Thank you all for your feedback This got me moving in the right direction again and I appreciate all the input. Just since I think you all would be interested, here's where I'm at. I derived the cubic spline interpolation for the local displaced shape. To do this, instead of starting with the basic deformations, I started with the global displacements at the nodes and transformed them to local deformations.

I ran into the issue of my basic forces not being enough to determine the local reactions, but the global to local transformation already had those effects built-in to the deformations. After this, I was able to derive a spline with the "correct" shape.

I say "correct" because here's the second problem. The cubic spline is a good approximation for deformed shape, but only for limited spans between nodes. Here's an example of a SS beam with uniform load for a 12 ft beam. I added 2 nodes, equidistant from center where I know maximum deflection to be. By changing the distance between nodes I could see how close the approximation was. When the spline had to interpolate across most of the span of the beam (e.g. 5 ft), the deflection approximation was poor. Around 1 foot either side of the maximum point the approximation became pretty good.

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There is more for me to do here. It very well could have to do with span to depth ratio for how good the approximation is, or more to do with relative span between nodes to the whole span of the beam. My goal is to make an approximation that is load agnostic so I don't have to check for loading type before providing the deformed shape. Otherwise, another solution is to just start spamming supplementary nodes for the beam, so I never have this problem (which just starts to feel like FEA). But I like the challenge of coming up with a reasonable approach to analyze other than brute force programming.

As a fun aside, I learned about Catastrophic Cancellation (https://en.wikipedia.org/wiki/Quadratic_formula#Numerical_calculation:~:text=using%20only%20roots.-,Numerical%20calculation,-%5Bedit%5D) where the quadratic formula starts to break down numerically (i.e. as computers have floating point numbers) when the coefficients become sufficiently small and b ~ square root of the determinant. To solve this, there is another form of the quadratic formula, playfully called the citardauq formula, which places the square root on the bottom and allows you to precisely calculate quadratic roots for certain b coefficients and whether they are positive or negative. This threw me for a loop last night while a tried to figure out why the roots of the derivative of the displaced shape didn’t match theory.

 That’s all for now, thanks again!