AtlasCalcsUtility Calculation Suite
2025-08-21 14 min

Moment of Inertia Calculator

Calculate area moment of inertia for rectangle, circle, I-beam, and hollow sections.

Calculator

Unit System
Cross-Section Shape
Width (b)(mm)
mm
Height (h)(mm)
mm
Decimal Precision
Results update live as you type

Results

Normal

Ix (x-axis)

66666666.67mm^4

Iy (y-axis)

16666666.67mm^4

Area

20000.00mm^2

Section Modulus Sx

666666.67mm^3

Visualizations

Results Overview

66,666,667mm^4Ix
16,666,667mm^4Iy
20,000mm^4Area
666,667mm^4Section Modulus Sx

Formula

The area moment of inertia measures resistance to bending. It is the second moment of area about the neutral axis.

I = integral(y^2 dA)

Alternative Form

Rectangle: Ix = bh^3/12 | Circle: Ix = pi*d^4/64

Variables

SymbolNameUnitDescription
IxMoment of Inertia (x-axis)mm^4Resistance to bending about x-axis
IyMoment of Inertia (y-axis)mm^4Resistance to bending about y-axis
ACross-Sectional Areamm^2Total area of the cross-section

Explanation

I depends on how area is distributed from the neutral axis. Area farther from the axis increases I cubically.

Derivation

From I = integral(y^2 dA), integrating over the cross-section gives formulas for standard shapes.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Identify the cross-section shape and dimensions

Result: Shape determined
Step 2

Locate the centroid (neutral axis)

Result: Centroid found
Step 3

Apply the appropriate I formula

Result: I calculated
Step 4

Use parallel axis theorem for composite sections

Result: Composite I computed
Step 5

Calculate section modulus S = I/c

Result: S computed
Final Answer
66666666.67

Worked Examples

See how the calculator works with different scenarios.

Scenario

100 mm wide x 200 mm deep.

Inputs

Width (b):100 mm
Height (h):200 mm

Calculation

Ix = 100*200^3/12 = 66.67E6 mm^4. Iy = 200*100^3/12 = 16.67E6 mm^4.

Result

Ix = 66,670,000 mm^4

Interpretation

Ix is 4x Iy because depth contributes cubically.

Understanding Your Result

Your Result: Ix (x-axis): 66666666.67 mm^4

Typical range: Small: 10,000-100,000 mm^4. Building beams: 10-100M mm^4. Girders: 100M-10B mm^4.

What This Means

Moment of inertia quantifies cross-section resistance to bending and deflection.

Why It Matters

I directly determines beam deflection and bending stress. Most important property for beam design.

Professional Interpretation

Compare I against required I from deflection limits. Use minimum I for buckling.

Recommendations

  • •Check both Ix and Iy
  • •Use minimum I for column buckling
  • •Apply parallel axis theorem for composites

Educational Content

Introduction

Moment of inertia is the fundamental section property for bending analysis.

Concept Explanation

I = integral(y^2 dA) measures area distribution from the neutral axis.

Applications

▸Beam deflection
▸Bending stress
▸Column buckling
▸Structural optimization
▸Machine design

Benefits

  • •Directly relates to deflection and stress
  • •Enables section comparison
  • •Foundation for beam formulas

Limitations

  • •Assumes linear elastic behavior
  • •Does not account for shear deformation
  • •Valid only for prismatic sections

Best Practices

  • •Use consistent units (mm^4)
  • •Check minimum I for asymmetric sections
  • •Apply parallel axis theorem correctly

Common Misconceptions

Myth: Larger I always means better beam
Truth: I only relates to bending. Shear, buckling, and deflection may govern.

Practical Tips

  • •I proportional to h^3 for rectangles
  • •I-beams maximize I per weight
  • •Parallel axis theorem for built-up sections
  • •J = Ix + Iy for torsion

Common Mistakes

Wrong axis for I
Beam bends about axis perpendicular to load.
Forgetting parallel axis theorem
I = sum(Ii + Ai*di^2) about composite centroid.
Confusing I with J
I is for bending, J is for torsion. J = Ix + Iy.

Assumptions

This calculator makes the following assumptions to provide estimates:

Elastic Behavior: Formulas valid in linear elastic range.
Prismatic Section: Cross-section constant along beam length.

Limitations

Be aware of these limitations when using this calculator:

Non-Prismatic: Variable sections require integration.
Cracked Sections: Reduced effective I for cracked concrete.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the area moment of inertia?
I = integral(y^2 dA) measures resistance to bending. It depends on shape and area distribution from the neutral axis.
What is the difference between Ix and Iy?
Ix resists bending about the x-axis, Iy about the y-axis. For beams, Ix typically governs.
How do you calculate I for a rectangle?
Ix = bh^3/12, Iy = hb^3/12. h is the dimension parallel to the bending axis.
How do you calculate I for a circle?
Ix = Iy = pi*d^4/64. Equal about any diameter due to symmetry.
How do you calculate I for an I-beam?
Ix = bf*H^3/12 - (bf-tw)*hw^3/12. Subtract voids from the outer rectangle.
What is the parallel axis theorem?
I = I_centroid + A*d^2. Shifts I from the centroid to any parallel axis.
How does I affect deflection?
Deflection is inversely proportional to I. Doubling I halves deflection.
What are the units of moment of inertia?
Length^4: mm^4, cm^4, or in^4. Ranges from 1000 to 1 billion mm^4.
How do you calculate I for composite sections?
Use parallel axis theorem for each component: I = sum(Ii + Ai*di^2).
What is the radius of gyration?
r = sqrt(I/A). The equivalent distance for concentrated area. Used in buckling analysis.
What is polar moment of inertia?
J = Ix + Iy. Measures torsion resistance. For circle: J = pi*d^4/32 = 2*Ix.
How does I change with depth?
I is proportional to h^3 for rectangles. Doubling depth increases I by 8x.
What is the moment of inertia of a triangle?
Ix = bh^3/36 about the centroid, which is at h/3 from the base.
How do you calculate I for a hollow tube?
Hollow circle: I = pi*(D^4-d^4)/64. Hollow rectangle: I = (BH^3-bh^3)/12.
What is the effective moment of inertia?
For reinforced concrete (ACI), Ie accounts for cracking: Ie = (Mcr/Ma)^3*Ig + [1-(Mcr/Ma)^3]*Icr.
How do you measure I experimentally?
Methods: pendulum, torsional vibration, deflection tests, strain gauges on beams.
What is I for a semicircle?
About base diameter: Ix = pi*r^4/8. About centroid: Ix = (pi/8 - 8/(9*pi))*r^4.
How does I relate to bending stress?
sigma = My/I. Higher I means lower stress for the same moment and distance y.
Hollow vs solid section I?
Hollow has less I than solid of same outer dims, but much higher I per unit weight.
How do you calculate I for built-up sections?
Find composite centroid first, then apply parallel axis theorem for each component.
What is transfer moment of inertia?
The Ad^2 term in parallel axis theorem due to axis shift from centroid.
What is second moment of area?
Another name for moment of inertia. Integral of y^2 or x^2 over the cross-section.
How does I affect column buckling?
Pcr = pi^2*EI/(KL)^2. Higher I increases buckling resistance. Use minimum I.
What is product moment of inertia?
Ixy = integral(xy dA). Measures asymmetry. Zero for symmetric sections.
How do you find principal moments?
I1,2 = (Ix+Iy)/2 +/- sqrt(((Ix-Iy)/2)^2 + Ixy^2).
How does I change with temperature?
Geometric I doesn't change. Thermal gradients cause internal stresses in indeterminate structures.
Minimum I for floor beams?
Depends on span and deflection limit. Typical: 10-100 million mm^4. Use L/360 limit.
How do you compare I of different shapes?
Compare I per unit area or I/weight. I-beams have highest efficiency for major axis bending.
What is I of a hollow rectangle?
Ix = (BH^3 - bh^3)/12 where B,H are outer and b,h are inner dimensions.
What is the parallel axis theorem used for?
Finding I of composite sections and shifted axes. Essential for built-up beams.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Moment of Inertia
I = integral(y^2 dA), second moment of area.
Neutral Axis
Axis where bending stress is zero.
Parallel Axis Theorem
I = I_centroid + A*d^2.
Radius of Gyration
r = sqrt(I/A).
Polar Moment
J = Ix + Iy, torsion resistance.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

AISC 360Eurocode 3ACI 318NDS for Wood

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Page Information

Last Updated:

2025-08-21

Version:

2.0.0

License:

Free to use for educational and professional purposes.