Beams & Columns — Bending & Buckling
Second moment of area, bending stress, beam deflection, and Euler column buckling — with the cross-section drawn live and the moment diagram plotted.
Beams are everywhere a load has to span a gap — floor joists, bridge girders, an aircraft wing. Two numbers decide a beam: how much it bends (stress, which can break it) and how much it sags (deflection, which annoys everyone). Both trace back to a single geometry property of the cross-section.
- 01 Second moment of area — why shape beats size ELI5: a ruler is rigid on edge and floppy flat — same material, same area, wildly different stiffness. The reason is the second moment of area I, which rewards…
- 02 Bending stress — the load fibre furthest out feels ELI5: when a beam bends, the top squashes and the bottom stretches — and the outermost fibres feel the most. Bending stress σ = M·c / I, where M is the bending…
- 03 Deflection — how far it sags ELI5: even a beam that won't break can sag enough to crack plaster or feel bouncy. For a central point load the midspan sag is δ = P·L³ / (48·E·I) — the…
- 04 Columns — buckling, the sneaky failure ELI5: a slender column doesn't crush — it suddenly bows out sideways at a load far below its crushing strength. Euler's formula gives that critical…
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