"Correction: in any
human structural analysis every component is
usually considered separately (to simplify the problem at hand). Software assembles element stiffness matrices into a global structure stiffness matrix and solves the whole."
So, what else analyzes structures, besides humans?
"Only if the connection between the shelf and sides is continuous. That is not the case here."
That's why I mentioned it. In this case, if the shelf deflects too much, it will bypass one of the supports. Then, if it springs back in time, it'll jam against the support and....err......LOOK AT WHAT YOU HAVE DONE!!!!!!!!!!!!!!!!!!!!!
"It is not the moment of inertia that is resisted. It is flexure that is resisted by the moment of inertia."
You're right, except it's rotation that's resisted- flexure is caused by the inability of the material or member to remain rigid. However, "there is strength in flexibility", as my professors used to tell us.
"Why not less than 1/3rd?"
Because I said so. No, it was just a generalization, really. If the two ribs are at one edge and we assume somewhat even weight distribution, the opposite edge can flex. If they're in the middle, it would be possible for both outer edges to flex if the weight is concentrated mid-span. I guess they could be located 25% in from the front and back.