Explanation Calculation: Sketch the cross section as shown in Figure 1. Find the moment of inertia ( I ¯ x ) 1 using the relation as shown below: ( I ¯ x ) 1 = 1 12 b 1 h 1 3 (1) Here, b 1 is width of the section 1 and h 1 is height of the section 1. Substitute 100 mm for b 1 and 8 mm for h 1 in Equation (1). ( I ¯ x ) 1 = 1 12 ( 100 ) × 8 3 = 4 , 266.67 mm 4 Find the moment of inertia ( I ¯ x ) 2 and ( I ¯ x ) 3 using the relation as shown below: ( I ¯ x ) 2 = ( I ¯ x ) 3 = 1 12 b 2 h 2 3 + ( b d ) y ¯ 2 (2) Here, b 2 is width of the section 2 and h 2 is height of the section 2. Substitute 8 mm for b 2 , 20 mm for y ¯ 2 , and 32 mm for h 2 in Equation (2). ( I ¯ x ) 2 = ( I ¯ x ) 3 = 1 12 ( 8 ) ( 32 ) 3 + ( 8 × 32 ) ( 20 ) 2 = 21 , 845.333 + 102 , 400 = 124 , 245.333 mm 4 Find the total moment of inertia ( I ¯ x ) using the relation as shown below: ( I ¯ x ) = ( I ¯ x ) 1 + ( I ¯ x ) 2 + ( I ¯ x ) 3 (3) Substitute 4 , 266.67 mm 4 for ( I ¯ x ) 1 , 124 , 245.333 mm 4 for ( I ¯ x ) 2 , and 124 , 245.333 mm 4 for ( I ¯ x ) 3 in Equation (3). ( I ¯ x ) = 4 , 266.67 + 124 , 245.333 + 124 , 245.333 = 252 , 757 mm 2 Find the moment of inertia ( I ¯ y ) 1 using the relation as shown below: ( I ¯ y ) 1 = 1 12 b 1 3 h 1 (4) Substitute 100 mm for b 1 and 8 mm for h 1 in Equation (4). ( I ¯ y ) 1 = 1 12 ( 100 3 ) × 8 = 666 , 666.667 mm 4 Find the moment of inertia ( I ¯ y ) 2 and ( I ¯ y ) 3 using the relation as shown below: ( I ¯ y ) 2 = ( I ¯ y ) 3 = 1 12 b 2 3 h 2 + ( b d ) y ¯ 2 (5) Substitute 8 mm for b 2 , 46 mm for y ¯ 2 , and 32 mm for h 2 in Equation (5). ( I ¯ y ) 2 = ( I ¯ y ) 3 = 1 12 ( 8 ) 3 ( 32 ) + ( 8 × 32 ) ( 46 ) 2 = 1 , 365.333 + 541 , 696 = 543 , 061.3 mm 4 Find the total moment of inertia ( I ¯ y ) using the relation as shown below: ( I ¯ y ) = ( I ¯ y ) 1 + ( I ¯ y ) 2 + ( I ¯ y ) 3 (6) Substitute 666 , 666

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ISBN: 9781264095032

Author: BEER

Publisher: MCGRAW-HILL HIGHER EDUCATION

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Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provi…

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Chapter 9.3, Problem 9.88P

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Find the orientation of the principal axes at the origin and the corresponding value of moment of inertia.

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Chapter 9.3, Problem 9.87P

Chapter 9.3, Problem 9.89P

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Find the orientation of the principal axes at the origin and the corresponding value of moment of inertia.

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