Howdy vSalonistas, need some advice on a book shelf I'm building for the wife. I'll likely build another just like it after this one is finished. She's got a TON of books.

Approx. Dimensions: 5ft tall, 4ft wide, 1ft deep

I'm building it out of 3/4" plywood with a red oak face frame to hide the edges of the plywood, and add stiffness. And that's where my question is. It's a wide book case, the shelves are 48" long. They'll be fixed to the sides, sitting in 1/8" dados and screwed to the sides with pocket screws. The backing is a 1/4" thick ply that's nailed in place, it's not really a load bearing piece.

I don't want these shelves to sag, but I also don't want to put a divider down the middle if I can avoid it.

So, I'm mounting a piece of (actual dimensions) 3/4" x 1.5" red oak along the front edge as part of the face frame, and to add stiffness. There will be lotsa pocket screws used to attach it. Will that be enough, or should I put another piece under the back edge of each shelf as well? Or should I bite the bullet and go ahead and put a divider down the center?

Basically a larger version of this record player stand I just finished: