Implementation — FEniCSx tutorial (jsdokken.com)
f = fem.Constant(domain, default_scalar_type((0, 0, rho * g)))
So a negative force is applied on the zaxis from what it looks like so far. In terms of the beam there would also be moment due to the z displacements. Does the dolfinx software account for this already someplace in the background or would the moment need to be computed and also added in to the sample?
Once in a while we get asked these sort of questions on this forum “does dolfinx account for some physical principle automatically or not?”.
The answer is always no: dolfinx builds the problem starting from the weak formulation of the partial differential equation. If the weak formulation accounts for the physical principle, then dolfinx will take into account, otherwise there is no way that the problem will include that physical principle.
Therefore, the task that you have to do to answer your question is to look at the weak form of the PDE in the tutorial, and see if “moment due to the z displacements” are included in there or not. Unfortunately I can’t help you with that because I lack knowledge in that specific application domain.
Not sure this is exactly what you need but you can check that you indeed get bending moments on cantilever beams as expected, see Computing consistent reaction forces — Computational Mechanics Numerical Tours with FEniCSx
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There is a u and v or mu and nu of the weak form. Nu is integrated somehow with f. The weak form is described as a formulation that relates to Greene’s theorem. A \delta u is talked about. It is maybe unclear to me so far if that is a symbolic that is solved for or if it represents a scalar quantity. v or Nu is talked about as either a symbolic quantity or a value in some places between 0 and 1. As far as:
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
They are so declared and initialized however beyond that I have very little to go from what they actually are are become. There is a u that is talked about as a coordinate or location of mesh and then there is a \delta u which is a rate of change of that location as I take it so far. f is clearly defined however what actually happens with v or nu “under the hood” is a situation of inquiry for me in the case of the highlighted tutorial.
Nu is described in depth by Mark S. Gockenback in many chapters throughout his book. It is a little bit difficult to take proper context with so many occurences of it in text and what they mean.
I can’t teach a whole mechanics course in the comments of one post on a finite element software forum posts, but it might be worth clarifying one or two of your statements above to add a bit of “application specific” expertise here. Hopefully this is helpful!:

Moments in a beam model actually only arise from accounting for the various stresses through a beam’s crosssection. In a 3D solid mechanics formulation (such as the one you linked to in Jorgen’s excellent dolfinx tutorial), we don’t need to account for moments. Their effect is “built into” the 3D solid mechanics formulation.

The u and v you mentioned are the displacement trial and test functions, used to solve for what the displacement field looks in a solid model given specific loads and boundary conditions. \nu is a material parameter commonly used in mechanics for relating stress to strain. There are many “constitutive models”, but a common one is linear elasticity, which so happens to require only two material parameters to relate stresses to strains. One instance is the pair of E and \nu, the Elastic modulus and poisson ratio. There are other forms such as Lamé parameters, \lambda and \mu, which are used in the FEniCSx tutorial. You can see in this wikipedia page how these parameters relate to each other.
I would suggest having a look at https://www.continuummechanics.org/ and http://solidmechanics.org/ as they are both free and high quality resources to learn a bit more about solid mechanics.
P.S. if you are interesting in only transverse deflection calculations and extracting moments/forces, there are a few implementations of beam models on the forums, such as here.
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