3D internal force diagrams are very similar to 2D, just we need to draw 3 additional graphs for Vy, Mz and Mx. If you’ve mastered the 2D case, this should be quite an intuitive next step.
[1] Firstly, let's discuss how to get Vy. It is exactly the same as Vz, but in the y direction. The exact same conventions hold, so here you see that when the force points into y, it is negative. Then the force at the very end is opposite to F, so it jumps up. In case any of you don’t know, the dot here stands for out of page, and a circle with a cross is into page.
Here we can quickly see from this image that when a z internal force crosses a lever arm in the x direction, we get an internal moment in the positive y direction. When there’s a force in the y direction that crosses the lever arm in x, it causes a moment in the negative z direction. Due to this, we need to flip the signs in our Mz diagram. Here we see that Vy is a negative constant, so in Mz this corresponds to a positive linear curve. Ensure you remember this detail when drawing your diagrams.
[2] A moment in the x direction is known as a torsion. Torsion can only be caused by a moment in the y or z direction being transferred over to the x direction due to geometry, or a direct torsional moment being applied. The torsion has nothing to do with the normal force; remember this, it’s very important. In early statics, we always just assume the beam to be a bar and ignore the exact cross-sectional properties; the torsional moment is only a result of the previously mentioned forces. In the future, when we analyse cross sections and more stiffness-based mechanics, point forces can also cause torsional moments, in this case here with the trapezoid. This, however, is not our concern when dealing with these 3D internal force diagrams, which are concerned with the properties of the beam as a whole.
Torsion, for those unaware, is when you have a twisting motion on the beam. Imagine, as kids, when someone would twist a part of your arm in opposite directions, or how you open a bottle cap. The moment generated like this is known as torsion. If we want to see this more mechanics like, imagine you just have a rod like this clamped to the wall, and you apply a moment like this. The clamp opposes this motion, so there is an internal rotational force in the axial direction.
[3] Here I’ll use a similar figure to the one in [1], just without the point torsional moment, such that I end with a non-zero value at point P. I do the same analysis as earlier to get to Mz. Here on bar 2, I see that x2 = z1 from the local coordinate systems. This means that my Mx bar has to start with a value of F L, the value at the end of z1. This M(x) graph will now stay constant unless a distributed or point torsional moment is applied in the x2 direction. The N graph of section 2 here does not affect Mx.
Since in 3D all the forces and moments in the x,y, and z directions are so coupled, it is absolutely vital that you solve all 3 diagrams simultaneously; there is no other way around it. Here, I will solve a full task I came up with that’s much more complex; this is very much how a difficult exam task could look.
