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

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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.

**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.

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_{e }= 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 L_{e} = L

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

in case of Column with both end Fixed L_{e} = L / 2

**(iv) For one end fixed and other hinged**

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

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

slenderness (s) = L_{e }/ k = Effective length of member / Least radius of gyration

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_{R }= 1 / P_{C} + 1 / P_{E}

P_{c} = σ_{C}. A = ultimate load for a strut

Eulerian crippling load for the standard case

Where a = Rankine's constant =

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