Identified content errors were corrected on 2026-10-25. Sources and scope for this page: Formulas as shown on this page (standard engineering relationships, no single governing revision), covering every value this calculator produces. No separate source check is recorded for the correction; the figures rest on the sources listed and on the working shown on the page. Full sources, values and limits on this page.
Page last changed 2026-11-08 · based on n/a — standard engineering relationships, no single governing revision
Quick Answer
Euler Column Buckling
The Formulas Used
Euler critical load: Pcr = π²·E·I ÷ (K·L)²
Critical stress: σcr = Pcr ÷ A Slenderness ratio: K·L ÷ r
Effective length factors: pinned-pinned K = 1.0 · fixed-pinned K = 0.7 · fixed-fixed K = 0.5 · fixed-free K = 2.0
Effective length and ideal buckling
Effective length is K·L. Changing K from 1 to 2 doubles effective length and reduces the ideal Euler load to one quarter, with E, I and L unchanged. For L = 60 in, the effective lengths are 60 and 120 in.
The familiar ideal values are 1 for pinned–pinned, approximately 0.7 for fixed–pinned, 0.5 for fixed–fixed and 2 for fixed–free. Actual restraint and bracing require an appropriate structural model.
Euler’s result assumes a straight, slender, centrally loaded member with linear-elastic response. It is not an allowable design load. Inelastic response, imperfections, residual stress, eccentricity and local or torsional buckling require separate treatment.
Frequently Asked Questions
- What is Euler's buckling formula?
- Pcr = π²·E·I/(K·L)², where E is the modulus of elasticity, I is the moment of inertia about the weak axis, L is the unbraced length and K is the effective length factor. It gives the axial load at which a slender column becomes unstable.
- Does higher yield strength increase the Euler load?
- Yield strength does not appear in the ideal elastic formula. E, section geometry and effective length control this calculation; yielding can still limit a real column.
- What is the effective length factor K?
- A multiplier on the actual length that accounts for end conditions. K = 1.0 for pinned ends, 0.7 for fixed-pinned, 0.5 for fixed-fixed and 2.0 for fixed-free. Because it is squared in the formula, going from 1.0 to 2.0 cuts the capacity to a quarter.
- Which axis governs?
- Compare the applicable buckling modes and EI/(KL)² for each direction. The smaller I does not necessarily govern if bracing or effective length differs between axes.
- When does Euler’s formula stop applying?
- The elastic model is unsuitable when the material response at the calculated stress is outside its linear proportional range or when another failure mode governs. There is no universal one-half-yield-strength cutoff. Use the applicable design method for real capacity.
Related
Value Sources
References and calculation methods used for this page are listed below. Numbered markers connect table columns where present. A reference listing does not itself establish that every value has been independently checked.
| # | Source | Type | Revision / method |
|---|---|---|---|
| [1] | Values computed in your browser | derived | Evaluated locally from the formulas shown on the page. No data leaves the device. |
Data Sources
| Standard | Revision | What it covers on this page |
|---|---|---|
| Formulas as shown on this page | n/a — standard engineering relationships, no single governing revision | every value this calculator produces |
Verification scope
No page-specific independent source review is recorded here. The listed references and formulas explain the basis of the compilation; automated consistency checks do not verify the original sources or establish product conformity.
Derived values — the following values on this page are calculated, not taken directly from the standard:
| Value | How it is derived |
|---|---|
| All outputs | Computed in the browser from the formulas above. No data leaves the device. |
Outputs are computed from the formulas shown. Verify against the governing standard for design or acceptance work.
Cite This Page
Read the source notes, calculation methods and verification scope above before reusing a value. To cite this page: