Assessment of Theories for Free Vibration Analysis of Homogeneous and Multilayered Plates

This paper assesses classical and advanced theories for free vibrational response of homogeneous and multilayered simply supported plates. Closed form solutions are given for thick and thin geometries. Single layer and multilayered plates made of metallic, composite and piezo-electric materials, are considered. Classical theories based on Kirchhoff and Reissner-Mindlin assumptions are compared with refined theories obtained by enhancing the order of the expansion of the displacement fields in the thickness directionz. The effect of the Zig-Zag form of the displacement distribution inzas well as of the Interlaminar Continuity of transverse shear and normal stresses at the layer interface were evaluated. A number of conclusions have been drawn. These conclusions could be used as desk-bed in order to choose the most valuable theories for a given problem

Laminated Beam Analysis by Polynomial, rigonometric, Exponential and Zig-Zag Theories

A number of refined beam theories are discussed in this paper. These theories were obtained by expanding the unknown displacement variables over the beam section axes by adopting Taylor's polynomials, trigonometric series, exponential, hyperbolic and zig-zag functions. The Finite Element method is used to derive governing equations in weak form. By using the Unified Formulation introduced by the first author, these equations are written in terms of a small number of fundamental nuclei, whose forms do not depend on the expansions used. The results from the different models considered are compared in terms of displacements, stress and degrees of freedom (DOFs). Mechanical tests for thick laminated beams are presented in order to evaluate the capability of the finite elements. They show that the use of various different functions can improve the performance of the higher-order theories by yielding satisfactory results with a low computational cost

Multi-layered plate finite element models with node-dependent kinematics for smart structures with piezoelectric components

Abstract This article presents a type of plate Finite Element (FE) models with adaptive mathematical refinement capabilities for modeling laminated smart structures with piezoelectric layers or distributed patches. The p-version shape functions are used in combination with the higher-order Layer-Wise (LW) kinematics adopting hierarchical Legendre polynomials. Node-Dependent Kinematics (NDK) is employed to implement local LW models in the regions with piezoelectric components and simulate the global substrate structure with the Equivalent Single-Layer (ESL) approach. Through the proposed NDK FE models, the electro-mechanical behavior of smart structures can be predicted with high fidelity and numerical efficiency, and various patch configurations can be conveniently modeled through one set of mesh grids. Moreover, the effectiveness and efficiency of the NDK FE approach are assessed through numerical examples and its application is demonstrated

Refined Beam Elements with only Displacement Variables andPlate/Shell Capabilities

This paper proposes a refined beam formulation with displacement variables only. Lagrange-type polynomials, in fact, are used to interpolate the displacement field over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) polynomials are considered which lead to linear, quasi-linear (bilinear), and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Unified Formulation (UF). With UF application the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section in terms of the implemented L3, L4, and L9 elements. A number of numerical problems have been solved and compared with results given by classical beam theories (Euler-Bernoulli and Timoshenko), refined beam theories based on the use of Taylor-type expansions in the neighborhood of the beam axis, and solid element models from commercial codes. Poisson locking correction is analyzed. Applications to compact, thin-walled open/closed sections are discussed. The investigation conducted shows that: (1) the proposed formulation is very suitable to increase accuracy when localized effects have to be detected; (2) it leads to shell-like results in case of thin-walled closed cross-section analysis as well as in open cross-section analysis; (3) it allows us to modify the boundary conditions over the cross-section easily by introducing localized constraints; (4) it allows us to introduce geometrical boundary conditions along the beam axis which lead to plate/shell-like cases

Free vibration analysis of variable stiffness composite laminated beams and plates by novel hierarchical differential quadrature finite elements

The present work deals with the free vibration behavior of the variable stiffness composite laminates (VSCLs)featured by spatially varyingfibre orientation angles via novel quasi‐three‐dimensional solutions. The CarreraUnified Formulation (CUF) is employed to construct such novel models, where cross‐section kinematics aredescribed with the improved hierarchical Legendre expansion (IHLE) of primary mechanical variables. Theproposed expansions not only maintain the hierarchical properties of the HLE model but also become less sen-sitive to the numbering sequence of expansion terms. As a result of these enhanced kinematics, EquivalentSingle Layer (ESL) and Layer‐Wise (LW) models can be formulated more robustly. The weak form differentialquadraturefinite element method (DQFEM) is employed to solve the governing equations derived by the prin-ciple of virtual displacements. Based on CUF‐based DQFEM, even a single beam element is sufficient to tacklemany complex issues with high accuracy. Compact VSCL beams and plates with variousfibre paths, boundaryconditions, lamination schemes, and thickness‐to‐width ratios have been studied in several numerical exam-ples. The proposed method’s accuracy and effectiveness are validated by comparing results to published dat

Analysis of reinforced and thin-walled structures by multi-line refined 1D/beam models

This paper focuses the attention on the use of appropriate combinations of refined one-dimensional (1D) beam theories to analyze thin-walled, reinforced structures. The cross-section of a slender body is seen as the sum of different sub-domains. Each sub-domain is subsequently used as the cross-section of a beam discretization. Displacement variables are then expanded around the beam axis of each subdomain by using refined 1D models which are based on the Carrera Unified Formulation. The order of the beam elements can vary in different sub-domains. This subdivision has been called "multi-line" as opposed to the "one-line" approach of classical beam theories. 1D compatibility conditions of the displacements at selected points of the sub-domain interface boundaries are imposed by using Lagrange multipliers. Various problems have been analyzed to highlight the advantages and disadvantages of the present multi-line approach. It is concluded that the multi-line approach appears very effective in the case of thin-walled sections made by locally connected walls as well as in the case of reinforced structures

Buckling and post-buckling of anisotropic flat panels subjected to axial and shear in-plane loadings accounting for classical and refined structural and nonlinear theories

Abstract This article investigates the large deflection and post-buckling of composite plates by employing the Carrera Unified Formulation (CUF). As a consequence, the geometrically nonlinear governing equations and the relevant incremental equations are derived in terms of fundamental nuclei, which are invariant of the theory approximation order. By using the Lagrange expansion functions across the laminate thickness and the classical finite element (FE) approximation, layerwise (LW) refined plate models are implemented. The Newton–Raphson linearization scheme with the path-following method based on the arc-length constraint is employed to solve geometrically nonlinear composite plate problems. In this study, different composite plates subjected to large deflections/rotations and post-buckling are analysed, and the corresponding equilibrium curves are compared with the results in the available literature or the traditional FEM-based solutions. The effects of various parameters, such as stacking sequence, number of layers, loading conditions, and edge conditions are demonstrated. The accuracy and reliability of the proposed method for solving the composite plates' geometrically nonlinear problems are verified

Carrera Unified Formulation for Free-Vibration Analysis of Aircraft Structures

Advanced structural models, based on variable one-, two-, and three-dimensional kinematics, are proposed in this paper and applied to the analysis of the free vibration of reinforced aircraft shell structures. The used models go beyond classical structural theories, that is, Euler–Bernoulli (for one-dimensional beams) and Kirchhoff (for two-dimensional plates) type assumptions. The order of the expansion of the displacement fields over the cross section (one-dimensional case) and along the plate thickness (two-dimensional case) is, in fact, a free parameter of the problem. In this paper, Lagrange polynomials are used to build such expansions, and as a consequence, only displacements are used as the problem unknowns (no rotations or derivatives of displacements, which are typical of one-dimensional/two-dimensional classical theories, are introduced). The finite-element method is used to provide numerical solutions. The related arrays and the governing dynamical equations are written in terms of a few fundamental nuclei according to the Carrera unified formulation. Classical three-dimensional finite-element solid models are also considered. One-, two-, and three-dimensional finite elements are easily connected to each other to make the most appropriate computational model of the reinforced shell structures. The capability to use the same fundamental nucleus to derive finite-element matrices of one-, two-, and three-dimensional elements of the present model is unique because it is usually not available in other finite-element formulations, that is, no ad hoc techniques are required in the present case to couple finite elements with different kinematics. Three main benchmarks have been analyzed: a plate stiffened by means of bidirectional I-stiffeners, a simplified model of a complete aircraft, and a fuselage–wing connection. Comparison with commercial finite-element software (MSC Nastran) is provided for most of the quoted numerical investigations. The modal assurance criterion has been used to compare the free-vibration modes of the different models. The present mathematical models appear closer to reality and cheaper, from the computational point of view, than those of other existing formulations. Carrera unified-formulation-based finite elements do not require the definition of virtual lines (beam axes) or virtual surfaces (plate reference surfaces), and only physical lines/surfaces are therefore used

Refined shell elements for the analysis of functionally graded structures

The present paper considers the static analysis of plates and shells made of Functionally Graded Material (FGM), subjected to mechanical loads. Refined models based on the Carrera's Unified Formulation (CUF) are employed to account for grading material variation in the thickness direction. The governing equations are derived from the Principle of Virtual Displacement (PVD) in order to apply the Finite Element Method (FEM). A nine-nodes shell element with exact cylindrical geometry is considered. The shell can degenerate in the plate element by imposing an infinite radius of curvature. The Mixed Interpolation of Tensorial Components (MITC) technique is extended to the CUF in order to contrast the membrane and shear locking phenomenon. Different thickness ratios and orders of expansion for the displacement field are analyzed. The FEM results are compared with both benchmark solutions from literature and the results obtained using the Navier method that provides the analytical solution for simply-supported structures subjected to sinusoidal pressure loads. The shell element based on refined theories of the CUF turns out to be very efficient and its use is mandatory with respect to the classical models in the study of FGM structures