INSTRUCTOR WITH 2 STUDENTS RUNS OFF END OF RWY ON LNDG ROLL.

Narrative

I WAS ON A TRAINING FLT WITH 2 STUDENTS (BOTH PVT PLTS). WE WERE DOING A FULL STOP LNDG AT CROOKSTON ARPT ON RWY 24 -- WHICH IS A TURF RWY. AT CROOKSTON THERE ARE 2 TURF RWYS THAT INTERSECT ONE ANOTHER AND ONE OF THEM (17-35) CONTINUES ACROSS THE PAVED RWY 13-31 AND ITS ASSOCIATED TXWY. AFTER LNDG I RECALL LOOKING TO THE L OF THE NOSE OF THE ACFT AND SEEING THE TXWY UP AHEAD. I THOUGHT TO MYSELF THAT WE COULD EITHER TURN OFF ON THE TXWY OR CONTINUE AND TURN OFF ON RWY 31 BEYOND THE TXWY. NO BRAKES WERE APPLIED ON THE AFTER LNDG ROLLOUT BECAUSE I WANTED TO CONTINUE UP TO THE TXWY. I THEN SAW A TXWY LIGHT DIRECTLY IN FRONT OF US APPROX 15-20 FT AHEAD. THIS IS WHEN I REALIZED I WAS NOT ON THE TURF RWY THAT CROSSED THE TXWY AND RWY 31. IMMEDIATELY AFTER THAT I SAW THE DITCH THAT WE ENDED UP GOING IN; CAUSING THE NOSEWHEEL TO COLLAPSE AND BENDING THE PROP BLADES. THERE WAS NO TIME TO REACT. I BELIEVE IF THERE WERE SOME KIND OF RWY END MARKERS THAT THIS INCIDENT COULD HAVE BEEN AVOIDED. I HAVE OPERATED IN AND OUT OF CROOKSTON ARPT MANY TIMES. THIS OCCURRENCE HAS DEFINITELY MADE ME AWARE OF THE IMPORTANCE OF PERIODICALLY REVIEWING THE ARPT LAYOUT -- WHICH I THINK MIGHT HAVE PREVENTED THIS. I AM CONCERNED OF THIS SAME THING HAPPENING AGAIN AND THINK IT WOULD BE BENEFICIAL TO ALL PLTS OPERATING AT CROOKSTON IF THE END OF THE RWY WAS CLRLY MARKED ESPECIALLY BECAUSE OF THE DITCH AT THE END. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATES SHE IS AWARE THAT THIS IS HER FAULT; SHE SHOULD HAVE BEEN MORE ORIENTED TO HER RWY. BUT -- THIS DITCH IS RIGHT AT THE END OF THE RWY AND IS ABOUT 2 FT DEEP. SHE HAS SINCE HEARD OF OTHERS WHO HAVE GONE OFF INTO THE DITCH. THIS INCIDENT CAUSED NOSEWHEEL TO COLLAPSE AND DAMAGED ON PROP BLADE. FAA FOLLOW UP OCCURRED AND SHE HAS TAKEN A 609 RIDE. RPTR FEELS STRONGLY THAT SOME SIGNS SHOULD BE PLACED WHERE THEY ARE QUITE VISIBLE TO INDICATE RWY END. APPARENTLY THERE ARE SOME MARKERS NOW; BUT THEY ARE WAY OFF TO THE SIDE IN THE GRASS AND NOT VISIBLE TO THE PLT.