Free Moment of Inertia Calculator – Instant MOI Results

What Is Moment of Inertia?

Second moment of area (often called the moment of inertia in structural contexts and written as I) measures how a cross-section�s area is distributed about an axis. The more area that lies far from the axis, the larger the resistance to bending.

I = ? r� dA   (second moment of area, for beam bending)

Do not confuse this with the mass moment of inertia used in dynamics, which integrates mass instead of area. Both quantify resistance: one to bending (area), the other to angular acceleration (mass).

Why Moment of Inertia Matters

• Beam deflection and stress: Higher I reduces bending stress and deflection under load.
• Section selection: Guides whether to choose a rectangular, circular, or I-beam section for stiffness and weight.
• Safety margins: Crucial in bridges, frames, machine elements, and aerospace components.
• Cost optimization: Achieve target stiffness with minimal material by pushing area away from the neutral axis.

Formulas for Common Shapes (Centroidal Axes)

These expressions assume the neutral (centroidal) axis perpendicular to the shape�s plane unless stated otherwise. All results are for the second moment of area in SI units (m⁴).

For composite or cut-out sections (e.g., a plate with a circular hole), compute each part�s MOI about the same reference axis; subtract voids.

Parallel Axis Theorem

Most hand formulas are about a shape�s centroid. When the actual bending axis is offset, apply the parallel axis theorem:

I_axis = I_centroid + A d�

where A is area and d is the perpendicular distance between axes. Use this for built-up sections, stiffeners, or when the neutral axis shifts after transformations.

Polar vs. Area Moment of Inertia

Area moment of inertia (I_x, I_y) governs bending stiffness. The polar moment of inertia (J) relates to torsional stiffness for circular sections:

J = I_x + I_y

For a solid circle, J = (? r^4)/2. For thin-walled noncircular sections, torsion requires shear-flow methods; J alone may not predict angle of twist accurately.

Units and Conversions

• Second moment of area (structural): SI = m⁴, common engineering subunits = mm⁴, cm⁴.
• Mass moment of inertia (dynamics): SI = kg�m².
• Watch exponents: m ? mm implies factors of 10³; raising to the 4th power changes values by 10¹².

1 cm^4 = 1e-8 m^4    |    1 mm^4 = 1e-12 m^4

Worked Examples

1) Rectangular Beam About Its Strong Axis

Given: b = 120 mm, h = 300 mm. Compute I_x about centroidal x-axis.

I_x = b h^3 / 12
= (120 mm) (300 mm)^3 / 12
= (120)(27,000,000) / 12 mm^4
= 270,000,000 mm^4
= 2.70 � 10^8 mm^4 = 2.70 � 10^-4 m^4

2) Circular Tube (Hollow Round)

Given: R_o = 50 mm, R_i = 40 mm. Compute I about a centroidal diameter.

I = (?/4) (R_o^4 - R_i^4)
= (?/4) (50^4 - 40^4) mm^4
= (?/4) (6.25e6 - 2.56e6)
= (?/4) (3.69e6) mm^4
? 2.897 � 10^6 mm^4
= 2.897 � 10^-6 m^4

3) Built-Up I-Section (Outline)

Split into two flanges and one web, compute each I about its centroid, then shift to the section centroid using A d�, and add. This approach also works for channels, tees, and custom profiles.

Common Mistakes and How to Avoid Them

• Mixing units: Keep dimensions consistent. Convert to m, mm, or in and stick to one system.
• Diameter vs. radius: Formulas use r. If you have diameter D, use r = D/2 before raising to the 4th power.
• Forgetting parallel axis theorem: If the axis is not through the centroid, apply I_axis = I_c + A d�.
• Confusing area MOI with mass MOI: Area MOI is for bending (m⁴); mass MOI is for rotational dynamics (kg�m²).
• Assuming circular torsion rules for noncircular sections: Noncircular thin-walled sections require advanced torsion theory.

FAQ

What units does this page use?

For second moment of area, we present results in m⁴ (SI). If you work in mm, convert carefully: 1 mm⁴ = 10^?12 m⁴.

Is second moment of area the same as mass moment of inertia?

No. Second moment of area (m⁴) is geometric and used in bending. Mass moment of inertia (kg�m²) describes a body�s resistance to angular acceleration in dynamics.

Which axis do the table formulas refer to?

They refer to centroidal axes. Use the parallel axis theorem to shift to another axis as needed.

Can I combine multiple shapes?

Yes. Compute each component�s MOI about the same reference axis (shifting with the parallel axis theorem) and add. Subtract holes or cutouts.

How do I increase stiffness efficiently?

Move material away from the neutral axis (e.g., taller sections, flanges). An I-beam achieves high I with less mass than a solid rectangle of the same depth.