Moment of Inertia of a Beam Calculator

Beam Section Input

Choose a section shape. Enter all dimensions in the same unit.

Section shape

Unit

Width or flange width

Overall height or depth

Inner rectangle width

Inner rectangle height

Outer diameter

Inner diameter

Web thickness

Flange thickness

Horizontal axis offset

Vertical axis offset

Formula Used

The calculator uses standard second moment of area formulas. Composite shapes use area addition, subtraction, and the parallel axis theorem.

A = total cross-sectional area
Ixx = second moment of area about the horizontal centroidal axis
Iyy = second moment of area about the vertical centroidal axis
Ishifted = Icentroid + A d^2
S = I / c
r = sqrt(I / A)

How to Use This Calculator

Select the beam section shape.
Choose one unit and keep all dimensions consistent.
Enter only the dimensions required for your selected shape.
Add optional axis offsets when using the parallel axis theorem.
Click Calculate to show results below the header.
Use CSV or PDF buttons to save the same result.

Example Data Table

Shape	Main dimensions	Area	Ixx	Common use
Rectangle	b = 200 mm, h = 300 mm	60,000 mm^2	450,000,000 mm^4	Timber and plate beams
Solid circle	D = 250 mm	49,087.385 mm^2	191,747,598.486 mm^4	Round shafts and posts
I section	b = 200 mm, h = 300 mm, tw = 12 mm, tf = 20 mm	11,120 mm^2	174,642,666.667 mm^4	Floor and roof members

Beam Inertia Basics

Moment of inertia describes how a beam section resists bending. It is a geometric property. It does not depend on steel, wood, or concrete strength. A larger value means the section spreads material farther from the neutral axis. That usually lowers bending stress and deflection.

Why Section Shape Matters

Depth often controls stiffness more than width. A deep rectangle can be much stiffer than a shallow wide piece. Circular sections behave the same about both main axes. Hollow tubes can save weight while keeping good stiffness. I sections place flanges far apart. That makes them efficient for floor beams, roof members, and frames.

Using Design Results

This calculator estimates area, centroid, second moment of area, section modulus, and radius of gyration. Area helps with weight estimates. The centroid locates the neutral axis for bending. Ixx is used for bending about the strong horizontal axis. Iyy is used for bending about the vertical axis. Section modulus links bending moment to stress. Radius of gyration helps compare slenderness and column behavior.

Practical Accuracy Tips

Use one unit system for every dimension. Do not mix inches with millimeters in one calculation. Check that wall thickness and web thickness are smaller than outer dimensions. For hollow shapes, confirm the inner opening is centered. For T sections, note that the centroid is not at mid depth. This changes top and bottom section modulus values.

Common Input Mistakes

Many wrong results come from simple entries. A flange thickness must not exceed half the full depth. An inner diameter must be smaller than the outer diameter. Web thickness should stay within the flange width. Negative offsets are not needed here. Enter distance as a positive separation from the centroidal axis. Compare both axes before choosing a beam. A member may be strong one way and weak the other way.

When to Review Further

These results are useful for early sizing and study work. They do not replace a full structural design. Real beams also need load checks, shear checks, lateral stability checks, connection design, and code limits. Material defects and fabrication tolerances can change performance. Always review critical beams with accepted design standards and qualified engineering judgment. Save exported records with each design file.

FAQs

What is moment of inertia of a beam?

It is a geometric property of the beam cross section. It shows how strongly the section resists bending about an axis.

Which axis should I use for beam bending?

Use Ixx for bending about the horizontal centroidal axis. Use Iyy when bending occurs about the vertical centroidal axis.

Can I mix units in the form?

No. Use one unit system for every entered dimension. Mixed units will give incorrect area and inertia values.

What does section modulus mean?

Section modulus equals inertia divided by the distance to the extreme fiber. It helps estimate bending stress from bending moment.

Why is a deep beam stiffer?

Depth appears as a cubed term in many inertia formulas. Extra depth can greatly increase resistance to bending.

What is the parallel axis theorem?

It adjusts inertia from a centroidal axis to another parallel axis. The added value is area times offset distance squared.

Does material strength change the inertia value?

No. Moment of inertia depends on shape and dimensions only. Material properties affect stress, deflection, and safety checks later.

Is this enough for final structural design?

No. Use it for section property checks. Final design should include loads, supports, codes, connections, and professional review.