Explanation Given information: The force of 30 kN acts at the free end of the beam. Calculation: Find the moment of inertia of the rectangular cross section using the relation: I = 1 12 b d 3 Substitute 50 mm for b and 150 mm for d . I = 1 12 ( 50 mm ) ( 150 mm ) 3 = 14.0625 × 10 6 mm 4 = 14.0625 × 10 6 mm 4 ( 10 − 3 m 1 mm ) 4 = 14.0625 × 10 − 6 m 4 At point A , The internal shear force V is as follows: Σ F y = 0 V − 30 = 0 V = 30 kN The internal moment M is, Σ M O = 0 − M − 30 × 0 .3 = 0 M = − 9 kN ⋅ m Find the area at point A . A ' = ( 50 mm ) ( 50 mm ) = 2.5 × 10 3 mm 2 = 2.5 × 10 3 mm 2 ( 10 − 6 m 2 1 mm 2 ) = 2.5 × 10 − 3 m 2 The centroid distance y ¯ to point A is y ¯ = 0.05 m . Find the first moment of area at point A ( Q A ) using the relation. Q A = y ¯ A ' Substitute 0.05 m for y ¯ and 2.5 × 10 − 3 m 2 for A ' . Q A = 0.05 m × 2.5 × 10 − 3 m 2 = 0.125 × 10 − 3 m 3 Find the normal stress at point A using the relation: σ A = − M y A I Substitute − 9 kN ⋅ m for M , 25 mm for y A , and 14.0625 × 10 − 6 m 4 for I . σ A = − ( − 9 kN ⋅ m ) ( 25 mm ) 14.0625 × 10 − 6 m 4 = ( 9 kN ⋅ m × 10 3 N 1 kN ) ( 25 mm × 10 − 3 m 1 mm ) 14.0625 × 10 − 6 m 4 = 16 × 10 6 N m 2 = 16 MPa Find the shear stress at point A using the relation: τ A = V Q A I t Substitute 30 kN for V , 0.125 × 10 − 3 m 3 for Q A , 14.0625 × 10 − 6 m 4 for I , and 50 mm for t . τ A = ( 30 kN ) ( 0.125 × 10 − 3 m 3 ) ( 14

Statics and Mechanics of Materials Plus Mastering Engineering with Pearson eText - Access Card Package (5th Edition)

5th Edition

ISBN: 9780134301006

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.4, Problem 11FP

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Find the principal stresses at point A on the cross section of the beam.

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Chapter 14.4, Problem 10FP

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Chapter 14 Solutions

Statics and Mechanics of Materials Plus Mastering Engineering with Pearson eText - Access Card Package (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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Find the principal stresses at point A on the cross section of the beam.

Solution Summary: The author calculates the moment of inertia of the rectangular cross section using the relation: lSigma

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Find the principal stresses at point A on the cross section of the beam.

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Chapter 14 Solutions