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Strain energy in a solid circular shaft under torsion: expression in terms of τ and C A solid circular shaft is subjected to torsion; let τ denote maximum shear stress and C the shear modulus (modulus of rigidity). Express the total strain energy stored in the shaft in the form (factor) × Volume of shaft. Choose the correct factor.

Difficulty: Medium

τ² / (2 C)
τ² / (3 C)
τ² / (4 C)
τ² / C

Correct Answer: τ² / (4 C)

Explanation:

Approach (energy method)
Total strain energy U = ∫ (τ² / (2 C)) dV. For a solid circular shaft in pure torsion, τ varies linearly with radius: τ(r) = τ_max(r/R).

Step-by-step integration
Let R = outer radius, L = length. dV = 2π r L dr.U = ∫₀^R [ (τ_max² r² / R²) / (2 C) ] · (2π r L dr )= (τ_max² /(2 C R²)) · 2π L ∫₀^R r³ dr= (τ_max² /(2 C R²)) · 2π L · (R⁴/4)= (τ_max² /(4 C)) · (π R² L)= (τ² /(4 C)) × Volume.

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