Area moment de inertia — typical cross sections

Formulas et values pour area moment de inertia de common cross sections.

areamomentinertia

Overview

The area moment of inertia (also called second moment of area) is a geometric property of a cross section that describes its resistance to bending. It appears in the flexure formula:

$\sigma = \frac{M \cdot y}{I}$

where:

$\sigma$ = bending stress (Pa)
$M$ = bending moment (N·m)
$y$ = distance from the neutral axis (m)
$I$ = area moment of inertia (m⁴)

A larger moment of inertia means a stiffer beam — more resistant to deflection under load.

Parallel Axis Theorem

The moment of inertia about any axis parallel to the centroidal axis:

$I = I_{centroid} + A \cdot d^2$

where:

$A$ = cross-sectional area
$d$ = distance between the two parallel axes

Section Modulus

The elastic section modulus relates moment of inertia to the maximum bending stress:

$S = \frac{I}{c}$

where $c$ is the distance from the neutral axis to the extreme fiber.

Common Cross Sections

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Area moment of inertia formulas for common cross sections
Cross Section	Area (A)	Moment of Inertia Ix	Section Modulus Sx
Rectangle (b × h)	b·h	b·h³ / 12	b·h² / 6
Square (a × a)	a²	a⁴ / 12	a³ / 6
Solid Circle (diameter d)	π·d² / 4	π·d⁴ / 64	π·d³ / 32
Hollow Circle (D outer, d inner)	π(D² − d²) / 4	π(D⁴ − d⁴) / 64	π(D⁴ − d⁴) / (32·D)
I-Beam (B×H outer, b×h web)	B·H − b·h	(B·H³ − b·h³) / 12	(B·H³ − b·h³) / (6·H)
Triangle (base b, height h)	b·h / 2	b·h³ / 36	b·h² / 24
Semicircle (radius r)	π·r² / 2	r⁴(π/8 − 8/(9π))	—
Thin-Wall Tube (r mean, t wall)	2·π·r·t	π·r³·t	π·r²·t
Ellipse (semi-axes a, b)	π·a·b	π·a·b³ / 4	π·a·b² / 4

Standard Steel Shapes — Example Values

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Properties of standard European IPE and HEB steel sections
Section	Depth (mm)	Width (mm)	Ix (10⁶ mm⁴)	Sx (10³ mm³)	Mass (kg/m)
IPE 100	100	55	1.71	34.2	8.1
IPE 200	200	100	19.4	194	22.4
IPE 300	300	150	83.6	557	42.2
IPE 400	400	180	231	1160	66.3
IPE 500	500	200	482	1930	90.7
IPE 600	600	220	921	3070	122
HEB 100	100	100	4.50	89.9	20.4
HEB 200	200	200	57.0	570	61.3
HEB 300	300	300	252	1680	117
HEB 400	400	300	577	2880	155
HEB 500	500	300	1072	4290	187

Beam Deflection

The maximum deflection of a simply supported beam with uniform load:

$\delta_{max} = \frac{5 \cdot w \cdot L^4}{384 \cdot E \cdot I}$

where:

$w$ = load per unit length (N/m)
$L$ = span (m)
$E$ = modulus of elasticity (Pa)
$I$ = moment of inertia (m⁴)

Applications

Structural beam and column design
Machine component sizing
Bridge and building analysis
Crane and hoist engineering
Aerospace structural analysis

Source: engineeringtoolbox.com