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.
3–68* to 3–71*
A countershaft carrying two V-belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tensions shown. The power is transmitted through the shaft and delivered to the belt on pulley B. Assume the belt tension on the loose side at B is 15 percent of the tension on the tight side.
(a) Determine the tensions in the belt on pulley B, assuming the shaft is running at a constant speed.
(b) Find the magnitudes of 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.
Problem 3–68*
The factor of safety for yielding from maximum-shear-stress theory.
The factor of safety for yielding from distortion-energy theory.
Answer to Problem 39P
The factor of safety for yielding from maximum-shear-stress theory is
The factor of safety for yielding from distortion-energy theory is
Explanation of Solution
The Free body diagram of pulley
Figure-(1)
The free body diagram of pulley
Figure-(2)
The given assumption is that the belt tension on the loose side at
Write the relationship between tension on the loose side with respect to tension on the tight side.
Here, the tension on the loose side is
Write the equation to balance the tension on the counter shaft.
Here, the tension on the tight side of pulley
Substitute
Write the tension on the loose side.
Write the magnitude of bearing reaction force at
Here, the magnitude of the bearing force at
Write the magnitude of bearing reaction force at
Here, the magnitude of bearing reaction force at
Write the shear force at
Here, the shear force at
Write the shear force at
Here, the shear force at
Write the shear force at
Here, the shear force at
Write the shear force at
Here, the shear force at
We know that, the moment at the supports of the simply supported beam is zero.
Write the moment at
Here, the moment at
Write the moment at
Here, the moment at
Write the moment at
Here, the moment at
It is clear from the shear force diagram and bending moment diagram of the shaft that the point of maximum bending moment is at
Write the torque acting at pulley
Here, the torque acting on pulley
Write the bending stress.
Here, the bending stress is
Write the shear stress.
Here, the shear stress is
Write the maximum principal stress.
Here, the maximum principal stress is
Calculate the minimum principal stress.
Here, the minimum principal stress is
Write 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
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
The shear force diagram and bending moment diagram for the shaft is as follows.
Figure-(3)
Substitute
Substitute
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” to obtain the yield strength of
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
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