COLUMNS

DEFINITION: - It is a vertical member subjected to axial compressive load is called Column.

TYPES:

1.   Short column: - (L/d) < 8 or α <32

2.   Medium Column: - 8≤ (L/d) ≤ 30 or 32≤ α ≤ 120

3.   Long Column: - (L/d) > 30 or α > 120

EULER’S EQUATION: - (Used mainly for Long Column)

Critical Load Pe = (Ï€2EI)/ (Le)2

Where, I =Ak2

[A= Cross sectional area of column

 K = Radius of gyration]

Le =Equivalent length of the column.

[NOTE: The vertical column has two moment of inertia (Ixxand Iyy). The column will tend to buckle in the direction of least moment of inertia. Therefore least MI is to be used]

Equivalent Length of column in various end conditions: -

SL.   NO.

END CONDITION OF COLUMN

CRIPPLING LOAD

RELATION BETWEEN EFFECTIVE LENGTH AND ACTUAL LENGTH

1

Both end hinged

(Ï€2EI)/ L2

Le = L

2

One end fixed and one end free

(Ï€2EI)/ 4L2

Le = 2L

3

Both end fixed

4(Ï€2EI)/ L2

Le = L/2

4

One end fixed and one end hinged

2(Ï€2EI)/ L2

Le = L/√2

FREE HAND SKETCH OF DIFFERENT END CONDITION OF COLUMN:
Equivalent Length of both end hinged column.
Fig.1 | Equivalent Length of both end hinged column.

Equivalent length of one end fixed other end free
Fig. 2 |  Equivalent length of one end fixed other end free.

Equivalent length of both end fixed column
Fig. 3 | Equivalent length of  both end fixed column

Equivalent length of one end fixed other end hinged
Fig. 4 | Equivalent length of one end fixed other end hinged


SLENDERNESS RATIO: -

Pe= (Ï€2EI)/ (Le)2

Pe/A = π2E (k2/Le2)

Pe= π2EA / (Le/k)2

The ratio of (Le/k) is known as Slenderness Ratio.

ASSUMPTIONS MADE IN EULER’S THEORY:-

1.   The column is initially perfectly straight and is axially loaded.

2.   The section of the column is uniform.

3.   The column material is perfectly elastic, homogeneous, and isotropic and obeys the Hooke’s Law.

4.   The length of the column is very large compared to the lateral dimension.

5.   The direct stress is very less compared with the bending stress corresponding to the bulking condition.

6.   Self-weight of the column is ignored.

7.   The column will fail by bulking alone.

 

LIMITATION OF EULER’S THEORY:-

1.   The value of ‘I’ in the column formula is always least MI of the cross section. Thus any tendency to buckle occurs about the least axis of inertia of the cross section.

2.   Euler’s formula also shows that critical load only depends upon modulus of elasticity and dimension, not strength of the materials.

3.   Euler’s formula determines Critical Loads, not working loads.

 

RANKINE FORMULA OR EMPERIAL FORMULA:-

1/PR = 1/PC + 1/Pe

Where, PC = FC.A = Crushing Load.

Pe =Bulking load according to Euler’s formula.

 

Therefore Rankine’s crippling load PR = (FC.A) / [1+ α’ (L/k)2]

Where, α’ = FC/ Ï€2E

[NOTE: α and FC are constant for a given material]

VALUE OF α AND FC FOR DIFFERENT MATERIAL –

Cast Iron-               FC =550 N/mm2 and α’ = 1/1600

Mild Steel-              FC =320 N/mm2 and α’ = 1/7500

Wrought Iron-         FC =250 N/mm2and α’ = 1/9000

Strong Timber-       FC =50 N/mm2and α’ = 1/750

[NOTE: Rankine’s formula as well as Euler’s formula does not include factor of safety.]