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

Elastic column stability with P_cr = π²EI/(KL)²: choose end conditions, effective length factor K, first mode shape, and load ratio P/P_cr.

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

About this model

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

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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Euler Column Buckling

Live graphs

About this model

How it works

Key equations

Frequently asked questions

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Live graphs

About this model

How it works

Key equations

Frequently asked questions