Erik Kostson
Ansys Employee

I never looked into this in detail, since most designs perhaps avoid to get into that regime of post buckling. Having said that though, some structure say like a curved plate (snap through example), might snap through and then have considerable post snap through strength.

So below are two curves, one (black line) showing a considerable post buckling strength after the max load (F) has been reached, and showing a stiffening post buckling response. The second (red line) shows a softening post buckling response after the max. load (F) has been reached, and does not have any post buckling strength. So there might be some situations a designer might want to use that post response if it is giving enough strength to the structure.

That is my five cents - as I said I have not looked into this and do not know much more about it, and especially in terms of how this is used in design of structures - there should be plenty of material out there - something I found: