Free vibration of axially FG curved beam on orthotropic Pasternak foundation via mixed FEM

The main objective of this study is to present an accurate and consistent original formulation to describe the interaction between planar straight/curved beam structure and orthotropic foundation by highlighting the problematic orthotropic foundation models currently used in the literature. The proposed formulation introduces a consistent reduction of a 2D stress state of an elastic orthotropic foundation to a 1D form suitable for interaction with a beam. The formulation is also extended for curved beams resting on an arbitrarily orthotropic generalized Pasternak foundation by including a rocking effect. The beam is assumed to be made of a two-phase composite material of metal-matrix and ceramic-inclusion varying continuously through the parabolic beam axis. The effective material properties of the axially functionally graded material are predicted through a cubic local representative volume elements scheme. A two-noded curved beam element is employed based on a mixed finite element formulation based on Timoshenko beam theory by considering cross-sectional warping and the tests are performed over free vibration analyses. The new formulation of orthotropic Pasternak foundation for straight beam interaction is firstly verified by the results of a plate resting on orthotropic foundation. The flaws of existing formulations in the literature are explained, and the erroneous results produced by some of them are shown with comparison examples. The formulation is capable of modelling straight beams as well as general curved planar beams and orthotropic Pasternak foundation interaction problems. It is believed that the derived expression be safely implemented for further analyses, where beam like structures interacting with orthotropic foundation or substrate. Furthermore, this study presents free vibration results of several original benchmark examples using the mixed finite element formulation, e.g. straight and parabolic beams resting on an arbitrarily orthotropic generalized Pasternak foundation.