Formula used
This calculator uses the Euler elastic buckling equation:
- Pcr: critical buckling load
- E: Young’s modulus of the material
- I: least second moment of area about the buckling axis
- L: unsupported column length
- K: effective length factor determined by end restraints
How to use this calculator
- Enter the unsupported column length and choose its unit.
- Select an end condition (or enter a custom K).
- Pick a material; optionally override the modulus.
- Provide section properties by direct I or by shape.
- Set a safety factor and, optionally, an applied load.
- Click calculate; results appear above the form.
- Download results as CSV or PDF for documentation.
Example data table
| # | L (m) | End | E (GPa) | Section | I (m⁴) | K | Pcr (kN) |
|---|---|---|---|---|---|---|---|
| 1 | 3 | Pinned–Pinned | 200 | Rect 50×100 mm | 5.208e-6 | 1 | 1143 |
| 2 | 2.5 | Fixed–Fixed | 69 | Solid Ø50 mm | 3.068e-7 | 0.5 | 268 |
| 3 | 4 | Fixed–Free | 193 | Hollow Ø60/40 mm | 4.909e-7 | 2 | 29.2 |
What the critical buckling load represents
Euler buckling predicts the axial load where a slender, straight column becomes laterally unstable under ideal alignment. The calculator reports Pcr using the selected modulus, end restraint, and stiffness about the weakest axis. Because the formula scales with 1/L², doubling the unsupported length reduces capacity to one quarter. Assumptions include concentric loading, prismatic geometry, small deflections, and elastic behavior, so treat results as a screening value before final code checks.
Role of end restraint and effective length factor
Boundary conditions change the effective buckling length through K. Typical values include pinned–pinned K=1.0, fixed–free K=2.0, fixed–pinned K≈0.699, and fixed–fixed K=0.5. The tool computes K·L so you can compare bracing or connection upgrades directly. If end restraint is uncertain, conservative practice is to select a higher K or verify stiffness using connection details and lateral translation limits.
Section stiffness and least-axis inertia
Column stability depends strongly on the second moment of area I. For rectangles the weaker axis governs, so the calculator evaluates both Ix and Iy and uses the smaller value. For circular sections it applies I=πd⁴/64 (solid) and I=π(D⁴−d⁴)/64 (hollow), ensuring consistent SI conversions. You may also enter a published I directly (for I‑beams, channels, or built‑up sections) using mm⁴, cm⁴, in⁴, or m⁴.
Slenderness ratio, stresses, and interpretation
When area A is available, the calculator adds radius of gyration r=√(I/A), slenderness λ=(K·L)/r, and stresses σ=P/A. High λ indicates a buckling-driven design, while lower λ may require inelastic or code-based checks. Designers often track λ to compare alternatives quickly, such as increasing section depth, reducing unbraced length, or adding intermediate bracing. Results are intended for preliminary sizing, not final certification.
Practical workflow, safety factor, and exports
Use the safety factor to convert elastic capacity into an allowable load: Pallow=Pcr/SF. If you enter an applied load, the utilization ratio highlights margin quickly. Downloads produce a clean CSV for spreadsheets and a compact PDF summary for reviews, site records, or design notes. Each export includes the inputs and the computed outputs from your successful run for consistent documentation.
FAQs
1) When should I use Euler buckling?
Use it for long, slender columns that remain elastic and have small initial imperfections. For short or stocky members, use material yield and code provisions instead.
2) Why does the rectangle option use the weaker axis?
Buckling occurs about the axis with the smallest second moment of area. The tool computes both principal inertias and conservatively selects the smaller value.
3) What does the safety factor change?
It reduces the theoretical critical load to an allowable load for design screening. Choose a factor that matches your standards, uncertainties, and the consequence of failure.
4) Can I mix metric and imperial units?
Yes. Length, modulus, inertia, and force units can be selected independently. Internally everything is converted to SI before calculation, then converted back to your output unit.
5) Why is my result extremely high or low?
Check that length and section dimensions use the intended units, and confirm the correct buckling axis inertia. Small changes in length or diameter can shift results dramatically due to the squared and fourth-power relationships.
6) What is included in the CSV and PDF exports?
Exports include inputs, effective length, critical and allowable loads, and derived metrics when available. They are based on the most recent successful calculation stored in your session.