Problem 9.187 y Determine the moments of inertia of the shaded area x y Determine the moments of inertia of the shaded area shown with respect to the x and y axes when a = 20 mm.

a x y Solving Problems on Your Own Determine the moments of inertia of the shaded area shown with respect to the x and y axes when a = 20 mm. Problem 9.187 C 1. Compute the moments of inertia of a composite area with respect to a given axis. 1a. Divide the area into sections. The sections should have a shape for which the centroid and moments of inertia can be easily determined (e.g. from Fig. 9.12 in the book).

a x y Solving Problems on Your Own Determine the moments of inertia of the shaded area shown with respect to the x and y axes when a = 20 mm. Problem 9.187 C 1b. Compute the moment of inertia of each section. The moment of inertia of a section with respect to the given axis is determined by using the parallel-axis theorem: I = I + A d2 Where I is the moment of inertia of the section about its own centroidal axis, I is the moment of inertia of the section about the given axis, d is the distance between the two axes, and A is the section’s area.

a x y Solving Problems on Your Own Determine the moments of inertia of the shaded area shown with respect to the x and y axes when a = 20 mm. Problem 9.187 C 1c. Compute the moment of inertia of the whole area. The moment of inertia of the whole area is determined by adding the moments of inertia of all the sections.

Divide the area into sections. x y Problem 9.187 Solution Divide the area into sections. C y 4 a 3 p 2 C’ x’ B B x C 1 4 a 3 p x’’ C’’ 3

( Ix’)2 = (IBB)2 _ A d2 = p a4 _ p a2 ( )2 = p a4 _ a4 y y Problem 9.187 Solution 4 a 3 p 2 C’ x’ Compute the moment of inertia of each section. B B C x C 1 4 a 3 p x’’ C’’ 3 Moment of inertia with respect to the x axis (IBB)2 = p a4 1 8 For section 2: (IBB)2 = ( Ix’)2 + A d2 1 8 1 2 4 a 3 p 1 8 8 9 p ( Ix’)2 = (IBB)2 _ A d2 = p a4 _ p a2 ( )2 = p a4 _ a4 (Ix)2 = ( Ix’)2 + A d2 = p a4 _ a4 + p a2 ( a + )2 4 a 3 p 1 8 8 9 p 1 2 5 8 4 3 (Ix)2 = pa4 + a4

Ix = (Ix)1 + (Ix)2 + (Ix)3 = a4 + p a4 + a4 + p a4 + a4 4 3 y y Problem 9.187 Solution 4 a 3 p 2 C’ x’ B B C x C 1 4 a 3 p x’’ C’’ 3 (Ix)1 = (2a) (2a)3 = a4 1 12 4 3 For section 1: 5 8 4 3 For section 3: (Ix)3 = (Ix)2 = p a4 + a4 Compute the moment of inertia of the whole area. Moment of inertia of the whole area: Ix = (Ix)1 + (Ix)2 + (Ix)3 = a4 + p a4 + a4 + p a4 + a4 4 3 5 8 (For a = 20 mm) Ix = 4 a4 + pa4 = 1.268 x 106 mm4 5 4 Ix = 1.268 x 106 mm4

Iy = (Iy)1 + (Iy)2 + (Iy)3 = a4 + p a4 + p a4 = a4 + p a4 4 3 1 8 x y y Problem 9.187 Solution 4 a 3 p 2 C’ x’ B B C x C 1 4 a 3 p x’’ C’ 3 Moment of inertia with respect to the y axis 1 12 4 3 For section 1: (Iy)1 = (2a) (2a)3 = a4 1 8 For section 3: (Iy)3 = p a4 1 8 For section 2: (Iy)2 = p a4 Moment of inertia of the whole area: Iy = (Iy)1 + (Iy)2 + (Iy)3 = a4 + p a4 + p a4 = a4 + p a4 4 3 1 8 Iy = 339 x 103 mm4 (For a = 20 mm)