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Euler's Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

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Columns and Struts

  • A structural member subjected to an axial compressive force is called 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 analysed with the Euler’s column formulas can be given as

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

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

(i) For both end hinged

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineeringin case of Column hinged at both end Le = L

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

(ii) For one end fixed and other free

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

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

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

(iii) For both end fixed

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

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

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering
(iv) For one end fixed and other hinged

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineeringin case of Column with one end fixed and other hinged Le = L / √2

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

(v) Effective Length for different End conditions

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

Slenderness Ratio (S)

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

slenderness (s) = L/ k = Effective length of member / Least radius of gyration

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

Modes of failure of Columns

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

Rankine’s Formula

Rankine proposed an empirical formula for columns which coven all Lasts ranging from very short to very long struts. He proposed the relation
1 / P= 1 / PC + 1 / PE

Pc = σC. A = ultimate load for a strut
Eulerian crippling load for the standard case

Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering
Where a = Rankine's constant = Euler`s Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering

The document Euler's Theory of Columns Notes | Study Strength of Materials (SOM) - Mechanical Engineering is a part of the Mechanical Engineering Course Strength of Materials (SOM).
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