Parallel Axis Theorem (Steiner's Theorem)
Computes the moment of inertia of any structural area or rigid body about any parallel axis displaced by distance d from the centroidal axis.
\[ I_z = I_{cm} + A \cdot d^2 \]
Plain Text Formula:
I_z = I_cm + A * d^2
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ I_z \] | Moment of Inertia about Axis z | m⁴ or cm⁴ | Second moment of area about offset parallel axis |
| \[ I_{cm} \] | Centroidal Moment of Inertia | m⁴ or cm⁴ | Moment of inertia about the centroidal neutral axis |
| \[ A \] | Cross-Sectional Area | m² or cm² | Total area of the geometric component |
| \[ d \] | Perpendicular Offset Distance | m or cm | Distance between centroidal axis and target parallel axis |
💡 Worked Engineering Example
Problem Statement:
A flange plate (area A=50 cm², centroidal I_cm=4.17 cm⁴) has its center located 15 cm from the overall beam neutral axis. Find I_z.
Step-by-Step Solution:
- d = 15 cm, d^2 = 225 cm²
- Transfer term A * d^2 = 50 * 225 = 11,250 cm⁴
- I_z = 4.17 + 11,250 = 11,254.17 cm⁴
🎯 Result: Parallel axis transfer yields an effective inertia of 11,254.17 cm⁴.