I really hate people like this. This guy pops up on a woodworking forum begging for help. He bought a house with a built in bookcase with 4-foot long shelves and the shelves are sagging. He wants suggestions how he can stop this from happening. Lots of people jump in, but in the end, he says “I have minimal time, budget, and materials to make this happen. Adding dividers, building fancy torsion box shelves or multi ply shelves with rebar or other metal components is beyond what I can produce right now. ” Then why the hell did you ask?
Seriously, what were you expecting? Magic? The shelves are too long. To fix it, you either have to reinforce the shelves, you have to build more support. Those are your choices. There aren’t any others. To say that you can’t do any of those things is to say you don’t want to fix the problem.
There’s a tool out there called the Sagulator that tells you how much your shelves are going to sag, based on the materials and dimensions. Now I know this guy didn’t build the shelves, but he inherited the problem and he can either live with the problem or deal with it. Physics doesn’t go away because he doesn’t want to do what’s necessary to correct the issue.
So he’s stuck with three possible things to do. Either get stronger shelves or add additional shelf supports in the middle, or just don’t put heavy stuff on the shelves. I suggested that he go buy some 4′ steel rods, route a groove in the bottom side of his shelves and epoxy in the rods, then cover it with a strip of matching wood. Nobody would ever know he had done anything and the shelves would stand up to much more weight. But nope, he just can’t be bothered. Nobody had a magic wand to fix his problems without having to do any actual work.
So stop asking if you don’t want help.