On July 20, 1998, at 1642 central daylight time, a Cessna R182, N9142C, operated by an airline transport pilot, received substantial damage following an in-flight fire while maneuvering in the traffic pattern at Bismarck Municipal Airport, near Bismarck, North Dakota. Visual meteorological conditions prevailed at the time of the accident. The 14 CFR Part 91 instructional flight was not operating on a flight plan. The pilot and student reported no injuries. The local flight originated Bismarck Municipal Airport at 1415. In a written statement, the instructor reported that they had returned to the to the traffic pattern to perform takeoff and landings after conducting various air work maneuvers for approximately 1-1/2 hours. During their fourth takeoff in the traffic pattern, the instructor noticed abnormal engine performance brought on by no fuel pressure at an altitude of approximately 500' above ground level. The auxiliary boost pump was actuated but it did not rectify and was subsequently turned off. The instructor requested an immediate landing from the tower, lowered the landing gear and throttled back the engine. The instructor stated that when the engine was throttled back, the engine backfired and caused an explosion in the engine compartment. The instructor proceeded to shut down aircraft systems and reported that fire was visible on the right side of the airplane. The airplane was then "dead sticked" to a landing on a taxiway where fire personnel extinguished the fire. A fractured electric fuel pump to carburetor T-connector, Part Number AN917-20, fitting was found during inspection of the engine compartment by a Federal Aviation Administration Inspector. Metallurgical analysis of the separated fitting by the National Transportation Safety Board's Material Laboratory revealed clamshell shaped crack arrest marks and step-like features to be present. Closer examination of the step-like features revealed a large number of markings similar to river patterns, which converged at numerous locations on the outside diameter (at the root of the thread) of the fitting. No mechanical damage was noted at any of the crack origin locations. The last airplane inspection was a 100-hour inspection performed in March of 1998.