Abstract
This paper aims to present a finite element (FE) formulation for the study of the natural frequencies of functionally graded orthotropic laminated plates characterized by cross-ply layups. A nine-node Lagrange element is considered for this purpose. The main novelty of the research is the modelling of the reinforcing fibers of the orthotropic layers assuming a non-uniform distribution in the thickness direction. The Halpin–Tsai approach is employed to define the overall mechanical properties of the composite layers starting from the features of the two constituents (fiber and epoxy resin). Several functions are introduced to describe the dependency on the thickness coordinate of their volume fraction. The analyses are carried out in the theoretical framework provided by the first-order shear deformation theory (FSDT) for laminated thick plates. Nevertheless, the same approach is used to deal with the vibration analysis of thin plates, neglecting the shear stiffness of the structure. This objective is achieved by properly choosing the value of the shear correction factor, without any modification in the formulation. The results prove that the dynamic response of thin and thick plates, in terms of natural frequencies and mode shapes, is affected by the non-uniform placement of the fibers along the thickness direction.
Highlights
The finite element (FE) method currently represents the most-utilized computational approach to solve several engineering problems and in applications whose solutions cannot be obtained analytically [1]
This paper aims to present a finite element (FE) formulation for the study of the natural frequencies of functionally graded orthotropic laminated plates characterized by cross-ply layups
The analyses are carried out in the theoretical framework provided by the first-order shear deformation theory (FSDT) for laminated thick plates
Summary
The finite element (FE) method currently represents the most-utilized computational approach to solve several engineering problems and in applications whose solutions cannot be obtained analytically [1]. At this point, the nodal degrees of freedom can be collected in a sole vector u(e) to simplify the nomenclature: u(e) = ux(e) u(ye) uz(e) φx(e) φ(ye) T = ux(e,1) · · · ux(e,9) u(ye,1) · · · u(ye,9) uz(e,1) · · · uz(e,9) φx(e,1) · · · φx(e,9) φ(ye,1) · · · φ(ye,9) T ,. Where the vectors Bξ, Bη collect the derivatives of the shape functions (9) with respect to ξ, η At this point, the compatibility equations can be presented to define the strain components in each element. Such terms are needed to compute the stress resultants in each element by means of the constitutive relation shown below in matrix form: Nx(e) N(ye)
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