Shear and moment diagram

Definition

Let a beam occupy $0 \le x \le L$ and carry a distributed transverse load $q(x)$, point forces and point couples, together with its support reactions, in equilibrium. Let $V(x)$ be the shear force and $M(x)$ the bending moment at the section $x$. The shear diagram is the graph of $V$ on $[0, L]$, and the moment diagram is the graph of $M$ on $[0, L]$. Between point loads they satisfy $$\frac{dV}{dx} = -q, \qquad \frac{dM}{dx} = V.$$ At a point force $P$ in the direction of $q$, the function $V$ jumps by $-P$. At a point couple, $M$ jumps by the value of the couple, with the sign given by the convention.