Bending moment

Remarks

Here $y$ points in the direction of the load, so that a moment that makes the beam sag, with tension on the side $y > 0$, is positive. Example: a simply supported beam of length $L$ with a point load $P$ at mid-span has support reactions $P/2$. For $0 \le x \le L/2$ the forces to the left of $x$ are only the reaction $P/2$ at $0$, with lever arm $x$, so $M(x) = Px/2$. The largest value is $M(L/2) = PL/4$.

Non-example: the integral $\int_A \sigma_{xx} \, dy \, dz$ is the axial force, not the bending moment; it is $0$ in pure bending, while $M$ is not.

In the Euler–Bernoulli theory the stress is $\sigma_{xx} = -E y w''$, where $w$ is the deflection in the direction of $y$, so $M = -E I w''$, where $I$ is the second moment of area. Texts differ in the sign conventions for $M$.