# Buckling Calculator (Euler and Johnson)

URL: https://speedycalculator.com/calculators/physics/buckling/
Last updated: 2026-10-30

> The buckling calculator finds the critical load at which a column buckles, using Euler's formula for long columns and Johnson's parabola for intermediate ones, from the end conditions, length, section properties and material.

## Inputs

- End conditions (effective length factor K)
- Column length L (m, cm, mm, km, in, ft, yd, mi)
- Area moment of inertia I (mm⁴, cm⁴, m⁴, in⁴)
- Cross-section area A (mm², cm², m², in²)
- Material
- Young's modulus E (GPa, MPa, Pa, psi, Mpsi)
- Yield strength σ_y (MPa, kPa, Pa, GPa, psi, ksi)

## Results

- Radius of gyration r
- Slenderness ratio K·L/r
- Transition slenderness
- Euler critical load
- Johnson critical load
- Critical buckling load
- Governing formula

## Formula

```
Euler: P_cr = π²·E·I / (K·L)² · Slenderness λ = K·L/r, r = √(I/A)
Johnson (λ < λ_c): P_cr = A·[σ_y − (σ_y·λ/(2π))²/E],  λ_c = √(2π²E/σ_y)
```

## Explanation

A slender column fails not by crushing but by suddenly bowing sideways – buckling. Euler's formula predicts this for long columns; for stockier ones yielding interacts with buckling and Johnson's formula is used instead.

### How to use the Buckling Calculator

- **End conditions (effective length factor K)**
- **Column length L**
- **Area moment of inertia I**
- **Cross-section area A**
- **Material**
- **Young's modulus E**
- **Yield strength σ_y**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose the end conditions, enter the length, the area moment of inertia (about the weak axis) and the cross-section area, and choose a material.

### Inputs explained

FieldWhat to enterEnd conditions (effective length factor K)Choose one: Fixed–fixed (theory 0.5), Fixed–fixed (design 0.65), Fixed–pinned (theory 0.7), Fixed–pinned (design 0.8) …Column length LUnits: m, cm, mm, km, in, ft …Area moment of inertia IUnits: mm⁴, cm⁴, m⁴, in⁴Cross-section area AUnits: mm², cm², m², in²MaterialChoose one: Aluminium 6061-T6, Aluminium 2024-T3, Aluminium 1100-H12, Steel AISI 4340 …Young's modulus EUnits: GPa, MPa, Pa, psi, MpsiYield strength σ_yUnits: MPa, kPa, Pa, GPa, psi, ksi

### What each result means

- **Radius of gyration r** = sqrt([Area moment of inertia I]÷[Cross-section area A])×1000
- **Slenderness ratio K·L/r** = [End conditions (effective length factor K)]×[Column length L]÷sqrt([Area moment of inertia I]÷[Cross-section area A])
- **Transition slenderness** = sqrt(2×pi^2×[E]÷[sy])
- **Euler critical load** = pi^2×[E]×[Area moment of inertia I]÷([End conditions (effective length factor K)]×[Column length L])^2
- **Johnson critical load** = [Cross-section area A]×([sy] - ([sy]×[Slenderness ratio K·L÷r]÷(2×pi))^2÷[E])

### Example values

How the main result changes when **Column length L** changes and everything else stays at the example values above:

Column length L (m)Critical buckling load (N)1.5471,503.41712.25435,882.68853386,013.66843.75321,896.35694.5243,693.93586137,077.8389

## Worked example

Inputs — End conditions (effective length factor K): Pinned–pinned (theory 1.0); Column length L: 3 m; Area moment of inertia I: 2,500,000 mm⁴; Cross-section area A: 2,000 mm²; Material: Structural steel A36.

Results — Radius of gyration r = 35.3553 mm; Slenderness ratio K·L/r = 84.8528; Transition slenderness = 125.6637; Euler critical load = 548,311.35562 N; Johnson critical load = 386,013.6684 N; **Critical buckling load = 386,013.6684 N**; Governing formula = Johnson (intermediate column).

## Key terms

- **Slenderness ratio**: Effective length divided by radius of gyration.

## FAQ

**What is the effective length factor?**
A multiplier on the length that accounts for how the ends are supported.

**Why use design K values?**
Real connections are never perfectly fixed, so codes recommend slightly larger K.

**Which axis should I use for I?**
The weakest axis – the column buckles about it.

**Why switch to Johnson's formula?**
For short columns Euler overestimates the load; yielding governs instead.


## Sources

- [MIT OpenCourseWare – Mechanics of Materials](https://ocw.mit.edu/courses/2-001-mechanics-materials-i-fall-2006/)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)