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*

*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

Ïƒ

_{d}=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

Ïƒ

_{C}=P_{C }/A
Where,

Ïƒ

_{C}=Â Crushing stress developed in the column
P

_{C}= Crushing load###
*Buckling
or bending stress*

*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 (Ïƒ

_{b}) developed in the column due to axial compressive load could be written as mentioned here.
Ïƒ

_{b}= (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

Image Courtesy: Google

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