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Cross-section and I-beam

The bending stress $\sigma = My/I$ depends on the second moment of area $I$ , which grows quickly with distance from the neutral axis. An I-beam puts most of its material in the flanges, far from the middle, giving a large $I$ per kilogram compared with a solid rectangle.

I_{rect} = \dfrac{bh^3}{12}

second moment of area of a rectangle

Symbols

$b$	width	m
$h$	height	m
$I$	second moment of area	m⁴

Example

A rectangle with $b = 50$ mm, $h = 100$ mm:

$I = 50\cdot 100^3/12 \approx 4.17\cdot10^6$ mm⁴.

Material far from the neutral axis counts the most – that is why I-beams are so efficient in bending.

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Part of Statics and Dynamics: Bending of beams.