Vibration analysis of thin-walled pipes with circular axis using the Generalized Beam Theory

This paper presents the dynamic analysis extension of the Generalized Beam Theory (GBT) formulation developed in [1] for curved thin-walled pipes. The new proposed dynamic curved GBT element formulation goes beyond the current GBT?s analysis of straight pipes, not only concerning the coupling and behavior of pipe bends or toroidal shells, but also concerning the effects of the transverse shear modes. In the preceding paper, the stresses and the deformations of a pipe bend under static conditions have been determined through the potential energy formulation for different loading and support conditions. In this paper, the formulation of the kinetic energy is presented to determine the inertia of the GBT element. The consistent element mass matrix derived from the GBT dynamic formulation involves the coupling of certain types of GBT deformation modes resembling that of the element stiffness matrix. Here, to illustrate the application and capabilities of the developed GBT formulation, numerical examples concerning undamped free vibration analysis of truncated and closed toroidal shells with different support conditions are considered involving a combination and coupling of bending, warping, torsional, axisymmetric and local deformation modes. For the purpose of validation, these examples are compared with refined shell finite element models.