Most likely not a composite system as that concept wasn't very prevalent back then, if at all.  The concrete was traditionally used as a fire protection material and if it is spalling off then I would assume that you have some sort of rusting going on due to moisture.

Tyipcally, you can remove the poor concrete, clean up the beam, and re-patch or replace the encasement - but some attention should be directed toward the root cause of the rusting.  Is the concrete exposed to chlorides?  Would some sort of protection be required to alleviate this?  

Do you have many closely spaced steel joists (say about 2 feet between beams)?

If this is the case the concrete is likely to act as an arch between the steel joists. Timber shuttering would have been placed tight to the underside of the steel joists and the concrete poured over the top and (usually) levelled off with the top of the steel joists. There would typically be no reinforcement.

Sand/Cement and plaster would be directly applied to the underside to form the ceiling finishes and the top surface would likely be screed or direct bonded timber parquet type flooring.

I've seen may floors of this type. Usually the spans are quite short and the steel joists check out as OK for the loads...

Just returned yesterday from the facility.  Of five buildings in question, two have the construction year in the facade.  Those two were 1907 and 1910.  The other three are presumably from about the same time.

For close to 100 years old, the buildings are mostly in pretty fair shape.  The exception is Bldg C, where you find the condition I described earlier.  The joists are closer to 6-8 feet apart, with the rusted bottom flange now exposed.  Between joists, the slab is reinforced with #3 or #4 bars at 3-4 inches o.c.  I know, because I could see the bars.  In fact, I was able to remove pieces of the bars.  There are whole sections where the bars have completely rusted through.

I can see using JAE's recommendation for the beams--I've done something similar in the past.  But how does one go about restoring a slab that has lost its reinforcement?  You can't exactly clean the rust off of the bars when the bars are nothing but rust.  I want to demo and rebuild the floor (actually, the whole building should be bulldozed).  However, as it is a public project, with historical significance, the state will likely want more of a restoration option.

"But how does one go about restoring a slab that has lost its reinforcement?  You can't exactly clean the rust off of the bars when the bars are nothing but rust. "

I'm working on an old building with similar problems, although my building is cast-in-place concrete.  as with many old buildings, the basement extends beyond the building footprint under the sidewalk and alleys.  The slab and beam reinforcing is severely corroded, to the extent that many of the reinforcing bars no longer exist.  We are in the process of removing all of the corrosion.  I required that eEverything be shored and it's difficult to work.  For the slabs, I am considering four things:  score grooves into the bottom of the slabs between the beams and install new reinforcing to replace the old;  install reinforcing and pour new concrete below the existing slabs increasing the overall slab thickness; carbon fiber reinforcing;  and adding steel beams between the concrete beams.  Any thoughts about one method or the other?  Since many of the existing concrete beams need to be restored, I can do whatever essentially based on economics.  the various options are driving the contractor crazy, but I don't know what else to do.  So far, he has not liked anything I proposed.  Sorry to hijack your thread rholder, but we have a similar problem.  one of us, or someone else, might have an idea that helps one of both of us.

Regarding details of the original construction; many of the early construction and structural books have very good descriptions and sketches.  Look for books like:

Kidder-Parker Architects'and Builders' Handbook
by Frank Kidder and Harry Parker
John Wiley & Sons, Inc., Editions of 1884, 1892, 1897, 1904
(I have the 18th Edition from 1936)

Handbook of Building Constrution, Vol 1 & 2
by George A. Hool, Editor
McGraw-Hill Book Co., 2nd Ed. 1920

There are many others. There are also some recent books showing old construction details which you should be able to find on Amazon.com or PowelBooks.com

Have you considered FRP on the bottom side of the slab? Just an idea?

"All I know is that it's a cementitious wrap of some sort, and the concern is that it is spalling off.  If it's all part of a structural system, there will be repairs to be made."

I run into a lot of buildings like the in NY. Have you taken a sample of the material? It could very well be an asbestos for fireproofing. Or it could just a mortar fill. I would first take a sample to determine the wrap material before analyzing the structural system.

Thanks for the tip, jheidt.  I now have a Hool and a Kidder-Parker on the way.  I suspect these will come in extremely handy, for this, as well as other projects I have.

I saw no joint to indicate that the slab was a different pour from what covered the beams.  I've also seen some old construction details (1906 Sweet's Catalog) with similar--though not identical--construction techniques, where the beam is, in fact, encased in the slab pour.

I have been in touch with a Sika rep.  Hopefully, he'll be able to tell me about FRP.

Right now, I'm trying to make recommendations for repair.  The tough part (the tough part?) is walking that line between making repairs to the structure, and asserting that the original design was okay in the first place.