Reading through the posts above - several very good posts, BTW - I see that no one chose to highlight what was likely the cause: lack of proper drainage of the balcony surface. Several of you cited lack of treatment of the joists (pressure treated), and some of you questioned proper use of the building code design live load. However, if the balcony had drainage properly installed and well maintained, then the serviceable life of the joists supporting the balcony would have been greatly enhanced.
The photo posted above does suggest lack of a proper drainage system. Rainwater was likely allowed to pond on the surface of the balcony, spreading over its surface. I think this is why we see fairly uniform deterioration of all of the joists; if there was localized deterioration, then I would have blamed a leaking drainage system.
To compound the situation, I question whether there was a proper moisture barrier in place that would have prevented - or at least limited - the amount of water infiltration into the joists.
Furthermore, one (or two) of you suggested that the header directly below should also be investigated for possible deterioration due to water infiltration. I agree that this possibility cannot be ruled out. This brings to light something about this type of problem that is so often overlooked: we need to open pandora's box, so to speak, and question the current state of all the other balconies in this structure. After all, the intent of the building code is life safety!
Richard L. Flower, P. E., LEED Green Associate
Senior Structural Engineer
Complere Engineering Group, Inc.