Euler's Theory of Columns Study Notes for Mechanical Engineering Exams

By Akhil Gupta|Updated : August 13th, 2021

Euler's Theory of Columns plays a significant role while preparing for GATE ME 2022 Exam. Below we have given comprehensive Euler's Theory of Columns Study Notes for Mechanical Engineering. Let's take a look and understand the Theory deeply. 

Table of Content

Columns and Struts:

  • A structural member subjected to an axial compressive force is called a strut. As per definition strut may be horizontal, inclined, or even vertical.
  • The vertical strut is called a column.

Euler’s Column Theory

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

  • There is always crookedness in the column and the load may not be exactly axial.
  • This formula does not take into account the axial stress and the buckling load is given by this formula may be much more than the actual buckling load.

Euler’s Buckling (or crippling load)

  • 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 analyzed with the Euler’s column formulas can be given as  


where, E = Modulus of elasticity, Le= Effective Length of the column, and I = Moment of inertia of column section.

  For both end hinged:


in case of Column hinged at both end L= L



 For one end fixed and other free:


in case of column one end fixed and other free: L= 2L


 For both end fixed:


in case of Column with both end Fixed L= L/2


For one end fixed and other hinged:


in case of Column with one end fixed and other hinged L= L/√2


Effective Length for different End conditions


Slenderness Ratio ( S)

The slenderness ratio of a compression member is defined as the ratio of its effective length to least radius of gyration.



Modes of failure of Columns


 Rankine’s Formula:

Rankine proposed an empirical formula for columns that coven all Lasts ranging from very short to very long struts. He proposed the relation


   Pc = σC. A = ultimate load for a strut

Eulerian crippling load for the standard case




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