Explanation Given information: Diameter of the shaft is 6 in. Calculation: Let the moment T and the force be N are applied at the upper segment of the shaft. Sketch the free body diagram of the rotor shaft’s upper segment as shown in Figure 1. Refer Figure 1. Apply Equilibrium equations. Sum of vertical forces is equal to 0. Σ F y = 0 N − 50 = 0 N = 50 kip Sum of moments is equal to 0. Σ M y = 0 T − 10 = 0 T = 10 kip ⋅ ft Find the cross sectional area of the rotor shaft’s cross section ( A ) with radius of 3 in.: A = π ( 3 2 ) = 9 π in 2 Find the polar moment of inertia of the rotor shaft’s cross section ( J ) with the radius of 3 in.: J = π 2 ( 3 4 ) = 40.5 π in 4 Calculate the normal stress ( σ ) acting on the surface of the shaft using the relation: σ = N A Substitute 50 kip for M D and 9 π in 2 for I σ = 50 kip 9 π in 2 = 1.768 ksi Find the shear stress ( τ ) acting on the surface of the shaft using the relation: τ = T c J Here, c is the radius of the shaft. Substitute 10 kip ⋅ ft for T , 3 in for c , and 40.5 π in 4 for J . τ = ( 10 kip ⋅ ft ) ( 3 in ) 40.5 π in 4 = ( 10 kip ⋅ ft × 12 in . 1 ft ) ( 3 in ) 40.5 π in 4 = 2.829 ksi Sketch the state of stress on the differential element as shown in Figure 2. Refer Figure 2. Apply the sign conventions: The value of normal stresses are σ x = 0 and σ y = 1.768 ksi . The value of shear stress is τ x y = 2.829 ksi . The procedure to construct Mohr’s circle is shown below. The coordinate systems are the horizontal axis represents the normal stress with positive towards right and the vertical axis represents shear stress with positive downwards. Plot the center of the circle C which is located σ avg = σ x + σ y 2 from the origin. Plot the reference points A having coordinates A ( σ x , τ x y ) . This point represents that the x ′ axis coincides with the x axis. Hence, θ = 0 ° . Connect the point A with the centre C and find CA by trigonometry which represents the radius of the circle R

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 67P

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Find the principal stresses at a point located on the surface of the shaft.

Find the maximum in-plane shear stress at a point located on the surface of the shaft.

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Chapter 14.4, Problem 66P

Chapter 14.4, Problem 68P

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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 a point located on the surface of the shaft. Find the maximum in-plane shear stress at a point located on the surface of the shaft.

Solution Summary: The author calculates the principal stresses and the maximum in-plane shear stress at a point located on the surface of the shaft.