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 strut,Â Difference between long columns and short columns
andÂ Failure of a columnÂ withÂ the help of
our previous posts.

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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 hinged 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 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 column
after application of crippling load or when column buckles.

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 as displayed 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 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 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.

C

_{1}and C_{2}are the constant 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 with
both the ends hinged, we will have following end conditions as mentioned here.

At x = 0, deflection y = 0

At x = L, deflection y = 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}.
At x = 0, deflection y = 0 and hence after
using these value in lateral deflection equation for column, we will have constant
of integration i.e.C

_{1}= 0
Similarly, we will use second end
condition i.e. at x = L, deflection y = 0 and constant of integration i.e.C

_{1}= 0 in above lateral deflection equation for column and we will have value of constant of integration i.e.C_{2}.
At x = L, deflection y = 0 and we have already
determined constant of integration i.e.C

_{1}= 0. Therefore, constant of integration i.e.C_{2}will be determined as displayed here in following figure.
Let us assume that C

_{2}is zero, we have already concluded that C_{1}= 0 and in this situation lateral deflection of column i.e. y will also be zero.Â
We can also say from here that column
will not be bend after application of crippling load P and as we know that this
statement will never be true and hence our assumption of taking C

_{2}= 0 is wrong.
So only one condition is left as
displayed here

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

Do you have suggestions? Please write in
comment box.

We will now derive the expression for crippling load when one end of the column is fixed and other end is free, in the category of
strength of material, in our next post.

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**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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