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you need to determine how much weight will be on the shelves. then put the numbers into the
Sagulator to determine the sag.
Books can be very heavy if they are hardcovers, paperback not as much.
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Quite simply, no. Way too much leverage on the shelves. The other thing to consider is making some way to conceal the fasteners into the studs and spacing the parts so they hit the stud centres.
Blackhat
Bad experiences come from poor decisions. So do good stories.
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Assuming that the stiles are supporting the shelf at the rear the sagulator is irrelevant as it does not address this problem. The shelves are to small for follio sized books and a little large for quatro (9½x11"). The follio size will run ~60lbs/liner foot. The leverage will be too much for the upper piece to support. The rails would be fine but the attachment to the rails from the top of the shelf would be too small. Also unless it is screwed to a wall stud it pull out easily. You could narrow the shelves to fit octavo (9x11) and probably get 3 shelves in the same space but there would still be too much torque on the supporting structure.
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Your design isn't that bad, but the choice of MDF is. I see why you put the brackets on top of the shelf, to act as a book stop, and it should work fine as long as you mortise it into both the shelf and stile with the grain running vertically. If you made foxed tenons and under cut the mortises at the ends it would never come apart. I would move the rails up so the tops of them are above the shelf. That would eliminate the need to notch the shelf, make it easy to secure the shelf to the rail with screws through the back, and act as a stop to prevent books from hitting the wall. And it would look better, IMHO. If you switched to 1" hardwood shelves I think it would be fine.
John
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Use plywood or solid lumber instead of MDF.As another post mentioned it will sag with the weight of the books. Maybe you could do a half circle instead of the quarter circle under the shelves.This would give you the support needed and the book stop also.
Mortise and tenon for the rail to stile will be plenty strong.
I would not use biscuits for the connection,not sure if it is fail safe like M&T.
mike