International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

The dynamic stability of a cracked pipeline resting on a Winkler elastic foundation and immersed in fluid that is moving with a particular velocity is investigated. The Galerkin method is employed to approach numerically the problem. Conclusions are drawn on the influence of the rigidity of the Winkler elastic foundation on the critical flow velocity of the pipe.

The dynamic stability of a cracked pipeline resting on a Winkler elastic foundation and immersed in fluid that is moving with a particular velocity is investigated. The Galerkin method is employed to approach numerically the problem. Conclusions are drawn on the influence of the rigidity of the Winkler elastic foundation on the critical flow velocity of the pipe.

The Energy method is widely applied to determine the critical loads in elastic systems. A widely used variant of the method applies the Rayleigh-Ritz approach where the approximation of the buckling mode of the column is a function that satisfies the boundary conditions. A numerical example of a two-storey column is considered. An important aspect in the problem is the solution the complex integrals that emerge during the solution process. That problem could be overcome by the use of math software. The investigated column is hinged at its both ends and has an additional lateral support in the middle. It is loaded with a compressive distributed load alongside its length.

The Picard’s successive iteration method is applied to determine the Euler critical force for a slender column. The initial approximation of the buckling mode of the column is a polynomial function that satisfies the boundary conditions. A numerical example is solved and the obtained result is compared with that obtained by the Euler formula. An assessment of the accuracy of the solution is made at each iteration step. In the paper is shown that the Picard’s iteration method could also be used for obtaining the closed form exact solution of the buckling load. The investigated column is hinged at its upper end and supported by a Q- apparatus at the other. It is loaded with a compressive force at the upper end.