Ratio of Maximum Bending Stress to Shear Stress in a Shaft Subjected to Bending Moment and Torsion

When a shaft is subjected to bending moment and then to torsion, the maximum bending stress and the maximum shear stress occur at different points on the cross-section of the shaft. The maximum bending stress occurs at the outermost fiber of the shaft, where the bending moment is maximum. On the other hand, the maximum shear stress occurs at the neutral axis of the shaft, where the torsional moment is maximum.

If the magnitude of bending moment is found to be the same as that of torque, then the ratio of maximum bending stress to shear stress can be calculated as follows:

Let M be the magnitude of bending moment and T be the magnitude of torque.

The maximum bending stress in the shaft is given by:

σ_b = M*y/I

where y is the distance from the neutral axis to the outermost fiber of the shaft and I is the moment of inertia of the shaft cross-section.

The maximum shear stress in the shaft is given by:

τ_max = T*r/J

where r is the radius of the shaft and J is the polar moment of inertia of the shaft cross-section.

Since M = T, we can equate the expressions for σ_b and τ_max and obtain:

σ_b/τ_max = (M*y/I)/(T*r/J)

Simplifying this expression, we get:

σ_b/τ_max = (M/I)/(T/r*J)

Since M/I is the bending stress factor (f_b) and T/r*J is the torsion stress factor (f_t), we can write:

σ_b/τ_max = f_b/f_t

For a circular shaft, the torsion stress factor can be calculated as:

f_t = 16/π*d^3

where d is the diameter of the shaft.

Substituting the values of f_b and f_t, we get:

σ_b/τ_max = (32/π)*d^3/(M*y)

For M = T, the above expression simplifies to:

σ_b/τ_max = (16/π)*d^3/(M*y)

Therefore, if the magnitude of bending moment is found to be the same as that of torque, the ratio of maximum bending stress to shear stress in the shaft is given by:

σ_b/τ_max = 2*(d/y)^3

where d/y is the ratio of the diameter to the distance from the neutral axis to the outermost fiber of the shaft.

Thus, the ratio of maximum bending stress to shear stress is proportional to the cube of the diameter to distance ratio. If this ratio is large, the shaft is more prone to failure due to bending stress than due to shear stress. If this ratio is small, the shaft is more prone to failure due to shear stress than due to bending stress.

We have formula for circular area(shaft has circular area) 1) sigma=M/Z. Then Z=I/y.. So sigma= 32*M/(pi*d*d*d) 2) shear stress= 16T/(pi*d*d*d) Now sigma/shear stress=2*M/T But given M=T So answer is 2