This presentation summarizes key aspects of columns for engineering. It defines a column as a structural member subjected to axial compression. It classifies columns as short, medium, or long based on their slenderness ratio. It describes the failure of columns by crushing (for short columns) or buckling (for long columns). Euler's theory of buckling is summarized, which models buckling based on a column's effective length, modulus of elasticity, and moment of inertia. Different effective lengths are defined for varying end conditions. Finally, Rankine's empirical formula is presented as applicable to both short and long columns.

1. SCHOOL OF ENGINEERING & TECHNOLOGY
PRESENTATION ON
“COLUMNS”
SUBMITTED BY: SUBMITTED TO:
RANICK KUMAR Mr. GAURAV VERMA
(BETN1ME14055) (Dept of Mech. Engg)

2. OVERVIEW
This presentation will cover:
 What is column?
 Classification of column
 Failure of column
 Euler’s theory of Buckling of column
 Effective length of a column
 Rankine’s formula for column

3. COLUMNS
A Bar or a member of structure inclined at 90˚ degree to the
horizontal and carrying an axial compressive load is called as
COLUMN.
SLENDERNESS RATIO
The ratio of the actual length od a column to the least radius
gyration of the column is known as slenderness ratio.
𝑠𝑙𝑒𝑛𝑑𝑒𝑟𝑛𝑒𝑠𝑠 𝑟𝑎𝑡𝑖𝑜 =
𝑎𝑐𝑡𝑢𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ(𝑙)
𝑙𝑒𝑎𝑠𝑡 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑔𝑦𝑟𝑎𝑡𝑖𝑜𝑛(𝑘)

4. CLASSIFICATION OF COLUMNS
 SHORT COLUMN
Columns which have length less than 8 times with their respective
diameter or slenderness ratio less than 32 are called as short column.
 MEDIUM COLUMN
Columns which have their length vary from 8 to 30 times to their
diameter or slenderness ratio lie between 33 to 120 are called as
medium column.
 LONG COLUMN
Columns having their length more than 30 times with their diameter or
slenderness ratio more than 120 are called as long column.

5. FAILURE OF A COLUMN
The failure of a column takes place due to anyone of the following stresses
set up in the columns:
i. Direct compressive stresses
ii. Buckling stresses
iii. Combined direct compressive and buckling stresses.
BUCKLING LOAD
The minimum axial load at which the column tends to have
lateral displacement or buckle is called buckling load.

6. FAILURE OF A SHORT COLUMN
A short column of uniform cross sectional area A, subjected to an axial
compressive load P. The compressive stress induced is
given by
p =
𝑃
𝐴
If the compressive load on the short column is gradually increased,
a stage will reach when the column will be on point of failure by
crushing. Let,
Pc = Crushing load, 𝜎c = Crushing stress
A= Area of cross-section
Then , 𝜎=
𝑃 𝑐
𝐴

7. EULER’S THEORY OF BUCKLING OF COLUMN
ASSUMPTIONS:
 The column is initially perfectly straight and the load is applied
axially.
 The cross-section of the column is uniform throughout its length.
 The column material is perfectly elastic, homogenous and isotropic
and obeys Hooke’s law.
The length of the column is very large as compared to its lateral
dimension.
The direct stress is very small as compared to the bending stress.
 The column will fail by buckling alone.
The self weight of column is negligible.

8. EFFECTIVE LENGTH OF A COLUMN
The effective length of a given column with given end conditions is the length of
an equivalent column of the same material and cross-section with hinged ends,
and having the value of the crippling load equal to that of the given column. It is
also called as equivalent length.
Let Le= Effective length of the column
l= Actual length of the column
P= Crippling load for the column.
E= Young’s modulus
I= Moment of inertia
Then the crippling load for any type of end condition is given by
P=
𝞹2
𝐸𝐼
𝐿 𝑒
2

9. THE CRIPLLING LOAD IN TERMS OF ACTUAL LENGTH AND EFFECTIVE LENGTH
ALSO THE RELATION BETWEEN EFFECTIVE LENTH AND ACTUAL LENGTH
S.No END CONDITIONS OF COLUMN CRIPPLING LOAD IN TERMS OF RELATION BETWEEN
EFFECTIVE LENGTH
AND ACTUAL LENGTHACTUAL LENGTH EFFECTIVE
LENGTH
1. Both ends hinged 𝜋2 𝐸𝐼
𝑙2
𝜋2 𝐸𝐼
𝑙 𝑒
2
Le=l
2. One end fixed and other is
free
𝜋2 𝐸𝐼
4𝑙2
𝜋2 𝐸𝐼
𝑙 𝑒
2
Le=2l
3. Both ends fixed 4𝜋2 𝐸𝐼
𝑙2
𝜋2 𝐸𝐼
𝑙 𝑒
2
Le=
𝑙
2
4. One end fixed and other is
hinged
4𝜋2 𝐸𝐼
𝑙2
𝜋2 𝐸𝐼
𝑙 𝑒
2
Le=
𝑙
√2

10. NOTE:
In the previous slide there are two values of moment of inertia i.e.,
Ixx and Iyy.
The value of I (moment of inertia) in given expressions should be taken
as the least value of the two moments of inertia as the column will tend
to bend in the direction of least moment of inertia

11. RANKINE’S FORMULA
We have seen that Euler’s formula gives correct results only for very
long column. But what happens when the column is a short or the
column is not very long. Rankine established an empirical formula which
is applicable to all columns whether they are short or long. The
Rankine’s formula is given by:

13. THANK YOU FOR YOUR
ATTENTION!!!
I HOPE YOU ENJOYED THE
PRESENTATION.