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. Rankine’s formula for columns with the help of this
post.

###
**Rankine’s
formula for columns**

We are very well aware with Euler’s formula and its limitations too and we have seen it in our recent post.

Let us think that we need to analyze and determine the
value of crippling load and crippling stress for a short column.

Can we think to use Euler’s formula? Certainly your
answer will be no, as Euler’s formula is not valid for short column.

Rankine’s had suggested one empirical formula, on
the basis of results secured from experiments performed by him, which will be
valid for all types of column i.e. for long columns and also for short columns.

###
**Let
us first see here the Rankine’s formula **

Empirical formula, suggested by Rankine, which is
applicable for short columns and long columns will be termed as Rankine’s
formula.

Where,

P = Crippling load from Rankine’s formula

P

_{C }= Crushing load
P

_{E}= Crippling load from Euler’s formula
P

_{C }= σ_{c}x A
σ

_{c}= Ultimate crushing stress
A = Area of cross-section of given column

Let us think that we have one column with given material
of construction and area of cross-section.

Value of ultimate crushing stress (σ

_{c}) and cross-sectional area (A) will be constant for a given column and therefore crushing load will also be constant for a given column.
In simple, we can say that crippling load from Rankine’s
formula i.e. P will be dependent over the value of P

_{E}i.e. crippling load from Euler’s formula.

*Let us recall here the Euler’s formula*
Where,

E = Young modulus of elasticity and it will be
constant for given material of column

I = Moment of inertia of given column and it will
also be constant for a given cross-section of column.

Le = Effective length of the column

###
*Let
us consider the case of short column and long column one by one*

*Let us consider the case of short column and long column one by one*

###
**Short
column**

We are considering here first the case of short
column and we will try to figure out here the crippling load and also failure
mechanism of short column.

Effective length of the column i.e. Le will be small
and its square value will be quite small and hence we can easily say that value
of crippling load from Euler formula will be very large.

We can also write that value of (1/P

_{E}) will be very small and hence we can neglect it with respect to the value of crushing load 1/P_{C}.
Recall the empirical formula given by Rankine, we
can write here following equation of crippling load for short column.

**P = P**

_{C}
We can explain it in very simple way that crippling
load from Rankine’s formula will be equal to the crushing load P

_{C}.###
**Long
column**

We are considering here now the case of long column
and we will try to figure out here the crippling load and also failure
mechanism of long column.

_{E}will be very small.

We can also write that value of (1/P

_{E}) will be very large and hence we can neglect the term (1/P_{C}) with respect to the term (1/P_{E}).
Recall the empirical formula given by Rankine, we
can write here following equation of crippling load for long column.

**P = P**

_{E}
We can explain it in very simple way that crippling
load from Rankine’s formula will be equal to the crippling load P

_{E}.
Do you have suggestions? Please write in comment
box.

We will now discuss the basic concepts of thin and
thick cylinders, in the category of strength of material, in our next post.

### Reference:

Strength of material, By R. K. Bansal

Image Courtesy: Google