How to use the Euler Buckling Load Calculator
- 1. Choose the endsPinned, fixed, fixed-pinned or fixed-free. The effective length is set for you.
- 2. Enter the sectionSolid or hollow circle, rectangle, or your own I and A.
- 3. Compare loadsEuler load, Rankine load and the safe load after the factor of safety.
Euler and Rankine formulas
Euler's crippling load for a long column is P = π²EI / Le², where I is the least moment of inertia and Le is the effective length.
- Both ends pinned: Le = L
- Both ends fixed: Le = L/2
- One fixed, one pinned: Le = L/√2
- One fixed, one free: Le = 2L
Slenderness ratio λ = Le / k, with k = √(I/A). Euler is valid only when λ is above π√(E/σc). For shorter columns, use the Rankine-Gordon formula P = σc·A / (1 + αλ²), which blends crushing and buckling.
Worked example and exam tips
A solid round bar, 3 m long and 50 mm in diameter, has both ends hinged and E = 200 GPa. I = π × 50⁴/64 = 306 796 mm⁴. P = π² × 200 000 × 306 796 / 3000² = 67.29 kN. If one end is fixed and the other free, Le = 6 m and the load falls to a quarter, 16.8 kN.
- Always use the smaller I. A rectangular strut buckles about its thinner side.
- Rankine constants are typical textbook values: 1/7500 for mild steel, 1/1600 for cast iron, 1/9000 for wrought iron.
- If σcr comes out above the crushing stress, the column fails by crushing, not buckling.
Frequently asked questions
What is the effective length of a column?
It is the length of an equivalent pin-ended column with the same buckling load. Fixed ends shorten it and a free end lengthens it.
When is Euler's formula not valid?
For short and intermediate columns, where the slenderness ratio is below π√(E/σc). Then Euler predicts a load higher than the crushing load.
Why does the tool also give a Rankine load?
Rankine's formula works for both short and long columns, so most textbooks ask for both and compare them.
Is this calculator free?
Yes. You fill in a short form once to see your result, and the other Uncodemy tools then open without it. Every calculation runs in your browser and nothing you type is sent anywhere.