Moment of Inertia Calculator
Calculate area moment of inertia (second moment of area) for various geometric shapes
Calculate Area Moment of Inertia
Choose the geometric shape to calculate its moment of inertia
Width along x-axis
Height along y-axis
Moment of Inertia Results
Example Calculation
Rectangle: 12 cm × 8 cm
Given: Width (b) = 12 cm, Height (h) = 8 cm
Ix calculation: Ix = b × h³ / 12 = 12 × 8³ / 12 = 12 × 512 / 12 = 512 cm⁴
Iy calculation: Iy = h × b³ / 12 = 8 × 12³ / 12 = 8 × 1728 / 12 = 1152 cm⁴
Polar moment: J = Ix + Iy = 512 + 1152 = 1664 cm⁴
Area: A = b × h = 12 × 8 = 96 cm²
Moment of Inertia Basics
X-axis Moment
Resistance to bending about horizontal axis
Y-axis Moment
Resistance to bending about vertical axis
Polar Moment
Resistance to torsion (J = Ix + Iy)
Applications
Structural engineering and beam design
Mechanical engineering stress analysis
Cross-sectional strength calculations
Bridge and building design
Deflection and vibration analysis
Understanding Area Moment of Inertia
What is Area Moment of Inertia?
The area moment of inertia (also called second moment of area) is a geometric property that describes how area is distributed about an axis. It's crucial for calculating bending stress and deflection in structural elements.
Key Concepts
- •Units: Length to the fourth power (m⁴, cm⁴, in⁴)
- •Ix: Moment about horizontal (x) axis
- •Iy: Moment about vertical (y) axis
- •Centroid: Formulas assume axes pass through centroid
Parallel Axis Theorem
I = I₀ + A × d²
- I: Moment of inertia about new axis
- I₀: Moment of inertia about centroidal axis
- A: Cross-sectional area
- d: Distance between parallel axes
Note: Use parallel axis theorem when the axis doesn't pass through the centroid.
Engineering Applications
Beam Design
Higher moment of inertia = greater resistance to bending and less deflection
Stress Analysis
Used in flexural stress formula: σ = M×y/I
Deflection Calculations
Critical for determining beam deflection under loads