BIZOMAT MAKEOS · Strength
Column Formula Comparison (Rankine, Euler and Classic Safe Load)
How do the Rankine, Euler and classic working-stress column formulas compare for this strut, and does it carry my load by each?
Comparison of column formulas for a concentrically loaded strut: slenderness ratio, Euler buckling load, Rankine–Gordon crippling load and safety factor (as the Rankine column tool), and the classic working-stress safe unit stress and safe load for a main or secondary member (as the classic column safe stress tool), with checks of the Rankine factor and the classic safe load.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Cross-section area | mm² | Area of the strut section. |
| Least radius of gyration | mm | Least radius of gyration of the section. |
| Length | mm | Length between end restraints. |
| Effective length factor | Effective length factor for the end conditions. | |
| Crushing (compressive) strength | MPa | Crushing strength of the material (for the Rankine formula). |
| Elastic modulus | GPa | Elastic modulus. |
| Rankine constant (0 = theoretical) | Empirical Rankine constant; 0 uses S_c ÷ π²E. | |
| Member type (classic formula) | choice | Whether the strut is a main member or bracing (classic formula). |
| Applied concentric load | kN | Concentric load on the strut. |
| Required Rankine safety factor | Safety factor you need by the Rankine formula. |
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: Cross-section area
- What it is
- The cross-sectional area of the strut, 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 1500 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All the loads (Euler, Rankine, classic and crushing) rise in direct proportion to area.
- Get it right
- Enter mm², not cm² (1 cm² = 100 mm²).
What this means: Least radius of gyration
- What it is
- The least radius of gyration of the section, the square root of the smaller second moment of area divided by the area.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger radius lowers the slenderness ratio and raises every buckling load.
- Get it right
- Use the least value, about the weak axis, unless the strut is braced against buckling that way.
What this means: Length
- What it is
- The length of the strut between the points that restrain it 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 2500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Slenderness rises in proportion to length; the Euler load falls with its square.
- Get it right
- Use the distance between restraints, not the overall member length if it is braced part way.
What this means: Effective length factor
- What it is
- The effective length factor that converts the actual length to the buckling length for the end conditions.
- 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
- Slenderness rises in direct proportion to this factor.
- Get it right
- Choose it for the real end restraint from your structural code; 1 is pinned at both ends.
What this means: Crushing (compressive) strength
- What it is
- The crushing (compressive) strength of the material used in the Rankine formula, from the material data sheet.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 320 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The Rankine crippling load rises with the crushing strength; with the theoretical constant the effect fades for slender struts.
- Get it right
- It is used only by the Rankine formula and the theoretical Rankine constant, not by the classic formula.
What this means: Elastic modulus
- What it is
- The elastic (Young’s) modulus of the strut material, used for the Euler load and the theoretical Rankine constant.
- 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.
- Effect on the result
- The Euler load rises in direct proportion to the modulus.
- Get it right
- Enter gigapascals, not megapascals.
What this means: Rankine constant (0 = theoretical)
- What it is
- The empirical constant in the Rankine–Gordon formula for the material and end conditions; zero tells the tool to use the theoretical value from crushing strength and modulus.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 1. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger constant lowers the Rankine crippling load for slender struts.
- Get it right
- 0 means theoretical (Sc ÷ π²E). Take an empirical value only from a handbook or standard for the same material.
What this means: Member type (classic formula)
- What it is
- Whether the strut is a main load-bearing member or a bracing or secondary member, which selects the version of the classic safe-stress formula.
- Units and format
- Choose one of 2 options from the list: Main member; Bracing or secondary member.
- Example
- The form opens with “Main member” selected as a worked example. Choose the option that matches your case.
- Get it right
- Main members above a slenderness of 200 are outside the classic formulas and are rejected.
What this means: Applied concentric load
- What it is
- The axial compressive load applied through the centre of the strut section.
- Units and format
- Enter a number in kN: kilonewtons (1 kN = 1000 N). The form accepts 0 kN to 10000000 kN. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 kN as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The Rankine safety factor falls in inverse proportion to the load.
- Get it right
- Enter kilonewtons. The tool assumes the load is concentric; an eccentric load needs a combined check.
What this means: Required Rankine safety factor
- What it is
- The safety factor you require on the Rankine crippling load, from your design code or company practice.
- 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 3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from your design standard; it is compared with crippling load ÷ applied load.
Method and formulas
- Slenderness = K L ÷ r; Euler load = π² E A ÷ (K L ÷ r)².
- Rankine crippling load = S_c A ÷ [1 + a (K L ÷ r)²] (as the Rankine column tool).
- Classic safe unit stress from the slenderness by the working-stress formulas, × area for the safe load (as the classic column safe stress tool).
- Example: KL/r = 100; Euler 296.09 kN, Rankine 183.13 kN (factor 3.0521), classic safe load 125.66 kN.
Limitations
- Concentric load and straight member; eccentric loads need the secant formula.
- The classic formula is a historic working-stress method; use the current steel code for new designs.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check an existing strut by more than one method
- Teach how column formulas compare
- Check old steelwork designed by working stress
Who it is for
- Structural designers
- Maintenance engineers
- Students
What you get
- Slenderness ratio
- Euler and Rankine loads and factors
- Classic safe stress and load
Interpreting the result
- Euler overestimates short columns; Rankine blends crushing and buckling, so it is lower than both at every length.
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
When yield meets buckling
What it covers: Why Euler's formula overestimates stocky members, and how the Johnson parabola, the Rankine–Gordon formula and classic column formulas bridge the gap between squashing and buckling.
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.
Open the Column Formula Comparison (Rankine, Euler and Classic Safe Load) calculator