Dynamic Pipe Stresses During Water Hammer: IV — Elastoplasticity in Thick Walled Tubes

Closed form equations were used to approximate the elastoplastic stresses in a thick walled pipe subjected to water hammer. The elastoplastic problem was considered for both the stresses during the initial elastic expansion and the ensuing plastic stresses. The stresses in the pipe can be expressed as a function of the static elastic stresses, static plastic stresses, and time dependent vibration equations. Up to the yield stress, existing elastic stress equations govern the pipe response. At the yield stress, plastic deformation is expected during static loading conditions, but the extent of plastic deformation during dynamic loading may be affected by the dynamic yield stress of the material. The effect of the dynamic yield stress was shown to eliminate plastic deformation for time spans shorter than those typically encountered in water hammer events. If the load is sustained after the initial elastic yielding, the pipe will plastically deform. The pipe will also plastically deform if initially loaded to a stress beyond the dynamic yield stress, regardless of the water hammer duration. In either case, a plastic stress zone is formed at the bore of the pipe when a pipe is internally pressurized beyond the dynamic yield stress. As the pressure increases, the plastic zone expands toward the outer pipe wall. Static stress equations are available to fully describe the elastoplastic stresses as the plastic zone slowly expands. These static stress equations were substituted into the appropriate vibration equations to obtain the elastoplastic dynamic stress for sudden pipe wall expansion. The resulting equations were simplified to obtain the maximum plastic stress at a point on the pipe wall.