In our previous topics, we have seen some important
concepts such as concept of
eccentric loading , Assumptions made in the Euler’s column theory and difference between long column and short column with the help of our previous posts.

Today we will see here one very important topic in
strength of material i.e. limitations of Euler's formula in columns with the
help of this post.

###
**Limitations
of Euler's formula in columns **

Before going ahead to see the
limitations of Euler's formula in columns, we must have to understand here the significance of crippling stress and
slenderness ratio of column first.

####
*
Crippling stress*

*Crippling stress*

When a column will be subjected to axial
compressive loads, there will be developed bending moment and hence bending
stress in the column. Column will be bent due to this bending stress developed
in the column.

Load at which column just bends or
buckles will be termed as buckling or crippling load.

Crippling stress, developed in the given
column due to crippling load, could be easily determined by securing the ratio
of crippling load to the area of cross-section of the given column.

Crippling stress = Crippling load / area
of cross-section

####
**Slenderness
ratio of column**

Slenderness ratio of column is basically defined as
the ratio of effective length of the column to the least radius of gyration. Slenderness
ratio will be given in numbers because it is one ratio and hence slenderness
ratio will not have any unit.

Slenderness ratio is usually displayed by Greek
letter λ

Slenderness ratio = Effective length of the column/
Least radius of gyration

λ = Le / k

###
**
Let
us come to the main topic i.e. limitations of Euler's
formula in columns**

We have seen above the formula for crippling stress,
where slenderness ratio is indicated by λ. If
value of slenderness ratio (λ = Le / k) is small
then value of its square will be quite small and therefore value for crippling
stress developed in the respective column will be quite high.

For any column material, we must note it here that
value of crippling stress must not be greater than the crushing stress. If crippling
stress exceeds the crushing stress, in that situation, Euler's formula will not
be applicable for that column.

Therefore there must be one certain limit of slenderness
ratio for a column material so that crippling stress could not exceed the
crushing stress.

In order to secure the limit of slenderness ratio for
a column material, we will have to follow the following equation.

Crippling stress = Crushing stress,

This equation is only written for securing the limit
of slenderness ratio for a column material, we will equate the crippling stress
with crushing stress.

###
**Example
for better understanding of limit of slenderness ratio for a column material**

Let us see here one example and let us solve for
limit of slenderness ratio for a column material.

Let us consider that we have one column AB of mild
steel with hinged at both ends carrying a crippling load P. For mild steel,
column AB

Crippling stress = 330 MPa

Young’s modulus of elasticity, E= 2.1 x 10

^{5}MPa
Now we will follow the above mentioned equation in
order to secure the limit of slenderness ratio for a column material.

Crippling stress = Crushing stress

From here, slenderness ratio will be approximate equal
to 80 or we can say it is 80. Now what is the importance of saying that limit
of slenderness ratio 80?

From here we will conclude for a column AB of mild steel with hinged at both ends, if slenderness ratio falls below 80 then in that case crippling stress will be high as compared to crushing stress and therefore in that case Euler's formula will not be applicable for that column AB.

Do you
have suggestions? Please write in comment box.

We will
now discuss Rankine's formula for columns, in the category of strength of
material, in our next post.

###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google