why Dc Adapter get faulty on UPS?

Two possibilities.

One is if it uses a standard power transformer. If the UPS puts out square waves or "sine" waves that are full of harmonics it may cause enough inefficiency in the transformer to overheat it.

Two, if it uses a switchmode convertor, it might not like square waves. If a 120 V square wave is input you get 120V DC rectified. If it's a sine wave it would be 160 V DC. If it only has 120 V DC it's the same as only feeding it 85 V AC sine wave. It might not like less than 90 V AC (effectively) because to remain in regulation the peak current will be higher in the chopper output device(s).

UPSes are only supposed to be used for a short time. In fact at one time I was thinking of connecting some big big truck batteries to one and having a longer use time but thought about it for a minute and decided it would be a bad idea. Not only do I think the UPS itself may burn up, there is also the issue of what is plugged into it. If you want to stay off the grid for a long time either get a generator or...........

It converts to DC right ? Well just cut the cord, get a battery of the proper voltage and set it up with a charging ballast resistor and a diode to feed the thing when the power goes out. I have seriously been considering doing this to cordless phones.