# STRENGTH OF MATERIAL:THEORY OF COLUMNS

## DEFINITION OF COLUMN

COLUMN:- It is a compression member, that is so slender compare to its length that under gradually increasing loads.

it fails by buckling at loadsome considerably less than those required to cause failure by crushing.

Column-compression &       Buckling.

Compression Member:- A compression member is a structural member which is straight and subjected to two equal and opposite compressive force applied at its end.

• A compression member is generally considered  a long column. when its more than 10 times its latest lateral dimension.

### MODES OF FAILURE OF A COLUMN:-

• It is subjected to axial load,may fail under the following modes:
1. Crushing
2. Buckling
3. Mixed mode of Buckling and Crushing

### Euler’s Theory(Buckling Failure)

#### Assumption involved in Euler’s Theory:-

• Axis of the column is perfectly straight when unloaded.
• The line of thrust coincides exactly with the unrestrained axis of the column.
• Material is isotropic and homogeneous.
• It is long and prismatic and it fails only in buckling.

Limitation of Euler’s Formula

1. The critical load that causes buckling depends not on strength of the material, but only on its dimensions and modules of Elasticity.
2. this converts the strut problem from the problem of instability to a problem of stress of which Euler’s Theory takes into account.

In order for Euler’s formula to be applicable, the critical stress must not exceed the proportional limit.

Ideal end conditions and effective length

• Depending upon various combinations of restrains, there may be the following four cases of end condition:
• Case-1: Both end hinged
• Case-2: one end fixed and other end free
• Case-3: Both ends fixed
• Case-4: one end fixed and other end hinged