I've used single pile caps for industrial stuff and utility stuff on some occasions. It's generally either lightly loaded or large diameter cast piles. Even then, though, I'd normally frame the superstructure so there's a tie beam somewhere near the bottom. This hasn't *always* been the case though.
I'd probably feel uncomfortable using a single pile (except possibly for monopole construction, but that's a special case) if normal operation/service loads put the resultant force outside the cross section of the pile even if the numbers worked. I'd be concerned about long term settling releasing some of that lateral restraint on the piles. Short term loads I'd be less concerned about. Also, you may not be allowed to use them without grade beams depending on your code and seismics in the area.
For actual design, just draw your free body diagrams at your critical sections and locations.
- Check anchorage of your column into the cap
- Make sure your cap can transfer the forces between the column and the pile. Either treat it as a column, a beam, a combination of the two or strut and tie depending on the situation. I've never really seen much that addresses this specific issue, but you can use a lot of the same concepts you'd use for a column pedastal so looking up design guides for that would likely be a good plan.
- Check anchorage of your pile into the cap
But first make sure your assumptions actually make sense. If you're transferring moment into the pile and it's a single pile cap you really have to make sure that your structure is relatively flexible in comparison to the pile including the geotechnical side or that you've actually modeled the whole system including foundation springs and things. Depending on the framing on top, you could easily have a structure that's an order of magnitude stiffer than the foundation. In that case, regardless of whether you design the piles to take that moment capacity, the connection will only be partially fixed. If that's true, you may be underdesigning your superstructure based on the assumption of a fixed cap.
While a direct welded connection is preferred from the engineering side of things, there are reasons to avoid it depending on the conditions you've got. It can be crappy for making fitup work, especially when you're talking about a single column on a single steel pile. You tend to have to screw around a lot on the welds and they're generally in terrible weld positions. Compression primary loads aren't terrible, because you can shim but if it's a reasonably significant moment connection or there's uplift, it can get ugly.