
Concept explainers
The factor of safety for yielding from distortion-energy theory.
The factor of safety for yielding from maximum-shear-stress theory.

Answer to Problem 45P
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 arrangement of shafts.
Figure (1)
The free body diagram of the arrangement of shafts is as follows.
Figure (2)
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
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 of maximum bending stress on the critical stress element.
Here, the maximum bending stress on the critical stress element is
Write the expression of principal stresses on the critical stress element.
Here, the principal stresses on the critical stress element are
Write the expression of maximum shear stress on the critical stress element.
Here, the maximum shear stress on the critical stress element 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 reaction at
Substitute
Thus, the reaction at
Substitute
Thus, the reaction at
Substitute
Thus, the reaction at
Substitute
Thus, the reaction at
Substitute
Substitute
Substitute
Substitute
Substitute
The figure below shows the shear force and bending moment diagram in
Figure (3)
Substitute
Substitute
Substitute
Substitute
The figure below shows the shear force and bending moment diagram in
Figure (4)
Substitute
Substitute
Substitute
Thus, the torsional shear stress for critical stress element is
Substitute
Thus, the bending stress for critical stress element is
Substitute
Thus, the axial stress for critical stress element is
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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