Beams & Columns — Bending & Buckling

Deflection — how far it sags

continues from lesson 2 — values defined earlier in the course stay live here

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 span is cubed, so doubling the span sags it eightfold.

E_{beam} := 200 GPa

= 200 GPa steel stiffness

\delta _{mid} := P_{load} * L_{span}^3 / (48 * E_{beam} * I_{rect}) in mm

= 6.75 mm midspan sag ≈ 6.75 mm

✓ pass

\delta_{mid} < \frac{L_{span}}{250}

inside the common span/250 serviceability limit

Real-world hook: the span-cubed law is why long floors feel bouncy, why bridges need deeper girders as they get longer, and why bookshelves sag in the middle.

Try it yourself: the peak bending moment for a 20 kN central load on a 4 m span? (M = P·L / 4, answer in kN·m.)

Mmax_{you} := 20 kN * 4 m / 4

= 20 kN m ✏️ Your turn: multiply the load by the span and divide by 4.

✓ pass

abs(Mmax_{you} - 20 kN * 4 m / 4) < 1e-6 kN*m

green when your peak moment is correct