**Introduction:**

Due to fixed ends, there will be fixed end moments say M0 at the ends A and B. The fixed end moments will be acting in such direction so that slope at the fixed ends becomes zero.

**Description:**

Consider a column AB of length I and uniform cross-sectional area fixed at both ends A and B as shown in Fig. Let P is the crippling load at which the column has buckled. Due to the crippling load P, the column will deflect as shown. Due to fixed ends, there will be fixed end moments say M0 at the ends A and B. The fixed end moments will be acting in such direction so that slope at the fixed ends becomes zero.

Consider a section at a distance x from the end A. Let the deflection of the column at the section is y. As both the ends of the column are fixed and the column carries a crippling load, there will be some fixed end moments at A and B.

Let M0 = Fixed end moments at A and B.

Substituting the value x = 0 and y = 0 in equation no (i), we get.

Differentiating equation (i), with respect to x, we get.