Duty of Power Transformer:

The question doesn't make much sense.

Why would you ever want to run a transformer in overload? Not only would you shorten the service life of the transformer but everything else would suffer from undervoltage problems.

Sounds like a good way to destroy expensive equipment!

Power transformers usually do have an overload rating, in some percent over base rating. Utilities work with this all the time, when in an emergency they have to switch loads over from one transformer to another. There are often overload ratings for both "long-term emergency" (usually in intervals of hours, such as 8 hours) and "short-term emergency" (usually in minutes). The overload ratings could be 10%, 20% or even as high as 50% or more.

Yes, the extra thermal load on the transformer does shorten its lifetime. That is all figured into the equation when originally specifying the unit: How much overload capacity do you pay to have built in so that you still get a normal lifespan (30-40 years) out of it, assuming you have some statistically-determined length of emergency overload time on it during its life.

Can a K-rated transformer take a load exceeding its normal rated load?.

Ah yes! Now that you mention that I seem to remember having heard something like that before. I suppose I thought it was a foolish notion but I realize that sometimes it is a necessity. I guess I would tend to over-design a little anyway due to expanding needs. That was something I learned to expect a long time ago.

Duty rating is not related to overload.

Continuous duty means that the device is designed to operate continuously at the nameplate load rating. The alternative design is intermittent duty.

Intermittent duty equipment is only supposed to operate under load for relatively short periods. Examples are the field flashing circuit on a generator (typically loaded for 3-5 seconds during startup) or a doorbell transformer (only loaded when the door chime is making a sound). Because the device is loaded for only a short time, no significant heat is generated in the windings. Intermittent duty devices lack the heat dissipation and higher temperature rating of continuous duty equipment.