Explanation Given: The width of the rectangular area is 120 mm . The height of the rectangular area is 30 mm . The angle is 30 ° . Show the beam’s rectangular area as in Figure (1). From Figure 1, As the rectangular area is symmetrical about the x and y axes, write the product of inertia for the rectangular area with respect to the x and y axes as 0 . I x y = 0 Compute the moment of inertia for the rectangular area with respect to the x -axis. I x = 1 12 a b 3 (I) Compute the moment of inertia for the rectangular area with respect to the y -axis. I y = 1 12 b a 3 (II) Compute the moment of inertia for the rectangular area with respect to the u -axis. I u = [ I x + I y 2 ] + [ I x − I y 2 ] cos 2 θ − I x y sin 2 θ (III) Compute the moment of inertia for the rectangular area with respect to the v -axis. I v = [ I x + I y 2 ] − [ I x − I y 2 ] cos 2 θ + I x y sin 2 θ (IV) Compute the moment of inertia for the rectangular area with respect to the u and v axes. I u v = [ I x − I y 2 ] sin 2 θ + I x y cos 2 θ (V) Conclusion: Substitute 120 mm for a and 30 mm for b in Equation (I). I x = 1 12 ( 120 ) ( 30 ) 3 = 324 ( 10 4 ) 12 = 0.27 ( 10 6 ) mm 4 Substitute 120 mm for a and 30 mm for b in Equation (II). I y = 1 12 ( 30 ) ( 120 ) 3 = 5184 ( 10 4 ) 12 = 4.32 ( 10 6 ) mm 4 Substitute 0.27 ( 10 6 ) mm 4 for I x , 4.32 ( 10 6 ) mm 4 for I y , 0 for I x y , and 30 ° for θ in Equation (III). I u = { [ 0.270 + 4.32 2 ] + [ 0.270 − 4.32 2 ] cos 2 ( 30 ° ) − ( 0 ) sin 2 ( 30 ° ) } ( 10 6 ) = { [ 0.270 + 4.32 2 ] + [ 0.270 − 4.32 2 ] ( cos 60 ° ) − 0 } ( 10 6 ) = [ 2

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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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 10.7, Problem 70P

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The moments of inertia Iu, Iv and the product of inertia Iuv for the rectangular area.

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Chapter 10.7, Problem 69P

Chapter 10.7, Problem 71P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The moments of inertia I u , I v and the product of inertia I u v for the rectangular area.

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