A long structural member subject to a compressive load is called a strut.   Struts with large cross sections compared with the length generally fail under compressive stress and the conventional failure criteria apply.  When the cross section area is not large compared to the length i.e the member is slender, then the member will generally fail by buckling well before the compressive yield strength is reached.

The notes below relate to uniform straight members made from homogeneous engineering materials used within the elastic operating range. It is assumed that an end load is applied along the centroid of the ends.  The strut will remain straight until the end load reaches a critical value and buckling will be initiated.  Any increase in load will result in a catastrophic collapse and a reduction in load will allow the strut to straighten.   The value of the critical load depends upon the slenderness ratio and the end fixing conditions.  The slenderness ratio (λ )is defined as the effective length =L_e / the least radius of gyration = k of the section  The principal end fixing conditions are listed below

The principal end fixing conditions are listed below

The simple analysis below is based on the pinned-pinned arrangement.  The other arrangements are derived from this by replacing the length L by the effective length L_e.  

For the pinned-pinned case the effective length L_e = L.
For the Fixed -Fixed case the effective length L_e = L/2.
For the Fixed-Free case the effective length L_e = L x 2.
For the Fixed-Pinned case the effective length L_e approx. L x 0,7.

Quick derivation for curvature (1/R)

Note: The derivation below is based on a strut with pinned ends. A similar method can be used to arrive at the Euler loads for other end arrangements which will confirm the basis for the factors in arriving at the equivalent length b.

M / I = σ / y = E / R

When x = 0 y = 0 and therefore A cos μ.0 + B sin μ.0 = A = 0 therefore A = 0
When x = b , y = 0 and so B sin μb = 0.
B cannot be 0 because there would be no deflection and no buckling which is contrary to experience.
Hence sin μ L_e = 0. therefore μ L_e = 0, π, 2π, 3 π etc

(W/EI) L_e ² = π ², 4.π ², 9.π ² etc
therefore
W = π ² E.I / L_e² , π ² E.I / (L_e / 2) ², π ² E.I / (L_e / 3) ²
The lowest value of W results from π ² E.I / L_e²

The lowest value of W resulting from this procedure is called the Euler load (W_e ) and failure of long slender beams due to buckling results from this much earlier than failure due pure compression.
As the moment of inertia I = A.k ² and the end force W = σ A. The formula can be rewritten

Important Note: The value of I and the equivalent value of k are assumed to be the minimum values for the section under consideration

This theory takes no account of the compressive stress.   For a metal with a compressive strength of less than 300 N/mm² and a Young's Modulus of about 200 kN/mm².    The strut will tend to fail in compression if the slenderness ratio (L_e/ k) is less than 80.    Therefore for steel Eulers equation is not reliable for slenderness ratios less than 80 and really should not be used for slenderness ratios less than 120.

This criteria is based on experimental results.

This criteria suggests that the strut will fail at a load given by.

1 / W _R = 1 / W_c + 1 / W_e

W_c = Compressive failure Load
W_e = Euler Load

Substituting c = σ _c / ( π ² E) - A constant for each material

This design criteria provides more accurate buckling loads than the euler theory especially at lower slenderness ratios.  At higher slenderness ratios the two methods yield similar results. The experimental values for c are not in direct agreement with the theoretical values. BS 449-2:1969 includes tables for the safe working stresses for all slenderness ratios and a range of steel specifications.

Important ..The notes and equation and table below is provided for general guidance. For detail structural design it is important to refer to the identified standards.   The information below is only a trivial relative to the level of detail provided in the standard.

The equation below is used as the basis for the allowable design stresses as provided in the relevant tables in BS 449 and is considered the most reliable of the methods available for buckling loads for long slender struts..The equation below is similar to that provided in appendix B of BS 449 part 2 :1969

p_c = Permissible average compressive stress
k₂ = Load coefficient ( normally 1,7)
σ _y = Yield stress N/m²
σ _y = Euler stress N/m²
η = 0,3 ( L_e / 100k )²