You could build it as Mel suggests, out of conduit, but instead of using the rebar to stake it into the ground, you could use those U-clamps they sell at the hardware store..just use several of them per "leg" and screw them directly to the bed frame.
I haven't tried it, but in theory it should work--maybe not for melons or super heavy crops, but certainly for beans and moderately pruned tomatoes I would think.

I assume you're using conduit or something similar, so you can just use the piece of hardware (whose name is escaping me at the moment) that secures said conduit to a piece of lumber, and just screw the bottom of the trellis to the box itself. The SFG book recommends that you go no higher than 4 feet, but it's up to you I guess. Hopefully you can understand what I'm talking about.

I have my trellis secured to the sides of the box with U shaped brackets. Use two, evenly spaced, per side. They work great.

Thanks! I'm not the engineer but I know my husband will know exactly what you mean and I'll pass it on to him.

Mel suggests that you can also use a strong wire or two to attach the top of the trellis to the south wall of the box, thus creating a strong, supportive triangle. (Assuming of course that your trellis is on the north wall of your box).

Or you could just build a frame out of 2x2 lumber and screw it to the boxes. Probably cost about $3 per box to do it that way.

Here is a link that might be useful: Garden blog