Basic Course on finite element analysis (Zeng) Note Two-beam elements equation derivation

Is the diagram in the basic course of finite element analysis.

This is the diagram given in the mechanics of materials (Sun Xunfang).

The reason for these two pictures is that when the formula is deduced, the first picture let me misunderstand: the red Arrow Callout of the micro end of the external load $\bar{p (x)}$, looks like a surface load , the actual derivation of the formula, $\bar{p (x)}$ is a line load . Since some of the symbols in the two pictures are inconsistent, the following deduction is used in the Zeng teacher's book of the symbol, and sometimes $\bar{p (x)}$ abbreviated to $\bar{p}$.

We take the deflection equation of the beam $v (x) $ as the entire derivation of the basic amount, but also the entire derivation of the core, all the derivation is actually around this $v (x) $ to unfold. As for why to put $v (x) $ as the basic amount, according to my personal understanding:

1. First of all, for the beam, the most concern is of course under the external force deformation, that is, the deflection along the length of the beam change law, that is, $v (x) $.
2. If we establish the deflection of the differential equation $v (x) $, then the deflection at both ends of the beam is known, i.e. there is a very definite boundary condition.

Since the beam is only subjected to external force load $\bar{p (x)}$, the key to establishing the deflection equation $v (x) $ is to establish the relationship between the deflection $v (x) $ and the external load $\bar{p (x)}$. The establishment of this relationship can be very intuitive to think of the micro-section using the $y$ direction of the resultant balance, namely

\begin{equation}
(Q+DQ) +\bar{p (x)}-q=0
\end{equation}

After simplifying, you can get

\begin{equation} \label{y directional equilibrium equation}
(Q+DQ) +\bar{p (x)}-q=0
\end{equation}

Basic Course on finite element analysis (Zeng) Note Two-beam elements equation derivation