B767 CREW HAD ENG FAILURE ON TKOF.

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SCHEDULED PAX FLT BRU-JFK. FO FLYING; SECOND FO IN JUMP SEAT. TKOF GROSS WT 379000 LBS. TKOF FUEL 113000 LBS. UPON ROTATION (APPROX 170 KTS); MAIN GEAR STILL ON RWY; WE FELT AND HEARD A LOUD THUMP AND THE ACFT YAWED TO THE R. ALL 3 PLTS INITIALLY THOUGHT WE HAD BLOWN A TIRE. WE THEN NOTICED ENG #2 EGT IN THE RED. AFTER A BRIEF DISCUSSION; WE DECIDED AGAINST A BLOWN TIRE GOING THROUGH THE ENG (GEAR IS WELL BEHIND ENGS) AND RETRACTED THE GEAR. NO PROBS. WE CONTINUED CLB TO 1000 FT AGL. DEP WS CONTACTED AND AN EMER WAS DECLARED. AT 1000 FT AGL; WE LEVELED OFF; ACCELERATED TO FLAPS 5 DEG SPD AND STARTED TO CLB. FO (PNF) ADVISED ME THAT ALL OF THE ENG STALL/SURGE CHKLIST HAD BEEN ACCOMPLISHED. EGT STILL IN RED. I DECIDED TO SHUT ENG DOWN. THIS WAS ACCOMPLISHED USING FAA APPROVED EMER CHKLIST. WE REQUESTED FUEL DUMPING. 30000 LBS WAS DUMPED AT 4000 FT MSL. FLT ATTENDANTS RPTED A LOT OF VIBRATION IN CABIN. WE DECIDED TO DO A FLY-BY TO MAKE SURE TIRES/MAIN GEAR OK. THIS WAS ACCOMPLISHED. WE WERE SATISFIED NO GEAR/TIRE PROB EXISTED. WE WERE VECTORED FOR AN APCH AND LANDED ON RWY 25L; BRUSSELS WITHOUT INCIDENT. AFTER BEING CHKED BY CFR (WHILE STOPPED ON RWY) WE TAXIED TO PARKING SPOT AND DEPLANED PAX. FLT TIME WAS 48 MINS. BRU RADAR WAS EXCELLENT. CFR WAS EXCELLENT. WHILE INFLT WE ACCOMPLISHED ALL NORMAL CHKLISTS; THE 2 CHKLISTS PREVIOUSLY MENTIONED; PLUS LNDG WITH 1 ENG INOP CHKLIST. WE ACARS'ED COMPANY WHILE IN AIR; BUT RECEIVED NO REPLY. FO FLEW ENTIRE TIME AND DID AN EXCELLENT JOB. THE FO (PNF) DID AN EXCELLENT JOB RUNNING CHKLISTS AND KEEPING COM FLOWING BTWN THE COCKPIT AND THE FLT ATTENDANT CREW. FLT ATTENDANT CREW KEPT PAX INFORMED AND CALM. KUDOS TO ALL! I AM VERY PROUD OF THE ENTIRE CREW. SUPPLEMENTAL INFO FROM ACN 454203: ON TKOF ROLL; AT ROTATION; THERE WAS A 'THUMP; THUMP; THUMP;' FOLLOWED BY STRONG AIRFRAME VIBRATIONS. VIBRATIONS SEEMED TO ORIGINATE FROM THE R SIDE OF ACFT. AIRFRAME VIBRATIONS DISSIPATED SOON AFTER LIFTOFF. IN SAME INSTANT; ACFT YAWED; R ENG EGT WAS NOTED VERY HIGH (OUT OF LIMITS); ROTATION DECREASING. BASED ON VIBRATIONS; WE FELT A POSSIBILITY THAT WE HAD GEAR/TIRE DAMAGE WITH THE POSSIBILITY OF TREAD INGESTION INTO THE ENG; OR TREAD SEPARATION DAMAGING ENG CASING. WE FELT CONFIDENT THIS DID NOT HAPPEN; SO WE RAISED GEAR. DUE TO USING FULL RUDDER TRIM PLUS ADDITIONAL RUDDER PEDAL INPUT; IT WAS FELT ENG WAS PUTTING OUT VERY LITTLE THRUST. AT 1000 FT AGL; AIRSPD WAS INCREASED; WE CLBED OUT AT 180 KTS/FLAPS 5 DEGS. EMER DECLARED; WE FLEW A L PATTERN. CAPT AND RELIEF PLT RAN APPROPRIATE CHKLISTS; ENG WAS SHUT DOWN. AT 4000 FT; FUEL DUMPING INITIATED. DURING CONVERSATIONS WITH FLT ATTENDANTS; THEY WERE QUITE ADAMANT THAT BASED ON WHAT THEY FELT; THAT THE LNDG GEAR OR TIRES WERE DAMAGED. WE DECIDED TO DO A FLY-BY FOR TWR TO CHK GEAR. ONE INDIVIDUAL FELT THE GEAR LOOKED FINE; ANOTHER FELT THE GEAR LOOKED DAMAGED. WE CAME AROUND FOR LNDG. I LANDED WITH THE GND SPOILERS NOT ARMED AND WITH NO AUTOBRAKES; AIMING TO TOUCHDOWN SLIGHTLY L OF CTRLINE. TOUCHDOWN WAS SMOOTH. WE STOPPED ON RWY; EMER PERSONNEL INSPECTED ACFT AND ESCORTED US TO REMOTE PARKING PAD. THE INITIAL VIBRATIONS WERE QUITE DIFFERENT THAN ANYTHING ANY OF US HAD FELT BEFORE; THUS THE INITIAL 'CONFUSION' OVER POSSIBLE GEAR/TIRE DAMAGE. AS A RESULT; WE LEFT THE GEAR DOWN FOR SEVERAL SECONDS DURING THE INITIAL CLBOUT WHILE WE DID SOME QUICK ANALYSIS. AFTER GETTING FEEDBACK FROM THE FLT ATTENDANTS; WHO FELT THERE WAS GEAR DAMAGE; WE DECIDED TO DO THE FLY-BY. UNCONVENTIONAL; BUT WE WERE STILL OVER OUR MAX LNDG WT; AND WITH THE ACFT FLYING WELL WE THOUGHT IT PRUDENT TO INCREASE OUR SITUATIONAL AWARENESS. AS THE FLY-BY WAS INCONCLUSIVE; I DECIDED TO LAND L OF CTRLINE AS A PRECAUTION. I HAND FLEW THE ACFT DUE TO RUDDER TRIM NOT BEING SUFFICIENT TO TRIM ACFT; THUS I DID NOT WANT TO ENGAGE THE AUTOPLT. THIS TOOK ME OUT OF THE LOOP A BIT IN TERMS OF CHKLIST WORK; BUT WITH A RELIEF PLT THERE WERE NO PROBS. CRM WORKED VERY WELL. CAPT AND RELIEF PLT RAN ALL CHKLISTS AND SHUT DOWN ENG. RELIEF PLT COMMUNICATED WITH FLT ATTENDANTS AND PAX; CAPT WITH ATC. TKOF WT WAS APPROX 380000 LBS; WE DUMPED APPROX 35000 LBS; LNDG WT WAS APPROX 330000-338000 LBS.