I've stored hay on pallets and had lots more rodent damage than leaving it on the dirt. I recognize that for horses you don't want the little bit of rot that you get on the bottom.
Filling with soil, sand, gravel will work, then put a vapor barrier down and put six inches of course rock, I would use three inch rock and roll it down. You don't want to sell rocks in your hay, and large rock will not stick to the hay.
Your tray design is OK, but place the deck boards tight to keep the vermin out. Hay stores very good on elevated lumber floors. Calculate the load on each column, you'll know why everyone puts down footings. You may need more legs or a TEE on the bottom of the legs. You probably should not plan on more than 1,000 lbs per square foot ground bearing to avoid settling. 500 lbs would be better. A 4x4 (3-1/2"x3'1/2") is only 0.085 square feet, so at 1,000 lbs per square foot, the end of a 4x4 is safe holding 85 lbs. One bale of hay at 67 lbs leaves 18 lbs for the structure. As was suggested, if you bury the 4x4 with a footing underneath, even one square foot, gives you 1,000 lbs to work with. 18" square footing gives you a ton per leg, 30 bales per leg, so for 200 bales you need 8 legs with 18" footings.
You can argue over how much ground load is acceptable, but unless you know, 1,000 lbs is safe. Really good solid is only about 2,500 psf.
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Today's Featured Article - Hydraulics - Cylinder Anatomy - by Curtis von Fange. Let’s make one more addition to our series on hydraulics. I’ve noticed a few questions in the comment section that could pertain to hydraulic cylinders so I thought we could take a short look at this real workhorse of the circuit. Cylinders are the reason for the hydraulic circuit. They take the fluid power delivered from the pump and magically change it into mechanical power. There are many types of cylinders that one might run across on a farm scenario. Each one could take a chapter in
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