COLUMNS

 

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 P_e = (π²EI)/ (L_e)²

Where, I =Ak²

[A= Cross sectional area of column

 K = Radius of gyration]

L_e =Equivalent length of the column.

[NOTE: The vertical column has two moment of inertia (I_xx and I_yy). 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	(π2EI)/ L2	Le = L
2	One end fixed and one end free	(π2EI)/ 4L2	Le = 2L
3	Both end fixed	4(π2EI)/ L2	Le = L/2
4	One end fixed and one end hinged	2(π2EI)/ L2	Le = L/√2

FREE HAND SKETCH OF DIFFERENT END CONDITION OF COLUMN:

Fig.1 | Equivalent Length of both end hinged column.

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

Fig. 3 | Equivalent length of  both end fixed column

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

SLENDERNESS RATIO: -

P_e = (π²EI)/ (L_e)²

P_e/A = π²E (k²/L_e²)

P_e = π²EA / (L_e/k)²

The ratio of (L_e/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/P_R = 1/P_C + 1/P_e

Where, P_C = F_C.A = Crushing Load.

P_e =Bulking load according to Euler’s formula.

 

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

Where, α’ = F_C/ π²E

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

VALUE OF α AND F_C FOR DIFFERENT MATERIAL –

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

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

Wrought Iron-         F_C =250 N/mm² and α’ = 1/9000

Strong Timber-       F_C =50 N/mm² and α’ = 1/750

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