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Why exactly do you want to frame it that way. It seems like you're introducing a lot of steel that doesn't need to be there.
You can wrap the joists that are exposed and cap the ends to keep moisture out. I'll size the steel for you.
I apologize for the technical difficulty. I posted the size a couple days ago. I will repost it when I get home later.
It's a W8x18 or W6x20 ASTM A992