Euler critical load
P_cr = π²EI/(KL)²
Elastic buckling load depends on flexural rigidity and effective length.
Column Buckling Calculator
Screen a compression member with Euler buckling and the Johnson parabola using geometry, material properties, end conditions, and applied load.
Access: Free to use, no installation, and No account required.
Inputs
This is a preliminary ideal-column screen. Real columns require section-specific local buckling, residual stress, connection, imperfection, bracing, code, and load-combination checks.
Results
Euler versus Johnson
Euler buckling applies to sufficiently slender elastic columns. The workbench engine switches to the Johnson parabola below its transition slenderness so short-column capacity is not overstated by Euler’s elastic formula.
Engineering reference
Column Buckling Calculator: background and worked detail
Use the Column Buckling Calculator to estimate effective length, slenderness ratio, Euler critical load, Johnson inelastic-column load, the governing idealized capacity, and factor of safety for a prismatic column.
Euler buckling and slenderness
Euler buckling describes elastic instability of sufficiently slender columns. When the Euler stress would exceed the material yield region, the engine switches to the Johnson parabola as a simple short/intermediate-column correction.
Real compression-member design is strongly affected by imperfections, residual stress, local buckling, section shape, bracing, connection restraint, eccentricity, code resistance factors, and load combinations.
Equations used by this calculator
Radius of gyration
r_g = √(I/A)
Section area and second moment define a characteristic buckling radius.
Slenderness ratio
λ = KL/r_g
Effective length divided by radius of gyration indicates whether a column is slender.
End-condition checks
Changing end conditions should move the critical load by the expected factor.
End restraint
Case: Compare pinned-pinned with fixed-fixed using identical E, I, A, and L.
Expected: Fixed-fixed should have a shorter effective length and a much higher Euler critical load.
Length scaling
Case: Double L in an Euler-governed case.
Expected: Euler critical load should fall by approximately a factor of four.
3 m pinned steel column
Use E = 200 GPa, I = 100 cm⁴, A = 10 cm², L = 3 m, Sy = 250 MPa, 50 kN applied load, and pinned-pinned ends.
- The effective length is 3 m because K = 1.
- The radius of gyration is √(I/A) and the slenderness is based on KL/r_g.
- The engine compares Euler and Johnson regimes and reports the governing design load and factor of safety.
Result: Use the returned governing model as a preliminary stability screen, not as a code-compliant member capacity.
Assumptions and model boundaries
Assumptions
- Straight prismatic column with idealized end-condition factors.
- Material is represented by E and yield strength.
- Applied load is concentric in the focused model.
Limitations
- Does not evaluate local buckling, torsional/flexural-torsional buckling, initial crookedness, residual stress, connection stiffness, eccentricity, bracing, or code factors.
- Johnson/Euler switching is a preliminary mechanics model, not a jurisdiction-specific structural design method.
Column Buckling Calculator FAQ
What does K mean in column buckling?
K is the effective-length factor representing end restraint. Euler capacity varies with 1/(KL)².
Why can Johnson govern instead of Euler?
For stockier columns, Euler can predict elastic buckling stress above the physically relevant yield region; the Johnson parabola provides an inelastic transition model.
Is the factor of safety a code design result?
No. It is the ratio of the model's governing idealized load to the entered applied load. Code design requires prescribed resistance and load factors plus additional limit states.
Where this calculation comes from
Shared with the Statics and Strength Workbench, which adds inelastic buckling and combined axial-bending cases.
Open the source workbench →Read calculation and source methodology →