Explanation Given: The total height of the cross-sectional area is 840 mm and the thickness is 20 mm. To determine the moment of inertia for the cross-sectional area about the x and y axis, and subdivide the cross-sectional area into 3 segments as: Part 1: rectangle Part 2: rectangle Part 3: rectangle Show the free body diagram of the subdivided cross-sectional area as in Figure 1. From Figure (1), Write the height and width of the rectangle in part 1. b 1 = 100 mm h 1 = 20 mm Here, the height of rectangle in part 1 is h 1 and width of rectangle in part 1 is b 1 . Write the height and width of the rectangle in part 2. b 2 = 840 mm h 2 = 20 mm Here, the height of rectangle in part 2 is h 2 and width of rectangle in part 2 is b 2 . Write the height and width of the rectangle in part 3. b 3 = 100 mm h 3 = 200 mm Here, the height of rectangle in part 3 is h 3 and width of rectangle in part 3 is b 3 . Conclusion: Calculate the distance between centroid and y axis about x axis for rectangle in part 1. d x 1 = b 1 2 + h 1 2 (I) Substitute 20 mm for h 1 and 100 mm for b 1 in Equation (I). d x 1 = 100 2 + 20 2 = 50 + 10 = 60 mm Calculate the distance between centroid and x axis about y axis for rectangle in part 1. d y 1 = 400 mm + h 1 2 (II) Substitute 20 mm for h 1 in Equation (II). d y 1 = 400 mm + 20 2 = 410 mm Calculate the product of inertia about x and y axis for rectangle in part 1. I x y 1 = ( b 1 h 1 ) ( d x 1 ) ( d y 1 ) (III) Substitute 20 mm for h 1 , 60 mm for d x 1 , 410 mm for d y 1 , and 100 mm for b 1 in Equation (III). I x y 1 = [ ( 100 ) ( 20 ) ] ( 60 ) ( 410 ) = 49.2 ( 10 6 ) mm 4 Calculate the distance between centroid and y axis about x axis for rectangle in part 2. d x 2 = 0 mm Calculate the distance between centroid and x axis about y axis for rectangle in part 2. d y 2 = 0 mm Calculate the product of inertia about x and y axis for rectangle in part 2. I x y 2 = ( b 2 h 2 ) ( d x 2 ) ( d y 2 ) (IV) Substitute 20 mm for h 2 , 0 mm for d x 2 , 0 mm for d y 2 , and 840 mm for b 2 in Equation (IV)

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ISBN: 9780134780955

Author: HIBBELER

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Chapter 10, Problem 66P

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The product of inertia of the cross-sectional area with respect to the x and y axes.

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Chapter 10, Problem 65P

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The product of inertia of the cross-sectional area with respect to the x and y axes.

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