No-sag shelf solution?

I am building a sofa table out of 3/4" walnut using mortise and tenon construction for the back and side aprons. The stretchers connecting the legs on the left and right sides (front to back) lower will also use mortise and tenon joinery.
I plan on using a piece of 3/4" walnut approximately 10" wide by 45" long to act as both a lower shelf and to act as a center stretcher connecting the two lower leg stretchers. There will be a cleat on the interior of both the left and right lower leg stretchers which will support the shelf.
I am not planning on placing anything heavy on the shelf but I am concerned about possible sag.
I thought about gluing lips on the front and back of the shelf which would span the entire length. I know that this is effective on bookshelves but I rejected this idea since I want the shelf thickness to equal the table top thickness (3/4")
Since the bottom of the lower shelf will be about 7" above the floor and not readily visible I thought about glueing two 3/4" runners to the shelf the same length as the lower shelf on either side of the cleats supporting the shelf. Or milling two shallow dadoes about 1/4 deep by 3/8 wide and inserting splines which would sit about 1/2" proud of the bottom surface
Any one have any comments on this approach? Any one see any problems?
All the best
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A 45 inch unsupported span certainly has the potential for sag under any significant load. I think you're right to add some reinforcement.
I have some commerial bookcases with 36 inch span. Each shelf has dado about 1/2in deep and 1/8in wide. A steel spline is stapled therein and flush with the underside of the shelf. 36 inch span, a ton of large/heavy reference books, and no signs of sagging after 3 years.
I like the design. When I next make something like this I'm planning to cut two dados and epoxy the steel splines in them.
However, since the underside of your shelf will not be visible, why not simply attach one or two strips of steel T-section to it? You could use epoxy or lots of screws.
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Try: http://www.woodbin.com/calcs/sagulator.htm
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Hope this is of use to you:
http://www.woodbin.com/calcs/sagulator.htm
Tom
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According to the sagulator, your shelf would start to show a noticeable sag at around a 60 pound uniform load. Since you said you're not going to put anything heavy on this, perhaps you don't need to add any support. You said are worried about sag, however we don't know why.
I would guess single stretcher run under the center of the lower shelf should a) be hidden and b) take care of any sag you might encounter - according to your description. The other options you mentioned would also provide additional stiffening. If in doubt, you can set the shelf board on two end supports and try it out. -- JeffB remove no.spam. to email
toolista wrote:

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snipped-for-privacy@no.spam.san.rr.com says...

I'd be inclined to put a single stretcher also, but instead of putting it under the lower shelf, I'd put it over -- between the lower shelf and the table top. You don't even have to run it through to the end; stay back from the end of the bottom shelf and just use an 8" long offcut in the centre of the table ...
-P.
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toolista wrote:

I glued a single runner along the bottom of 4' x 11" x 3/4" bookshelf, along with a front lip, both 3/4" wide by about 2" wide. This not only provides support for the shelf, it also provides (via braces) support for the shelf above it, which supports the shelf above it, which supports the top. It's been there a couple of years and no sag yet.
I imagine a single runner would do in your case.
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Inset cleats will bear weight just as well as cleats set flush with the edges.
Any thoughts of building a box beam under the shelf? You could set your brick collection on the shelf without it sagging.
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