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

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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?

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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 $x+dx$.

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  • $\begingroup$ Thanks for your very detailed explanation, especially by mentioning some keywords that are missing from my textbook, e.g., pure moment or moment of a couple. I can now search them online for a better understanding. $\endgroup$
    – Saint Paul
    Commented Aug 19, 2021 at 13:15

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