Mechanics of Materials
Polar Moment of Inertia Simulator
A variable-controlled mechanics laboratory studying shaft dimensions, J calculations, shear stresses under applied torque loads, materials, and twisting profiles.
Shaft Profile
Hollow Circular
Shear profile
Nodes Simulated
41 diameter steps
Torsional curve
Output
CSV
Data exportable
Parameters
Material Preset
Animated torque sweep
Governing Equations
Polar Moment (Solid Shaft)
J = π * D⁴ / 32
Polar Moment (Hollow Shaft)
J = π * (D⁴ - d⁴) / 32
Torsional Shear Stress
τ = T * r / J
Torsional Angle of Twist
θ = T * L / (G * J)
With an applied torque of 1000 N*m, the maximum shear stress of 5.43 MPa occurs at the outer surface (r = 50.0 mm).
For hollow shafts, setting an inner diameter d removes material from the center where shear stress is lowest, improving structural weight efficiency.
Linked Diagram
Polar Moment (J)
9203884.7273mm⁴
Max Shear Stress
5.43MPa
Cross-sectional Area
5890.5mm²
Angle of Twist
0.156degrees
Stress and Stiffness vs Outer Diameter
Simulation Performance Table
| Outer Dia (mm) | Polar Moment J (mm⁴) | Cross Area (mm²) | Max Stress (MPa) |
|---|---|---|---|
| 55 mm | 2.848e+5 | 412.3 | 96.57 |
| 75 mm | 2.493e+6 | 2454.4 | 15.04 |
| 95 mm | 7.383e+6 | 5124.7 | 6.43 |
| 115 mm | 1.656e+7 | 8423.4 | 3.47 |
| 135 mm | 3.200e+7 | 12350.4 | 2.11 |
| 155 mm | 5.605e+7 | 16905.7 | 1.38 |
| 175 mm | 9.146e+7 | 22089.3 | 0.96 |
| 195 mm | 1.413e+8 | 27901.3 | 0.69 |
| 215 mm | 2.092e+8 | 34341.5 | 0.51 |
| 235 mm | 2.988e+8 | 41410.1 | 0.39 |
| 255 mm | 4.145e+8 | 49107.0 | 0.31 |