Calculate polar moment of inertia J for solid/hollow shafts, square and rectangular tubes. Compute shear stress, twist angle, and torsional stiffness.
The polar moment of inertia (J) quantifies a cross-section's resistance to torsional deformation. It is the key property for shaft design: higher J means lower shear stress and less twist for a given torque.
For a solid circular shaft, J = πd⁴/32. For a hollow shaft, J = π(d₀⁴ − dᵢ⁴)/32. Hollow shafts are remarkably efficient — a tube with the same weight as a solid shaft can have much higher J because material is distributed far from the center.
This calculator computes J for five common cross-sections: solid circle, hollow circle, solid square, square tube, and rectangular tube. It then uses J to calculate maximum shear stress (τ = Tc/J), angle of twist (θ = TL/GJ), and torsional stiffness (GJ/L). A weight comparison shows the structural efficiency of each shape.
Presets cover common shaft and tube sizes. The formula reference table lists J equations for all shapes, making this tool essential for mechanical design, structural engineering, and machine element analysis.
Shaft and tube torsion analysis is fundamental to mechanical design. This calculator provides instant J values and stress analysis for the five most common cross-sections.
It eliminates tedious hand calculations and provides a weight-efficiency comparison that helps optimize shaft design. The note above highlights common interpretation risks for this workflow. Use this guidance when comparing outputs across similar calculators. Keep this check aligned with your reporting standard.
Solid circle: J = πd⁴/32. Hollow circle: J = π(d₀⁴ − dᵢ⁴)/32. Shear stress: τ_max = Tc/J (c = distance to outer fiber). Twist angle: θ = TL/(GJ). Torsional stiffness: k = GJ/L.
Result: J = 533,146 mm⁴, τ = 23.5 MPa, θ = 0.068°
J = π(0.05⁴ − 0.03⁴)/32 = 5.33×10⁻⁷ m⁴. τ = 500 × 0.025 / 5.33×10⁻⁷ = 23.5 MPa. θ = 500 × 1 / (79×10⁹ × 5.33×10⁻⁷) = 0.0012 rad = 0.068°.
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I (area moment of inertia) resists bending about a single axis. J (polar moment) resists torsion and equals Ix + Iy for any cross-section. For circular sections, J = 2I.
Material near the center contributes little to J (r⁴ weighting), but adds weight. Moving material to the outside increases J dramatically. A hollow shaft with 60% bore can have 87% of the J at 64% of the weight.
Non-circular sections experience warping under torsion. For squares, rectangles, and open sections (I-beams, channels), the actual behavior is more complex than τ = Tc/J. This calculator uses appropriate approximations.
Steel: 79-82 GPa. Aluminum: 26 GPa. Copper: 44 GPa. Titanium: 42 GPa. Cast iron: 41 GPa. The shear modulus relates to Young's modulus by G = E/(2(1+ν)).
T_max = τ_allow × J / c. For steel shafts, typical allowable shear stress is 40-80 MPa (with safety factor). For dynamic/fatigue loads, use 20-40 MPa.
When torque combines with bending, use equivalent torque: T_eq = √(M² + T²) for maximum shear stress theory, or T_eq = √(M² + 0.75T²) for Von Mises.