Euler Buckling Load⚠ unverified
Aerospace / Structures · Critical Euler buckling load of a column
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E | E | Pa | 70000000000.0 | Young's modulus |
| I | I | m^4 | 1e-06 | Second moment of area |
| L | L | m | 2.0 | Length |
| K | K | — | 1.0 | Effective-length factor |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| Pcr | Pcr | N | Critical load |
The science & history
Understanding the Parameters
- $E$, $I$ — stiffer material/section → higher $P_{cr}$.
- $L$, $K$ — effective length $KL$; doubling $KL$ quarters $P_{cr}$.
- $P_{cr}$ — valid only in the elastic slender range; short columns yield before Euler buckling.
Derivation (Approaching a Proof)
Solve the beam-column equation $EI y'' + P y = 0$ with appropriate BCs. Lowest eigenvalue is $P_{cr} = \pi^{2} EI / L_{\mathrm{eff}}^{2}$ with $L_{\mathrm{eff}} = KL$.
History
Leonhard Euler (1744) derived the critical load; effective-length factors extend it to practical ends.
Related Concepts: Column Slenderness Ratio, Factor of Safety, Section Modulus
Notes: Registry calculator materials-euler-buckling-load (unverified). Ideal elastic column;
no eccentricity or residual stress.