As per the maximum principal stress theory, when a shaft is subjected to a bending moment M and torque T and if σ is the allowable stress in axial tension, then the radius “r” of the shaft is given by
- Ar³ = (2/πσ) [M + √(M² + T²)]Correct
- Br³ = (4/πσ) [M + √(M² + T²)]
- Cr³ = (1/2πσ) [M + √(M² + T²)]
- Dr³ = (1/πσ) [M + √(M² + T²)]
Solution
Concept
Maximum Principal Stress Theory for Shafts
Formula
r³ = (2/πσ) [M + √(M² + T²)]
Step-by-step solution
- Calculate equivalent bending moment using Me = 0.5 \* \[M + √(M² + T²)\] - Apply maximum bending stress formula σ = 4 \* Me / (π \* r³) - Substitute Me into the stress equation to get σ = (2 / (π \* r³)) \* \[M + √(M² + T²)\] - Rearrange the equation to solve for r³ = (2 / πσ) \* \[M + √(M² + T²)\]
Explanation
According to the maximum principal stress theory, the equivalent bending moment on a shaft is given by Me = 0.5 × [M + √(M² + T²)]. The maximum bending stress σ induced in a solid circular shaft of radius r is calculated as σ = 4 × Me / (π × r³). Substituting Me into this stress equation gives σ = (2 / (π × r³)) × [M + √(M² + T²)]. Rearranging the equation to solve for r³ gives r³ = (2 / πσ) × [M + √(M² + T²)].
Shortcut
To quickly remember this formula, use these key memory hooks: - Factor Rule: For radius (r³), the constant multiplier is 2 (i.e., 2/πσ). For diameter (d³), the constant multiplier is 16 (i.e., 16/πσ). - Equivalent Bending Moment: Remember that Rankine's theory relies on equivalent bending moment Me = 1/2 \[M + √(M² + T²)\]. - Derive in 2 steps: Bending stress σ = Me / Z. Since section modulus Z = πr³/4, dividing Me by Z flips 1/2 over 1/4 to give 2/πσ.
Common mistake
Students often confuse the formula written in terms of diameter d with radius r, or forget the 1/2 factor in the equivalent bending moment Me = 1/2[M + √(M² + T²)], leading them to incorrectly select option B or D instead of accounting for section modulus Z = πr³/4.
Correct answer: A) r³ = (2/πσ) [M + √(M² + T²)]