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,Â Failure of a column
and expression for crippling load when one end of the column is fixed and other end is free withÂ 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 both the ends of the column are fixed 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 also fixed. In simple, both the ends of the column are fixed.

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 ACB. Therefore after
application of crippling load or when column buckles, ACB will indicate the
position of the column.

Now, we
will consider one section at a distance x from end A and let us consider that y
is the lateral deflection of the column at considered section.

Column is
fixed at both ends and also carrying a crippling load P and hence there will be
some fixed end moments at end A and end B. let us think that this fixed end
moment is M

_{0}.
Now we
will determine the bending moment developed across the section and we can write
it as mentioned here

Bending
Moment, M = M

_{0}â€“ P. y
We have
taken negative sign here for bending moment developed due to crippling load
across the section and we can refer the post for securing the information about
theÂ

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

*deflection equation*Â 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 both the ends are fixed and column is subjected with crippling load P.

C

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

*End conditions for long columns*Â and we will secure the value of constant of integration i.e. C_{1}Â and C_{2}Â by using the respective end conditions.
As we know
that for long column, when both the ends of the column are fixed and column is
subjected with crippling load P, we will have following end conditions as
mentioned here.

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

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

Deflection
y will be zero and slope dy/dx will also be zero, i.e. y = 0 and dy/dx = 0

####
*At other fixed end B of the column,
i.e. at x =L*

*At other fixed end B of the column, i.e. at x =L*

Deflection
y will be zero and slope dy/dx will also be zero, i.e. y = 0 and dy/dx = 0

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}= - M_{0}/P
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 dy/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 dy/dx = 0 in above slope equation, we will have
value of C

_{2}and it will be zero or C_{2}= 0.
Now itâ€™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 other fixed end or end B.

From here
we will have expression for crippling load, when when both the ends of the
column are fixed and we have displayed it in following figure.

Do you
have suggestions? Please write in comment box.

We will
now derive expression for crippling load when one end of the column is fixed
and other end is hinged, 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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