Mechanics of Materials
Mechanics of Materials
9th Edition
ISBN: 9780133409321
Author: HIBBELER, R. C.
Publisher: Pearson College Div
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Question
Chapter 9, Problem 9.89RP
To determine

The principal stresses at any point located on the surface of the shaft.

Expert Solution & Answer
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Answer to Problem 9.89RP

The principal stresses at any point located on the surface of the shaft are σ1=10.7MPa andσ2=35.8MPa_ .

Explanation of Solution

Given information:

Rotation of the propeller is ω=15rad/s .

The engine develops the power of 900 kW.

Thrust on the shaft is F=1.23MN .

The outer diameter of the shaft is 250 mm.

Calculation:

Find the power transmission as shown below.

T0=Pω

Here, P is the power transmission and ω is the rotation.

Substitute 900 kW for P and 15rad/s for ω .

T0=900kW×103Nm/s1kW15rad/s=60×103Nm

Sketch the internal torque and forces for the propeller shaft as shown in Figure 1.

Mechanics of Materials, Chapter 9, Problem 9.89RP , additional homework tip  1

Find the section properties as shown below.

Find the cross sectional area as shown below.

Mechanics of Materials, Chapter 9, Problem 9.89RP , additional homework tip  2

Here, d is the diameter of the shaft.

Substitute 250 mm for d.

A=π(250mm×1m1,000mm)24=0.04909m2

Find the polar moment of inertia as shown below.

J=πd432

Substitute 250 mm for d.

J=π(250mm×1m1,000mm)432=3.835×104m4

Find the normal stress as shown below.

σ=NA

Here, N is the normal force and A is the cross sectional area.

Substitute 1.23MN for N and 0.04909m2 for A.

σ=1.23MN×106N1MN0.04909m2=25.056×106N/m2

Convert the unit from N/m2 to MPa.

σ=25.056×106N/m2×1MPa106N/m2=25.06MPa

Find the shear stress as shown below.

τ=TcJ

Here, T is the torque, J is the polar moment of inertia, and c is the centroidal distance between the neutral axis and the extreme fibre.

Substitute 60×103Nm for T, 0.125 m for c, and 3.835×104m4 for J.

τ=60×103×0.1253.835×104=19.557×106N/m2

Convert the unit from N/m2 to MPa.

τ=19.557×106N/m2×1MPa106N/m2=19.56MPa

Sketch the state of stress as shown in Figure 2.

Mechanics of Materials, Chapter 9, Problem 9.89RP , additional homework tip  3

Refer to Figure 2.

σx=25.06MPaσy=0τxy=19.56MPa

Find the in-plane principal stresses as shown below.

σ1,2=σx+σy2±(σxσy2)2+τxy2

Here, σx is the normal stress along x direction, σy is the normal stress along y direction, and τxy is the shear stress.

Substitute 25.06MPa for σx , 0 for σy , and 19.56 MPa for τxy .

σ1,2=25.06+02±(25.06+02)2+19.562=12.53±539.5945=12.53±23.23

The principal stresses are,

σ1=12.53+23.23=10.7MPaσ2=12.5323.23=35.76MPa35.8MPa

Therefore, the principal stresses at any point located on the surface of the shaft are σ1=10.7MPa andσ2=35.8MPa_ .

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Chapter 9 Solutions

Mechanics of Materials

Ch. 9.3 - Determine the stress components acting on the...Ch. 9.3 - Determine the normal stress and shear stress...Ch. 9.3 - Determine the normal stress and shear stress...Ch. 9.3 - 9-6. Determine the normal stress and shear stress...Ch. 9.3 - 9-7. Determine the normal stress and shear stress...Ch. 9.3 - *9-8. Determine the equivalent state of stress on...Ch. 9.3 - 9-9. Determine the equivalent state of stress on...Ch. 9.3 - Determine the equivalent state of stress on an...Ch. 9.3 - Determine the equivalent slate of stress on an...Ch. 9.3 - *9-12. Determine the equivalent state of stress on...Ch. 9.3 - 9-13. Determine the equivalent state of stress on...Ch. 9.3 - 9-14. 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The state of stress at a point in a member...Ch. 9.3 - 9-31. Determine the principal stress at point A on...Ch. 9.3 - 9-32. Determine the maximum in-plane shear stress...Ch. 9.3 - 9-33. The clamp bears down on the smooth surface...Ch. 9.3 - 9-34. Determine the principal stress and the...Ch. 9.3 - 9-35. The square steel plate has a thickness of 10...Ch. 9.3 - *9-36. The square steel plate has a thickness of...Ch. 9.3 - The shaft has a diameter d and is subjected to the...Ch. 9.3 - Prob. 9.38PCh. 9.3 - Prob. 9.39PCh. 9.3 - The wide-flange beam is subjected to the 50-kN...Ch. 9.3 - Solve Pro b. 9-40 for point B located on the web...Ch. 9.3 - Prob. 9.42PCh. 9.3 - Prob. 9.43PCh. 9.4 - Use Mohrs circle to determine the normal stress...Ch. 9.4 - Also, find the corresponding orientation of the...Ch. 9.4 - Draw Mohrs circle and determine the principal...Ch. 9.4 - Determine the principal stresses at a point on the...Ch. 9.4 - Determine the principal stresses at point A on the...Ch. 9.4 - Point A is just below the flange.Ch. 9.4 - Solve Prob.93 using Mohrs circle. 93. Determine...Ch. 9.4 - 9-45. Solve Prob. 9-6 using Mohr’s circle. 9-6....Ch. 9.4 - 9-46. Solve Prob. 9-14 using Mohr’s circle. 9-14....Ch. 9.4 - Solve Prob.911 using Mohrs circle. 911. Determine...Ch. 9.4 - *9-48. Solve Prob. 9-12 using Mohr’s...Ch. 9.4 - Solve Prob.916 using Mohrs circle. 916. Determine...Ch. 9.4 - Mohrs circle for the state of stress is shown in...Ch. 9.4 - Prob. 9.51PCh. 9.4 - Prob. 9.52PCh. 9.4 - 9-53. Determine the equivalent state of stress if...Ch. 9.4 - Prob. 9.54PCh. 9.4 - Prob. 9.55PCh. 9.4 - Prob. 9.56PCh. 9.4 - Determine (a) the principal stresses and (b) the...Ch. 9.4 - 9-58. Determine the equivalent state of stress if...Ch. 9.4 - Prob. 9.59PCh. 9.4 - Prob. 9.60PCh. 9.4 - 9-61. Draw Mohr’s circle that describes each of...Ch. 9.4 - The grains of wood in the board make an angle of...Ch. 9.4 - The post is fixed supported at its base and a...Ch. 9.4 - Determine the principal stresses, the maximum...Ch. 9.4 - The thin-walled pipe has an inner diameter of 0.5...Ch. 9.4 - 9-66. Determine the principal stress and maximum...Ch. 9.4 - Prob. 9.67PCh. 9.4 - The rotor shaft of the helicopter is subjected to...Ch. 9.4 - The pedal crank for a bicycle has the cross...Ch. 9.4 - A spherical pressure vessel has an inner radius of...Ch. 9.4 - The cylindrical pressure vessel has an inner...Ch. 9.4 - Determine the normal and shear stresses at point D...Ch. 9.4 - Determine the principal stress at point D, Which...Ch. 9.4 - If the box wrench is subjected to the 50 lb force,...Ch. 9.4 - If the box wrench is subjected to the 50-lb force,...Ch. 9.4 - Prob. 9.76PCh. 9.5 - Draw the three Mohrs circles that describe each of...Ch. 9.5 - Draw the three Mohrs circles that describe the...Ch. 9.5 - 9-79. The stress at a point is shown on the...Ch. 9.5 - Determine the principal stresses and the absolute...Ch. 9.5 - 9-81. 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