In our previous topics, we have seen some important concepts such asÂ Assumptions made in the Eulerâ€™s column theory,Â Difference between column and strutÂ and difference between long columns and short columnsÂ with the help of our previous posts.

Today we will see here one very important topic in strength of material i.e. Failure of a column with the help of this post.

We will first understand here the basic concept of columns and after that we will discuss the failure of column due to stresses developed in the column.

So let us first understand here the meaning and characteristics of columns.

### What is column?

Column is basically defined as a vertical member of a structure and it will be subjected with vertical compressive load. Line of action of compressive load will pass through the axis of the column or sometime also parallel to the axis of the column.

In simple, a member of structure will be termed as column if it is vertical and itâ€™s both ends are fixed rigidly and also subjected with axial vertical compressive load.

Example
Vertical structural member between roof and floor could be considered as the best example of column.

### Failure of column

Failure of column usually occurs due to anyone of the following stresses developed in the column.
Direct compressive stress
Buckling or bending stress
Combined of direct compressive stress and buckling stress

### Direct compressive stress

Short column will fail only because of direct compressive stress developed in the column or we can also say that short column will usually fail due to direct compression or crushing.

Let us consider one column as displayed here in following figure. Let us consider that A is the area of uniform cross section of the column and L is the length of the column. Let us consider that column is subjected with compressive load P and hence there will be developed one direct compressive stress in the column.
Direct compressive stress will be given by following formula as mentioned here
Direct compressive stress = Direct compressive load/Area of cross section of the column
Ïƒd =Â  P/A

Let us consider that we are increasing the direct compressive load gradually and hence direct compressive stress, developed in the column, will also increase gradually. As we are increasing the direct compressive load gradually and therefore it will reach up to a point at which column will be failed due to crushing.

Direct compressive load corresponding to this point will be termed as crushing load and stress developed in the column will be termed as crushing stress.

Mathematically, we can write here the crushing load and crushing stress as mentioned here
ÏƒC =PC /A
Where,
ÏƒC =Â  Crushing stress developed in the column

### Buckling or bending stress

In case of failure of long column, we will not consider the direct compressive stress because direct compressive stress will be negligible as compared to the bending stress developed in the column due to the application of axial compressive load.

Therefore, we can say here that long columns will fail only because of buckling or bending stress developed in the column. Long column is basically defined as the column in which the ratio of effective length of the column to the least lateral dimension of the column is more than 12.

Let us consider that column displayed below is long column and we are studying here the bending stress developed in the long column due to the application of axial vertical compressive load P.
Let us consider that A is the area of uniform cross section of the column and L is the length of the column. Let us consider that column is subjected with axial vertical compressive load P and hence there will be developed one bending stress in the column.

Let us consider that e is the maximum bending of the column at the center of the column and Z is the section modulus about the axis of bending.

Bending stress (Ïƒb) developed in the column due to axial compressive load could be written as mentioned here.
Ïƒb = (P x e)/Z

Axial compressive load at which column just buckles will be termed as buckling load and corresponding bending stress developed in the column will be termed as buckling stress.

A long column will fail once buckling stress developed in the column reaches to crushing stress.

Do you have suggestions? Please write in comment box.

We will now discuss end conditions for long columns in detail in the category of strength of material in our next post.

### Reference:

Strength of material, By R. K. Bansal