TWI Knowledge Summary
Buckling
by Graham Slater
Most structural analysis is concerned with small strains and distortions in a stressed material or structure. Under certain conditions, usually involving compressive stresses, a structural member may develop large distortions under critical loading conditions. This condition is described as buckling instability, or simply buckling.For columns and struts with pinned ends, the theoretical critical compressive load for buckling is given by the EULER FORMULA:
| where | P
crit
L E I |
= critical load
= length of strut = Young's Modulus = second moment of area |
Other similar equations can be found for other load combinations and end conditions. The theory is only applicable at large slenderness ratios and real struts will fail by yielding at lower loads, due to eccentricity of loading, initial curvature, etc.
Empirical formulae and codes have been established for design of real structural columns, and can be found in appropriate standards and textbooks. Three examples are:
i) Rankine-Gordon
where P
r
A
a= allowable load
= radius of gyration
= area of cross section
= allowable compressive stress
= constant dependent on end condition and materialThe formula applies for very short columns as well as larger slenderness ratios. Typical values of the constant 'a' for pin-ended struts are 10 -4 for steel and timber, and 6 x 10 -4 for cast iron.
ii) Linear
P =where c is a constant and equals 0.005 for steel and 0.008 for cast iron, with pinned ends.
iii) Parabolic
P =This is intended to agree with the Euler formula for long columns. For pinned ends and L/r < 150, the constant c = 2.3 x 10 -5 for mild steel.
Some section shapes may be more prone to torsional buckling than flexural buckling, especially if the shear centre is not coincident with the centroid. Torsional loading itself may develop buckling instability, in a thin-walled cylinder for example.
Problems of structural instability are not restricted entirely to compression members. Deep beams for example may fail by lateral buckling involving torsion and bending perpendicular to the plane of the depth of the beam. Pressure vessels may fail by buckling under external pressure, and sometimes under internal pressure in formed ends.
Further reading:
Case J and Chilver A H: 'Strength of materials and structures', Edward Arnold Limited, Second Edition, 1971.Benham P P and Crawford R J: 'Mechanics of engineering materials', Longman Scientific and Technical, 1987.
BS5950:1990: 'Structural use of steelwork in building, Part 1: Code of practice for design in simple and continuous construction: hot rolled sections'.