Polar Moment of Inertia Calculator
A polar moment of inertia calculator finds J = π × diameter⁴ ÷ 32 for a solid round shaft: a 2 in diameter shaft has J of about 1.571 in⁴.
Polar moment of inertia calculator formula
A polar moment of inertia calculator gives J for a solid round shaft from its diameter. The formula is J = π × d⁴ ÷ 32, where d is the diameter in inches and J comes out in inches to the fourth power (in⁴).
The calculator takes one input, the shaft diameter. The default of 2 inches gives J = π × 16 ÷ 32 = 1.571 in⁴, shown as 1.57 in⁴.
Worked examples
Example 1 is a solid steel shaft, 2 inches in diameter.
- d⁴ = 2 × 2 × 2 × 2 = 16
- π × 16 = 50.27
- 50.27 ÷ 32 = 1.571 in⁴
Example 2 is a larger shaft, 3 inches in diameter.
- d⁴ = 3 × 3 × 3 × 3 = 81
- π × 81 = 254.47
- 254.47 ÷ 32 = 7.95 in⁴
The second shaft is 50 percent wider, and its J is 5.06 times higher (1.5⁴ = 5.0625). That fourth-power rule is the most useful thing to remember about this property.
Using J in a torsion check
J on its own is only a section property. It gets used in two formulas.
- Shear stress at the surface: τ = T × r ÷ J, where T is torque in lb·in and r is the radius in inches.
- Angle of twist: θ = T × L ÷ (G × J), in radians, where L is the shaft length and G is the shear modulus of the material.
Say a shaft carries 1,000 lb·in of torque.
- On the 2 in shaft, τ = 1,000 × 1 ÷ 1.571 = 636.6 psi
- On the 3 in shaft, τ = 1,000 × 1.5 ÷ 7.95 = 188.6 psi
The thicker shaft carries the same torque at about 30 percent of the stress. For the twist, steel has a shear modulus of roughly 11.5 million psi. A 2 in steel shaft 48 inches long under 1,000 lb·in twists about 0.15 degrees (48,000 ÷ (11,500,000 × 1.571) = 0.00266 radians).
These are textbook formulas. Real shafts also see bending, keyways, shock loads and fatigue, so a design needs a safety factor and a review by a licensed professional engineer.
J for common solid round shafts
| Diameter | J (in⁴) |
|---|---|
| 1 in | 0.10 |
| 1.5 in | 0.50 |
| 2 in | 1.57 |
| 2.5 in | 3.83 |
| 3 in | 7.95 |
| 4 in | 25.13 |
| 6 in | 127.23 |
Going from 1 inch to 2 inches multiplies J by 16. A shaft of 0.5 inch has J of 0.006 in⁴, which the calculator shows as 0.01 since it rounds to two places.
How to measure the shaft
- Use calipers or a micrometer on the bare shaft. Paint and rust change the reading.
- Measure at a few spots along the length. A worn shaft is smaller where the bearing rides.
- Use the smallest diameter along the stressed length. That is where stress is highest.
- If a shaft has a keyway or a spline, the effective diameter is smaller than the nominal one. Ask the shaft maker for guidance.
- For a hollow tube, measure the outside diameter and the wall thickness, then work out the inside diameter.
Common mistakes
- Using radius where the formula wants diameter. The formula above is written with the diameter. If you have the radius r, J = π × r⁴ ÷ 2.
- Mixing up J with the section modulus, which is a different property used in stress formulas.
- Squaring or cubing the diameter instead of raising it to the fourth power.
- Using the formula for square or hexagonal shafts. It applies only to round cross-sections.
- Treating a calculated stress as safe without checking the material’s yield strength and a safety factor. That is a job for an engineer.
Related calculators
For bending resistance, use the moment of inertia calculator. Once you have J, the shear stress calculator turns a load into stress, and the bending stress calculator covers the other main case.
Frequently asked questions
What is polar moment of inertia used for?
It measures how well a round cross-section resists twisting. You use it in the torsion formulas for shear stress, which is torque × radius ÷ J, and for angle of twist, which is torque × length ÷ (shear modulus × J). Drive shafts, axles and PTO shafts are common examples.
How is polar moment of inertia different from regular moment of inertia?
Regular moment of inertia (I) measures resistance to bending about one axis. Polar moment (J) measures resistance to twisting about the shaft's center line. For a solid circle, J is exactly twice I, so a 2 in shaft has I of 0.785 in⁴ and J of 1.571 in⁴.
Why does shaft diameter matter so much for torsional stiffness?
Because J grows with the fourth power of the diameter. A 3 in shaft is 1.5 times the diameter of a 2 in shaft, but its J is 5.06 times larger (7.95 vs 1.57 in⁴). A small increase in diameter buys a big gain in twisting stiffness.
Does this formula work for hollow shafts?
No, this calculator is for solid round shafts. For a hollow tube, J = π × (D⁴ − d⁴) ÷ 32, where D is the outside diameter and d is the inside diameter. Subtract the missing core from the solid shaft's J. Non-round sections, such as squares and channels, need different formulas.
Estimates only. Verify quantities with your supplier before purchasing.