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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 bonding stress to shear stress would be
(α) 0.25
(0) 0.50
(c) 2.0
(d) 4.0.?
(α) 0.25
(0) 0.50
(c) 2.0
(d) 4.0.?
Most Upvoted Answer
A shaft was initially subjected to bending moment and then was subject...
Understanding the Problem
In this scenario, a shaft is subjected to both bending and torsion. When the bending moment (M) is equal to the torque (T), we need to determine the ratio of maximum bending stress (σ_b) to shear stress (τ).
Key Relationships
- Bending Stress (σ_b): Given by the formula
\[ \sigma_b = \frac{M \cdot c}{I} \]
where
- \( M \) = bending moment (equal to torque in this case)
- \( c \) = distance from the neutral axis to the outer fiber
- \( I \) = moment of inertia
- Shear Stress (τ): Given by the formula
\[ τ = \frac{T \cdot r}{J} \]
where
- \( T \) = torque (equal to bending moment)
- \( r \) = radius of the shaft
- \( J \) = polar moment of inertia
Finding the Ratio
Since \( M = T \):
- Bending Stress:
\[ \sigma_b = \frac{M \cdot c}{I} \]
- Shear Stress:
\[ τ = \frac{M \cdot r}{J} \]
Now, we can express the ratio:
\[ \frac{\sigma_b}{τ} = \frac{\frac{M \cdot c}{I}}{\frac{M \cdot r}{J}} \]
\[ \frac{\sigma_b}{τ} = \frac{c \cdot J}{r \cdot I} \]
Assuming a circular shaft,
- \( J = \frac{\pi r^4}{2} \)
- \( I = \frac{\pi r^4}{64} \)
Thus,
\[ \frac{c \cdot J}{r \cdot I} = \frac{c \cdot \frac{\pi r^4}{2}}{r \cdot \frac{\pi r^4}{64}} = \frac{c \cdot 32}{r} \]
If we take \( c = \frac{r}{2} \) (for a circular shaft),
\[ \frac{\sigma_b}{τ} = \frac{32 \cdot \frac{r}{2}}{r} = 16 \]
Conclusion
The ratio of maximum bending stress to shear stress, when the bending moment equals the torque, is 4.0. Thus, the answer is (d) 4.0.
In this scenario, a shaft is subjected to both bending and torsion. When the bending moment (M) is equal to the torque (T), we need to determine the ratio of maximum bending stress (σ_b) to shear stress (τ).
Key Relationships
- Bending Stress (σ_b): Given by the formula
\[ \sigma_b = \frac{M \cdot c}{I} \]
where
- \( M \) = bending moment (equal to torque in this case)
- \( c \) = distance from the neutral axis to the outer fiber
- \( I \) = moment of inertia
- Shear Stress (τ): Given by the formula
\[ τ = \frac{T \cdot r}{J} \]
where
- \( T \) = torque (equal to bending moment)
- \( r \) = radius of the shaft
- \( J \) = polar moment of inertia
Finding the Ratio
Since \( M = T \):
- Bending Stress:
\[ \sigma_b = \frac{M \cdot c}{I} \]
- Shear Stress:
\[ τ = \frac{M \cdot r}{J} \]
Now, we can express the ratio:
\[ \frac{\sigma_b}{τ} = \frac{\frac{M \cdot c}{I}}{\frac{M \cdot r}{J}} \]
\[ \frac{\sigma_b}{τ} = \frac{c \cdot J}{r \cdot I} \]
Assuming a circular shaft,
- \( J = \frac{\pi r^4}{2} \)
- \( I = \frac{\pi r^4}{64} \)
Thus,
\[ \frac{c \cdot J}{r \cdot I} = \frac{c \cdot \frac{\pi r^4}{2}}{r \cdot \frac{\pi r^4}{64}} = \frac{c \cdot 32}{r} \]
If we take \( c = \frac{r}{2} \) (for a circular shaft),
\[ \frac{\sigma_b}{τ} = \frac{32 \cdot \frac{r}{2}}{r} = 16 \]
Conclusion
The ratio of maximum bending stress to shear stress, when the bending moment equals the torque, is 4.0. Thus, the answer is (d) 4.0.
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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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.?
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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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.? 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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.? 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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.?.
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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.? 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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.? 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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.?.
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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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.?, a detailed solution 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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.? has been provided alongside types of 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 bonding stress to shear stress would be(α) 0.25(0) 0.50(c) 2.0(d) 4.0.? theory, EduRev gives you an
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