More info: I've seen quite a few over the years of smaller dimensions that were nothing but corrugated decking and concrete. So there is a place for that, but that place wouldn't be 32 by 32. There was another thread on here where a guy built a shop recently, designed by an engineer, in that style, with an irregular footprint, and I think there was supposed to be I-beam framing. As often happens the contractor just did whatever they wanted to. They substituted the proper kind of decking, which is tall, to the improper kind. I am not sure just how unsound that was, but I would be concerned.
We've had lots of suggestions here over the years about post-tensioned, pre-tensioned, precast spans, etc. I am not a structural engineer, but I suppose none of those would be very attractive for mounting a lift upstairs. But your engineer may disagree. My shop was designed by an eminent structural engineer here locally, but I think his design was pretty original. I could certainly be wrong. I work in a factory, and I don't see that it's a typical way to hold up point loads.