The student pilot was performing a first supervised solo traffic pattern flight on runway 25 at Portland-Troutdale Airport (KTTD). After completing three touch-and-go landings, he flared prematurely and bounced the fourth landing and subsequently landed hard. According to the student pilot, the aircraft yawed to the left due to the impact and a low crosswind condition. In a subsequent discussion after the accident, the student pilot reported to his Certified Flight Instructor (CFI) that he was focused on the cockpit instruments during the initial bounced landing recovery efforts, and by the time he looked out again, the airplane had drifted to the left edge of the runway. He initiated a rejected landing, resulting in a climb-out heading 50 degrees to the left of the runway heading. This was followed by a traffic pattern with a full stop landing. The student pilot and CFI reported that earlier in the day they had completed 1 hour of dual instruction. The dual flight consisted of airwork and seven touch-and-go landings in the pattern. The CFI observed the solo flight from the ramp but did not witness the rejected landing because he had turned to walk back to the FBO facility. Later that day, a dispatcher observed damage to the aircraft and notified the student pilot and CFI. The damage looked like the horizontal stabilizer had been grazed by a pole-like object on the bottom side. An FAA inspector confirmed structural damage to the right horizontal stabilizer. The CFI requested to examine the runway and found a damaged runway light, with a ground track mark in the grass leading up to and away from it. The CFI stated that the airplane had no mechanical malfunctions or irregularities with the control system during their dual flight, and the student did not claim any.