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DEFree Moment of Inertia, Centroid and Section Modulus Calculator
Accurate Online Tool for Easy Moment of Inertia, Centroid, and Beam Section Calculations
Verify Cross-Sections in FEA with Confidence
Calculate section properties and perform standard-based checks (AISC, DNV, Eurocode, etc) directly in your FEA model using structural design and analysis software SDC Verifier.
What is Moment of Inertia , Centroid and Section Modulus Calculator
Free Moment of Inertia Calculator Functionality
SDC Verifier’s Free Moment of Inertia Calculator allows you to calculate properties for a wide range of standard profiles commonly used in structural and mechanical engineering:
- Rectangular
- Hollow Rectangular
- Circular
- Hollow Circular
- I-Shape
- T-Shape
- Channel
- L-Shape (Angle)
- Z-Shape
These options cover both solid and hollow sections, as well as open profiles, making the tool versatile for different engineering scenarios.
Parameters calculated
The calculator provides accurate results for key geometric and structural properties, including:
🔹 Geometrical Characteristics
- A – Area (mm²/in²): Total cross-sectional area.
- P – Perimeter (mm/in): Outline length of the section.
- Cy, Cz – Centroid coordinates (mm/in): Distance from origin to center of gravity along Y and Z axes.
- θ (Theta) – Rotation angle of the principal axes with respect to the original centroidal coordinate system.
🔹 Moment of Inertia
Geometric Axes
- Iy, Iz – Moments of inertia (mm⁴/in⁴): Resistance to bending about Y and Z axes.
- Iyz – Product of inertia (mm⁴/in⁴): Indicates asymmetry of the section.
- Ix – Second polar moment of area (mm⁴/in⁴): Resistance to torsion.
Principal Axes
- I1, I2 – Moments of inertia (mm⁴/in⁴): About major and minor principal axes.
- I12 – Product of inertia (mm⁴/in⁴): Should be zero for principal axes.
- I3 – Second polar moment (mm⁴/in⁴): Sum of I1 and I2.
🔹 Elastic Section Modulus
- Zy, Zz – Section modulus (mm³/in³): Indicates bending strength about Y and Z axes.
- Z1, Z2 – About major and minor principal axes.
🔹 Plastic Section Modulus
- Sy, Sz – Plastic section modulus (mm³/in³): Capacity of the section to resist plastic bending.
- S1, S2 – About major and minor principal axes.
🔹 Distance from Centroid to Extreme Fibres
- Top/Bottom and Right/Left fibre distances (mm/in): Used to calculate section modulus.
🔹 Radius of Gyration
- ry, rz – About geometric Y and Z axes (mm/in): Indicates efficiency of area distribution.
- rx – Radius of gyration about polar axis (mm/in), derived from polar moment of inertia.
🔹 Shear Area
- Ay, Az – Effective shear areas (mm²/in²): For shear stress calculations.
- r1, r2, r3
🔹 Radius of Gyration
• ry, rz – About geometric Y and Z axes (mm/in): Indicates efficiency of area distribution.
• rx – Radius of gyration about polar axis (mm/in), derived from polar moment of inertia.
• r1, r2 – About principal axes (mm/in): Same as for Y and Z axes but relative to principal axes.
• r3 – Polar radius of gyration about principal axes (mm/in).
🔹 Torsional and Warping Properties
- J – Torsional constant (mm⁴/in⁴): Measures resistance to pure torsion.
- Cw – Warping constant (mm⁶/in⁶): Indicates resistance to warping during torsion.
- A1, A2 – Shear areas about principal axes (mm²/in²).
How to Use the Free Moment of Inertia Calculator? Step-by-Step Instructions
How to Use the Shape Calculator
- Select the Unit System
- Choose between Metric or Imperial (default is Metric).
- Choose the Shape Type
- Pick one of the available cross-section shapes:
Rectangular
Hollow Rectangular
Circular
Hollow Circular
I-Shape
T-Shape
Channel
L-Shape
Z-Shape
3. Enter the Shape Parameters
- Fill in the required dimensions for your selected shape:
- Height (h) – e.g., 100 mm/in
- Width (b) – e.g., 55 mm/in
- Flange Thickness (tf) – e.g., 5.7 mm/in
- Web Thickness (tw) – e.g., 4.1 mm/in
(Note: Parameter names may vary depending on the selected shape.)
4. Get Instant Results
- The calculator will display detailed properties automatically.
5. Receive Results by Email
- Enter your email to get the results sent directly to your inbox.
Example: I-beam Section Output
Basic Properties
- Height (h) = 100 mm
- Width (b) = 55 mm
- Flange thickness (tf) = 5.7 mm
- Web thickness (tw) = 4.1 mm
Results:
Geometrical characteristics
- Angle of Major Principal Axis: 0.00°
- Area (A): 990.26 mm²
- Perimeter (P): 411.80 mm
- Centroid Coordinates:
- Y-axis (Cy): 27.50 mm
- Z-axis (Cz): 50.00 mm
Moments of Inertia (Geometric Axes)
- Moment of Inertia about Y-axis (Iy): 1,633,226.78 mm⁴
- Moment of Inertia about Z-axis (Iz): 158,565.12 mm⁴
- Product of Inertia (Iyz): 0.00 mm⁴
- Second Polar Moment of Area (Ix): 1,791,791.90 mm⁴
Moments of Inertia (Principal Axes)
- Major Axis (I₁): 1,633,226.78 mm⁴
- Minor Axis (I₂): 158,565.12 mm⁴
- Product of Inertia (I₁₂): 0.00 mm⁴
- Second Polar Moment of Area (I₃): 1,791,791.90 mm⁴
Elastic Section Modulus
- About Y-axis (Zy): 32,664.54 mm³
- About Z-axis (Zz): 5,766.00 mm³
- Major Axis (Z₁): 32,664.54 mm³
- Minor Axis (Z₂): 5,766.00 mm³
Plastic Section Modulus
- About Y-axis (Sy): 37,609.26 mm³
- About Z-axis (Sz): 8,993.59 mm³
- Major Axis (S₁): 37,609.26 mm³
- Minor Axis (S₂): 8,993.59 mm³
Distances from Centroid to Extreme Fibers
- Top Fiber: 50.00 mm
- Bottom Fiber: 50.00 mm
- Right Fiber: 27.50 mm
- Left Fiber: 27.50 mm
Radius of Gyration
- About Y-axis (ry): 40.61 mm
- About Z-axis (rz): 12.65 mm
- Polar Radius (rx): 42.54 mm
Shear Areas
- About Y-axis (Ay): 522.50 mm²
- About Z-axis (Az): 410.00 mm²
- Major Axis (A₁): 522.50 mm²
- Minor Axis (A₂): 410.00 mm²
Torsional and Warping Properties
- Torsional Constant (J): 8,956.83 mm⁴
- Warping Constant (Cw): 351,378,405.64 mm⁶
See the Formula for Moment of Inertia of a Rectangle
Let’s calculate the moment of inertia of a rectangle with the following dimensions:
- Width (b): 55 mm
- Height (h): 100 mm
The formula for the moment of inertia (I) is:
Substitute the given values:
Result:
This value represents the moment of inertia about the base of the rectangle (horizontal axis). It’s a key parameter in assessing the bending resistance of a section.
Key Benefits of SDC Verifier’s Moment of Inertia Calculator
- No Registration: Access all the features instantly without the need to sign up. Start calculating right away with full functionality.
- Instant Results: Receive results in real time—quick, reliable, and accurate calculations without any delays.
- Intuitive Interface: The user-friendly interface makes it easy to perform calculations, even for those with minimal technical experience.
- Accuracy of Calculations: Calculations are based on analytical formulas for standard shapes and assume ideal geometry.
What Can You Do After Receiving Results on Moment of Inertia?
- Saving Results, PDF, Export: Save your calculations, export them to PDF for reports, or via email.