**Introduction:**

The free end will sway sideways when load is applied at free end and curvature in the length I will be similar to that of upper half of the column whose both ends are hinged.

**Description:**

Consider a column AB, of length l and uniform cross-sectional area, fixed at the end A and free at the end B. The free end will sway sideways when load is applied at free end and curvature in the length I will be similar to that of upper half of the column whose both ends are hinged.

Let P is the crippling load at which the column has just buckled. Due to the crippling load P, the column will deflect as shown in Fig., in which AB is the original position of the column and AB', is the deflected position due to crippling load P.

Consider any section at a distance x from the fixed end A.

Let y = Deflection (or lateral displacement) at the section

a = Deflection at the free end B'

Then moment at the section due to the crippling load = P (a - y)

( ve. sign is taken due to sign convention)

The solution of the Differential Equation is:

Where C1 and C2 are the constants of integration and the values are obtained from the boundary conditions, which are as follows:

i. At fixed end, the deflection as well as slope will be zero.