BIZOMAT MAKEOS · Strength
Column Buckling (Euler and Johnson)
What load will buckle this column or strut, and is the safety factor enough?
Critical buckling load of a straight, centrally loaded column using Euler’s formula for slender members and the J.B. Johnson parabola for intermediate ones, with the safety factor against an applied load.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Elastic modulus | GPa | Elastic modulus from the material data sheet or an approved materials database (steel is typically about 200 GPa). |
| Yield strength | MPa | Minimum yield strength from the material certificate or specification, not a typical value. |
| Cross-section area | mm² | Area of the section, from a section table or the section properties calculator. |
| Least second moment of area | mm⁴ | Smallest second moment of area of the section (the weak axis). |
| Unsupported length | mm | Length between points that are held against sideways movement. |
| Effective length factor K | Theoretical values: 1.0 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2.0 fixed–free. Design codes often use larger values for real end conditions. | |
| Applied axial load | N | Largest compressive load on the member. |
| Required safety factor | Factor of safety required by your design code, customer specification or company standard. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Elastic modulus
- What it is
- The elastic modulus (Young’s modulus) of the column material: its stiffness in tension and compression, from the material data sheet.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Yield strength
- What it is
- The minimum yield strength of the column material, the stress at which it starts to deform permanently, from the material certificate or specification.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 5000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 250 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the specified minimum, not a typical test value.
What this means: Cross-section area
- What it is
- The cross-section area of the column or strut, the area of material cut across its length, from a section table or a section properties calculation.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0.000001 mm² to 100000000 mm². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the net area of the actual section, in mm².
What this means: Least second moment of area
- What it is
- The smallest second moment of area of the section, about its weak axis. It describes how the material is spread from that axis and resists bending sideways.
- Units and format
- Enter a number in mm⁴: millimetres to the fourth power (second moment of area). The form accepts 0.000001 mm⁴ or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 400000 mm⁴ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the least value of the section; a column buckles about its weakest axis unless that direction is braced.
What this means: Unsupported length
- What it is
- The length of the column between the points that are held against sideways movement.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- In the slender (Euler) range the buckling load falls with the square of the length.
- Get it right
- Measure between actual lateral supports, not the overall member length if it is braced partway.
What this means: Effective length factor K
- What it is
- The effective length factor: a multiplier on the unsupported length that accounts for how the ends are held (pinned, fixed or free).
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger factor increases the slenderness and lowers the buckling load.
- Get it right
- Design codes often use larger values than the theoretical ones for real end conditions; use your code’s value.
What this means: Applied axial load
- What it is
- The largest compressive axial load the member must carry along its length.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The safety factor falls in proportion as this load rises.
- Get it right
- Include all load cases that put the member in compression and use the largest.
What this means: Required safety factor
- What it is
- The minimum ratio of critical buckling load to applied load that the member must achieve, from your design code, customer specification or company standard.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- This is a requirement you set; the tool only compares the calculated factor with it.
Method and formulas
- r = √(I ÷ A); slenderness λ = K L ÷ r; transition C_c = π√(2E ÷ S_y).
- Euler (λ ≥ C_c): σ_cr = π²E ÷ λ².
- Johnson (λ < C_c): σ_cr = S_y − (S_y λ ÷ 2π)² ÷ E.
- P_cr = σ_cr × A; safety factor = P_cr ÷ P.
- Example: the default strut has λ = 100, below C_c = 125.66, so Johnson gives P_cr = 170.84 kN and a factor of 3.4169.
Limitations
- Straight, centrally loaded, prismatic members; eccentricity and initial crookedness reduce capacity (see the eccentric column method or your code).
- Local buckling of thin walls and lateral-torsional buckling are not checked.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check push rods, cylinder rods, props and frame members in compression
- Choose a section to resist buckling
- Compare end fixity options
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Slenderness ratio and transition slenderness
- Method used (Euler or Johnson)
- Critical load
- Safety factor
Interpreting the result
- Structural steelwork must also be checked to the applicable code (e.g. AS 4100), which includes imperfection factors.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 281–287. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Small holes and punch strength
In Blanking and Piercing Dies by the Numbers
What it covers: Why small punches fail, Hodgson's 1903 limit on hole size restated in SI, crushing and buckling checks for MP-40's punches, guided punches, and a Ø3 punch that broke on its first stroke.
Method explained here · BIZOMAT original book
Why straight members go sideways
What it covers: Why a straight member in compression fails by bending sideways at a load set by its stiffness, not its strength: Euler's load, the energy view and the fork between straight and bent.
Method explained here · BIZOMAT original book
Check stiffness, frequency and buckling
In Design Simulation by the Numbers
What it covers: Use modal analysis to keep the bracket away from resonance, check each mode against a hand estimate, test sensitivity to the frame and use buckling results with caution.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.