What is the difference between J and I?

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.

Why are hollow shafts more efficient?

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.

When does the J formula not apply?

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.

What shear modulus should I use?

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)).

How do I find maximum allowable torque?

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.

What about combined loading?

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.

Polar Moment of Inertia Calculator

Calculate polar moment of inertia J for solid/hollow shafts, square and rectangular tubes. Compute shear stress, twist angle, and torsional stiffness.

Cross-Section Shape

Outer Diameter (mm)

Applied Torque (N·m)

Shaft Length (mm)

Shear Modulus G (GPa)

Polar Moment J

613,592.3 mm⁴

6.1359e-7 m⁴

Max Shear Stress

20.37 MPa

τ = Tc/J

Angle of Twist

0.5910°

0.010315 rad

Torsional Stiffness

48,474 N·m/rad

GJ/L

Weight (steel)

15.413 kg/m

ρ = 7850 kg/m³

J per Weight

39.8 mm⁴/(g/m)

Structural efficiency

Shear Stress (typical yield shear ≈ 200 MPa steel)

20.4 / 200 MPa

Polar Moment Formulas by Shape

Shape	Formula	Notes
Solid Circle	J = πd⁴/32	Most efficient for torsion
Hollow Circle	J = π(d₀⁴ − dᵢ⁴)/32	Much more efficient per weight
Solid Square	J = a⁴/6 (approx)	Less efficient than circle
Thin-walled Circle	J ≈ 2πR³t	For t << R
Thin-walled Rectangle	J ≈ 2t(b−t)²(h−t)² / (b+h−2t)	Approximation

Planning notes, formulas, and examples

About the Polar Moment of Inertia Calculator

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 calculator essential for mechanical design, structural engineering, and machine element analysis.

When This Page Helps

Shaft and tube torsion analysis is fundamental to mechanical design. It computes 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.

How to Use the Inputs

Select the cross-section shape from the dropdown.
Enter the dimensions (diameter, width, wall thickness) in millimeters.
Enter the applied torque in Newton-meters.
Enter the shaft length and shear modulus (79 GPa for steel).
Read J, maximum shear stress, angle of twist, and torsional stiffness.
Compare the J-to-weight ratio for structural efficiency.

Formula used

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.

Example Calculation

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°.

Tips & Best Practices

A 50mm hollow shaft with 30mm bore has 87% of the J of a solid 50mm shaft at only 64% the weight.
For minimum-weight design at a given J, the optimal wall thickness is approximately R/10.
The angle of twist in degrees is very small for most practical shafts — if θ > 1°/m, the shaft is probably too slender.
Keyways reduce J by roughly 25% — account for this in shaft design.
For non-circular sections, use the Bredt-Batho formula for thin-walled closed sections.

When To Use This Calculator

Calculate polar moment of inertia J for solid/hollow shafts, square and rectangular tubes. Compute shear stress, twist angle, and torsional stiffness. Use it when you need a repeatable calculation in the physics / general category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.

How To Check The Result

Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.

Practical Notes

Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

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.