Theory of Columns

Basic definitions

Compression member: A compression member is a structural member which is straight and subjected to two equal and opposite compressive forces applied at its ends.

Column: A column is a vertical compression member. When a compression member is sufficiently slender compared with its length, it fails by lateral instability or buckling at a load much less than the crushing (compressive) strength of the material. Such a member is called a column.

• A compression member is generally considered to be a long column when its unsupported length is more than 10 times its least lateral dimension.
• Stanchion (post): vertical compression member supporting floors or girders in a building.
• Principal rafter: top chord member in a roof truss (may be in compression in certain arrangements).
• Boom: the principal compression member in a crane.
• Strut: a compression member in a roof truss; it may be vertical or inclined.

Modes of failure of a column

• Crushing (material failure): failure by compressive yielding or crushing of the material when compressive stress reaches material strength.
• Buckling (elastic instability): lateral deflection occurring at a load lower than crushing load for slender members.
• Mixed mode: combination of buckling and crushing for intermediate slenderness.

Euler's theory (elastic buckling)

Euler's formula predicts the critical load at which an ideal, perfectly straight, elastic column will buckle. The formula applies to long, slender columns where elastic buckling occurs before material yield.

Assumptions of Euler's theory

• The unloaded axis of the column is perfectly straight.
• The load is perfectly axial and coincides with the centroidal axis (line of thrust) of the unstrained column.
• Flexural rigidity EI is constant along the length.
• The material is homogeneous and isotropic and obeys Hooke's law (linear elasticity) up to the critical stress.
• Buckling occurs while the material is still in its elastic range (i.e., critical stress does not exceed the proportional limit).
• The length is large relative to the cross section so that buckling governs failure.

Limitations of Euler's formula

• The critical load depends only on geometry and modulus of elasticity and not on the material strength; therefore Euler's formula is invalid when the critical stress exceeds the proportional (elastic) limit of the material.
• Imperfections (initial curvature, eccentricity of load, residual stresses) reduce the buckling load compared with the ideal Euler value.
• For intermediate and short columns, a combination of crushing and buckling occurs and Euler's formula overestimates load capacity.

Derivation outline and Euler's critical load for a pinned-pinned column

For a slender column of length L, flexural rigidity EI, and axial compressive load P, small lateral deflection y(x) satisfies the differential equation:

EI (d²y/dx²) + P y = 0.

Solving this with boundary conditions y = 0 at x = 0 and x = L (pinned-pinned), non-trivial solution exists when P = (π² EI)/(L²).

Euler critical load (pinned-pinned): P_cr = π² EI / L²

General form using effective length

For other end conditions, the Euler critical load is written in terms of an effective length L_e:

P_cr = π² EI / (L_e)²

Effective length L_e = K L, where K is an effective length factor depending on end restraints.

Ideal end conditions and effective length factors

Common ideal end conditions and their corresponding effective length factors K and equivalent effective lengths L_e are:

• Case 1 - Both ends pinned (hinged): K = 1.0, L_e = L.

• Case 2 - One end fixed, other end free (cantilever): K = 2.0, L_e = 2L.

• Case 3 - Both ends fixed: K = 0.5, L_e = 0.5L.

• Case 4 - One end fixed, other end pinned: K ≈ 0.7, L_e ≈ 0.7L.

These ideal K values are used for theoretical calculations. For practical (real) end restraints, effective length factors are given in codes (for example IS:800) and depend on relative stiffness of connected members. Use code tables for design where available.

Radius of gyration and slenderness ratio

• Radius of gyration (r): r = √(I / A), where I is the least moment of inertia of the cross-section about centroidal axis and A is the area of the section.

• Slenderness ratio (λ): λ = L_e / r (dimensionless). A higher slenderness ratio indicates a more slender column and greater susceptibility to buckling.

Classification by slenderness is qualitative: very slender columns fail by elastic buckling, very short columns fail by crushing, and intermediate slenderness columns show interaction between buckling and crushing.

Practical end conditions and effective length

Codes (for example IS:800) provide guidance and tables to determine effective length factors for various practical end restraints. Use these tables to obtain L_e when end restraints deviate from the ideal cases.

Rankine's formula (empirical for all column lengths)

Rankine proposed an empirical formula that can be used for columns of all lengths, combining the effects of buckling and crushing. A convenient and commonly used form is:

1 / P = 1 / P_e + 1 / P_a

where

• P = allowable or predicted ultimate load on the column according to Rankine.
• P_e = Euler critical (buckling) load = π² EI / (L_e)².
• P_a = crushing load = A f_c, where f_c is the crushing (or ultimate compressive) stress of the material.

This relation gives P less than both P_e and P_a and yields realistic values in intermediate slenderness ranges. Various equivalent algebraic forms and empirically determined Rankine constants are used in different references.

In the expression above, f_c or σ_c denotes a material constant representing crushing strength; values must be taken from material property data or code recommendations.

Design checks and applicability

• Use Euler's formula (elastic buckling) only when the critical stress from Euler does not exceed the material's proportional (elastic) limit.
• For very short columns, perform crushing (material strength) checks using σ = P / A and compare with allowable compressive stress.
• For intermediate slenderness, use empirical formulas (for example Rankine) or code-based interaction curves to evaluate capacity.
• Consider imperfections, residual stresses, eccentricities and slenderness when selecting a design value; code recommended reduction factors and safety factors must be applied in practical design.

Worked example (Euler buckling for a pinned-pinned steel column)

Given: a steel column of length 3000 mm (L = 3000 mm) is pinned at both ends. Cross-section area A = 2000 mm². Moment of inertia about the weak axis I = 8 × 10⁶ mm⁴. Take E = 200 × 10³ N/mm². Find the Euler critical load.

Sol.

Compute L² = (3000 mm)² = 9.0 × 10⁶ mm².

Compute numerator π² E I = π² × 200000 N/mm² × 8.0 × 10⁶ mm⁴.

Compute the factor E I = 200000 × 8.0 × 10⁶ = 1.6 × 10¹² N·mm².

Multiply by π² ≈ 9.8696 to get π² E I ≈ 9.8696 × 1.6 × 10¹² = 1.5791 × 10¹³ N·mm².

Divide by L²: P_cr = 1.5791 × 10¹³ / 9.0 × 10⁶ = 1.7546 × 10⁶ N.

Therefore P_cr ≈ 1.75 MN (approx.).

Check units: E (N/mm²) × I (mm⁴) / L² (mm²) → N, as required.

Applications and practical notes

• Columns are widely used in buildings, bridges, towers, cranes and machinery supports. Correct buckling analysis is essential for safety and economy.
• For steel design, use code provisions (IS, Eurocode, BS, etc.) for effective lengths, slenderness limits, allowable stresses and design interaction checks.
• Consider lateral bracing, cross-section choice, orientation (weak/strong axis), and connections to control effective length and increase buckling strength.
• Use the radius of gyration to compare different sections: larger r reduces slenderness and increases buckling resistance for given length.

Summary

Columns fail either by crushing or by buckling. Euler's theory gives the elastic buckling load for ideal slender columns; effective length factors account for end restraints. Radius of gyration and slenderness ratio are key geometric parameters affecting buckling. For all-length design, Rankine's empirical formula provides a usable estimate combining buckling and crushing effects. For safe design, use code tables and include effects of imperfections and eccentricities.