Euler Buckling Calculator: Critical Load, Slenderness Ratio and Johnson Parabola
Enter the column length, end conditions, material and cross-section. The calculator returns the slenderness ratio, the Euler and Johnson critical loads, the governing value and a safety factor, in mm or inches.
Source: ASTM A36/A36M; E per AISC 360-22. Editing E or Sy switches to Custom.
Results
Critical load
Above load- Governing method
- Euler (elastic)
- Critical load
- 228.2 kN
- Critical stress
- 100.2 MPa
- Safety factor
- 2.28
- Euler load
- 228.2 kN
- Johnson load
- n/a above (KL/r)_T
- Squash load A × Sy
- 569.4 kN
Ideal column theory. Not a code check: AISC 360-22 and EN 1993-1-1 include imperfections and give lower values.
Slenderness
- Effective length factor K
- 1.00
- Effective length KL
- 7200.000 mm
- Slenderness ratio KL/r
- 140.4
- Transition slenderness
- 125.7
KL/r 140.4 is above the transition 125.7: slender, Euler governs.
Section
- Area A
- 2277.65 mm²
- Second moment I
- 5,993,079 mm⁴
- Radius of gyration r
- 51.296 mm
Column and mode shape
Critical stress against slenderness
- Johnson parabola
- Euler (dashed where it overestimates)
- Applied stress P/A
Every value below uses the inputs above and updates as you edit them. The lines work in mm, mm², mm⁴, MPa and N (1 MPa × 1 mm² = 1 N).
1. Section properties
The column bends about the axis with the smaller second moment of area. The radius of gyration turns that stiffness into a length that can be compared with the column length.
Inside diameter
Area
Second moment
Radius of gyration
2. Effective length and slenderness
The end conditions set how much of the column bends as one half-wave. K times L is the length of an equivalent pinned column. The transition slenderness is where the Euler stress falls to half the yield strength.
Effective length factor
Effective length
Slenderness ratio
Transition slenderness
Range
3. Critical load
Above the transition the column stays elastic and the Euler load applies. Below it the Johnson parabola, tangent to Euler at the transition and equal to the yield strength at zero length, gives the inelastic value.
Euler load
Euler stress
Johnson stress
Johnson load
Governing
- Johnson parabola
- Euler (dashed where it overestimates)
- Applied stress P/A
4. Safety factor
The safety factor is the governing critical load divided by the applied load.
Applied stress
Safety factor
5. EN 1993-1-1 reduction factor
Eurocode 3 reduces the squash load A fy by χ, which already includes the initial bow and residual stresses of real steel members (clause 6.3.1.2).
Choose a buckling curve (steel only) to see the EN 1993-1-1 check.
Method
Section properties. The area A and the second moments of area I about both axes come from the dimensions. I is πd⁴/64 for a solid round, π(D⁴ − d⁴)/64 for a tube and bh³/12 for a rectangle. A rectangular tube is outer minus inner (sharp corners), and an I-section is flanges plus web (no root fillets). A and I can also be entered directly from a section table. The column buckles about the axis with the smaller I, and the radius of gyration is r = √(I / A).
Effective length and slenderness. The effective length is KL, where L is the column length and K, the effective length factor, depends on the end conditions. The theoretical values (0.5 fixed-fixed, 0.7 fixed-pinned, 1.0 pinned-pinned, 1.0 fixed-guided, 2.0 fixed-free, 2.0 pinned-guided) assume perfectly rigid or perfectly free ends. The default is the recommended design values of AISC 360-22 Commentary Table C-A-7.1 (0.65, 0.80, 1.0, 1.2, 2.1 and 2.0). These allow for real connections not being fully fixed. The slenderness ratio is KL/r.
Critical load. For a slender column the Euler load is P_cr = π²EI / (KL)², or as a stress σ_cr = π²E / (KL/r)², where E is the elastic modulus. Euler assumes the column stays elastic, so it overestimates the strength of stockier columns whose critical stress approaches yield. Let Sy be the yield strength. Below the transition slenderness (KL/r)_T = √(2π²E / Sy), the calculator uses the Johnson parabola σ_cr = Sy − (Sy × KL/r)² / (4π²E) instead (Shigley, Section 4-14). The two curves meet at σ = Sy / 2, and the calculator reports whichever governs.
Safety factor and EN 1993-1-1. With an applied axial load P, the safety factor is P_cr / P. For steel, the calculator can also show the flexural buckling reduction factor χ of EN 1993-1-1:2005 clause 6.3.1.2. It uses the non-dimensional slenderness λ̄ = √(A fy / N_cr), Φ = 0.5 [1 + α (λ̄ − 0.2) + λ̄²] and χ = 1 / (Φ + √(Φ² − λ̄²)) ≤ 1. Here fy is the yield strength, N_cr the Euler load and α the imperfection factor: 0.13, 0.21, 0.34, 0.49 or 0.76 for buckling curves a0, a, b, c and d (Table 6.1). The buckling resistance is N_b,Rd = χ A fy / γ_M1 with γ_M1 = 1.0.
Assumptions
- Ideal column theory: a straight, prismatic, centrally loaded member with no initial bow, eccentricity or residual stress. Real columns buckle below the Euler and Johnson values; design codes (AISC 360-22 Chapter E, EN 1993-1-1 6.3) include these effects and give lower strengths.
- Flexural buckling only, about the minor axis, with the same K about both axes. Torsional, flexural-torsional and local buckling of thin walls or flanges are not checked.
- Linear elastic material up to yield for Euler. The Johnson parabola is an empirical fit for the inelastic range, using the compressive yield strength.
- Material properties are typical minimums for the named product form. Check the procurement specification, the thickness range and MMPDS or the relevant standard.
- Rectangular tubes have sharp corners and I-sections have no root fillets, so A and I differ slightly from section tables. Enter table values with the direct A and I option.
- The EN 1993-1-1 reduction factor uses γ_M1 = 1.0 and applies to steel members. Choose the buckling curve from Table 6.2 for the section and axis.
Standards and references
| Standard | Note |
|---|---|
| ANSI/AISC 360-22, Specification for Structural Steel Buildings, Commentary Appendix 7, Table C-A-7.1 | Theoretical and recommended design values of the effective length factor K for six idealised end conditions. |
| EN 1993-1-1:2005 + A1:2014, Eurocode 3: Design of steel structures, Part 1-1, clause 6.3.1 | Flexural buckling reduction factor χ, imperfection factors (Table 6.1) and buckling curve selection (Table 6.2). |
| Budynas, R. G. and Nisbett, J. K., Shigley’s Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015, Sections 4-12 to 4-14 | Euler column, end-condition constants, the Johnson parabolic formula and the tangent-point slenderness. |
| Hibbeler, R. C., Mechanics of Materials, 8th ed., Pearson, 2011, Chapter 13 | Euler critical load and stress, effective length; Examples 13.1 and 13.2 are reproduced in the tests. |
| Gardner, L. and Nethercot, D. A., Designers’ Guide to EN 1993-1-1, Thomas Telford, 2005 | Example 6.7, flexural buckling of a 244.5 × 10 CHS in S355, reproduced in the tests. |
| EN 10025-2:2019, ASTM A36/A36M, MMPDS (Metallic Materials Properties Development and Standardization) | Sources of the preset elastic modulus and yield strength values. |
Worked example: 150 mm steel tube, 7.2 m, pinned both ends
A round A36 steel tube, 150 mm outside diameter and 5 mm wall, is 7.2 m long and pinned at both ends (K = 1.0). E = 200 GPa and Sy = 250 MPa. This is Hibbeler’s Example 13.1, loaded by default with an applied load of 100 kN.
| Result | Value |
|---|---|
| Area A | 2277.65 mm² |
| Second moment I | 5,993,079 mm⁴ |
| Radius of gyration r | 51.296 mm |
| Slenderness ratio KL/r | 140.4 |
| Transition slenderness | 125.7, so Euler governs |
| Euler critical load | 228.2 kN (Hibbeler: 228.2 kN) |
| Critical stress | 100.2 MPa |
| Safety factor at 100 kN | 2.28 |
Frequently asked questions
What is the slenderness ratio?
The slenderness ratio is the effective length divided by the radius of gyration, KL/r. It measures how prone a column is to buckling, whatever its size: the Euler critical stress is π²E / (KL/r)². Steel columns above about 120 are slender and governed by Euler. Below that, yield starts to interact with buckling.
What is the Euler buckling formula?
P_cr = π²EI / (KL)², where E is the elastic modulus, I the smaller second moment of area, L the length and K the effective length factor. It gives the load at which an ideal, straight, elastic column becomes unstable. Doubling the length divides the critical load by four.
Which effective length factor K should I use?
Use the recommended design values of AISC 360-22 Table C-A-7.1 unless the end restraint is genuinely rigid. They are 0.65 for fixed-fixed, 0.80 for fixed-pinned and 1.0 for pinned-pinned. A fixed column with a guided sway end takes 1.2, a flagpole 2.1, and a pinned column with a guided sway end 2.0. The theoretical values (0.5, 0.7, 1.0, 1.0, 2.0, 2.0) assume ideal restraint that real connections rarely give.
When does the Johnson parabola apply instead of Euler?
When KL/r is below the transition slenderness √(2π²E / Sy), about 108 for S355 steel and 126 for A36. There the Euler stress would exceed half the yield strength, and inelastic behaviour lowers the real strength. Hibbeler’s W200×46 example (KL/r 78.5) gives 1888 kN by Euler but 1485 kN by Johnson.
Why is the EN 1993-1-1 value lower than Euler?
The Eurocode buckling curves are calibrated on tests of real members with initial bow and residual stress. The reduction factor χ already includes these imperfections, so N_b,Rd = χ A fy is lower than both the Euler load and the squash load A fy. For the Designers’ Guide 244.5 × 10 CHS over 4 m, Euler gives 6572 kN and EN 1993-1-1 gives 2296 kN.
What safety factor is typical for column buckling?
Mechanical design commonly uses 2 to 4 on the Euler or Johnson load. Go higher when end fixity, load eccentricity or straightness are uncertain. Structural codes instead apply resistance factors to the code curves with factored loads, for example φ = 0.90 in AISC 360-22 or γ_M1 = 1.0 in EN 1993-1-1.
Does the calculator check local buckling of thin walls?
No. It checks overall flexural buckling only. Thin-walled tubes, box sections and slender I-section flanges can buckle locally first. Check the width-to-thickness limits of AISC 360-22 Table B4.1a or the cross-section classification of EN 1993-1-1 Table 5.2.
Changelog
| Date | Change |
|---|---|
| First published. |