In-plane vibration analysis of homogeneous curved pipes conveying internally pressurized fluid with localized valve and elastic support effects

This article investigates the in-plane vibrational behavior of a homogeneous curved pipe conveying internally pressurized fluid. The model is extended to incorporate a discrete point mass, representing a valve, and a localized spring support. The inertial effect of the valve is introduced as a concentrated mass term using a Dirac delta function at the corresponding axial location, while the elastic support is modeled through a distributed stiffness contribution in the governing equation of motion. The originality of the approach lies in the adoption of quintic Hermite shape functions within a finite element framework, enabling accurate representation of both displacement and slope continuity. Special attention is devoted to the influence of the pipe curvature angle on the fundamental natural frequency under varying flow velocities, as well as the effect of the valve-to-pipe mass ratio. Parametric studies reveal that increasing curvature and flow velocity significantly alter the dynamic response, while the mass ratio strongly governs frequency modes. In addition, a convergence analysis is performed to ensure numerical stability, and the results are validated against reference solutions, confirming the accuracy and physical plausibility of the proposed formulation. The results provide valuable insights for the design and reliability assessment of curved piping systems in engineering applications.