Question
I have a 1/2 hp Sta-rite shallow jet pump(220v).Elevation head is 20'. Static pr=57, design pr=39, pumps 20gpm at the pump discharge. Running hot, shuts off after +- 20 min. Pulling 14.8 to 15.6 amps dep on load. Spec plate on pump warns max of 13.4a. Pump, press switch wiring, O.K. Wire size fr source/<100'=#12. Cause?: air leak, input side? Press switch out of adjustment? Suggestions?

Answer
Thanks for the question.
If it's drawing over the name plate max. amps it will over heat. Most motor have an internal thermal protection circuit which protects the motor and "surrounding building" from fire. In your case the thermal protection is working fine.
The problem is your drawing too much water from the pump. Generally, the more water you ask for the more electricity (amps) the pump asks for.
20 GPM from a 1/2hp is too much water. You will need to lower your water demand on the pump to resolve the problem. If you want to test this, install a gate valve on the discharge of the pump. Restrict the flow and watch the amps drop. If nothing else it's an interesting "physic" experiment.
Good luck.

Matt