(a) Explanation Given information: The moment ( M ) is 80 kip ⋅ in . . The height ( h ) of the cross-section is 6 in . The cross-section of the machine component is a trapezoid. Calculation: Show the cross section of the machine component as shown in Figure 1. Refer Figure 1. Calculate the area of the triangle 1 and 2 as follows: Substitute 3.5 in . for a and 6 in . for h . A 1 = 1 2 a h = 1 2 × 3.5 × 6 = 10.5 in . 2 Substitute 2.5 in . for a and 6 in . for h . A 2 = 1 2 b h = 1 2 × 2.5 × 6 = 7.5 in . 2 Calculate the mean radius of triangle 1 as follows: Substitute 4 in . for r 1 and 6 in . for h . r ¯ 1 = r 1 + 1 3 h = 4 + 1 3 × 6 = 6 in . Calculate the mean radius of triangle 2 as follows: Substitute 4 in . for r 1 and 6 in . for h . r ¯ 2 = r 1 + 2 3 h = 4 + 2 3 × 6 = 8 in . Calculate the mean radius ( r ¯ ) of the cross-section using the relation: r ¯ = A 1 r ¯ 1 + A 2 r ¯ 2 A 1 + A 2 (1) Substitute 10.5 in . 2 for A 1 , 7.5 in . 2 for A 2 , 6 in . for r ¯ 1 , and 8 in . for r ¯ 2 in Equation (1). r ¯ = 10.5 × 6 + 7.5 × 8 10.5 + 7.5 = 123 18 = 6.8333 in . The bottom width and top width of the trapezoid cross-section are b 1 = 3.5 in . and b 2 = 2.5 in . Show the formula for the radius ( R ) of the neutral surface for the trapezoidal cross-section as follows: R = 1 2 h 2 ( b 1 + b 2 ) ( b 1 r 2 − b 2 r 1 ) ln r 2 r 1 − h ( b 1 − b 2 ) (2) Substitute 3 (b) To determine Find the stress at point B .

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ISBN: 8220100257063

Author: BEER

Publisher: YUZU

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Chapter 4.10, Problem 183P

(a)

To determine

Find the stress at point A.

(b)

To determine

Find the stress at point B.

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Chapter 4.10, Problem 182P

Chapter 4.10, Problem 184P

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Find the stress at point A .

Solution Summary: The author shows the cross-section of the machine component as shown in Figure 1.