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In this article, the influences of uniform velocity profile, mass ratio, length and Winkler elastic foundation on the static instability of pipe conveying incompressible fluid are investigated. The Euler-Bernoulli beam theory is employed to derive partial differential equation of pipes carrying fluid. The results were carried out using ANSYS Workbench program, where the analysis is based on the numerical solution; using Finite element method to formulate both the pipe structure and fluid flow equations. The numerical approach is based on some research and analytical models. The natural frequencies of the system are attained with respect to different boundary conditions, such as pinned-pinned ends, clamped-pinned ends and clamped-clamped ends. The numerical results show satisfactory agreement with the theory of many aspects of the pipe dynamical carrying incompressible fluid were observed numerically such as, the increase in flow velocity, mass ratio and length reduced from the rigidity of the system and consequently the proper modes. Winkler elastic foundation has a stabilizing effect on the system.

In this study, the natural frequencies of the pipe transporting an fluid resting on an elastic Winkler-type and the critical velocities of instabilities are obtained by the standard finite element method. A dynamic characteristic of a pipe carrying internal fluid undergoes mechanical load due to inertia effect of fluid, Coriolis force, fluid kinetic force due to fluid flow velocity, dynamic load due to inertia effect on the pipe structure. A numerical modal analysis is realized in the fluid-structure interaction configuration. One dimensional beam finite element is used for investigating the dynamic behavior of the thin pipe. According to the approved method, the different elementary matrices were extracted, which were including to a code called Matlab. We developed a program under Matlab with R2017b version, where computations are in the complex planes. The initial approach is based on some research and analytical models. The numerical results show satisfactory agreement with the analytical results. The increase in flow velocity, mass ratio and length reduced from the rigidity of the system. Regions and range of instabilities are presented by numerical aspects. We determined the influence of the different parameters on the static and dynamic instabilities of the system.