Bending moment M and torque T are applied on a solid circular shaft. ...
Bending stress due to bending moment
σ = 32M/πd^{3}
Shear stress due to twisting moment/ torque:
τ = 16T/πd^{3}
If the maximum bending stress equals maximum shear stress developed:
View all questions of this test
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.