# Derivation of the fundamental principle in the mechanics of beams

Can someone provide the derivation for the following equation:

$$\int_l Q_Z(x) dx = M_y (x) \tag1$$

or equivalently :

$$\frac{dM_y}{dx}=Q_Z\tag2.$$

I am interested in the $$1D$$ isotropic material version.

Appendix

When drawing the shear and moment diagram for this case, we would get something like this:

Since the $$Q_z$$ is positive, the slope of $$M_y$$ has to be positive, because of equation $$(2).$$

But if $$M_y$$ has to have a positive slope and has to be zero at the free end, than it has to look like in the picture, i.e. it has to have a negative value.

Can that be a priori recognized from the equation $$(1)$$ or is it just a convention that when a beam curves downwards that it will be negative, and when it curves upwards, it will be positive, like explained here?

• Draw a free body diagram. Mar 12 at 21:44

We just integrate and evaluate $$x$$ from $$0$$ to $$l$$.
$$\int_l Q_Z(x) dx = Q_z(x) \ x\biggr\rvert _o^l = Q_z(x)l= M_y(x)$$