On August 31, 2011, about 1630 eastern daylight time, a Cessna 172N, N739RU was substantially damaged during landing at Plymouth Municipal Airport (PYM), Plymouth, Massachusetts. The private pilot was not injured. The airplane was registered to a private owner, and operated by Alpha One Flight Service under the provisions of 14 Code of Federal Regulations Part 91. Visual meteorological conditions prevailed for the local flight and no flight plan had been filed. The pilot reported that she was planning to stay in the airport traffic pattern to conduct takeoff and landing practice. During the takeoff roll, forward pressure was required on the control column to prevent the airplane from pitching up prior to rotation speed. She reported that she was not concerned about the controllability of the airplane and elected not to abort the takeoff. During the climb, nose down trim was applied; however, forward pressure was still required to maintain the correct pitch attitude. However, she reported to a Federal Aviation Administration (FAA) inspector that once in flight everything was normal. During the landing flare, the pilot reduced engine power; however, she was unable to pull back on the control column. The airplane bounced several times, each time impacting the nose landing gear prior to the main landing gear. According to a written statement by a certificated flight instructor who flew the airplane prior to the accident flight; he "did not find any discrepancies with the aircraft during preflight and the flight was uneventful." The accident was not reported to the FAA or the NTSB, but the damaged airplane was discovered at the airport approximately two weeks after the accident by an FAA inspector. Prior to the discovery of the accident,one of the flight school's mechanics removed the engine cowling and noticed the firewall was damaged. He further noted that the elevator trim was in a slight nose-up setting. He moved the trim wheel and noted the elevator trim tab operated. He then disconnected the elevator push/pull rod that connected to the bellcrank and was able to move the elevator and reported the freedom of movement. He further noted slack in the elevator secondary control cables which he attributed to the damage to the structure. Examination of the airplane by an FAA inspector and an aircraft mechanic revealed that the firewall was buckled and the floor beneath the pilot and co-pilot's seat was damaged. The elevator push/pull rod, which connected to the bellcrank, was disconnected and the elevator moved unobstructed. The primary flight control cables remained connected at the elevator and the bellcrank. There was no evidence of damage to the elevator control "U." Examination behind the instrument panel revealed that there was nothing that would interfere with the travel of the control column. There was no indication of preimpact binding or damage. The aluminum channels, which were connected at the instrument panel and the firewall, remained attached at both ends by the rivets; however, were bent down causing binding to the aileron chain sprockets. The instrument panel had shifted aft of its normal position. The elevator control cables were void of tension and an accurate deflection indication of the elevator could not be obtained. The bearings and pulleys were free and showed no signs of previous damage. In a written statement, a representative of the flight school reported no mechanical malfunctions or failures that would have precluded normal operation of the airplane.