Dynamic behavior of piping as a beam system has been analyzed with the use of the dynamic stiffness method. According to this method, the equations describing the relation between unknown parameters are written by the method of initial parameters, therefore, the solution procedure is similar to that for a static problem. It is shown that for curvilinear beams it is simpler and more efficient to apply a model that consists of straight segments and inertia-free rotation elements. To determine natural frequencies of 3D beam systems, it is proposed to use a method of disconnection of displacements, which makes it possible to discern the frequencies corresponding to different vibration modes (transverse, longitudinal, etc.). The approach allows a correct simulation of the system behavior under forced vibrations induced by a harmonic exciting force.
The authors proposed an analytical method for the analysis of the end effect in a pipe bend loaded by a bending moment with consideration for the action of internal pressure. The method is based on the use of simplifying hypotheses and is reduced to the solution of a system of fourth-order differential equations along the axial coordinate with respect to unknown coefficients in the expansion for tangential displacements. An approximate analytical solution, which has a trapezoidal structure and is written in terms of Krylov's functions, has been obtained. Boundary conditions are formulated in terms of the tangential and longitudinal displacements and axial and shearing stress resultant. For the flexibility factor, analytical solutions are presented in the case where a bend is approximated by a rigid restraint on both ends. To verify the analytical solution and its applicability limits, two numerical procedures were developed, which are based on the finite difference method and the reduction to the Kochi problem by the expansion of the unknowns in the Fourier series over the circumferential coordinate. The authors compare the results obtained with data from the literature, discuss the advantages and disadvantages of the methods, and present recommendations for their practical application.
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