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
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 Le.
For the pinned-pinned case the effective length Le = L.
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
(W/EI) Le 2 = π 2,
4.π 2, 9.π 2 etc
The lowest value of W resulting from this procedure is called the Euler load
(We ) and failure of long slender beams due to buckling results from this
much earlier than failure due pure compression.
Important Note: The value of I and the equivalent value of k are assumed to be the minimum values for the section under consideration
Validity of Eulers theory
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 (Le/ 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.
Rankine - Gordon Criteria
This criteria is based on experimental results.
1 / W R = 1 / Wc + 1 / We
Wc = Compressive failure 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.
Perry Robertson formula (BS 449-2 )
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
pc = Permissible average compressive stress
The table is based on table 17 in BS 449 and relates to BS 4360 steels. Which is superseded by BS EN 10025 the grades of which are identified ().