🚀 Area Moment of Inertia Calculator & Simulator
- Select Cross-Section Shape: Choose your desired shape from Rectangle, Hollow Box, I-Beam, or Circle using the tabs at the top.
- Adjust Geometric Dimensions: Smoothly move the sliders for cross-sectional width (b), height (h), flange/web thickness (tf, tw), or circular diameter (D, d) to adjust dimensions.
- Observe Centroid and Inertia Axes in Real-Time: As the section size changes, observe in real-time how the centroid location and neutral axes (Neutral Axis X-X, Y-Y) projected on the 2D drawing move in sync.
- Analyze Section Stiffness Indicators: Evaluate bending stiffness in the final calculation area based on the measured cross-sectional area (A), second moment of area (Ix, Iy) relative to the neutral axis, and radius of gyration (rx, ry).
📚 View Detailed Formulas for Second Moment of Area by Shape & Physical Explanation of Centroidal Axes ▼
1. Physical Definition and Role of the Second Moment of Area (Moment of Inertia)
In the mechanical behavior of structural members, the Second Moment of Area (Moment of Inertia, I) is a purely geometric measure that represents a member's cross-sectional resistance to bending deformation when subjected to a bending moment. In other words, even if the material's inherent strength (elastic modulus) remains the same, varying the shape can increase bending resistance (stiffness) by dozens of times. The key constant contributed by the cross-sectional shape in this context is the second moment of area.
Mathematically, the second moment of area is calculated by integrating the infinitesimal area dA weighted by the square of its distance from the centroidal axis (y²) over the entire cross-section:
I_x = ∫ y² dA , I_y = ∫ x² dA [mm⁴]
2. Key Formulas for the Second Moment of Area for Representative Engineering Shapes
The formulas for the moment of inertia relative to the neutral axis for standard cross-sections used in practical design are as follows.
① Rectangular Section (Rectangle - Width b, Height h):
I_x = (b × h³) / 12 , I_y = (h × b³) / 12
② Circular Section (Solid Circle - Outer Diameter D):
I_x = I_y = (π × D⁴) / 64
③ I-Beam Section (I-Beam - Flange Width b, Total Height h, Web Thickness tw, Flange Thickness tf):
The moment of inertia is quickly derived by subtracting the two hollow rectangular spaces [width (b - tw), height (h - 2tf)] on both sides from the outer large rectangular area (width b, height h).
I_x = (b × h³) / 12 - (b - t_w) × (h - 2t_f)³ / 12 [mm⁴]
3. Parallel Axis Theorem and Strategies for Maximizing Sectional Performance
When calculating the moment of inertia for arbitrary composite cross-sections, the Parallel Axis Theorem must be applied to account for individual components whose centroidal axes do not align with the overall centroid.
I = I_c + A × d²
Here, I_c is the second moment of area about the component's own centroidal axis, A is the cross-sectional area, and d is the perpendicular distance from the component's centroid to the centroidal axis of the entire composite section. According to this theorem, even without increasing the amount of material (area A), intentionally placing mass (such as flanges) further away from the neutral axis (maximizing d) allows for optimal beam designs—such as H-beams or I-beams—that are lightweight yet possess exceptional bending resistance.