How to calculate Bending Moment

Lets consider the following beam under the action of the force $P$ at the middle of the length. The beam is supported by two fixed supports.

Now we can divide the beam into two parts based on the point where the force $P$ is applied, namely:-

1. Part A - where $x$ varies from $0$ to $\frac{L}{2}$ and
2. Part B - where $x$ varies from $\frac{L}{2}$ to $L$.

Now, let's calculate the bending moment in these two parts.

Part A

Assume a shear force $v$ acting along the direction shown.

Then on equating the forces and moments in equilibrium we have, $$v = -\frac{P}{2}$$ $$M_b = -v.x = \frac{P}{2}x$$

Part B

Similarly for this part the shear force and the bending moment obtained will be,

$$v = -\frac{P}{2}$$ $$ M_b = (L - x)\frac{P}{2}$$

Thus the bending moment is given as,

$$M_b = \frac{P}{2}x \hspace{1in} x \in (0, L)$$

$$M_b = (L - x)\frac{P}{2} \hspace{0.6in} x \in \left(\frac{L}{2}, L\right)$$

The bending moment diagram for the system will be,