Euler Buckling Load⚠ unverified
Physics / Materials · Compute the critical Euler buckling load of a column
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| youngs_modulus | youngsmodulus | Pa | 1.0 | Young's modulus of the column material |
| moment_of_inertia | momentofinertia | m^4 | 1.0 | Second moment of area of the cross-section |
| length | length | m | 1.0 | Unsupported length of the column |
| end_condition | endcondition | — | 1.0 | Effective-length factor K accounting for end constraints, dimensionless. Default is 1.0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Pcr | N | Critical buckling load, in newtons (N) |
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.