📐 Moment of Inertia Calculator

Moment of Inertia Calculator

Shape & Dimensions

Cross-Section Shape

Results

Ix- in⁴

Iy- in⁴

Sx- in³

Sy- in³

Calculator Description

The second moment of area (moment of inertia) is a geometric property representing the bending stiffness of a cross-section. Bending behavior of members depends on material and section shape, and it must be calculated in structural design.

What this calculator finds

Calculates the second moment of area (Ix, Iy) for various cross-section shapes including rectangular, circular, T-section, I-section, angle, and channel sections. Values are computed about the centroidal axis.

Why it matters

Essential section property for calculating bending stress and deflection of beams and supports
Used when designing equivalent sections for built-up members
Directly related to determining member stiffness in seismic design

Formula

Second Moment of Area for Basic Sections

Formulas for the most commonly used cross-section shapes. All values are about the centroidal axis.

Rectangle

I = \dfrac{b h^3}{12}

Circle

I = \dfrac{\pi d^4}{64}

Hollow circle

I = \dfrac{\pi\left(D^4 - d^4\right)}{64}

I-beam

I = \dfrac{B H^3 - b h^3}{12}

I — Second moment of area [mm⁴, cm⁴, m⁴]
b, h — Width and height of rectangle [mm]
d, D — Circle diameter [mm] (D: outer, d: inner)
B, H — I-beam outer dimensions [mm]
b, h (I형 내부) — I-beam internal dimensions excluding flanges [mm]

Relation to section modulus

S = \dfrac{I}{c}

\sigma = \dfrac{M}{S} = \dfrac{M c}{I}

S — Section modulus [mm³]
c — Distance from centroid to extreme fiber [mm]
M — Bending moment [N·mm]
σ — Bending stress [MPa]

Useful Tips

Practical tips

Ix and Iy values for I-beams can be looked up directly in steel specification tables (KS, JIS, ASTM)
The I of hollow rectangles (pipes) is calculated by subtracting the inner rectangle from the outer rectangle
When eccentric loads act, torsion moments must also be reviewed

Limitations & cautions

This calculator computes only the geometric properties of sections, not material behavior (yield, fracture)
Composite sections (multi-material) require effective section transformation, which this calculator cannot handle
Local buckling reduces effective section and requires separate review