Euler Column Buckling Calculator

The critical buckling load, slenderness ratio and critical stress of a slender column for the four standard end conditions.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

A slender column under compression can fail by buckling: it bows sideways suddenly and loses its stiffness. The Euler load is the load at which a perfectly straight elastic column buckles. It depends on the stiffness E I and the effective length, not on the yield strength.

The effective length is the real length times a factor K set by the end conditions: 1.0 for pinned ends, 0.5 for fixed ends, 0.7 for one fixed and one pinned, and 2.0 for one end fixed and one free. Real columns are never perfectly straight, so they buckle below the Euler load. Treat it as an upper limit.

Add the applied load and the yield strength to see the buckling factor of safety, whether the column is slender or stocky, and the second moment of area, length or load that would reach your target.

Formulas used

  • P_{cr} = \frac{\pi^2 E I}{(K L)^2}
  • \lambda = \frac{K L}{r},\quad r = \sqrt{I / A}
  • \sigma_{cr} = P_{cr} / A

Worked example

With the default numbers in the calculator above:

  1. Effective length: 3 m
  2. Critical (Euler) buckling load: 219 kN
  3. Radius of gyration: 25.8 mm
  4. Slenderness ratio: 116
  5. Critical stress: 146 MPa

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Does stronger steel make a slender column carry more?

No. The Euler load depends on E, not on yield strength, and E is about the same for all steels.

What if the critical stress is above yield?

Then the column is stocky and Euler's formula does not apply. It will crush by yielding or buckle inelastically.

Which I do I use?

The smaller one, because the column buckles about its weaker axis.

Related calculators

Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.

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