Euler's Theory of Columns
Columns and Struts
- Strut: A structural member subjected to an axial compressive force. A strut may be horizontal, inclined or vertical.
- Column: A vertical strut; a member whose principal function is to carry compressive loads in a vertical direction.
Euler's Column Theory
Overview: Euler's theory gives the elastic critical load (buckling load) for long, slender columns under axial compression. It assumes linear elastic behaviour and small lateral deflections so that buckling can be predicted from the elastic bending behaviour of the member.
Assumptions of Euler's theory
- Initially straight axis: The axis of the column is perfectly straight before loading.
- Perfect axial loading: The applied load is perfectly concentric and the line of action passes through the unstrained axis.
- Constant flexural rigidity: The bending stiffness EI is uniform along the length.
- Material homogeneity and isotropy: Material properties are uniform and elastic (Hooke's law applies).
- Small deflection theory: Lateral deflections are small so linearised curvature may be used.
Limitations of Euler's formula
- Real columns have initial crookedness and eccentricities; loads are seldom perfectly axial.
- Material yielding and inelastic behaviour are not accounted for; Euler's formula applies to elastic buckling only.
- For stocky (short) columns where compressive strength controls failure, Euler's value overestimates the actual crippling load.
- Intermediate columns (neither very long nor very short) may fail by inelastic buckling; such cases require empirical or inelastic theories (e.g., Rankine, Johnson).
Euler's critical (buckling or crippling) load
The elastic critical load for a long slender column (lowest buckling mode) is given by the Euler expression:
Pcr = π² EI / (Le)²
where:
- E = Modulus of elasticity of the material.
- I = Least moment of inertia of the cross-section about the axis of buckling.
- Le = Effective length of the column (depends on end conditions).
Derivation (outline)
The following gives a standard elastic derivation for the lowest mode (presented stepwise):
For small lateral deflection y(x), bending moment at a section is M(x) = -P·y(x).
By bending theory, EI·d²y/dx² = M(x).
Substitute M(x): EI·d²y/dx² + P·y = 0.
Let k² = P / (EI). Then d²y/dx² + k² y = 0.
General solution: y(x) = A·sin(kx) + B·cos(kx).
Apply boundary conditions appropriate to end supports to obtain characteristic values of k and thus P = (k²)·EI.
The smallest non-zero value of k gives the lowest critical load: k = π / Le, so Pcr = π² EI / (Le)².
Effective length and end conditions
The effective length Le is the length of an equivalent pinned-pinned column that has the same buckling load as the actual column with given end restraints. For many common end conditions:
- Both ends pinned (hinged): Le = L
- One end fixed and other free (cantilever): Le = 2L
- Both ends fixed: Le = L / 2
- One end fixed and the other hinged (fixed-pinned): Le = L / √2 ≈ 0.707L
Summary of effective lengths for different end conditions:
Slenderness Ratio (S)
Definition: The slenderness ratio is a nondimensional parameter that indicates the tendency of a compression member to buckle rather than to fail by compressive crushing.
S = Le / k
where k is the least radius of gyration of the cross-section given by k = √(I / A).
Columns with low slenderness ratios tend to fail by material crushing; columns with high slenderness ratios fail by elastic buckling. Exact limits between short, intermediate and long columns depend on material and code provisions.
Modes of failure of columns
- Crushing (material failure): Occurs when compressive stress reaches the material strength; typical of short, stocky columns.
- Buckling (elastic instability): Lateral deflection increases sharply at a critical axial load without local material failure; typical of long, slender columns.
- Inelastic buckling: For intermediate slenderness, material yields locally before global buckling; elastic Euler formula is not accurate here.
Rankine's Formula (empirical)
Purpose: Rankine proposed an empirical relation to estimate the safe axial load over the entire range from very short to very long columns by combining the effects of crushing and elastic buckling.
Rankine's relation (inverse form):
1 / PR = 1 / PC + 1 / PE
- PR = Predicted crippling load by Rankine.
- PC = Crushing (short-column) load = σC · A, where σC is the crushing (ultimate compressive) stress and A is the cross-sectional area.
- PE = Euler elastic critical load for given effective length = π² EI / (Le)².
Rankine's constant: Rankine introduced a constant a (sometimes called Rankine's constant) related to material and geometry; many expressions for Rankine formula use a in an equivalent algebraic form. The constant is often obtained from experimental data or empirical tables for common materials and sections.
Design considerations and practical notes
- Use Euler's formula only for long slender columns where stresses remain elastic and initial imperfections are small.
- For short columns use material strength (σC · A) to estimate failure load.
- For intermediate columns use empirical formulas such as Rankine or inelastic buckling theories (Johnson, Engesser) and follow relevant code provisions.
- Consider initial crookedness, residual stresses, eccentric loading, and imperfect end restraints; these reduce the actual buckling load below the ideal Euler prediction.
- Use the least radius of gyration and the correct effective length factor corresponding to support conditions when calculating slenderness and critical load.
Worked example (illustrative)
Given a pin-pin steel column of length L = 3.0 m, with E = 200 GPa, cross-sectional area A = 2000 mm², and least moment of inertia I = 8.0×10⁶ mm⁴, find the Euler critical load and slenderness ratio. (Units must be consistent.)
Convert units:
L = 3000 mm.
E = 200 × 10³ N/mm².
Compute radius of gyration k:
k = √(I / A) = √(8.0×10⁶ / 2000) = √(4000) = 63.2456 mm.
Effective length for pin-pin: Le = L = 3000 mm.
Slenderness ratio S = Le / k = 3000 / 63.2456 = 47.43.
Euler critical load:
Pcr = π² E I / (Le)².
Pcr = π² × (200 × 10³) × (8.0×10⁶) / (3000)² N.
Evaluate numerically (routine arithmetic omitted):
Pcr ≈ 1.11 × 10⁶ N (example numeric result; perform final arithmetic as required).
Interpretation: With slenderness ≈ 47, check code limits; if within elastic buckling range Euler result is applicable; otherwise use appropriate empirical method.
Summary
Euler's theory provides the elastic critical load formula Pcr = π² EI / (Le)² for long slender columns and requires correct choice of effective length according to end restraints. The slenderness ratio S = Le/k helps classify columns. Rankine's empirical relation combines crushing and buckling effects for design across the full range of column slenderness.
FAQs on Euler's Theory of Columns
| 1. What is Euler's formula for critical buckling load and when do I use it? |  |
| 2. How do end conditions affect the effective length of a column in Euler's theory? | |
| 3. What's the difference between elastic buckling and plastic buckling in column theory? | |
| 4. Why do thin, tall columns fail differently than short, thick columns? | |
| 5. How do I know if a column will actually follow Euler's buckling theory on an exam? | |
