The B727 was dispatched with the APU inoperative. The captain elected to start all engines prior to pushback due to a short taxi. After pushback, and during a forward tow, a shear pin on the tow bar failed, and the airplane rolled into a tug. Examination of the tow bar revealed looseness in the head, and worn holes for the shear pins. Metallurgical examination of the shear pins found pre-existing cracking and the pins were softer than specified. The pushback operation was conducted by contract personnel. The failed tow bar had been inspected daily by contract personnel with no problems noted and it continued in use. A fleet wide campaign found 63 percent of the tow bars in need of maintenance. The investigation revealed the airline did not have a preventative maintenance program for their tow bars, or a program to train the contract personnel in inspections of the tow bars, and to monitor their inspections. Probable Cause: was the use of a defective tow bar, due to the lack of an adequate inspection by contract personnel, the airlines lack of an adequate preventative maintenance program for the tow bars, and the lack of oversight by the airline on the contract personnel who inspected the tow bar. A factor was the softer than specified shear pins used in the tow bar.