In our previous topics, we have seen some important
concepts such as, difference
between column and strut, difference between long columns and short
columns with the help of our previous posts.

Today we will see here one very important topic in strength
of material i.e. effective length of column for different end conditions with
the help of this post.

Before
going ahead, we must have to understand here the significance of crippling load
or buckling load.

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.

### So, what is effective length of column?

Effective length of a given column is basically
defined as the distance between successive points of inflection or points of
zero movement. Effective length of the column will be dependent over the end
conditions of the given column.

####
*Let
us understand the concept of effective length of a given column in very simple
way*

*Let us understand the concept of effective length of a given column in very simple way*

We know very well the formula given by Euler’s for
buckling of a long column for any type of end conditions and it is given as
mentioned here.

Where,

P = Critical buckling load

E = Young’s modulus of elasticity of the material of
the given column

I = Moment of inertia

Le = Effective length of the given column with given
end conditions

Formula given by Euler’s for buckling is also termed
as formula for critical buckling load.

Let us consider a column AB of length L as displayed
in following figure. Let us consider that both the ends of the column are
hinged i.e. end A and end B are hinged.

Let us think that P is the load at which column just
bends or buckles or we can also say that crippling load is P and we have
displayed in following figure. Curve ACB indicates the condition of the column
after application of crippling load or when column buckles.

Recall here the expression
for crippling load when both the ends of the column are hinged. And we can
write here the equation or formula for buckling load or crippling load, as
mentioned here, for a column when both the ends of the column are hinged.

We can easily say that effective length of the column
will be equal to actual length of the column when both the ends of the column
are hinged.

L

_{e}= L
Considering the above statement, we can briefly
introduce the effective length in a more precise way and here it is.

Effective length of a given column for given end
conditions is basically defined as the length of an equivalent column of
similar material and cross-section with hinged at both ends and also subjected
with same value of crippling load by which given column was subjected.

###
**Effective
length of column for different end conditions**

####
*Both
end hinged*

*Both end hinged*

We have already seen the case of a column with hinged
at its both ends and in this case effective length of the column will be equal
to actual length of the column.

####
*Column
having one end fixed and other end free*

*Column having one end fixed and other end free*

Now we will see the effective length of the column
with one end is fixed and other end is free. We will recall here the previous
post

__expression for crippling load when one end of the column is fixed and other end is free__and we will write here the formula of crippling load in terms of actual length.
Let us compare above equation of crippling load which
is in terms of actual length with equation of crippling load in terms of effective
length.

We can easily write here the effective length of the
column with one end is fixed and other end is free and here it is.

L

_{e}= 2 L####
*Column
with both end fixed*

*Column with both end fixed*

Now we will see the effective length of the column when
it’s both ends are fixed. We will recall here the previous post i.e. expression for crippling load when both the ends of the column are fixed and we will write
here the formula of crippling load in terms of actual length.

Let us compare above equation of crippling load which
is in terms of actual length with equation of crippling load in terms of effective
length.

We can easily write here the effective length of the
column, when it’s both ends is fixed and here it is.

L

_{e}= 0.5 L####
*Column
having one end fixed and other end hinged*

*Column having one end fixed and other end hinged*

Now we will see the effective length of the column
with one end is fixed and other end is hinged.

We will recall here the previous post expression for crippling load when one end
of the column is fixed and other end is hinged and we will write here the
formula of crippling load in terms of actual length.

Let us compare above equation of crippling load which
is in terms of actual length with equation of crippling load in terms of effective
length.

We can easily write here the effective length of the
column with one end is fixed and other end is hinged and here it is.

L

_{e}= 0.7L
Do you
have suggestions? Please write in comment box.

We will
now derive the expression for crippling stress, in our next post.

###
**Reference:**

Strength
of material, By R. K. Bansal

Image Courtesy:
Google

Nice explanation

ReplyDeleteVery useful information,thanks.

ReplyDelete