(a) Explanation Given Information: Thickness of aluminum plate is 20 mm. The force acting on the bell crank is 80 N. Calculation: For Point D : Sketch the free body diagram of the crank as shown in Figure 1. Apply Equilibrium equation to find the force in the link BC . Sum of moments at point A is equal to 0. Σ M A = 0 F B C × 0.11 − 80 × 3 5 × 0.2 = 0 F B C × 0.11 = 9.6 F B C = 87.273 N Find the internal force at point D : Sketch the internal force diagram at point D as shown in Figure 2. Refer to Figure 2. Find the value of internal horizontal force ( N ) , internal vertical force ( V ) , and the internal moment ( M ) . Apply Equilibrium equations. Sum of vertical forces at point D is equal to 0. − V + 80 ( 3 5 ) = 0 V = 48 N Sum of horizontal forces at point D is equal to 0. − N + 80 ( 4 5 ) = 0 N = 64 N Sum of moments at point D is equal to 0. M − 80 × 3 5 × 0.15 = 0 M = 7.2 N ⋅ m Find the area ( A ) of the section at point D with the thickness of 20 mm. A = 40 mm × 20 mm = 800 mm 2 Find the moment of inertia ( I ) : I = 1 12 ( 20 mm ) ( 40 mm ) 3 = 0.1067 × 10 6 mm 4 Find the first moment of area at point D ( Q D ) using the relation: Q D = y ¯ ' A ' (1) Here, y ¯ ' is the centroid distance to point D and A ' is the area from centroid to point D . Find the area A ' using given Figure. A ' = 20 mm × 10 mm = 200 mm 2 Substitute 15 mm for y ¯ ' and 200 mm 2 for A ' in Equation (1). Q D = 15 mm × 200 mm 2 = 3 , 000 mm 3 = 3 × 10 − 6 m 3 Find the normal stress at point D using the relation: σ D = P A + M y I (2) Refer the given Figure. The value of y is − 10 mm . Substitute 64 N for, for P , 800 mm 2 for A , 7.2 N ⋅ m for M , − 10 mm for y , and 0.1067 × 10 6 mm 4 for I in Equation (2). σ D = 64 N 800 mm 2 + 7.2 N ⋅ m × ( − 10 mm ) 0.1067 × 10 6 mm 4 = 64 N 800 mm 2 − 7.2 N ⋅ m × 10 3 mm 1 m × 10 mm 0.1067 × 10 6 mm 4 = − 0.595 N mm 2 = − 0 (b) To determine Find the principal stresses at point E .

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.3, Problem 28P

(a)

To determine

Find the principal stresses at point D.

(b)

To determine

Find the principal stresses at point E.

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Chapter 14.3, Problem 29P

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Find the principal stresses at point D .

Solution Summary: The author explains how to find the principal stresses at point D by sketching the free body diagram of the crank as shown in Figure 1.