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Tuesday, 11 July 2017

RELATION BETWEEN CRIPPLING STRESS AND RADIUS OF GYRATION

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.

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.

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.k2
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.k2
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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