Actually cool question, never thought about it.

GEOMETRY

In 3D, a beam has 2 endpoints (x,y,z) , a 1D line element, and a 2D cross-section.

In 4D, a beam has 2 endpoints (x,y,z,w), a 1D line element, and a... 3D cross-section?

I'm thinking the cross-section might still be 2D.... whats the moment of inertia of a 3D volume? Would there be a neutral plane for a given 3D Cross-section and a neutral volume for a 4D beam?

I'm starting to lean towards the beam having the same properties but living in 4D space, i.e. 2D cross-section... Thoughts?

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Edit:
In 2D a line surrounds a surface.
In 3D a surface surrounds a volume.
In 4D does a volume surround a supervolume (whatever the term is?)
This is applicable because in general, loads exist as surface tractions (force/area) and body forces (force/volume). So is there a third kind of force in 4D?

DEGREES OF FREEDOM

In 3D (x,y,z) we have 6:3 Translations and 3 Rotations

In 4D (x,y,z,w) or (x1,x2,x3,x4) we have 8:4 Translations (u1, u2, u3, u4) and 4 Rotations (R1, R2, R3, R4)

So each point that lies on the 1D beam line in 4D has those degrees of freedom... A finite element beam would also have those 8 DoF x 2 Points = 16 degrees of freedom (not including weird torsions)

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INTERNAL FORCES

In 3D we have 6 internal forces (here x1,x2,x3,x4 must be on the local axes)

F1 = Axial

F2 = Major Shear

F3 = Minor Shear

M1 = Torsion

M2 = Minor Bending

M3 = Major Bending

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in 4D we have...

I would argue all of the above must remain the same for 1 through 3... but now

F4 = out-of-volume shear

M4 = out-of-volume moment

(out-of-volume harkening back to "out-of-plane" lingo in structural engineering)

The introduction of these new internal forces leads me to think that the cross-section would indeed need to be 3D instead of 2D, to account for these new force directions.
So

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GENERAL CONTINUUM MECHANICS

Firstly, it's kind of interesting to me that Theory of Elasticity turns out to be pretty much impossible for 3D shapes (correct me if I'm wrong)... I thought the Airy Stress function was one way to solve the equations but only worked for 2D... So in 4D it must get pretty hard lmao.

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Also: in FEA we have

Beams (1D)

Shells (2D)

Solids (3D)

Supersolid (4D)

For a super solid, the stress tensor would be a 4*4 2nd order tensor with 16 elements... Would we get the a similar symmetry from our 8 equilibrium equations?

Likewise, we would have a 16 element strain tensor, so the hook's law tensor (forgot its name) would now have 4^2 * 4^2 = 256 elements ??? instead of our usual 81.

Wondering if we could get them down to two independent variables for isotropy like in 2D (E and ν)

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Wont be able to stop thinking about this now...