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

Today we will see here one very important topic in
strength of material i.e. crippling stress in terms of effective length and
radius of gyration with the help of this post.

Before
going ahead, we must have to understand here the significance of crippling load
or buckling load, concept of effective length of the column and radius of
gyration of the column too.

###
**Crippling 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.

###
**E****ffective
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.

###
**Radius of gyration**

Radius of gyration of a body or a given
lamina is basically defined as the distance from the given axis up to a point
at which the entire area of the lamina will be considered to be concentrated.

We can also explain the radius of
gyration about an axis as a distance that if square of distance will be
multiplied with the area of lamina then we will have area moment of inertia of
lamina about that given axis.

I = A.k

^{2}
Do you want the detailed post based on radius of
gyration? Click here.

Now we will concentrate here with our main topic
i.e. crippling stress in terms of radius of gyration and effective length of
the column.

###
**Crippling
stress in terms of radius of gyration and effective length of the column**

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.

Crippling stress, developed in the given column due
to crippling load, could be easily determined by securing the ratio of
crippling load to the area of cross-section of the given column.

Crippling stress = Crippling load / area of
cross-section

As we have already seen above the equation of area
moment of inertia i.e. I = A.k

^{2}
Where,

A is the area of cross-section of the column

I is the least value of area moment of inertia

K is the least value of radius of gyration of the
given column

We will use the value of area moment of inertia in
equation of crippling stress displayed above and we will secure the formula for
crippling stress in terms of radius of gyration and effective length of the
column.

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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