Euler's Theory of Columns

# Euler's Theory of Columns | Strength of Materials (SOM) - Mechanical Engineering PDF Download

### 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

where, E = Modulus of elasticity, L= 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

(iv) For one end fixed and other hinged

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

(v) 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.

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

Modes of failure of Columns

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

Where a = Rankine's constant =

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

 1. What is Euler's Theory of Columns in mechanical engineering?
Ans. Euler's Theory of Columns is a concept in mechanical engineering that deals with the stability and strength of slender columns under compressive loads. It provides a mathematical framework to determine the critical load at which a column will buckle or fail.
 2. What are the assumptions made in Euler's Theory of Columns?
Ans. Euler's Theory of Columns makes certain assumptions for its application, including: - The column is slender and has a constant cross-section along its length. - The material of the column is homogeneous and isotropic, exhibiting linear-elastic behavior. - The column is subjected to an axial compressive load without any lateral forces or moments. - The column buckles in a single plane and does not experience any lateral deflection.
 3. How is the critical buckling load determined in Euler's Theory of Columns?
Ans. The critical buckling load in Euler's Theory of Columns can be determined using the formula: Pcr = (π^2 * E * I) / (l^2) Where Pcr is the critical load, E is the modulus of elasticity of the column material, I is the moment of inertia of the column's cross-section, and l is the effective length of the column.
 4. What is the significance of Euler's Theory of Columns in mechanical engineering?
Ans. Euler's Theory of Columns is of great significance in mechanical engineering as it helps engineers design and analyze slender columns to ensure their stability and prevent buckling failures. It provides a fundamental understanding of how loads and geometry affect the behavior of columns under compression, aiding in the safe and efficient design of various structural systems.
 5. Can Euler's Theory of Columns be applied to all types of columns?
Ans. Euler's Theory of Columns is primarily applicable to long, slender columns with relatively low slenderness ratios. It is most accurate for columns with pinned or hinged ends, as well as those with fixed or clamped ends. However, for columns with intermediate slenderness ratios or special boundary conditions, more advanced theories, such as the Rankine-Gordon formula or Perry-Robertson formula, may be required for accurate analysis.

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