# How to consider the reference point of a bending moment

When evaluating the relationship between shear force and bending moment, most books take look at the following bending segment with loading intencity of $$q$$:

In equilibrium,

sum of force: $$qdx+(V+dV)-V=0$$, so $$dV=-qdx$$.

sum of moment: $$M-(M+dM)+Vdx-qdxdx/2=0$$, dM=Vdx (ignore dxdx)

My question, why can we choose O as the reference? To my understanding, $$M$$ and $$M+dM$$ are bending moments that are referenced to the support of the beam (the end of the beam), not at O. Is it still valid to sum M, M+dM with moments that are referenced to O? How to understand the meaning of M and M+dM?

This extracted part of the continuum is in translational and rotational equilibrium. Rotational equilibrium means that torques balance about any point in space, whether it is a material point on the body or a non-material point off the body. For simplicity, the author chose to take moments about $$O$$ since there is no moment due to $$V+dV$$ about that point. The resulting equation $$V = \frac{dM}{dx}$$ would also be obtained taking any other point.
$$M$$ and $$M+dM$$ are bending moments due to the distribution of normal stress acting on the cross-section of the beam. This distribution yields a couple, meaning that, once again, it does not matter about what point the normal stress is integrated. The moments $$M$$ and $$M+dM$$ are only associated with cross-sections. Hence, they are functions of $$x$$, where $$x$$ indexes each cross-section. When you go to sum moments about $$O$$, both $$M$$ and $$M+dM$$ contribute as pure couples, not moments about any particular point.
In contrast, $$V$$ is a moment about $$O$$ due to a singular force. The moment caused by $$V$$ is not due to a pure couple, but due to a singular force acting at a distance from $$O$$. Hence, the moment arm for the torque produced by $$V$$ about $$O$$ is $$dx$$.
$$M$$ is the effect of the normal stress distribution acting on the cross-section at $$x$$, while $$M+dM$$ is the effect of the normal stress distribution acting on the cross-section at $$x+dx$$.