Finite element for the dynamic analysis of pipes subjected to water hammer

An axisymmetric finite element is developed for the dynamic analysis of pipes subjected to water hammer. The analysis seamlessly captures the pipe response due to water hammer by solving first the water hammer mass and momentum conservation equations, to recover the spatial–temporal distribution of the internal pressure, and then uses the predicted pressure history to form a time-dependent energy equivalent load vector within a finite element model. The study then determines the pipe response by solving the finite element discretized equations of motion. The formulation of the pipe response is based on Hamilton’s principle in conjunction with a thin shell theory formulation, and captures inertial and damping effects. The results predicted by the model are shown to be in agreement with those based on an axisymmetric shell model in Abaqus for static, free vibration and transient responses for benchmark problems. The water hammer and structural models are seamlessly integrated to enable advancing the transient solution well into the time domain to capture the effect of reflected pressure wave due to water hammer. The results indicate that the radial stress/displacement oscillations approach those of the quasi-steady response when the valve closure time exceeds eight times the period of radial vibrations obtained for the case of instantaneous valve closure.