A structural member loaded axially in compression is generally called a compression member. Vertical compression members in buildings are called columns, posts or stanchions. A compression member in roof trusses is called struts and in a crane is called a boom.

Columns which are short are subjected to crushing and behave like members under pure compression. Columns which are long tend to buckle out of the plane of the load axis.

**THEORY OF COLUMNS**

Euler’s formula for critical load for a pin-ended column subjected to axial load is

Where, L = length of column between the hinged ends,

E = modulus of elasticity, and

I = moment of inertia of the column section.

The column will become unserviceable if the loads are larger than . In the Euler equation, it is assumed that stress is proportional to strain, therefore,

Critical Stress =

Where, A= area of cross-section, and

r = radius of gyration about the bending axis

= slenderness ratio

**VARIOUS END CONDITIONS**

Columns with length L and effective length are shown in figure below:

*Strength of an Axially Loaded Compression Members*

Maximum axial compression load permitted on a compression member,

Where, P = axial compressive load (N),

= permissible stress in axial compression (MPa)

A = effective cross-sectional area of the member

**Indian Standard IS 800: 1984**

It stipulates that the direct stress on the cross-sectional area of axially loaded compression members should not exceed nor the permissible stress calculated using Merchant – Rankine formula.

Permissible stress in axial compression (MPa):

Where = yield stress of steel in MPa

= elastic critical stress in compression =

= slenderness ratio of the member

Where, = effective length of the member

r = appropriate radius of gyration of the member

E = modulus of elasticity = 200000 MPa, and

n = a factor assumed as 1.4

**EFFECTIVE LENGTH OF COMPRESSION MEMBER**

Table below gives the values of effective length recommended by the Indian Standard, IS 800. The actual length L of the compression member should be taken as the length from centre-to-centre of intersection of supporting members or the cantilevered length in the case of free standing struts.

**Table: Equivalent length for various end conditions**

Type | Effective length of member l | |

1 | Effectively held in position and restrained in direction at both ends. | 0.67 L |

2 | Effectively held in position at both ends restrained in direction at one end. | 0.85 L |

3 | Effectively held in position at both ends but not restrained in direction. | L |

4 | Effectively held in position and restrained in direction at one end and at the other end effectively restrained in direction but not held in position. | L |

5 | Effectively held in position and restrained in direction at one end and the other end partially restrained in direction but not held in position. | 1.5 L |

6 | Effectively held in position and restrained in direction at one end but not held in position or restrained in direction at the other end. | 2.0 L |

**Note:**

- L is the unsupported length of compression member.
- For battened struts, the effective length should be increased by 10%.