Column and Strut Buckling – Euler, Rankine and Perry-Robertson

Long compression members may become unstable and buckle at loads substantially below those required to cause simple compressive yielding. Buckling resistance depends upon member stiffness, length, cross-section, end restraint, material properties, imperfections and loading conditions.

This page covers classical elastic Euler buckling, the Rankine-Gordon relationship and legacy Perry-Robertson design information associated with BS 449-2.

Contents

• Introduction
• Symbols
• Effective Length and End Conditions
• Euler Buckling Theory
• Euler Buckling Stress and Slenderness
• Applicability of Euler Theory
• Rankine-Gordon Formula
• Perry-Robertson and Legacy BS 449 Data

Introduction

A strut or column is a structural member carrying a significant axial compressive load. Short, stocky compression members are generally governed by material strength. As the member becomes more slender, instability and buckling become increasingly important.

For a sufficiently slender member, the elastic buckling load can be substantially lower than the load corresponding to compressive yield of the cross-section.

The classical Euler analysis assumes:

• a straight, prismatic member;
• a homogeneous, linearly elastic material;
• constant Young's modulus E;
• an axial compressive load acting through the centroidal axis;
• idealised end restraints;
• small lateral deflection prior to buckling;
• no initial crookedness, residual stress or load eccentricity.

The tendency to buckle is characterised by the slenderness ratio:

where L_e is the effective buckling length and k is the radius of gyration about the relevant buckling axis.

For sections having different stiffness about their principal axes, buckling must be considered about the appropriate axes. Unless restraint prevents it, the member will normally be most susceptible to flexural buckling about the axis having the smallest relevant flexural rigidity EI.

Symbols

The radius of gyration is related to cross-sectional area and second moment of area by:

or equivalently:

Effective Length and End Conditions

Euler buckling can be written in terms of an effective length:

For the classical ideal end conditions:

Euler Buckling Theory

Consider an initially straight, perfectly elastic column with pinned ends subjected to a concentric axial compressive load P.

Curvature

The original RoyMech curvature construction is retained below.

For small deflections, beam curvature can be approximated by:

The elastic bending relationship is:

For a laterally deflected pin-ended column, the axial load produces a bending moment proportional to the lateral displacement. With a consistent sign convention:

Define:

The differential equation becomes:

with solution:

For pinned ends:

The first condition gives C₁ = 0. A non-zero buckled shape therefore requires:

so that:

The corresponding elastic buckling loads are:

The lowest eigenvalue, corresponding to n = 1, governs the first buckling mode:

Using L_e = KL, the same equation is:

Euler Buckling Stress and Slenderness

Since:

and:

the Euler critical stress is:

or, using the slenderness ratio λ:

Applicability of Euler Theory

Euler theory is an elastic buckling model. It is applicable only when the critical stress remains within the range in which the assumed elastic modulus is appropriate and the member is sufficiently slender for elastic instability to govern.

A useful theoretical transition can be obtained by setting the Euler critical stress equal to a chosen limiting material stress σ_lim:

giving:

This demonstrates why there is no universal slenderness ratio such as 80 or 120 below which Euler theory suddenly becomes invalid. The transition depends upon Young's modulus, material strength and the assumptions used in the design method.

At intermediate slenderness, inelastic behaviour, residual stresses and geometrical imperfections become increasingly significant. Design standards therefore generally use empirically calibrated column curves rather than the ideal Euler equation alone.

Rankine-Gordon Formula

The Rankine-Gordon formula is a classical empirical/semi-empirical method that combines the limiting effects of direct compression and Euler buckling. It has historically been used to estimate the strength of columns over a wider range of slenderness than Euler theory alone.

The relationship may be written:

where:

• P_R = Rankine-Gordon predicted load;
• P_c = limiting direct-compression load;
• P_e = Euler elastic buckling load.

With:

and:

the Rankine-Gordon load can be written:

where:

In practice, historical Rankine constants were often empirical rather than being calculated directly from this theoretical relationship.

Traditional Approximate Rankine Constants

Perry-Robertson Formula and Legacy BS 449 Data

BS 449-2 used permissible-stress design procedures for structural steelwork. The Perry-Robertson approach accounted for column imperfections and provided the basis for allowable compressive stresses tabulated for different slenderness ratios and steel grades.

The original RoyMech page reproduced a relationship based on Appendix B of BS 449-2:1969 together with allowable compressive stresses from Table 17. Because the standard is now legacy material, the tabulated data are preserved below with their original context rather than being mixed with current Eurocode design rules.

Legacy BS 449 Steel Grades

The original table relates to steel grades specified under BS 4360. Later European structural steel designations should not be assumed to be exact one-for-one replacements solely from the grade labels shown here. Material specification, thickness-dependent strength and the requirements of the applicable standard should be checked when assessing an existing structure.

Legacy Allowable Compressive Stress p_c

The following values are retained from the original RoyMech transcription of BS 449-2:1969 Table 17. Values of permissible average compressive stress p_c are in MPa.

Engineering Considerations Beyond Euler Buckling

The Euler equation describes ideal elastic flexural buckling. Practical compression-member design may additionally require consideration of:

• initial member crookedness;
• load eccentricity;
• residual stresses;
• inelastic material behaviour;
• connection and frame restraint;
• local plate buckling;
• torsional or flexural-torsional buckling;
• lateral restraint spacing;
• second-order P-Δ and P-δ effects;
• material and geometrical imperfections;
• appropriate partial factors or factors of safety.

For structural design, the requirements of the applicable design standard take precedence over the simplified classical equations on this page.

Related RoyMech Mechanics References

• Component Loading – direct, shear, bending and torsional loading.
• Stress and Strain – elastic stress-strain relationships and Young's modulus.
• Yield and Failure Criteria – yielding under combined stress states.
• Mohr's Circle – principal stress and stress transformation.
• Beam Design and Analysis – bending, shear and deflection of beams.
• Stress-Strain Curve – material yielding, proof stress and tensile behaviour.
• Thick Cylinders – stress analysis of thick-walled cylinders.
• Rotating Discs and Cylinders – stresses in rotating bodies.
• Torsion – torsional stress and deformation.
• Loaded Flat Plates – stress and deflection of flat plates.

Other Mechanics References

• Vectors
• Statics
• Kinematics
• Kinetics and Dynamics
• Mechanical Linkages
• Simple Machines

External Buckling References

1. DANotes – Buckling – detailed engineering notes on buckling and stability.
2. Buckling of Slender Struts – University of Plymouth – lecture notes on column buckling.
3. MITCalc – engineering calculation software including buckling calculations.