Classical Mechanics – Model Beam Using Differential Vectorial Formulation

Question

At the Wikipedia there are the differential formulation for Euler-Bernoulli Beam \eqref{1} and Timoshenko Beam \eqref{2}
$$
\begin{align}
&\dfrac{d^2}{dx^2}\left(EI\dfrac{d^2w}{dx^2}\right) = q(x) \label{1}\tag{1}\\
&\begin{cases}\dfrac{d^2}{dx^2}\left(EI\dfrac{d\varphi}{dx}\right) = q(x) \\
\dfrac{dw}{dx} = \varphi – \dfrac{1}{\kappa AG} \cdot \dfrac{d}{dx}\left(EI\dfrac{d\varphi}{dx}\right)
\end{cases} \label{2}\tag{2}
\end{align}
$$

Both of formulations supposes that the undeformed beam is at $(\vec{e}_x)$ direction, the distributed charge $q$ is at $(\vec{e}_z)$ direction and $w$ is the displacement at $(\vec{e}_z)$ direction.

The deduction uses, for example that

$$
Q = \dfrac{dM}{dx}
$$

But the shear force $\vec{Q}$ and the momentum $\vec{M}$ are not colinear:

$$
\vec{Q} = Q \cdot \vec{e}_{z} = \dfrac{dM}{dx} \cdot \vec{e}_z \ne \dfrac{dM}{dx} \cdot \vec{e}_{y} = \dfrac{d}{dx}\left(M \cdot \vec{e}_{y}\right) = \dfrac{d}{dx} \vec{M}
$$

Question: Then, is there a formulation $(1)$ and $(2)$ using vectorial notation? Like for example

$$
\vec{Q} = \nabla \times \vec{M}
$$

Cause

$$
\vec{Q} =
\begin{bmatrix}
0 \\ 0 \\ Q
\end{bmatrix}=
\begin{bmatrix}
\dfrac{-dM}{dz} \\ 0 \\ \dfrac{dM}{dx}
\end{bmatrix} =
\det \begin{bmatrix}
\vec{e}_x & \vec{e}_y & \vec{e}_z \\
\dfrac{d}{dx} & \dfrac{d}{dy} & \dfrac{d}{dz} \\
0 & M & 0
\end{bmatrix}
=
\begin{bmatrix}
\dfrac{d}{dx} \\ \dfrac{d}{dy} \\ \dfrac{d}{dz}
\end{bmatrix} \times
\begin{bmatrix}
0 \\ M \\ 0
\end{bmatrix}
$$

Motivation: I want an analytic model for a 3D beam which neutral line follows an arbitrary path $p(t) \in \mathbb{R}^{3}$. When I tried to get it, I could not use the scalar rotations cause the vectors' directions were not the same and I should use rotations.

Answer 1

The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the Encyclopedia of Physics entry [1] by Antman (which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698) presents two different classes of methods for solving the problem, namely projection methods ([1], §11, pp. 660-663) and the asymptotic method ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dated) survey of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.

Edit after the downvote. After I saw the OP and my answer downvoted, I decided to add a few words of explanation. According to Stuart Antman ([1], §1, p. 641),

A theory of rods² or, equivalently a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a curve and the time as the only independent variables.

_{²We use "rod" as a generic name for "arch", "bar", "beam", "column", "ring", "shaft", etc. We employ rod both in the intuitive sense of a slender body and in several precise mathematical senses. The meaning will be clearer from the context.}

Said that, by carefully reading the question and its motivation, one sees that the techniques shown in [1] exactly suit the needs of the Asker: his search for a particular 3D equation for the models \eqref{1} and \eqref{2} is probably useless since it would be unsuitable for the study of a non-rectilinear beam, while the general procedures described in [1] where the equations describing a 1D medium are deduced from the standard, general 3D equations by applying them to particular constitutive equations, are applicable for beams described by a general spatial curve.

Reference

[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) Festkörpermechanik/Mechanic of Solids, Handbuch der Physik/Encyclopedia of Physics, Vol. VI part 2, Springer-Verlag, 1972, 641-703.