These sorts of things are usually glued and screwed, but it's actually the glue that holds them together - the screws just hold everything tight while the glue cures. Screwing into hardboard or 1/4" ply is an exercise in futility, so I just used glue, and used my two 4x4's as long clamps. It would have been a bit easier, if I'd done this before I'd rough-cut the 4x4's, but it worked out.
With the shelf secure, get a couple of friends to come help, and stand the bench on its feet. I said earlier moving the top by yourself is dangerous. Trying to lift the entire bench is foolhardy. Of course, I already said I'm stubborn, so I did it myself by rigging a simple block-and-tackle using lightweight pulleys I got at the hardware store. (Not the lightest-weight pulleys, those are meant for flag poles and have a design load of something like 40 pounds. These had a design load of 420 pounds.)

I used the factory edge of the half-panel of hardboard as a guide for cutting the ply. I wanted to cut a 5-1/2" strip, and my saw cuts 4-1/2" from the edge of the shoe, so I wanted the edge of the hardboard 10" from the edge of the ply. So I set my combination square to 10", and used it to mechanically set the distance. Hint - if you need two things to be precisely the same length, try to avoid measuring them separately. Use some mechanical mechanism for setting the distance.
A better solution would have been to route a rabbet into the side, so that the added strip always had thickness. The way I did it means that the strip I glued in is very narrow, and hence very weak, at a certain point. In this case, that's not a problem, because it's going to be sitting under the countertop layer. I also noticed that because I had only clamped the strip down, and not into the edge, there was a noticeable glue gap where the strip butted up against the MDF. Again, in this application it isn't visible. But if I was doing something like this on the top of a table, I'd make sure to cut a clean rabbet, and to clamp both down and in.
Foundations of Woodworking runs every fall (September - December) and winter (January-March). The application process for each session opens 9 months in advance of the course start date. Applications will be considered for registration in the order in which they are received. There is no fee for applying. Accepted applicants will be invited to register. We ask for a deposit of $200 at the time of registration to ensure your place in the class. 
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