Concept explainers
6-37* to 6-46* For the problem specified in the table, build upon the results of the original problem to determine the minimum factor of safety for fatigue based on infinite life, using the modified Goodman criterion. The shaft rotates at a constant speed, has a constant diameter, and is made from cold-drawn AISI 1018 steel.
Problem Number | Original Problem, Page Number |
6-43* | 3–74, 152 |
3-74* In the figure, shaft AB transmits power to shaft CD through a set of bevel gears contacting at point E. The contact force at E on the gear of shaft CD is determined to be (FE)CD = –92.8i – 362.8j + 808.0k lbf. For shaft CD: (a) draw a free-body diagram and determine the reactions at C and D assuming simple supports (assume also that bearing C carries the thrust load), (b) draw the shear-force and bending-moment diagrams, (c) for the critical stress element, determine the torsional shear stress, the bending stress, and the axial stress, and (d) for the critical stress element, determine the principal stresses and the maximum shear stress.
The minimum factor of safety for fatigue based on infinite life.
Answer to Problem 43P
The minimum factor of safety for fatigue based on infinite life is
Explanation of Solution
The free body diagram of the arrangement of shafts is shown in the figure below.
Figure (1)
Write the expression of moment at
Here, the reaction at
Write the expression of moment at
Here, the reaction at
Write the expression of moment at
Here, the reaction at
Write the expression of moment at
Write the expression of net force at
Here, the reaction at
It is clear from the free body diagram of the shaft
The calculations for shear force diagram in
Write the expression of Shear force at
Here, the shear at
Write the expression of Shear force at
Here, the shear force at
Write the expression of Shear force at
Here, the shear force at
The calculations for bending moment diagram in
We known that, the bending moment at the supports of the simply supported beam is zero.
Write the bending moment at
Here, the bending moment at
Write the expression of bending moment at
Here, the bending moment at
Write the expression of bending moment at
Here, the bending moment at
The calculations for shear force diagram in
Write the expression of Shear force at
Here, the shear at
Write the expression of Shear force at
Here, the shear force at
Write the expression of Shear force at
Here, the shear force at
We known that, the bending moment at the supports of the simply supported beam is zero.
Write the bending moment at
Here, the bending moment at
Write the expression of bending moment at
Here, the bending moment at
It is clear from the bending moment diagram that the critical stress element is located at just right of
Write the expression of maximum torque acting on the shaft
Here, the maximum torque acting on the shaft
Write the expression of maximum bending moment acting on the shaft
Here, the maximum bending moment acting on the shaft
Write the expression of torsional shear stress for critical stress element.
Here, the torsional shear stress for critical stress element is
Write the expression of bending stress for critical stress element.
Here, the bending stress for critical stress element is
Write the expression of axial stress for critical stress element.
Here, the axial stress for critical stress element is
Write the expression for von Mises alternating stress.
Here, the amplitude component of the axial stress is
Write the expression for von Mises mid-range stress.
Here, the mid-range component of the axial stress is
Write the expression for von Mises maximum stress.
Here, the maximum component of the axial stress is
Write the expression for yield factor of safety.
Here, the yield strength of the material is
Write the expression for endurance limit for test specimen.
Here, the minimum tensile strength is
Write the surface factor for the countershaft.
Here, the constants for surface factor are
Write the size factor for the countershaft.
Write the endurance limit at the critical location of the machine part.
Write the modified Goodman equation.
Here, the fatigue factor of safety is
Conclusion:
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The figure below shows the shear force and bending moment diagram in
Figure-(2)
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The figure below shows the shear force and bending moment diagram in
Figure (3)
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Write the expression for the maximum stress.
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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 as
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Refer to Table 6-2 “Parameters for Marin Surface Modification Factor” to obtain
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Thus, the minimum factor of safety for fatigue based on infinite life is
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Chapter 6 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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