Quasi-3D beam models for the computation of eigenfrequencies of functionally graded beams with arbitrary boundary conditions

The present article deals with the free vibration analysis of three-dimensional metallic and functionally graded beams with arbitrary boundary conditions. The investigation is carried out by using refined variable-kinematics quasi-3D beam theories hierarchically generated by using the method of power series expansion of displacement components. Each displacement variable, in the displacement field, can be expanded at any desired order independently from the others and regarding to the results accuracy and the computational cost. The weak-form of the governing equations is derived via the Principle of the Virtual Displacements (PVD), while the Ritz method is used as solution technique. Algebraic Ritz functions, orthogonalised by using the Gram–Schmidt process, are employed in the analysis. Convergence and accuracy of the proposed formulation have been thoroughly examined. A comprehensive assessment of the developed beam models, for various boundary conditions, is also provided. The effect of significant parameters such as length-to-thickness ratio (slenderness ratio), volume fraction index and material properties, on the natural frequencies and mode shapes, is discussed

Some results on thermal stress of layered plates and shells by using Unified Formulation

This work presents some results on two-dimensional modelling of thermal stress problems in multilayered structures. The governing equations are written by referring to the Unified Formulation (UF) introduced by the first author. These equations are obtained in a compact form, that doesn't depend on the order of expansion of variables in the thickness direction or the variable description (layer-wise models and equivalent single layers models). Classical and refined theories based on the Principle of Virtual Displacements (PVD) and advanced mixed theories based on the Reissner Mixed Variational Theorem (RMVT) are both considered. As a result, a large variety of theories are derived and compared. The temperature profile along the thickness of the plate/shell is calculated by solving the Fourier's heat conduction equation. Alternatively, thermo-mechanical coupling problems can be considered, in which the thermal variation is influenced by mechanical loading. Exact closed-form solutions are provided for plates and shells, but also the applications of the Ritz method and the Finite Element Method (FEM) are presented

1D Hierarchical Ritz- and 2D GDQ-Based Approaches for the Mechanical Modeling of Thin-Walled Composite Structures

In the last decades, thin-walled composite structures with different geometries have been largely applied in many engineering applications because of their optimal strength-to-weight ratio. In a context where advanced numerical methods are required for an accurate estimation of their structural behaviour, especially when innovative constituents and materials are involved, we apply, in the present work, two different numerical approaches to treat the vibration response of some selected structural components with different shapes. In detail, we employ the 1D-Ritz formulation with 3D capabilities [1-3] and 2D Generalized Differential Quadrature (GDQ) method [4-6] both in the strong and weak form, whose results are compared to the ones based on a classical 3D finite element formulation. Thus, a systematic investigation is performed to study the sensitivity of the natural frequencies to different input parameters, namely, the boundary conditions, the length-to-thickness ratio, as well as different material and geometrical properties. The main advantages of the proposed solution techniques, are discussed comparatively, both in terms of convergence and accuracy, for each case study here analysed