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Bending moment M and torque T are applied on a solid circular shaft. If the maximum bending stress equals maximum shear stress developed, then M is equal to
  • a)
    T/2
  • b)
    T
  • c)
    2T
  • d)
    4T
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Bending moment M and torque T are applied on a solid circular shaft. ...
Bending stress due to bending moment
σ = 32M/πd3
Shear stress due to twisting moment/ torque:
τ = 16T/πd3
If the maximum bending stress equals maximum shear stress developed:
Most Upvoted Answer
Bending moment M and torque T are applied on a solid circular shaft. ...
Explanation:

The maximum bending stress and maximum shear stress occur at different points in a circular shaft.

When a bending moment is applied to a circular shaft, it causes the shaft to bend. This bending creates tensile and compressive stresses on the cross-section of the shaft. The maximum bending stress occurs at the outermost fibers of the shaft, farthest from the neutral axis.

When a torque is applied to a circular shaft, it causes the shaft to twist. This twisting creates shear stresses on the cross-section of the shaft. The maximum shear stress occurs at the outermost fibers of the shaft, perpendicular to the axis of the shaft.

Bending Moment and Shear Stress:

- Bending moment (M) is the product of the applied force and the distance from the axis of rotation.
- Torque (T) is the product of the applied force and the radius of the shaft.

In a circular shaft, the maximum bending stress (σ_b) is given by the formula:
σ_b = (M * c) / I

Where:
- M is the bending moment applied to the shaft
- c is the distance from the neutral axis to the outermost fiber of the shaft
- I is the moment of inertia of the shaft cross-section

The maximum shear stress (τ) is given by the formula:
τ = (T * r) / J

Where:
- T is the torque applied to the shaft
- r is the radius of the shaft
- J is the polar moment of inertia of the shaft cross-section

Equalizing the Maximum Bending Stress and Maximum Shear Stress:

To determine when the maximum bending stress equals the maximum shear stress, we can equate the formulas for bending stress and shear stress.

σ_b = τ

Substituting the formulas for bending stress and shear stress, we get:
(M * c) / I = (T * r) / J

Simplifying the equation, we find:
(M * c * J) = (T * r * I)

Since c, J, r, and I are constants for a given shaft, we can further simplify the equation:
M = T * (r * I) / (c * J)

From the equation, we can see that the bending moment (M) is proportional to the torque (T). Therefore, if the maximum bending stress equals the maximum shear stress, then M is equal to T divided by a constant factor.

Conclusion:

In this case, the correct answer is option 'A', M is equal to T/2.
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Bending moment M and torque T are applied on a solid circular shaft. If the maximum bending stress equals maximum shear stress developed, then M is equal toa)T/2b)Tc)2Td)4TCorrect answer is option 'A'. Can you explain this answer?
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