Taking into account the effect of internal fluid pressure, the chaotic vibration and suppression of a fluidconveying curved pipe subjected to motionlimiting constraint were investigated. With a fluidpressure term added, the equation of motion for the fluidconveying curved pipe was discretized by differential quadrature method. In the parameter region of fluid pressure, several motions (include chaotic vibration) were detected via numerical iterative calculations. It is shown that the fluid pressure factor greatly influences the dynamic responses of the pipe, which should be considered adequately in practical applications. Based on this, to suppress the chaotic vibration, a modified curved pipe model was undertaken, which can eliminate the chaotic vibration effectively.