Ultimate Guide & Calculator to Find Area Moment of Inertia in SolidWorks
A complete guide and calculator to understand and calculate the area moment of inertia and how to find this critical property in SolidWorks.
Area Moment of Inertia Calculator
The dimension parallel to the axis of bending.
The dimension perpendicular to the axis of bending.
The distance from the center to the edge of the circle.
Results
Iₓ = 4,166,666.67 mm⁴
Iᵧ = 1,041,666.67 mm⁴
Area = 5,000.00 mm²
J = 5,208,333.33 mm⁴
Formula for Rectangle: Iₓ = (base * height³) / 12
What is Area Moment of Inertia?
The Area Moment of Inertia, also known as the second moment of area, is a geometric property of a shape that indicates its resistance to bending. It’s crucial for engineers and designers because it directly relates to how much a beam or structural member will deflect under a load. The property depends entirely on the shape of the cross-section and the axis about which bending occurs.
It’s important not to confuse area moment of inertia with mass moment of inertia. Mass moment of inertia relates to a body’s resistance to rotational acceleration and involves mass, while area moment of inertia is a purely geometric property used for analyzing bending stress and deflection. For any complex shape, you can easily calculate the area moment of inertia using SolidWorks by accessing the “Section Properties” tool under the “Evaluate” tab. This calculator helps you understand the concept by computing it for basic shapes.
Area Moment of Inertia Formula and Explanation
The formula to calculate area moment of inertia varies depending on the shape’s geometry. The value is calculated with respect to a specific axis (usually ‘x’ or ‘y’). A larger value indicates a stiffer section that is more resistant to bending around that axis.
Common Formulas:
- Rectangle: The moment of inertia about the centroidal x-axis is
Iₓ = (base * height³) / 12. About the y-axis, it isIᵧ = (height * base³) / 12. - Circle: The moment of inertia is the same about any centroidal axis:
Iₓ = Iᵧ = (π * radius⁴) / 4.
| Variable | Meaning | Unit (Auto-Inferred) | Typical Range |
|---|---|---|---|
| Iₓ | Area Moment of Inertia about the x-axis | mm⁴, cm⁴, in⁴ | 0 to >10⁹ |
| Iᵧ | Area Moment of Inertia about the y-axis | mm⁴, cm⁴, in⁴ | |
| b | Base of a rectangle | mm, cm, in | 1 to >1000 |
| h | Height of a rectangle | mm, cm, in | 1 to >1000 |
| r | Radius of a circle | mm, cm, in | 1 to >1000 |
Practical Examples
Example 1: Rectangular Beam
Imagine a rectangular wooden plank used to bridge a small gap. To maximize its stiffness and minimize bending, you should orient it with its taller side vertical. Let’s see why.
- Inputs: A rectangular beam with a base (b) of 40 mm and a height (h) of 150 mm.
- Units: Millimeters (mm)
- Results:
- Iₓ = (40 * 150³) / 12 = 11,250,000 mm⁴
- Iᵧ = (150 * 40³) / 12 = 800,000 mm⁴
The result clearly shows that Iₓ is over 14 times larger than Iᵧ. This confirms the beam is far more resistant to bending when the force is applied against its shorter side (the base).
Example 2: Circular Shaft
Consider a solid circular steel shaft used in a machine.
- Inputs: A circular shaft with a radius (r) of 25 mm.
- Units: Millimeters (mm)
- Results:
- Iₓ = Iᵧ = (π * 25⁴) / 4 ≈ 306,796 mm⁴
For a circle, the stiffness is uniform in all directions, which is ideal for rotating shafts that experience loads from various angles.
How to Use This Calculator & Find the Value in SolidWorks
Using the Calculator:
- Select Shape: Choose between “Rectangle” and “Circle”.
- Choose Units: Select the measurement unit for your dimensions (mm, cm, or in).
- Enter Dimensions: Input the required values (e.g., base and height for a rectangle).
- Interpret Results: The calculator instantly provides the primary moment of inertia (Iₓ), as well as intermediate values like Iᵧ and the cross-sectional area. The chart helps visualize the difference in stiffness between the x and y axes.
Finding Area Moment of Inertia in SolidWorks:
For complex shapes, manual calculation is impractical. SolidWorks makes it simple:
- Create your part and ensure you have a sketch of the cross-section you want to analyze on a surface or plane.
- Go to the Evaluate tab in the CommandManager.
- Click on Section Properties.
- Select the sketch or face representing the cross-section.
- A dialog box will appear, showing the area, centroid, and moments of inertia (Ixx, Iyy, Ixy) about the centroid. For most bending calculations, you will use these principal moments of inertia.
For a deeper dive, check out our guide on Mass Moment of Inertia vs. Area Moment of Inertia.
Key Factors That Affect Area Moment of Inertia
- Height of the Section (h): This is the most significant factor. Since the height is cubed in the formula for a rectangle (h³), doubling the height increases the moment of inertia by a factor of eight, dramatically increasing bending resistance.
- Distribution of Area: The farther the area is from the bending axis, the more it contributes to the moment of inertia. This is why I-beams are so efficient; they place most of their material in the top and bottom flanges, far from the neutral axis.
- Axis of Bending: As seen in the rectangular beam example, the orientation of the part relative to the load determines which moment of inertia (Iₓ or Iᵧ) is relevant, and this can drastically change the stiffness.
- Shape Geometry: Different shapes have different inherent stiffness. A hollow tube can be nearly as stiff as a solid rod while using much less material, because its area is concentrated at the outer radius.
- Composite Shapes: For complex profiles like T-beams or C-channels, the total moment of inertia is calculated using the Parallel Axis Theorem, which combines the moments of inertia of several simpler shapes.
- Radius (for circular sections): The radius is raised to the fourth power (r⁴), making it a powerful factor. A small increase in radius leads to a very large increase in stiffness.
Frequently Asked Questions (FAQ)
Iₓ is the area moment of inertia about the horizontal x-axis, representing resistance to vertical bending. Iᵧ is about the vertical y-axis, representing resistance to horizontal bending. For a tall, thin rectangle, Iₓ will be much larger than Iᵧ.
The units are length to the fourth power, such as millimeters to the fourth power (mm⁴), inches to the fourth power (in⁴), or meters to the fourth power (m⁴).
It’s called the second moment because the formula involves the distance from the axis squared (r²). The first moment of area (which involves just ‘r’) is used to find the centroid of a shape.
No, this calculator is for simple solid rectangles and circles. To calculate the area moment of inertia for a composite shape like an I-beam, you must use the Parallel Axis Theorem, or more practically, use a CAD program like SolidWorks.
No. The area moment of inertia is a purely geometric property. However, material type (specifically its Modulus of Elasticity) is used alongside the moment of inertia to calculate the actual deflection of a beam.
The Polar Moment of Inertia (J) describes a shape’s resistance to torsion (twisting). For a 2D cross-section, it is the sum of the two perpendicular moments of inertia: J = Iₓ + Iᵧ. This calculator provides that value as well.
The values calculated by SolidWorks are extremely accurate, as they are based directly on the precise geometry of your 3D model. It is the industry-standard method for finding these properties for complex parts.
You can start by exploring topics like understanding beam deflection and the role of Young’s Modulus in material selection.
Related Tools and Internal Resources
Explore other engineering calculators and guides to expand your knowledge.
- Stress, Strain, and Young’s Modulus Calculator: Understand the relationship between stress, strain, and material properties.
- Beam Deflection Calculator: Calculate how much a beam will bend under various loads and supports.
- Engineering Unit Conversion Tool: Quickly convert between different units of length, force, pressure, and more.
- Parallel Axis Theorem Explained: A detailed guide on how to calculate the moment of inertia for composite shapes.
- Mass Moment of Inertia vs. Area Moment of Inertia: A clear comparison between these two important concepts.
- Introduction to SolidWorks Simulation: Learn how to set up basic structural analysis in SolidWorks.