Example Data Table
| Section |
Sample dimensions |
Area |
Centroidal Ix |
Centroidal Iy |
| Rectangle |
b = 200 mm, h = 100 mm |
20,000 mm2 |
16,666,666.67 mm4 |
66,666,666.67 mm4 |
| Circle |
d = 100 mm |
7,853.98 mm2 |
4,908,738.52 mm4 |
4,908,738.52 mm4 |
| Hollow circle |
Do = 120 mm, Di = 80 mm |
6,283.19 mm2 |
6,382,299.34 mm4 |
6,382,299.34 mm4 |
| I section |
bf = 150 mm, h = 250 mm, tf = 20 mm, tw = 10 mm |
8,100 mm2 |
89,285,000 mm4 |
11,254,250 mm4 |
Formula Used
The first moment of area is Qx = A × y and Qy = A × x. It uses area and centroid distance from the selected reference axis.
The second moment of area is Ix = ∫y² dA and Iy = ∫x² dA. Standard shapes use closed form centroidal equations.
The parallel axis theorem is Iref = Ic + A × d². Here, d is the centroid offset from the reference axis.
The polar estimate is J = Ix + Iy. The radius of gyration is r = √(I / A). Bending stress uses σ = M × c / I. Shear flow uses q = V × Q / I.
How to Use This Calculator
- Select the section type that matches your geometry.
- Choose a length unit. Keep all dimensions in that unit.
- Enter the required dimensions for the selected shape.
- Enter centroid offsets when you need reference axis results.
- For composite rectangles, enter each part centroid and action.
- Add optional bending or shear values for quick physics checks.
- Press the calculate button. The result appears above the form.
- Use the CSV or PDF button to save the result.
Understanding Moment of Area
Moment of area describes how area is distributed around an axis. It is not the same as mass moment of inertia. This calculator works with geometric area only. Engineers use it when checking beams, shafts, brackets, plates, and built sections. A larger second moment usually means stronger resistance to bending about that axis.
First And Second Area Moments
The first moment of area is area multiplied by a centroid distance. It is often written as Q. It helps with shear stress, shear flow, and centroid finding. The second moment of area is often written as I. It measures area spread using squared distance from an axis. It supports bending, deflection, buckling, and section comparison.
Why Shape Choice Matters
Each section has a different formula. A tall rectangle has a much larger Ix than a wide, shallow one with the same area. A hollow tube can keep high stiffness while reducing material. An I section moves material away from the neutral axis. That increases Ix without making the whole section solid.
Reference Axes And Offsets
Centroidal values describe axes passing through the section centroid. Reference axis values include the entered offsets. The calculator applies the parallel axis theorem for standard shapes. For composite rectangles, it finds the total centroid from each part first. Added parts use positive area. Cutouts use negative area. This makes the tool useful for channels, boxes, notches, and built plates.
Practical Physics Use
In physics and mechanics, moment of area connects geometry with load behavior. It helps predict which direction bends easily. It also explains why thin rulers bend more in one direction. The radius of gyration shows how far the area acts from a centroidal axis. The polar estimate adds Ix and Iy for a quick torsion related comparison.
Reading The Results
Always check units before using results in formulas. Area uses squared units. First moment uses cubed units. Second moment uses fourth power units. If input is in millimeters, Ix is in millimeters to the fourth power. The optional bending stress and shear flow checks are estimates. They depend on correct load, distance, and Q values. For critical design, confirm assumptions and use the required code. Document exported values for review.
FAQs
What is moment of area?
Moment of area describes how a shape area is distributed around an axis. First moment uses distance. Second moment uses squared distance. Both help explain centroid, bending, and shear behavior.
Is moment of area the same as mass inertia?
No. Moment of area uses geometry only. Mass moment of inertia also uses mass density. Use this tool for section properties, bending checks, and area based physics problems.
What does Ix mean?
Ix is the second moment of area about the x-axis. A larger Ix usually means the section resists bending more strongly around that axis.
What does Q mean?
Q is the first moment of area. It equals area multiplied by centroid distance from a chosen axis. It is often used in shear stress calculations.
When should I enter centroid offsets?
Enter offsets when your reference axes do not pass through the centroid. Leave them as zero when you only need centroidal section properties.
How do composite rectangles work?
Each rectangle has its own area and centroid. Added parts are positive. Cutouts are negative. The calculator combines them and then finds the net centroid and moments.
Why are units raised to powers?
Area uses length squared. First moment uses area times distance, so it is cubed. Second moment uses area times distance squared, so it is fourth power.
Can I export the result?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a simple report of the displayed values.