Sizing Shafts by Torque and Deflection Limits

When selecting a shaft diameter, you typically start with torque. Use the torsion formula τ = T*c/J to find a diameter that keeps shear stress below the material's yield strength. For a solid circular shaft, J = π*d^4/32 and c = d/2, so d = (16*T/(π*τ_allow))^(1/3). This ensures the shaft won't fail under static torque.

But torque alone isn't enough. Many shafts also need to limit angular deflection to prevent gear misalignment or bearing issues. The angle of twist θ = T*L/(G*J) must stay below a design limit, often 0.25 to 1 degree per foot. Rearranging gives d = (32*T*L/(π*G*θ))^(1/4). Notice the fourth root – deflection tends to govern for longer shafts.

Always check both criteria. Calculate the required diameter from stress, then check deflection. If deflection exceeds the limit, increase diameter to satisfy the deflection equation. For short, heavily loaded shafts, stress often drives the size. For long, lightly loaded shafts, deflection is the constraint.

A practical example: A 12-inch shaft transmitting 5000 in-lb with a 48-inch length, using steel (G=11.5e6 psi, τ_allow=8000 psi). Stress gives d=1.59 inches. Deflection at 0.5 deg total gives d=2.12 inches. Here deflection governs. Use 2.25 inches for standard stock. Always pick the larger diameter to cover both limits.

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