Can anything be said about the expected shape of a specific buckling mode? Or more precisely, is is true for example that for the first buckling mode the second derivative of lateral deflection w.r.t the length $d^2w / dx^2$ maintains its sign over the whole span $L$?
If yes, how can one show that mathematically? For a beam, one can easily work it out from $w(x) = A\sin(n\pi x/L)$ with $n=1$, but how about a plate for example?
Can any generalisation be made?
Thanks in advance!