To determine Find the moment of inertia ( I x ) and radius of gyration ( k x ) of the shaded area with respect to the x axis. Explanation Calculation: Show the asymmetrical I section as Figure 1. Consider the section 1. Calculate the area of the section 1 ( A 1 ) using the formula: A 1 = ( 12 + 12 ) × 6 = 144 mm 2 Calculate the moment of inertia of the section 1 ( I 1 ) using the formula: I 1 = 24 × 6 3 12 = 432 mm 4 Calculate the centroid of the section 1 ( y 1 ) from neutral axis using the relation: y 1 = 24 + 6 2 = 27 mm Calculate the moment of inertia of the section 1 ( I x ) 1 about x axis using the formula: ( I x ) 1 = I 1 + A 1 y 1 2 Substitute 432 mm 4 for I 1 , 144 mm 2 for A 1 and 27 mm for y 1 . ( I x ) 1 = 432 + ( 144 ) × 27 2 = 105408 mm 4 Consider the section 2. Calculate the area of the section 2 ( A 2 ) using the formula: A 2 = ( 24 + 24 ) × 8 = 384 mm 2 Calculate the moment of inertia of the section 2 ( I 2 ) using the formula: I 2 = 8 × 48 3 12 = 73728 mm 4 Calculate the centroid of the section 2 ( y 2 ) from neutral axis using the relation: Since the neutral axis is acting at the center, y 2 = 0 . Calculate the moment of inertia of the section 2 ( I x ) 2 about x axis using the formula: ( I x ) 2 = I 2 + A 2 y 2 2 Substitute 73728 mm 4 for I 2 , 384 mm 2 for A 2 and 0 mm for y 2 . ( I x ) 2 = 73728 + ( 384 ) × 0 2 = 73728 mm 4 Consider the section 3. Calculate the area of the section 3 ( A 3 ) using the formula: A 3 = ( 24 + 24 ) × 6 = 288 mm 2 Calculate the moment of inertia of the section 3 ( I 3 ) using the formula: I 3 = 48 × 6 3 12 = 864 mm 4 Calculate the centroid of the section 3 ( y 3 ) from neutral axis using the relation: y 3 = 24 + 6 2 = 27 mm Calculate the moment of inertia of the section 3 ( I x ) 3 about x axis using the formula: ( I x ) 3 = I 3 + A 3 y 3 2 Substitute 864 mm 4 for I 3 , 288 mm 2 for A 3 and 27 mm for y 3

Vector Mechanics for Engineers: Statics, 11th Edition

11th Edition

ISBN: 9780077687304

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek

Publisher: McGraw-Hill 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 9.2, Problem 9.31P

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Find the moment of inertia (Ix) and radius of gyration (kx) of the shaded area with respect to the x axis.

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Chapter 9.1, Problem 9.30P

Chapter 9.2, Problem 9.32P

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