Recent Updates

Monday, 10 July 2017

EFFECTIVE LENGTH OF COLUMN FOR DIFFERENT END CONDITIONS


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

Today we will see here one very important topic in strength of material i.e. effective length of column for different end conditions 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.

So, what is 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.

Let us understand the concept of effective length of a given column in very simple way

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.

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 the column after application of crippling load or when column buckles.
Recall here the expression for crippling load when both the ends of the column are hinged. And we can write here the equation or formula for buckling load or crippling load, as mentioned here, for a column when both the ends of the column are hinged.
We can easily say that effective length of the column will be equal to actual length of the column when both the ends of the column are hinged.
Le = L
Considering the above statement, we can briefly introduce the effective length in a more precise way and here it is.

Effective length of a given column for given end conditions is basically defined as the length of an equivalent column of similar material and cross-section with hinged at both ends and also subjected with same value of crippling load by which given column was subjected.

Effective length of column for different end conditions

Both end hinged

We have already seen the case of a column with hinged at its both ends and in this case effective length of the column will be equal to actual length of the column.

Column having one end fixed and other end free

Now we will see the effective length of the column with one end is fixed and other end is free. We will recall here the previous post expression for crippling load when one end of the column is fixed and other end is free and we will write here the formula of crippling load in terms of actual length.
Let us compare above equation of crippling load which is in terms of actual length with equation of crippling load in terms of effective length.

We can easily write here the effective length of the column with one end is fixed and other end is free and here it is.
Le = 2 L

Column with both end fixed

Now we will see the effective length of the column when it’s both ends are fixed. We will recall here the previous post i.e. expression for crippling load when both the ends of the column are fixed and we will write here the formula of crippling load in terms of actual length.
Let us compare above equation of crippling load which is in terms of actual length with equation of crippling load in terms of effective length.

We can easily write here the effective length of the column, when it’s both ends is fixed and here it is.
Le = 0.5 L

Column having one end fixed and other end hinged

Now we will see the effective length of the column with one end is fixed and other end is hinged.

We will recall here the previous post expression for crippling load when one end of the column is fixed and other end is hinged and we will write here the formula of crippling load in terms of actual length.
Let us compare above equation of crippling load which is in terms of actual length with equation of crippling load in terms of effective length.

We can easily write here the effective length of the column with one end is fixed and other end is hinged and here it is.
Le = 0.7L
Do you have suggestions? Please write in comment box.

We will now derive the expression for crippling stress, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

2 comments: