Yes, you're correct. It appears that I made a "typo" miscopying from my notes. However, the rest of it is correct - i.e that C1 = C3 =0. It turns out in this particular case that the sign error made no difference to the result. Thanks for catching that.

Correct. n refers to the mode number.

I will get there eventually, but for now I still have some more beam videos to make before moving on to plates.

Steel is way stronger and stiffer than concrete, however, it is also much more expensive to produce. Steel column tend to buckle because they are very slender in order to keep the cost down. If steel columns were made as thick as concrete columns, then they would surely be much harder to buckle than the equivalent concrete structure. From an engineering point-of-view, we would like the tensile strength properties of the steel AND the cheap and compressive properties of the concrete. On the other hand, steel is susceptible to fire damage and concrete is brittle and cracks easily (which severely reduces its tensile strength). This is why a combination of the two is used to make reinforced concrete in practice.

We are trying to get a static result out of a dynamic model. By setting the frequency to zero, we are finding the point when the effective stiffness of the column goes to 0. At this point, it cannot support a load.

I’m using an app called “Paper” by WeTransfer. Running on a 13 inch iPad Pro and using an Apple Pencil. Movies are captured on my Mac using QuickTime. The mic is a Blue Yeti. The voice is mine.

@@Freeball99 thanks!