CRIPPLING LOAD WHEN BOTH THE ENDS OF THE COLUMN ARE FIXED

In our previous topics, we have seen some important concepts such as expression for crippling load when both the ends of the column are hinged, Failure of a column and expression for crippling load when one end of the column is fixed and other end is free with the help of our previous posts.

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 fixed 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 end A of the column is fixed and other end i.e. end B of the column is also fixed. In simple, both the ends of the column are fixed.

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.

We have displayed, in above figure, the initial condition of the column as AB. We have also displayed here the deflected position of the column as ACB. Therefore after application of crippling load or when column buckles, ACB will indicate the position of the column.

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 at considered section.

Column is fixed at both ends and also carrying a crippling load P and hence there will be some fixed end moments at end A and end B. let us think that this fixed end moment is M₀.

Now we will determine the bending moment developed across the section and we can write it as mentioned here

Bending Moment, M = M₀ – P. 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 [d²y/dx²]

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 AB, whose both the ends are fixed and column is subjected with crippling load P.

C₁ and C₂ are the constants of integration, now next step is to determine the value of constant of integration i.e. C₁ and C₂.

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. C₁ and C₂ by using the respective end conditions.

As we know that for long column, when both the ends of the column are fixed and column is subjected with crippling load P, we will have following end conditions as mentioned here.

At fixed end A of the column, i.e. at x =0

Deflection y will be zero and slope dy/dx will also be zero, i.e. y = 0 and dy/dx = 0

At other fixed end B of the column, i.e. at x =L

Deflection y will be zero and slope dy/dx will also be zero, i.e. y = 0 and dy/dx = 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.C₁ and it will be as mentioned here.

C₁ = - M₀/P

Now, we will differentiate the lateral deflection equation with respect to x and we will have slope equation for column AB and it will be displayed by dy/dx.

As we have already discussed that at x = 0, slope will also be zero or dy/dx = 0 and therefore now we will use this end condition in above slope equation in order to secure the value of C₂.

After using the value of x =0 and dy/dx = 0 in above slope equation, we will have value of C₂ and it will be zero or C₂ = 0.

Now it’s time to analyze the lateral deflection equation after considering and implementing the value of both constants i.e. C₁and C₂.

Now we will consider the second end condition for this column AB i.e. end condition for other fixed end or end B.

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

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.