In our
previous topics, we have seen some important concepts such as Expression for crippling load when both the ends of the column are hinged, Difference between column and
strut, Difference between long
columns and short columns and Failure of a column with the help of our
previous posts.

Today we
will see here one very important topic in strength of material i.e. Expression
for crippling load when one end of the column is fixed and other end is free
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.

Let us
consider a column AB of length L as displayed in following figure. Let us
consider that end A of the column is fixed and other end i.e. end B of the
column is free.

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.

We have
displayed, in above figure, the initial condition of the column as AB. We have
also displayed here the deflected position of the column as AB’. Therefore after
application of crippling load or when column buckles, AB’ will indicate the
position of the column.

Now, we
will consider one section at a distance x from fixed end A and let us consider
that y is the lateral deflection of the column at considered section. Lateral
deflection at free end is displayed here by a as mentioned in above figure.

Now we
will determine the bending moment developed across the section and we can write
it as mentioned here.

Bending
Moment, M = P x (a-y)

We have
taken positive sign here for bending moment developed due to crippling load
across the section and we can refer the post for securing the information about
the

*sign conventions used for bending moment for columns*.
As we know
the expression for bending moment from

*deflection equation*and we can write as mentioned her.
Bending
Moment, M = E.I [d

^{2}y/dx^{2}]
We can
also write here the equation after equating both expressions for bending moment
mentioned above and we will have following equation.

Above
equation will also be termed as lateral deflection equation for column AB,
whose one end is fixed and other end is free.

C

_{1}and C_{2}are the constants of integration, now next step is to determine the value of constant of integration i.e. C_{1}and C_{2}.
We will
refer here one of our previous post i.e.

*End conditions for long columns*and we will secure the value of constant of integration i.e. C_{1}and C_{2}by using the respective end conditions.
As we know
that for long column, when one end of the column is fixed and other end is free,
we will have following end conditions as mentioned here.

####
*At fixed end of the column, i.e. at
x =0*

*At fixed end of the column, i.e. at x =0*

Deflection
y will be zero and slope dy/dx will also be zero.

####
*At free end of the column, i.e. at x
=L*

*At free end of the column, i.e. at x =L*

Deflection
y will be maximum and we have shown here the maximum lateral deflection of the
column at free end by a.

Slope at
free end i.e. at x = L, slope dy/dx will not be zero. Therefore, we have
following end conditions for free end of the column.

At x = L,
deflection y = a and slope dy/dx will not be zero.

Let us use
the first end condition i.e. at x = 0, deflection y = 0 in above lateral
deflection equation for column and we will have value of constant of
integration i.e.C

_{1}and it will be as mentioned here.
C

_{1}= - a
Now, we
will differentiate the lateral deflection equation with respect to x and we
will have slope equation for column AB and it will be displayed by dy/dx.

As we have
already discussed that at x = 0, slope will also be zero or d y/dx = 0 and
therefore now we will use this end condition in above slope equation in order
to secure the value of C

_{2}.
After
using the value of x =0 and d y/dx = 0 in above slope equation, we will have
value of C

_{2}and it will be zero or C_{2}= 0.
Now it’s
time to analyze the lateral deflection equation after considering and
implementing the value of both constants i.e. C

_{1 }and C_{2}.
Now we will
consider the second end condition for this column AB i.e. end condition for
free end.

From here
we will have expression for crippling load, when one end of the column is fixed
and other end is free and we have displayed it in following figure.

Do you
have suggestions? Please write in comment box.

We will
now derive the expression of crippling load when both the ends of the column are fixed, 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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