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A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be?
explain simply?
explain simply?
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A shaft was initially subjected to bending moment and then was subject...
Understanding Bending Moment and Torque in Shafts
When a shaft experiences both bending moments and torque, it is important to analyze the stresses produced by each type of load.
Key Concepts:
- **Bending Stress (\( \sigma \))**: The maximum bending stress in a shaft can be calculated using the formula:
\[
\sigma = \frac{M \cdot c}{I}
\]
where \( M \) is the bending moment, \( c \) is the distance from the neutral axis to the outermost fiber, and \( I \) is the moment of inertia.
- **Shear Stress (\( \tau \))**: The shear stress due to the torque (\( T \)) is given by:
\[
\tau = \frac{T \cdot r}{J}
\]
where \( r \) is the radius of the shaft, and \( J \) is the polar moment of inertia.
Equal Magnitudes of Bending Moment and Torque:
Given that the bending moment (\( M \)) is equal to the torque (\( T \)), we can denote both as \( P \).
Thus, both stresses can be rewritten as:
- Bending Stress:
\[
\sigma = \frac{P \cdot c}{I}
\]
- Shear Stress:
\[
\tau = \frac{P \cdot r}{J}
\]
Ratio of Maximum Bending Stress to Shear Stress:
To find the ratio of maximum bending stress to shear stress:
\[
\frac{\sigma}{\tau} = \frac{\frac{P \cdot c}{I}}{\frac{P \cdot r}{J}} = \frac{c \cdot J}{r \cdot I}
\]
Conclusion:
The ratio of maximum bending stress to shear stress is determined by the geometric properties of the shaft. For circular shafts, the relationship typically results in:
\[
\frac{\sigma}{\tau} = 3 \text{ (for solid circular shafts)}
\]
This indicates that the maximum bending stress is three times the shear stress when the bending moment equals the torque. Understanding this ratio is crucial for ensuring the shaft's integrity under combined loading conditions.
When a shaft experiences both bending moments and torque, it is important to analyze the stresses produced by each type of load.
Key Concepts:
- **Bending Stress (\( \sigma \))**: The maximum bending stress in a shaft can be calculated using the formula:
\[
\sigma = \frac{M \cdot c}{I}
\]
where \( M \) is the bending moment, \( c \) is the distance from the neutral axis to the outermost fiber, and \( I \) is the moment of inertia.
- **Shear Stress (\( \tau \))**: The shear stress due to the torque (\( T \)) is given by:
\[
\tau = \frac{T \cdot r}{J}
\]
where \( r \) is the radius of the shaft, and \( J \) is the polar moment of inertia.
Equal Magnitudes of Bending Moment and Torque:
Given that the bending moment (\( M \)) is equal to the torque (\( T \)), we can denote both as \( P \).
Thus, both stresses can be rewritten as:
- Bending Stress:
\[
\sigma = \frac{P \cdot c}{I}
\]
- Shear Stress:
\[
\tau = \frac{P \cdot r}{J}
\]
Ratio of Maximum Bending Stress to Shear Stress:
To find the ratio of maximum bending stress to shear stress:
\[
\frac{\sigma}{\tau} = \frac{\frac{P \cdot c}{I}}{\frac{P \cdot r}{J}} = \frac{c \cdot J}{r \cdot I}
\]
Conclusion:
The ratio of maximum bending stress to shear stress is determined by the geometric properties of the shaft. For circular shafts, the relationship typically results in:
\[
\frac{\sigma}{\tau} = 3 \text{ (for solid circular shafts)}
\]
This indicates that the maximum bending stress is three times the shear stress when the bending moment equals the torque. Understanding this ratio is crucial for ensuring the shaft's integrity under combined loading conditions.
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A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be? explain simply?
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A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be? explain simply? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be? explain simply? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be? explain simply?.
A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be? explain simply? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be? explain simply? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A shaft was initially subjected to bending moment and then was subjected to torsion. If the magnitude of the bending moment is found to be the same as that of the torque, then the ratio of maximum bending stress to shear stress would be? explain simply?.
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