Which formulas does the calculator use?

For long columns it uses Euler's formula, for intermediate-length columns Johnson's formula, and for eccentric loading the Secant formula. For short eccentric cases, it falls back to stress formulas.

What inputs do I need?

Length (L), area (A), second moment of area (I), modulus of elasticity (E), yield strengths (Sy and optionally Syc), applied force (P), end conditions (to set K), and optionally eccentricity (e) with distance to neutral axis (c).

What outputs are provided?

Critical buckling load (Fc), slenderness ratio (S), effective slenderness ratio (Seff), radius of gyration (r), effective length factor (K), allowable load (Fa), and factor of safety (fos).

AMESWEB

COLUMN BUCKLING CALCULATOR

Column buckling calculator for buckling analysis of compression members (columns). Includes Euler, Johnson, and Secant column formulas with unit presets and advanced per-field units.

Category	Solution with
Long columns with central loading	Euler column formula
Intermediate-length columns with central loading	J. B. Johnson formula
Columns with eccentric loading	Secant column formula
Struts or short columns with eccentric loading	Stress formulas

SAMPLE COLUMN	LOADING TYPE
SAMPLE COLUMN	CENTRAL LOADING	ECCENTRIC LOADING
	Concentric axial force through centroid	Eccentric axial force offset by e

End conditions

END CONDITIONS

Global Units

Preset Customize units per field

Choose a preset to set all units at once. Turn on Customize only if you need mixed units.

Input Parameters

Parameter	Value
Column length [L]	Unit: m
Cross-sectional area [A]	Unit: cm^2
Second moment of area [I]	Unit: cm^4
Distance to the neutral axis [c]	Unit: mm
Eccentricity [e]	Unit: mm
Design factor [n_d]**
Modulus of Elasticity [E]	Unit: GPa
Yield strength [S_y]	Unit: MPa
Compressive yield strength [S_yc]*	Unit: MPa
Applied Force [P]	Unit: N
End conditions

Note 1: Use dot "." as decimal separator.

Note 2: * If not known, use same value with yield strength.

Note 3: ** Select an appropriate design factor to cover material nonuniformity, uncertainty of service conditions, and calculation/analysis inaccuracy.

Results

Parameter	Value
Effective length factor [K] *	1.0
Radius of gyration of column [r]	--- mm
Slenderness ratio of column [S]	---
Effective slenderness ratio of column [S_eff]	---
Critical load for failure [F_c]	--- N
Allowable load (includes n_d) [F_a]	--- N
Factor of safety [fos]**	---
Column Category

Note 1: * Effective-length factor per end condition.

Note 2: ** Shall be larger than design factor. Green color means safe, red color means not safe according to input parameters.

Definitions

Allowable stress: The maximum working stress permitted for service conditions, typically the failure stress divided by a design factor. Here the related concept is allowable load F_a = F_c/n_d.
Buckling: Instability failure of a compression member with sudden lateral deflection at a load below yield. Capacity depends on slenderness, end conditions (K-factor), E, and I. Long columns → Euler; intermediate → Johnson; eccentric → Secant.
Failure stress: A limit state used for design—yield/proof/ultimate strength, or the stress corresponding to critical buckling load—depending on design basis and code.
Design factor (n_d): A specified ratio that reduces failure capacity to an allowable level to cover uncertainties. It is a requirement input, not a calculated result.
Eccentricity (e): Offset of the load line from the centroidal axis, creating M = P·e. Treated using the Secant (eccentric column) formula.
Effective length: Buckling length K·L between points of zero bending moment in the buckled shape; K is set by end conditions.
Effective slenderness ratio: S_eff = (K·L)/r with r = √(I/A). Governs the Euler/Johnson transition and imperfection sensitivity.
Factor of Safety (fos): fos = F_c / P. For a safe design, typically fos ≥ n_d. Unlike n_d, fos reflects the actual member state.
Modulus of elasticity (E): Elastic slope of the stress–strain curve (GPa or ksi). Higher E increases Euler buckling resistance: F_Euler ∝ E·I.
Neutral axis: Line in a cross-section under bending where normal stress is zero; distance to extreme fiber is c.
Neutral surface: Surface formed by neutral axes along the member; fibers on this surface carry no normal stress in pure bending.
Proportional limit: Largest stress at which stress and strain remain proportional (Hooke’s law holds).
Radius of gyration (area), r: r = √(I/A). Measures area distribution about an axis; drives slenderness S = L/r and buckling capacity.
Stability: Ability of a structure to sustain load without a sudden change in configuration (global elastic/inelastic buckling for columns).
Yield strength (S_y): Stress at which specified permanent strain occurs. Compressive yield S_yc may differ from tensile yield and is used for eccentric/short-column checks.

Examples

References

Young, W.C., Budynas, R. (2011). Roark's Formulas for Stress and Strain, 8th Ed.
Budynas, R., Nisbett, K. (2014). Shigley’s Mechanical Engineering Design
Beer, F.P., Johnston, E.R. (1992). Mechanics of Materials

Supplements

Support AMESWEB

If you find these engineering calculators helpful, you can support their maintenance and future development with a PayPal donation.

Donate via PayPal