Printer-Friendly - Centroid and Moment of Inertia

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Oct 30, 2023

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Centroid and Moment of Inertia Structures I R. Taher Centroids and Moment of Inertia 1 Warning The information contained in this lecture is protected by intellectual property laws. The content may be used only by the students enrolled in this course. Do not reproduce, transmit, publish, rewrite, sell or post the content on internet sites without a written permission from the instructor teaching this course. Centroids and Moment of Inertia 2 Textbook Reading Simplified Structural Analysis and Design for Architects, Revised 2 nd Edition, by Rima Taher, published by Cognella, 2021. Chapter 5 Centroids and Moment of Inertia Centroids and Moment of Inertia 3 Centroid and Moment of Inertia Center of Gravity of a Body Centroid Moment of an Area Centroid of Composite Areas Moment of Inertia Centroids and Moment of Inertia 4 Center of Gravity Gravity or the weight of a body, is the force of attraction between the body and the earth. The weight of a body is proportional to its mass. The line of action of a body will always pass through the center of gravity. Centroids and Moment of Inertia 5 If the body has an axis of symmetry, then the center of gravity will be located on it. If the body has two axes of symmetry then the center of gravity will be located at their intersection. Centroids and Moment of Inertia 6
Centroid It is the center of gravity of an area. The area is assumed to be weightless. Centroids and Moment of Inertia 7 Moment of an Area Centroids and Moment of Inertia 8 The moment of an area (A) about the Y-axis is called M Y and is given by: In a similar way, the moment of the area (A) about the X-axis is given by: Example Centroids and Moment of Inertia 9 Determine the moment of this area (A) about the X- and Y- axes. Solution Calculate the area (A): A = 4” x 6” =24 in 2 Centroids and Moment of Inertia 10 Calculate the moment of the area about both axes: Centroid of Composite Areas To locate the centroid of a composite area: 1- Divide the composite area into simple areas. 2- Sum up the area moments about the axis in question. 3- The moment of the composite area should be equal to the sum of the moments of its components. 4- If the centroid of the composite area is C then: Centroids and Moment of Inertia 11 Example Centroids and Moment of Inertia 12 Locate the centroid of the T- shape shown. This T-shape is symmetrical about its vertical axis a-a.
Solution Centroids and Moment of Inertia 13 Divide the composite area into two simple rectangular areas (flange and web) as shown. Centroids and Moment of Inertia 14 Calculate the coordinates of the two individual centroids: Centroids and Moment of Inertia 15 Calculate the coordinates of the centroid. Since the T-shape is symmetrical about its vertical axis, the centroid must be located on this axis. For the other coordinate, apply the previous formula: Centroids and Moment of Inertia 16 Moment of Inertia It is a sectional property that appears in some structural equations. For standard shapes, such as the standard steel sections, this property is tabulated. In other cases, it is necessary to calculate this property. Centroids and Moment of Inertia 17 Centroids and Moment of Inertia 18 Assume the simple area (A) is divided into some very small differential areas a 1 , a 2 , a 3 ,…, a n , as shown in the next figure. The moment of inertia (I X ) of the area (A) with respect to its centroidal axis X-X is given by: Y i : ordinate of the area a i with respect to the centroidal axis X-X.
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Centroids and Moment of Inertia 19 A few simple shapes are shown in the next figure . The formulas used to calculate the centroidal moments of inertia for these simple areas are given in the next table. Centroids and Moment of Inertia 20 Shape Moment of Inertia Rectangle 12 3 bh I X = Triangle 36 3 bh I X = Circle 64 4 d I X π = Example Determine the moment of inertia of a rectangular area with respect to both its centroidal axes. The width of the rectangle is 4 in and the depth is 8 in. Solution Centroids and Moment of Inertia 21

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