Why does end restraint change Pcr so much?

The buckled shape behaves as though the column has an effective length KL. Fixed ends shorten the effective half-wave and raise Pcr; a cantilever has K = 2 and therefore a much lower critical load.

Does P/Pcr > 1 mean the material has yielded?

Not necessarily. Euler buckling is a stability loss in an elastic idealization. Real columns may yield before buckling, buckle inelastically, or be governed by imperfections and code interaction curves.

Euler Column Buckling

Euler buckling is the elastic stability threshold for a slender ideal column under axial compression. The critical load is P_cr = π²EI/(KL)², where E is Young’s modulus, I is the second moment of area, L is the physical length, and K encodes end restraint through the effective length KL. This simulator compares pinned-pinned, fixed-free, fixed-pinned, and fixed-fixed columns, draws the first buckling mode, and shows how an initial crookedness is amplified as P approaches P_cr. It is a classroom stability model: material yielding, residual stresses, shear deformation, connection flexibility, load eccentricity, and design-code column curves are not included.

Who it's for: Mechanics of materials, structural analysis, and introductory steel/timber/concrete column design.

Key terms

Euler buckling
Critical load
Effective length factor
Slender column
Second moment of area

The first buckling mode is a stability threshold, not a material strength limit. Real design also checks yielding, residual stress, eccentricity, end fixity, and code column curves.

Live graphs

How it works

Euler elastic buckling for ideal columns: boundary conditions set the effective length factor K, and Pcr = π²EI/(KL)² gives the first-mode critical load.

Key equations

P_cr = π² E I / (K L)²

Pinned-pinned K=1, fixed-free K=2, fixed-pinned K≈0.699, fixed-fixed K=0.5

Frequently asked questions

Why does end restraint change Pcr so much?: The buckled shape behaves as though the column has an effective length KL. Fixed ends shorten the effective half-wave and raise Pcr; a cantilever has K = 2 and therefore a much lower critical load.
Does P/Pcr > 1 mean the material has yielded?: Not necessarily. Euler buckling is a stability loss in an elastic idealization. Real columns may yield before buckling, buckle inelastically, or be governed by imperfections and code interaction curves.

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Key equations

Frequently asked questions

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