# Euler Buckling Load Calculator

> Find the Euler critical buckling load π²EI ÷ (KL)² of a slender column, its slenderness ratio and critical stress, and check it against yield strength.

Category: Science & engineering · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/euler-buckling-load-calculator

A slender column under compression bows sideways and fails at the Euler load, π²EI ÷ (KL)², long before the material is crushed. The end conditions enter through the effective-length factor K.

Also called: Column buckling calculator, Critical load calculator.

## Inputs
- Elastic modulus E (unit GPa; default 200)
- Least second moment of area I (unit cm⁴; default 500)
- Unbraced length L (unit m; default 5)
- Effective-length factor K (default 1)
- Cross-sectional area A (unit cm²; default 20)
- Yield strength (optional) (unit MPa; default 250; optional)

## Results
- Critical buckling load
- Effective length KL
- Radius of gyration r
- Slenderness ratio KL/r
- Critical stress
- Euler validity check

## Formula

```
P_cr = π² E I ÷ (K L)²;  r = √(I ÷ A);  slenderness = K L ÷ r;  σ_cr = P_cr ÷ A
```

Where:
- E: elastic modulus
- I: least second moment of area
- K: effective-length factor
- L: unbraced length
- A: cross-sectional area
- r: radius of gyration

## Worked example
1. EI = 200 × 10⁹ × 500 × 10⁻⁸ = 1 × 10⁶ N·m².
2. P_cr = π² × 10⁶ ÷ (1 × 5)² = 9.8696 × 10⁶ ÷ 25 = 394,784 N = 394.78 kN.
3. r = √(500 ÷ 20) = 5 cm, so KL/r = 500 ÷ 5 = 100.
4. σ_cr = 394,784 N ÷ 2,000 mm² = 197.4 MPa, below fy = 250 MPa, so elastic buckling governs.

**Result:** A 5 m pinned-end steel column with I = 500 cm⁴ buckles elastically at about 395 kN.

## What the result means

The Euler load is an upper bound for a perfect column. If σ_cr approaches or exceeds the yield strength (stocky columns), failure is inelastic and design codes use column curves that give lower capacities. Apply a safety factor before using the load.

## Assumptions and limits
- Perfectly straight, centrally loaded, linear-elastic prismatic column.
- The effective-length factor K represents the end restraint you enter.
- Real columns have initial crookedness and residual stresses, so the true capacity is lower than P_cr, especially at intermediate slenderness.
- Not a code design check.

## Questions

### Which axis should I use?

The one with the smaller I (and its matching length and K), because the column buckles about its weakest axis.

### Why does K = 0.5 give four times the load?

Load varies with 1 ÷ (KL)², so halving the effective length quadruples P_cr.

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_Quantities are estimates. Confirm requirements with your engineer or supplier before ordering materials._

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/euler-buckling-load-calculator