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I don't really know much about building anything so I was trying to keep this as simple as possible (2x4s and OSB, wood glue and screws). I'm worried about the highlighted legs bowing.

This is for a grow box I wanted to build. The top part is a hood, and the front will be a hinged door. I'm trying to avoid another horizontal support across the two highlighted legs because I want to be able to have a completely open space from the front and top.

The hood would have an additional 10lb load from hanging lights. The inner table will hold a tote with up to 8 gallons of water.

enter image description here

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    They probably won't bow from load, but they might well warp from moisture, inside a dank box. Paint that thing VERY throughly. You might also want to use hardibacker (or similar) rather than OSB, and move the end panels INSIDE the legs. – Ecnerwal Nov 15 '14 at 1:19
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The legs will not bow at all, all the force acting on it is pushing straight down. There is nothing going on the inside to act upon it to push out, so this will hold anything you wish to place in it. It will also hold an incredible amount of weight on top without no bowing of the legs either, if you added an apron that is attached to the top the same way you have a front stretcher at the bottom. If you did the top a little differently and made it so the top blocks "keyed in, or just added a few removable screws, it would help hold the front together.That way you could still remove it when you need. Like this without the screws at the top, it will not bow, maybe spread a little if you don't fasten the top.

enter image description here

I added your modified picture to help show what would help

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    +1. There may be some minor flexing, but since the legs are short you'll probably never notice it. It would be a bit stronger if you tied the tops of the front two legs together, but the diagonal bracing you get from the sides and the top will help keep it fairly rigid. – keshlam Nov 7 '14 at 13:21
  • The legs will have a tendency to push apart on the top. There really should be another support in the front like you have on the bottom shelf. It may stay together if you have it on a flat hard surface and you never move it, but once you try, it may not hold together that well. You may want to invest in getting a pocket screw jig. Pocket screws will be a bit stronger than attaching through the face. You also would have less exposed screws. – Jason Hutchinson Nov 7 '14 at 13:52
  • Front apron is good. You might also add a horizontal at the top of the cabinet between the two back legs. That'll cut down on stuttering when you slide the cabinet sideways along the floor. – Wayfaring Stranger Nov 7 '14 at 14:39
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In order to reduce flex I would make frontal horizontal piece wider. Or add another below existing one (you may also place that behind front legs).

If you want to make sides even more rigid, you may turn legs 90deg around vertical axis.

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