Concept explainers
For the problem specified in the table, build upon the results of the original problem to determine the minimum factor of safety for yielding. Use both the maximum-shear-stress theory and the distortion-energy theory, and compare the results. The material is 1018 CD steel.
3–72* to 3–73*
A gear reduction unit uses the countershaft shown in the figure. Gear A receives power from another gear with the transmitted force FA applied at the 20° pressure angle as shown. The power is transmitted through the shaft and delivered through gear B through a transmitted force FB at the pressure angle shown.
(a) Determine the force FB, assuming the shaft is running at a constant speed.
(b) Find the bearing reaction forces, assuming the bearings act as simple supports.
(c) Draw shear-force and bending-moment diagrams for the shaft. If needed, make one set for the horizontal plane and another set for the vertical plane.
(d) At the point of maximum bending moment, determine the bending stress and the torsional shear stress.
(e) At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.
The factor of safety for yielding from distortion-energy theory.
The factor of safety for yielding from maximum-shear-stress theory.
Answer to Problem 43P
The factor of safety for yielding from distortion-energy theory is
The factor of safety for yielding from maximum-shear-stress theory is
Explanation of Solution
The figure below shows the free body diagram of pulley A.
Figure (1)
The figure below shows the free body diagram of pulley B.
Figure (2)
Calculate the force
Here, the force acting on pulley
Write the moment about bearing
Here, the reaction force at bearing
Write the equation to balance the forces in
Here, the reaction force at bearing
Write the moment about bearing
Here, the reaction force at bearing
Write the equation to balance the forces in
Here, the reaction force at bearing
Calculate the reaction forces at bearing
Here, the reaction force at bearing
Calculate the reaction forces at bearing
Here, the reaction force at bearing
The calculations for shear force and bending moment diagram in
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
The calculations for shear force and bending moment diagram in
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
Write the net moment at
Here, the net moment at
Write the net moment at
Here, the net moment at
Write the torque transmitted by shaft from
Here, the torque transmitted by shaft from
Calculate the bending stress.
Here, the bending stress is
Calculate the shear stress.
Here, the shear stress is
Calculate the maximum principal stress.
Here, the maximum principal stress is
Calculate the minimum principal stress.
Here, the minimum principal stress is
Calculate the maximum shear stress.
Here, maximum shear stress is
Calculate the factor of safety from maximum-shear-stress theory.
Here, the maximum yield stress for
Calculate the factor of safety from distortion-energy theory.
Here, the Von Mises stress is
Write the expression for von Mises stress.
Substitute
Conclusion:
Substitute
Thus, the force
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Substitute
The shear force and bending moment diagram in
Figure-(3)
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the shear force and bending moment diagram in
Figure-(4)
Substitute
Substitute
Since,
Substitute
Substitute
Thus, the bending stress at point of maximum bending moment is
Substitute
Substitute
Substitute
Substitute
Refer to the Table A-20 “Deterministic ASTM Minimum Tensile and Yield Strengths for Some Hot-Rolled (HR) and Cold-Drawn (CD) Steels” and obtain the yield strength as
Substitute
Thus, the factor of safety for yielding from maximum-shear-stress theory is
Substitute
Thus, the factor of safety for yielding from distortion-energy theory is
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Chapter 5 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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