Torsion of circular shafts – shear stress at the centre What is the shear stress at the centre of a solid circular shaft subjected to torsion?

Introduction / Context:
In torsion of circular shafts, shear stress varies linearly with radius. Knowing where the maximum and minimum shear occur is vital for design against yielding and fatigue.

Given Data / Assumptions:

• Solid circular shaft under torque T.
• Linear elastic behavior; Saint-Venant torsion theory.
• r = radial distance from the centre; R = outer radius.

Concept / Approach:
The torsion formula states τ(r) = T r / J, where J is the polar second moment of area (J = π R^4 / 2 for a solid circle). Shear stress is directly proportional to r, so it increases from the centre to the surface.

Step-by-Step Reasoning:
At r = 0 (the centre): τ(0) = T * 0 / J = 0.At r = R (outer surface): τ_max = T R / J, which is the maximum value.Therefore, the centre experiences zero shear stress under pure torsion.

Verification / Alternative check:
Shear strain distribution is γ(r) = r θ / L, also zero at the centre and maximum at the surface. Since τ = G * γ, τ follows the same radial linearity.

Why Other Options Are Wrong:
“Minimum but non-zero” conflicts with τ(0) = 0; “maximum” and “infinity” contradict the linear relation; “equal to average shear V / A” is a beam shear concept, unrelated to torsion at the centre.

Common Pitfalls:
Confusing torsional shear distribution with transverse shear in beams; forgetting that τ varies with radius in torsion.

Final Answer:
Zero