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Plates on the sides will add some strength, but you won't get the same bang for the buck as you get from more depth. It is up to you, how to do it, but I suggest more depth. Even if you just added a plate along the bottom of the frame, it would add considerable strength.

I agree with some of what you say, but IMHO the rails will be stiffer with 3" tall quarter-inch vertical plating than with quarter-inch tall 2.25-inch wide horizontal plating. I understand that generally, flanges are thicker than webs, but that applies better as the web-height to flange-width ratio increases.

Randii