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Other 196's:
How to determine the elastic tensor without any FEA tools?
Experiments and theory.
Build a bridge and see if it collapses!
...and then rebuild it.
I could swear there was a comic strip about this...
That's a valid approach, yet today, experiments are usually on a smaller scale.
Actually, a bridge like this is generally considered simple enough not to require FEA. Calculating it theoretically using trusses should be enough to make it. I'm sure FEA is still done but only because it's easy and cheap.
Magic & Voodoo
No, that's the FEA part.
If you have an actual engineering degree then you've done basic forms of these calculations by hand at some point, and have likely written a basic numerical solver in Matlab.