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.

Now as we have discussed that Euler’s formula is only applicable for long columns. We can secure the value of crippling load and crippling stress with the help of Euler’s formula for a long column.
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
PE = Crippling load from Euler’s formula
PC = σc x A
σc = 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 (σc) 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 PE 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

We can conclude here that for a given material and cross-section of column, value of crippling load from Euler’s formula will be dependent over the effective length of the column.

### 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/PE) will be very small and hence we can neglect it with respect to the value of crushing load 1/PC.

Recall the empirical formula given by Rankine, we can write here following equation of crippling load for short column.
P = PC
We can explain it in very simple way that crippling load from Rankine’s formula will be equal to the crushing load PC.

### 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.

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

We can also write that value of (1/PE) will be very large and hence we can neglect the term (1/PC) with respect to the term (1/PE).

Recall the empirical formula given by Rankine, we can write here following equation of crippling load for long column.
P = PE
We can explain it in very simple way that crippling load from Rankine’s formula will be equal to the crippling load PE.

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