It's actually some kind of "blanket". I think they look curved because the rig can only hold up 2 sides. You can tell these things are really heavy too, because the crane barge will be noticably tilting towards the load. It could be some kind of woven steel fabric inside some kind of plastic or composite wrap, at least that's what it looks like. The crane has a GPS antenna mounted on the top, so i'm guessing they are trying to be failry accurate where they lay these things down. Also, they go very, very slow. The one day i stood watching it took them a good 20 minutes to get it fully submerged.
It should be noted that Weeks Marine also moved four locomotives from a barge onto flat bed trucks over on staten island for the staten island railway not too long ago. They used a bigger crane for that operation i believe.
No I'm not a railroad professional, but that doesn't mean I can't provide first hand observational comments.