Yes! I want an analytical model (formulas) to compare with FEA to verify the FEA is computing what it should compute; to verify I'm using the elements / materials / properties in the FEM correctly.

I found an example of bolt pre-load and its impact on frequency (youtube.com/watch?v=kDiWkbYdMS8). However, this is really about contact, not internal stress. Is there an example of a hand-calc I can do to compare with a SOL 106 solution?

This description implies that it is independent of a pre-load. In other words, it doesn't matter what the tension is, the effect is the same. Is that true?

All the references I've seen so far all talk about geometric stiffness or stress stiffness in terms of FEA. Isn't there an analytical approach that describes this phenomena? I'd like to start with an analytical approach and then build a model and compare to make sure I understand how to use SOL 106.

It is clear to me that changing the stress-strain curve changes the modulus. However, the answer that intrigues me the most is that bolt preloads can affect the frequency of a structure without reaching the non-linearity in the stress-strain curve.

I'd love to understand this more explicitly. What do you recommend I read about in a mechanics text book (e.g. Rao / Mechanical Vibration does not capture this concept)? What do you recommend I read about for Nastran SOL 106 (I use Siemens Simcenter)?

It is old, but the math is good. I haven't found anything that is a really good replacement. I'm not sure I understand your terminology "stability of the system". Natural frequencies are resonances. If you divide the output by the input, you should see a peak where the system resonates. What arises from the theory of single degree of freedom simple harmonic oscillators is that there is a 180 deg phase shift comparing phase before and after a resonant peak. This may help in discriminating noise from resonance.

Nice direction. I tried to get the natural frequency in the vertical direction and was unsuccessful. I also tried to get displacement vs. load and was also unsuccessful. I think both can be done, but I need to spend more effort on fixturing.

It is designed to have both ends fixed (clamped in all 6 DoF's). One end slides laterally with respect to the other. As it does so, it forms and "s" shape. The pins in the end do curve. However, I'm not sure buckling is the correct term. I consider buckling to result in a dramatic drop in axial stiffness - like crushing a can. That does not appear to happen so I want to test for this condition.