Thursday, 13 July 2017

LIMITATIONS OF EULER'S FORMULA IN COLUMNS

LIMITATIONS OF EULER'S FORMULA IN COLUMNS

In our previous topics, we have seen some important concepts such as concept of eccentric loading , Assumptions made in the Euler’s column theory 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. limitations of Euler's formula in columns with the help of this post.

Limitations of Euler's formula in columns

Before going ahead to see the limitations of Euler's formula in columns, we must have to understand here the significance of crippling stress and slenderness ratio of column first.

 

Crippling stress

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.

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

Slenderness ratio of column

Slenderness ratio of column is basically defined as the ratio of effective length of the column to the least radius of gyration. Slenderness ratio will be given in numbers because it is one ratio and hence slenderness ratio will not have any unit.

Slenderness ratio is usually displayed by Greek letter λ
Slenderness ratio = Effective length of the column/ Least radius of gyration
λ = Le / k

 

Let us come to the main topic i.e. limitations of Euler's formula in columns

We have seen above the formula for crippling stress, where slenderness ratio is indicated by λ. If value of slenderness ratio (λ = Le / k) is small then value of its square will be quite small and therefore value for crippling stress developed in the respective column will be quite high.

For any column material, we must note it here that value of crippling stress must not be greater than the crushing stress. If crippling stress exceeds the crushing stress, in that situation, Euler's formula will not be applicable for that column.

Therefore there must be one certain limit of slenderness ratio for a column material so that crippling stress could not exceed the crushing stress.

In order to secure the limit of slenderness ratio for a column material, we will have to follow the following equation.
Crippling stress = Crushing stress,

This equation is only written for securing the limit of slenderness ratio for a column material, we will equate the crippling stress with crushing stress.

Example for better understanding of limit of slenderness ratio for a column material

Let us see here one example and let us solve for limit of slenderness ratio for a column material.
Let us consider that we have one column AB of mild steel with hinged at both ends carrying a crippling load P. For mild steel, column AB

Crippling stress = 330 MPa
Young’s modulus of elasticity, E= 2.1 x 105 MPa

Now we will follow the above mentioned equation in order to secure the limit of slenderness ratio for a column material.
Crippling stress = Crushing stress
From here, slenderness ratio will be approximate equal to 80 or we can say it is 80. Now what is the importance of saying that limit of slenderness ratio 80?

 
From here we will conclude for a column AB of mild steel with hinged at both ends, if slenderness ratio falls below 80 then in that case crippling stress will be high as compared to crushing stress and therefore in that case Euler's formula will not be applicable for that column AB.

Do you have suggestions? Please write in comment box.

We will now discuss Rankine's formula for columns, 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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