On September 8, 2022, about 1320 central daylight time, a Piper PA-28R-200 airplane, N29RM, was substantially damaged when it was involved in an accident near Meridianville, Alabama. The pilot under instruction and a flight instructor had minor injuries. The airplane was operated as a Title 14 Code of Federal Regulations Part 91 instructional flight.   The pilot under instruction was at the controls when the airplane entered the traffic pattern at Huntsville Executive Airport (MDQ), Meridianville, Alabama to practice takeoffs and landings. Shortly after turning onto final approach for runway 36, the engine began to run rough, as if a cylinder was “missing.” The flight instructor took over control of the airplane. He verified that the fuel boost pump was on and the throttle was advanced. The engine then stopped completely; however, the propeller continued to windmill. Unable to make it to the runway, the flight instructor landed the airplane in a bean field short of the runway. The pilots egressed the airplane and were met by first responders.   An inspector with the Federal Aviation Administration (FAA) reported that there was substantial damage to the fuselage and wings.   The wreckage was recovered to the owner’s facility, where the FAA inspector examined the engine and fuel system. The inspector reported that about 30 gallons of fuel were onboard, with no evidence of water or other contaminants in the fuel tanks. The engine contained 8 quarts of clean oil. The air induction system was unobstructed. Internal engine power and valvetrain continuity was established. Suction and compression were observed on all cylinders. Valve action was correct and there was no evidence of a stuck valve. Engine control linkages were secure with no binding observed. The magnetos and ignition leads were secure. The electric fuel boost pump was energized and pumped fuel. Fuel was observed in the line to the injector manifold. The top spark plugs were normal in color and wear. The fuel injector servo was removed; it passed a diaphragm leakage check per the Lycoming troubleshooting guide. There was no internal damage and all fuel injection nozzles were unobstructed.