Learn: Member slenderness
Slenderness compares a column’s effective length with the spread of its cross-sectional area. A long member with a small radius of gyration is more slender and generally more sensitive to buckling.
What the formula is saying
Radius of gyration is i = √(I/A); dividing effective length by i combines length and shape into one ratio. This lesson evaluates that ratio after i has already been found.
Read the symbols in plain language
- Lcr
- Effective length
Length of the equivalent pin-ended buckling half-wave, including the effect of end restraint; it is not automatically the physical member length.
mMetres measure length; 1 m = 1000 mm.
- i
- Radius of gyration
Square root of second moment divided by area, about the same axis used in the slenderness check.
mMetres measure length; 1 m = 1000 mm.
- λ
- Result to find
Member slenderness. Use matching length units so the final slenderness ratio is dimensionless.
ratio / no unit
Sort out the units first
L and i must both use the same length unit. Metres divided by metres gives a dimensionless number; a radius of gyration of 30 mm is 0.03 m.
Assumptions before calculating
The effective length already represents the end restraints, and i belongs to the buckling axis being considered. Both lengths must be positive.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find λ and explain the result in the stated output unit.
- Lcr · Effective length
- 3 m
- i · Radius of gyration
- 0.03 m
Identify the effective buckling length
This is the restraint-adjusted length for the buckling plane, not automatically the clear height.
(3) = 3 mDivide by radius of gyration
Use matching length units so the final slenderness ratio is dimensionless.
(3) ÷ (0.03) = 100
Does this worked answer make sense?
At fixed section, doubling effective length doubles slenderness. At fixed length, a larger i lowers slenderness.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- Lcr · Effective length
- 4 m
- i · Radius of gyration
- 0.05 m
Identify the effective buckling length
This is the restraint-adjusted length for the buckling plane, not automatically the clear height.
(4) = 4 mDivide by radius of gyration
Use matching length units so the final slenderness ratio is dimensionless.
(4) ÷ (0.05) = 80
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- Lcr · Effective length
- 5 m
- i · Radius of gyration
- 0.04 m
Find: Learn: Member slenderness
A hint, not the answer
Radius of gyration is i = √(I/A); dividing effective length by i combines length and shape into one ratio. This lesson evaluates that ratio after i has already been found.
L and i must both use the same length unit. Metres divided by metres gives a dimensionless number; a radius of gyration of 30 mm is 0.03 m.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Identify the effective buckling length
This is the restraint-adjusted length for the buckling plane, not automatically the clear height.
(5) = 5 mDivide by radius of gyration
Use matching length units so the final slenderness ratio is dimensionless.
(5) ÷ (0.04) = 125
Avoid the common trap
Do not divide metres by a radius still entered in millimetres. Do not use the section’s outside radius instead of its radius of gyration, or choose the wrong bending axis.
When this method applies — and when it does not
This is geometric slenderness, not reduced or nondimensional code slenderness. It does not include yield strength, buckling curves, local slenderness or lateral-torsional instability.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Member slenderness. See the Stresses in Beams and Beam Displacements modules; match the load and support conditions, not just the equation’s appearance.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.