia of Common Shapes:

5 //0 0. Parallel-is Theorem for an rea The parallel-ais theorem can be used to find the moment of inertia of an area about an ais that is parallel to an ais passing through the Centroid and about which the moment of inertia is known ( ). C The Mo for an area about an ais parallel to the Centroid ais is equal to its Mo about an ais passing through the area s Centroid plus the product of the area and the square of the perpendicular distance between the aes. d d O C d 0. Radius of Gration of an area For a given area and its Mo,, imagine that the entire area is located at distance k from the ais: k This k is called the radius of gration of the area about the ais. Similarl: k k O O k k 5

6 //0 Eample (a) (b) (c) Determine the moment of inertia for the rectangle area shown in the figure with respect to: The Centroid ais. The ais b passing through the base of the rectangle. The pole or z ais perpendicular to the - plan and passing through the Centroid C. Solution (a) Mo about ais: ntegration from =- h/ to = h/ d h h (b) Mo about b : b d h bd b d h (c) Polar Mo about point C: hb C bh h b bh h bh bh bh

7 //0 Eample Determine the moment of inertia for the shaded area shown in the figure about the ais. Solution differential element d parallel to the ais is chosen for integration: d 00 d ntegration with respect to, from = 0 to = 00 mm, ields: d 00mm mm 0 00 d 00m d d mm 7

8 //0 0. Moments of nertia for Composite reas composite area is made b adding or subtracting a series of simple shaped areas like rectangles, triangles, and circles. For eample, the area on the figure can be made from a rectangle minus a triangle and circle. The Mo of these simpler shaped areas about their Centroidal aes are found in most engineering handbooks. Using these data and the parallel-ais theorem, the Mo for a composite area can easil be calculated. The Mo of the composite area is equal to the algebraic sum of the moments of inertia of each of its parts. Eample Determine the moment of inertia of the area shown in the figure about the ais. 8

9 //0 Solution Composite Parts:. Divide the given area into its simpler shaped parts.. Locate the Centroid of each part and indicate the perpendicular distance from each Centroid to the desired reference ais. = Solution Parallel-is Theorem: (a) Circle: (b) Rectangle: d d mm mm lgebraic Summation: The Mo for the entire area is the algebraic summation of the individual Mo: mm.5 0 9