According to the maximum shear stress theory of failure, permissible twisting moment in a circular shaft is 'T'. The permissible twisting moment will the same shaft as per the maximum principal stress theory of failure will be:[IES-1998: ISRO-2008]a)T/2b)Tc)√2Td)2TCorrect answer is option 'D'. Can you explain this answer?

Question

According to the maximum shear stress theory of failure, permissible twisting moment in a circular shaft is 'T'. The permissible twisting moment will the same shaft as per the maximum principal stress theory of failure will be:[IES-1998: ISRO-2008]a)T/2b)Tc)&#8730;2Td)2TCorrect answer is option 'D'. Can you explain this answer?

Answer

Explanation:
The maximum shear stress theory of failure states that failure occurs when the maximum shear stress in a material exceeds the shear strength of the material. The formula for maximum shear stress in a circular shaft is:

τmax = T*c/J

where T is the twisting moment, c is the distance from the center of the shaft to the outer surface, and J is the polar moment of inertia of the shaft.

According to this theory, the permissible twisting moment is T.

The maximum principal stress theory of failure states that failure occurs when the maximum principal stress in a material exceeds the tensile or compressive strength of the material. The formula for maximum principal stress in a circular shaft is:

σmax = M*c/I

where M is the bending moment, c is the distance from the center of the shaft to the outer surface, and I is the moment of inertia of the shaft.

To relate this to twisting moment, we can use the torsion-bending equation:

M = T*(d/2)

where d is the diameter of the shaft.

Substituting this into the formula for maximum principal stress, we get:

σmax = T*c/(π/32)*d^3

The permissible twisting moment according to this theory is when σmax equals the tensile or compressive strength of the material.

Since the maximum shear stress theory gives a lower permissible twisting moment than the maximum principal stress theory, we can use the ratio of their formulas to find the permissible twisting moment according to the maximum principal stress theory:

(T/2)*c/J / (T*c/(π/32)*d^3) = 2/(π*d^2)

Multiplying both sides by T gives:

(T/2)*c/J = (2T*c)/(π*d^3)

Simplifying:

T/2 = π*d^3/(2J)

Multiplying both sides by 2 gives:

T = 2*π*d^3/J

Therefore, the permissible twisting moment according to the maximum principal stress theory is 2T, or option D.

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