Explanation Calculate the normal stress ( σ B ) acting at point B using the relation: σ B = M y z I y (1) Here, M y is the moment in y -direction, z is the distance in z -direction from centroid to point A , and I y is the moment of inertia. Sketch the internal forces and moment in free body diagram at point B as shown in Figure 1. Apply Equilibrium equations to find the value of moment at point B . Sum of moments in y -direction is equal to 0. Σ M y = 0 M y − ( 20 × 10 ) = 0 M y = 200 lb ⋅ in Sum of moments in x -direction is equal to 0. Σ M x = 0 T x + ( 20 × 12 ) = 0 T x = 240 lb ⋅ in Sum of forces in z -direction is equal to 0. Σ V z = 0 V z − 20 = 0 V z = 20 lb Find the moment of inertia of the section ( I ) : Outer radius of the pipe is 1.5 in. and the inner radius of the pipe is 1.375 in. I = π 4 [ ( 1.5 in ) 4 − ( 1.375 in ) 4 ] = 1.1687 in 4 Find the polar moment of inertia of the section ( J ) : J = π 2 [ ( 1.5 in ) 4 − ( 1.375 in ) 4 ] = 2.3374 in 4 Sketch the cross section at point B as shown in Figure 2. Find the first moment of area at point B ( Q B ) z using the relation: ( Q B ) z = Σ y ¯ ' A ' Here, y ¯ ' is the centroid distance to point B and A ' is the area from centroid to point B . Refer to Figure 2. ( Q B ) z = 0 . Substitute 200 lb ⋅ in for M y , 1.5 in . for z , and 1.1687 in 4 for I y in Equation (1). σ B = − ( 200 lb ⋅ in ) ( 1

Statics and Mechanics of Materials, Student Value Edition (5th Edition)

5th Edition

ISBN: 9780134382890

Author: Russell C. Hibbeler

Publisher: PEARSON

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Statics and Mechanics of Materials, Student Value Edition (5th Edition)

Ch. 14.3 - In each ease, the state of stress x, y, xy...Ch. 14.3 - Given the state of stress shown on the element,...Ch. 14.3 - Determine the normal stress and shear stress...Ch. 14.3 - Prob. 2FP Ch. 14.3 - Determine the equivalent state of stress on an...Ch. 14.3 - Prob. 4FP Ch. 14.3 - The beam is subjected to the load at its end....Ch. 14.3 - Prob. 6FP Ch. 14.3 - Prove that the sum of the normal stresses x+y=x+y...Ch. 14.3 - Determine the stress components acting on the...

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Solution Summary: The author calculates the principal stresses at point B using the relation: (sigma_B).

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