Assumptions of Euler's theory:
Euler's theory is based on the following assumptions:
(i) Axis of the column is perfectly straight when unloaded.
(ii) The line of thrust coincides exactly with the unstrained axis of the column.
(iii) Flexural rigidity El is uniform.
(iv) Material is isotropic and homogeneous.
Limitation of Euler’s Formula
The maximum load at which the column tends to have lateral displacement or tends to buckle is known as buckling or crippling load. Load columns can be analysed with the Euler’s column formulas can be given as
where, E = Modulus of elasticity, Le = Effective Length of column, and I = Moment of inertia of column section.
(i) For both end hinged
in case of Column hinged at both end Le = L
(ii) For one end fixed and other free
in case of column one end fixed and other free: Le = 2L
(iii) For both end fixed
in case of Column with both end Fixed Le = L / 2
in case of Column with one end fixed and other hinged Le = L / √2
(v) Effective Length for different End conditions
The slenderness ratio of a compression member is defined as the ratio of its effective length to least radius of gyration.
slenderness (s) = Le / k = Effective length of member / Least radius of gyration
Modes of failure of Columns
Rankine proposed an empirical formula for columns which coven all Lasts ranging from very short to very long struts. He proposed the relation
1 / PR = 1 / PC + 1 / PE
Pc = σC. A = ultimate load for a strut
Eulerian crippling load for the standard case
Where a = Rankine's constant =