(a) Explanation Given information: Refer to section 9.13A and 9.13B, “properties of rolled steel shapes” in the textbook. The properties of section W 14 × 38 as follows: The area of the section is 11.2 in 2 The moment of inertia ( I ¯ x ) about x axis is 5 , 385 in 4 The moment of inertia ( I ¯ y ) about x axis is 26.7 in 4 Calculation: Find the centroid x ¯ using the relation as shown below: x ¯ = 2 ( a 1 x 1 ) + a 2 x 2 a 1 + a 2 (1) Here, ( x 1 ) is centroid of plate section, a 1 is area of plate section, a 2 is area of W section and ( y 2 ) is centroid of W section. Substitute 26 in 2 for ( a 1 ) , a for ( x 1 ) , 11.2 in 2 for a 1 , and 0 for x 2 in Equation (1). x ¯ = 2 ( 26 ) ( a ) + 11.2 ( 0 ) 52 + 11.2 = 52 a 63.2 = 0.82279 a in . Find the area of the plate ( A plate ) as shown below: ( A plate ) = b d (2) Here, b is width of the plate section and d is the depth of the section. Substitute 26 in . for b and 1 in . for d in Equation (2). ( A plate ) = 26 × 1 = 26 in 2 Find the moment of inertia with respect to x axis for plate section: ( I x ) plate = I ¯ x + A d 2 = b d 3 12 + A d 2 (3) Here, A is cross section area of plate section and d is vertical distance from the centroid of the segment to the neutral axis. Substitute 26 in 2 for A , ( 7.05 + 0.5 ) in . for d 26 in . for b and 1 in . for d in Equation (3). ( I x ) plate = 26 × 1 3 12 + 26 × ( 7.05 + 0.5 ) 2 = 2.16677 + 1 , 482.065 = 1 , 484.23 in 4 Find the total moment of inertia ( I ¯ x ) and ( I ¯ y ) of entire section as shown below: ( I ¯ x ) = ( I ¯ y ) = ( I x ) W + 2 ( I x ) P L (4) Here, ( I x ) W is moment of inertia of x axis about W section and ( I x ) P L is moment of inertia of x axis plate section. Substitute 385 in 4 for ( I x ) W and 1484 (b) To determine Find the moment of inertia of the combined section with respect to x and y axis.

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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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 9.56P

(a)

To determine

Find the distance a in the combined section.

(b)

To determine

Find the moment of inertia of the combined section with respect to x and y axis.

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Find the distance a in the combined section.

Find the moment of inertia of the combined section with respect to x and y axis.

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