Module 10 · Lesson 10.2
Slenderness and second-order effects
When a column starts to notice its own deflection — and why that depends on how hard it is being pushed.
Why this matters
Everything in the previous lesson treated the column's geometry as fixed. It is not. A column under load deflects sideways, the axial force acts on that deflection, and the moment grows — which increases the deflection, which increases the moment. Whether this matters is not a property of the column alone: the same column is short under a light load and slender under a heavy one.
By the end of this lesson you should be able to
- Explain second-order effects as a feedback loop
- Calculate the slenderness ratio and the limiting slenderness
- Explain why the limit depends on the axial load
- Apply the minimum eccentricity
- Calculate a second-order moment by the nominal-curvature method
What you should already know
- The interaction diagram (previous lesson)
- Euler buckling and effective length (Structural Analysis Fundamentals, Module 18)
The feedback loop
Apply a moment to a column and it deflects sideways by some amount e. The axial force N is now acting at an eccentricity e from the section, so it applies an additional moment N·e. That extra moment causes extra deflection, which causes extra moment.
The loop either converges — the deflection settles at a value larger than the first-order one — or it does not, and the column buckles.
For a slender column the amplification can be large. For a stocky one it is negligible. The whole of slenderness design is deciding which case you are in, and then either ignoring the effect or calculating it.
This is the same mechanism as Euler buckling, but with an important difference: a real concrete column has an initial moment, is not straight, and cracks as it bends. It does not sit at zero moment waiting for a critical load. It amplifies from the beginning.
What it calculates: How slender the column is, as a pure number.
- l0
- Effective length — the actual length times the end-condition factor (mm)
- i
- Radius of gyration about the axis of bending (mm)
- h
- Section depth in the direction of bending (mm)
This assumes
- The effective length correctly reflects the real end restraints
- The section is uncracked for the purpose of computing i — a simplification
In plain terms: For a rectangle, i = h/√12 = 0.289h, so λ ≈ 3.46 l₀/h. A 400 mm column with a 4.2 m effective length has λ = 36. The number on its own means nothing until it is compared with the limit — and the limit is where the interesting part lies.
What it calculates: The slenderness below which second-order effects may be ignored.
- n
- Relative normal force — how hard the column is being squeezed (—)
- A
- Creep factor; 0.7 where the effective creep ratio is not calculated (—)
- B
- Reinforcement factor; 1.1 where the ratio is unknown (—)
- C
- Moment-ratio factor; 0.7 where the end moments are unknown (—)
This assumes
- The default A, B, C values are conservative substitutes for calculated ones
- Nationally determined; requires verification against the National Annex
In plain terms: The √n is the important part. Slenderness is not a property of the column — it is a property of the column and its load. Quadruple the axial load and the limiting slenderness halves. A column that was comfortably short at the top of a building can be slender at the bottom, at identical size.
Worked example
Is this column slender?
Given
- Column 300 mm × 400 mm, C30/37 (fcd = 17.0 N/mm²)
- Storey height 5.25 m, braced, with end conditions giving l₀ = 0.8 × 5.25 = 4.2 m
- Design axial load NEd = 1200 kN
- Bending about the 400 mm axis; d = 350 mm
Find
Whether second-order effects must be included, and the additional moment if so.
Assumptions
- A = 0.7, B = 1.1, C = 0.7 — the default values, all nationally determined
- Kr = Kφ = 1.0, which is conservative
Practice
A 350 mm square column has an effective length of 3.5 m. What is its slenderness ratio λ?
Practice
A column has Ac = 160,000 mm², fcd = 17.0 N/mm² and carries NEd = 1400 kN. What is the relative normal force n?
Practice
For that column, what is the limiting slenderness λlim? Take A = 0.7, B = 1.1, C = 0.7.
Summary
- A column deflects, the axial load acts on that deflection, and the moment grows
- λ = l₀/i, and for a rectangle λ ≈ 3.46 l₀/h
- λlim = 20ABC/√n — slenderness depends on the LOAD as well as the geometry
- Quadrupling the axial load halves the limiting slenderness
- Slender columns are the normal case in buildings, not an exception
- A minimum eccentricity of max(h/30, 20 mm) always applies
- The second-order moment moves the design point sideways on the interaction diagram
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint