Explanation Given information: The applied load P is 15 kN . The height ( h ) of the cross-section is 60 mm . The cross-section of the component is a trapezoid. The inner radius of curved component is r 1 = 40 mm . Calculation: Show the cross section of the machine component as shown in Figure 1. Refer Figure 1. Consider the area of the triangle 1 and 2 are denoted by A 1 and A 2 . Calculate the area of the triangle 1 and 2 as follows: A 1 = 1 2 b 1 h 1 = 1 2 × 35 × 60 = 1050 mm 2 A 2 = 1 2 b 2 h 2 = 1 2 × 25 × 60 = 750 mm 2 Calculate the mean radius of triangle 1 as follows: Substitute 40 mm for r 1 and 60 mm for h . r ¯ 1 = r 1 + 1 3 h = 40 + 1 3 × 60 = 60 mm Calculate the mean radius of triangle 2 as follows: Substitute 40 mm for r 1 and 60 mm for h . r ¯ 2 = r 1 + 2 3 h = 40 + 2 3 × 60 = 80 mm 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 750 mm 2 for A 2 , 1050 mm 2 for A 1 , 60 mm for r ¯ 1 , and 80 mm for r ¯ 2 in Equation (1). r ¯ = 1050 × 60 + 750 × 80 1050 + 750 = 123 , 000 1 , 800 = 68.333 mm The width of the triangle 1 and 2 are b 1 = 35 mm and b 2 = 25 mm 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 35 mm for b 1 , 25 mm for b 2 , 40 mm for r 1 , 100 mm for r 2 , and 60 mm for h in Equation (2)

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ISBN: 9780073398235

Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek

Publisher: McGraw-Hill Education

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Chapter 4.10, Problem 186P

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Find the largest tensile stress in section a-a.

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Mechanics of Materials, 7th Edition

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Find the largest tensile stress in section a - a .

Solution Summary: The author shows the largest tensile stress in section a-a, and the cross-section of the machine component, as shown in Figure 1.

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