Dynamic Pipe Stresses During Water Hammer: II — A Vibration Analysis

Dynamic pipe stresses result from pipe vibrations created by water hammer. Current vibration theory was adapted to derive equations that describe the pipe stresses in a pipe wall which occur at a distance from the pipe end. These original equations describe the stresses in a pipe expansion mode, which is sometimes referred to as a breathing mode. The breathing mode was assumed to result from a moving shock wave traveling along the length of a pipe, and the shock was modeled using a step pressure wave, using the classic single degree of freedom vibration equations. Pipe stresses affected by both undamped and damped vibration were modeled. To model the stresses at a point on the pipe wall, vibration equations were linearized, and existing step response equations were applied to create stress models. Essentially, the linear vibration equations were converted into closed form solutions for the maximum pipe stresses. The vibration model was shown to be comparable to an undamped FEA stress model. The vibration model was then used to evaluate the effects of damping on dynamic pipe stresses.