Euler Buckling Calculator
Euler’s formula gives the load at which a slender column buckles, π²EI/Le²: a 3 m pinned steel column with I = 8 × 10⁶ mm⁴ buckles at 1754.6 kN.
1 cm⁴ = 10 000 mm⁴; 1 m⁴ = 10¹² mm⁴.
For critical stress and slenderness ratio; 0 to skip.
1754.59633797144 kN
Full answer, then cut to fewer decimals (not rounded):
- 1754.59633797144 kN
- 1754.5963379714 kN
- 1754.596337971 kN
- 1754.59633797 kN
- 1754.5963379 kN
- 1754.596337 kN
- 1754.59633 kN
- 1754.5963 kN
- 1754.596 kN
- 1754.59 kN
- 1754.5 kN
- Effective length Le
- 3 m
- Critical stress σcr = Pcr/A
- 350.919 MPa
- Radius of gyration k = √(I/A)
- 40 mm
- Slenderness ratio Le/k
- 75
Step-by-step solution
- Pcr = π²EI / Le²
- Le = 1 × 3000 mm = 3000 mm
- Pcr = π² × 200000 MPa × 8000000 mm⁴ ÷ 3000² mm² = 1754600 N = 1754.6 kN
Euler Buckling Formula
Pcr = π²EI / Le²
Le effective length: L pinned, 2L fixed–free, L/2 fixed–fixed, L/√2 fixed–pinned
Worked Example
A 3 m pinned steel column with I = 8 × 10⁶ mm⁴ and A = 5000 mm²:
- Pcr = π²EI / Le²
- Le = 1 × 3000 mm = 3000 mm
- Pcr = π² × 200000 MPa × 8000000 mm⁴ ÷ 3000² mm² = 1754600 N = 1754.6 kN
Critical stress σcr = Pcr/A: 350.919 MPa · Radius of gyration k = √(I/A): 40 mm · Slenderness ratio Le/k: 75
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T/J = τ/r = Gθ/L
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Example: A 50 mm solid steel shaft (G = 80 GPa), 1 m long, carrying 1 kN·m: Maximum shear stress τmax = 40.7437 MPa.
Frequently asked questions
How does buckling load change with end conditions?
Relative to pinned ends: fixed–free gives 1/4, fixed–pinned 2 and fixed–fixed 4 times the load, because Pcr ∝ 1/Le².
When is Euler’s formula not valid?
For short columns whose critical stress π²E/(Le/k)² exceeds the proportional limit; they fail by crushing, and Rankine’s or Johnson’s formula is used instead.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Euler Buckling Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/euler-buckling-calculator/
IEEE
Scientific Converters, "Euler Buckling Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/euler-buckling-calculator/