The student pilot was conducting his final required solo cross-country flight. The flight to the first destination was uneventful, and the pilot entered the left-hand traffic pattern for the runway at the non-towered airport. The pilot reported that his base-to-final turn was slightly "wide," which placed the airplane to the right of the extended runway centerline. He also reported that he was "a little high" on the final approach. He corrected his flight path, and was satisfied that the airplane was positioned for a normal landing. The pilot used full flaps for this landing, as was his normal habit pattern. The airplane landed slightly beyond where the pilot had planned, but not far enough down the runway to cause him any concern. The touchdown was "hard," the airplane bounced, and the pilot "re-flared" to correct the situation. The airplane began "porpoising" and bounced at least two more times before the pilot was able to regain control and keep it on the ground. During the bounces, the pilot did not adjust the power or consider aborting the landing attempt. Although he had learned about porpoising in his ground training, he had never experienced it prior to this event, and did not immediately recall or implement the flight school's recommended corrective action, which was a go-around. The pilot taxied the airplane to a parking spot, and discovered that the airplane was substantially damaged. The firewall, cockpit floor, and several fuselage skins were buckled, and elevator travel was restricted. The pilot reported no preimpact mechanical malfunctions or failures with the airplane that would have precluded normal operation.