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A hybrid-mixed, four-node, quadrilateral element for the three-dimensional (3D) stress analysis of functionally graded (FG) plates using the method of sampling surfaces (SaS) is developed. The SaS formulation is based on choosing an inside the plate body N, not equally spaced SaS parallel to the middle surface, in order to introduce the displacements of these surfaces as basic plate variables. Such a choice of unknowns, with the consequent use of Lagrange polynomials of the degree N - 1 in the assumed distributions of displacements, strains, and mechanical properties through the thickness leads to a robust FG plate formulation. All SaS are located at Chebyshev polynomial nodes that permit one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and spurious zero-energy modes, the assumed natural strainmethod is employed. The proposed four-node quadrilateral element passes 3D patch tests for FG plates and exhibits a superior performance in the case of coarse distorted meshes. It can be useful for the 3D stress analysis of thin and thick metal/ceramic plates because the SaS formulation gives an opportunity to obtain the solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity. [ABSTRACT FROM AUTHOR]

Abstract In this work, a number of higher-order plate elements and an improved theory based on a zig-zag power function have been applied to metallic and composite layered structures with viscoelastic layers. The kinematic field is written by using an arbitrary number of continuous piecewise polynomial functions. The polynomial expansion order of a generic subdomain is a combination of zig-zag power functions depending on the plate thickness coordinate. Damped free-vibrations and frequency response analysis are performed modelling the viscoelastic properties with the complex modulus approach, and taking into account different damping laws for the viscoelastic layers included a five-parameter fractional-derivative model. The governing equations are derived from the Principle of Virtual Displacements and solved using the Finite Element (FE) method, and both full and reduced-order formulation are taken into account. Furthermore, the Mixed Interpolated Tensorial Components (MITC) method is employed to contrast the shear locking phenomenon. The Carrera Unified Formulation (CUF) has been employed to derive the governing FE equations for the various theories considered in this paper in an unified form. Several numerical investigations are carried out to validate and demonstrate the accuracy and efficiency of the present plate element. The results are reported in terms of frequencies, modal loss factors and frequency responses, and they are compared with solutions published in the literature and with solid finite element models. [ABSTRACT FROM AUTHOR]

Solid and higher-order plate finite elements are coupled for the analysis of composite structures by means of the Arlequin method. Plate and solid elements are derived by a unified formulation and they do not depend on the number of nodes. Higher-order and zig-zag plate elements are easily formulated regardless the approximation order along the through-the-thickness direction. Composite and sandwich plates are investigated. Multi-model solutions are assessed towards equivalent mono-model ones and commercial finite element software results. The numerical investigation shows that higher-order plate finite elements and solid ones are successfully coupled. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, the Carrera Unified Formulation and the generalized differential quadrature technique are combined for predicting the static deformations and the free vibration behavior of thin and thick isotropic as well as cross-ply laminated plates. Through numerical experiments, the capability and efficiency of this technique, based on the strong formulation of the problem equations, are demonstrated. The numerical accuracy and convergence are also examined. It is worth noting that all the presented numerical examples are compared with both literature and numerical solutions obtained with a finite element code. The proposed methodology appears to be able to deal not only with uniform boundary conditions, such as fully clamped or completely simply-supported, but also with mixed external conditions, that can be clamped, supported or free. [Copyright &y& Elsevier]

This article addresses the static deformations and free vibration analysis of laminated composite and sandwich plates by collocation with radial basis functions, according to a layerwise formulation. The present layerwise approach accounts for through-the-thickness deformations. The equations of motion and the boundary conditions are obtained by the Carrera’s Unified Formulation, and further interpolated by collocation with radial basis functions. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we combine the Carrera''s Unified Formulation CUF (E. Carrera. Theories and Finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking. Arch. Comput. Meth. Eng., 10:215–297, 2003) and a radial basis function (RBF) collocation technique for predicting the static deformations and free vibrations behavior of thin and thick cross-ply laminated plates. For the first time, the Reissner-Mixed Variational Theorem is used together with the RBF collocation to achieve a highly accurate technique. The accuracy and efficiency of this collocation technique for static and vibration problems are demonstrated through numerical examples. [Copyright &y& Elsevier]

This article considers the static analysis of multilayered piezoelectric plates subjected to mechanical pressures and to field loads such as electric potential and temperature. The Principle of Virtual Displacements has been extended to thermo-electro-mechanical cases by including the virtual internal electric and thermal works and by writing the opportune constitutive relations. The governing equations are given in terms of three main primary variables (displacements, electric potential and temperature) and they are solved in a closed-form solution. The primary variables are expanded in the thickness direction in both Equivalent Single Layer (ESL) and Layer-Wise (LW) form with higher orders of expansion. The models proposed enable to obtain a quasi-3D description of the static thermo-electro-mechanical analysis of multilayered plates and to analyze the effects of each physical field involved. The limitations of classical theories for such analyses have also been discussed. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the most popular failure criteria which are employed in the design and analysis of anisotropic materials/layered plates. In-plane stress (or strain) states are considered and both classical and advanced criteria are illustrated and compared. The advantages and disadvantages of different approaches are discussed through FEM numerical results, by referring to composite material data currently employed. Layer-wise plate models are adopted to provide quasi three-dimensional stress field and to limit the error of modeling. Failure envelopes emphasizing the in-plane shear influence in the case of biaxial traction/compression are proposed. The maximum strain, maximum stress, Tsai–Wu, Tsai–Hill, Hashin and LaRC03 failure criteria are described and compared for various benchmark problems. [Copyright &y& Elsevier]

Abstract: This paper evaluates a number of classical and refined two-dimensional theories for the analysis of metallic and composite layered plates. Thin-plate, shear deformation and higher order plate theories are compared for various plate problems related to different mechanical and geometrical boundary conditions (BCs), as well as geometries and staking sequence lay-out. The theories are implemented by referring to a Unified Formulation (UF) proposed by the first author. The UF allows displacement fields with any order N in the thickness plate direction to be introduced and any variables in the N-order displacement field to be discarded. The finite element method is applied to include anisotropy and complex BCs. The accuracy of given theories for each fixed problem is established in terms of displacement and stress fields. The best plate theories, that is the most accurate plate theories with few computational efforts, is then determined by exploring various possibilities and by selecting appropriate unknown variables upon application of genetic algorithms. A best plate curve is established which shows the best plate theories (number of terms and their meanings) in terms of accuracy. It is concluded that a best plate theory changes with changing geometry, lay-out and BCs. The genetic algorithm used allows the least expensive computational model of each given problem to be detected. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

This paper compares and evaluates various plate finite elements to analyse the static response of thick and thin plates subjected to different loading and boundary conditions. Plate elements are based on different assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner-Mindlin), refined (Reddy and Kant), and other higher-order displacement fields are implemented up to fourth-order expansion. The Unified Formulation UF by the first author is used to derive finite element matrices in terms of fundamental nuclei which consist of 3×3 arrays. The MITC4 shear-locking free type formulation is used for the FE approximation. Accuracy of a given plate element is established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates are implemented and compared using displacement and stress variables for various plate problems. Reduced models that are able to detect the 3D solution are built and a Best Plate Diagram (BPD) is introduced to give guidelines for the construction of plate theories based on a given accuracy and number of terms. It is concluded that the UF is a valuable tool to establish, for a given plate problem, the most accurate FE able to furnish results within a certain accuracy range. This allows us to obtain guidelines and recommendations in building refined elements in the bending analysis of plates for various geometries, loadings, and boundary conditions. [ABSTRACT FROM PUBLISHER]

Abstract: This work solves the Fourier heat conduction equation in the case of layered structures embedding orthotropic layers in order to obtain a calculated temperature profile in the thickness direction. The calculated temperature profile is obtained for both plate and shell geometries. Several comparisons between the calculated temperature profile and the usual assumption of linear form of in the thickness direction are given. A wrong temperature profile gives an erroneous thermal load for the static analysis of layered structures. Proposed results show the effect of a wrong use of on the static response of multilayered plates and shells. It is concluded that in the analysis of multilayered structures the correct solution of the heat conduction equation could be mandatory with respect to any further evaluation. [Copyright &y& Elsevier]

The free vibration analysis of composite thin and relatively thick plates accounting for uncertainty is addressed in this work. Classical and refined two-dimensional models derived via Carrera's Unified Formulation (CUF) are considered. Material properties and geometrical parameters are supposed to be random. The fundamental frequency related to the first bending eigenmode is stochastically described in terms of the mean value, the standard deviation, the related confidence intervals and the cumulative distribution function. The Monte Carlo Method is employed to account for uncertainty. Cross-ply, simply supported, orthotropic plates are accounted for. Symmetric and anti-symmetric lay-ups are investigated. Displacements based and mixed two-dimensional theories are adopted. Equivalent single layer and layer wise approaches are considered. A Navier type solution is assumed. The conducted analyses have shown that for the considered cases, the fundamental natural frequency is not very sensitive to the uncertainty in the material parameters, while uncertainty in the geometrical parameters should be accounted for. In the case of thin plates, all the considered models yield statistically matching results. For relatively thick plates, the difference in the mean value of the natural frequency is due to the different number of degrees of freedom in the model. [ABSTRACT FROM AUTHOR]

Abstract: This paper compares and evaluates various plate finite elements to analyze the static response of laminated plate when varying the thickness ratio, orthotropic ratio and the stacking sequence of the lay-out. Plate elements were created from different theoretical assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner–Mindlin), known refined (Reddy, Pandya, and Kant), and other higher-order displacement fields were then implemented up-to fourth-order expansion. Following this the Carrera Unified Formulation was used to derive finite element matrices in terms of fundamental nuclei which consist of 3×3 arrays. The accuracy of a given plate element was established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates were implemented and compared using displacement and stress variables for various plate problems. In this paper concluding results are presented as number and type of required displacement variables vs. accuracy on a given stress/displacement parameter. These diagrams are labelled as Best Plate Theories BPT curves, which differ by changing geometry, material data and lay-out as well as loading and boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: The present work evaluates the effect of thickness stretching in plate/shell structures made by materials which are functionally graded (FGM) in the thickness directions. That is done by removing or retaining the transverse normal strain in the kinematics assumptions of various refined plate/shell theories. Variable plate/shell models are implemented according to Carrera’s Unified Formulation. Plate/shell theories with constant transverse displacement are compared with the corresponding linear to fourth order of expansion in the thickness direction ones. Single-layered and multilayered FGM structures have been analyzed. A large numerical investigation, encompassing various plate/shell geometries as well as various grading rates for FGMs, has been conducted. It is mainly concluded that a refinements of classical theories that include additional in-plane variables could results meaningless unless transverse normal strain effects are taken into account. [ABSTRACT FROM AUTHOR]

Abstract: Linearized buckling analysis of laminated plates subjected to combined bi-axial/shear loading is described in this work. Isotropic, orthotropic and anisotropic plates are referred to. Two-dimensional plate modelling is considered. The Principle of Virtual Displacement (PVD) is applied. The Finite Element Method (FEM) is adopted in order to approximate the solution. The same cases-study are handled using different kinds of plate Finite Elements (FEs), with the purpose of providing some guidelines/benchmarks useful for identifying the most appropriate modelling for each class of buckling problem. The considered set of FEs is hierarchical in the sense that a variable kinematics approach is employed: both Equivalent Single Layer (ESL) and Layer-Wise (LW) variable descriptions are implemented, keeping the expansion order for thickness variables generic. FE matrices are obtained in compact form by referring to Carrera’s Unified Formulation (CUF). When available, exact solutions are referred to in order to assess the FEM results. [Copyright &y& Elsevier]

Abstract: In this paper functionally graded material (FGM) plates subjected to a transverse mechanical load are investigated. The unified formulation (UF) and the Reissner’s mixed variational theorem (RMVT) are extended to FGMs. RMVT permits to consider both displacements and transverse shear/normal stresses as primary variables. Significant improvements are obtained with respect to the classical models (based on the principle of virtual displacements (PVD)) where only the displacements are assumed. The proposed models consider various order of expansion for the primary variables through the thickness, and the description of the unknown variables can be equivalent single layer or layer wise. The material properties, that in a FGM layer change continuously in the thickness direction, are described via thickness functions that are a combination of Legendre’s polynomials. Results are related to bending problems and restricted to closed form cases. The proposed models are very general for the material properties, because they do not depend on the use of transition function in the thickness direction. The results are compared with 3D solutions and with PVD models. It is shown both the effectiveness of mixed theories to trace the 3D response as well as their superiority with respect to classical PVD applications. [ABSTRACT FROM AUTHOR]

This work deals with a refined Finite Element (FE) analysis of multilayered plates. Various two-dimensional axiomatic assumptions in the thickness direction are illustrated and discussed by considering: 1-Taylor type expansion; 2-combinations of Legendre polynomials; 3-Lagrange polynomials. Both cases of an equivalent single layer description (the whole plate is seen as an equivalent single layer) and a layer-wise description (each layer is seen as an independent plate) have been implemented. The order N of the thickness expansions is a free parameter of the present formulation. A large variety of plate theories are therefore obtained. Related standard serendipity-type quadrilateral FEs are considered in this paper. FE matrices are written in a concise form by referring to the Carrera Unified Formulation and in terms of a few fundamental nuclei, whose form does not depend on the through-the-thickness polynomial assumption, order N, variable description or element number of nodes. The advantages and disadvantages of the various FEs are discussed by considering static and dynamic problems related to significant multilayered plate problems. [ABSTRACT FROM AUTHOR]

This work deals with the linearized buckling analysis of laminated plates and shells. A variable-kinematics approach with hierarchical capabilities is considered to establish the accuracy of a large variety of classical and advanced plate/shell theories in order to evaluate buckling loads. So-called equivalent-single-layer as well as layerwise-variable descriptions are implemented. Interlaminar continuity of transverse shear and normal stresses are a priori fulfilled by referring to Reissner's mixed variational theorem. Stability equations are derived in compact form by referring to Carrera's unified formulation for the most general case of doubly curved shells. The eigenvalue problem is solved in the case of closed-form solutions related to simply supported boundary conditions and axially loaded multilayered plate/shells made of orthotropic layers. The cases of axial constant strains and constant stresses are considered and compared to available three-dimensional and two-dimensional results. Results related to Love and Donnell approximations are implemented for comparison purposes. The accuracy of various approximations is established fur significant multilayered plate and shell problems. [ABSTRACT FROM AUTHOR]

Abstract: This work deals with advanced finite element (FE) formulations for the analysis of multilayered structures in the case of multifield problems. The following four fields are considered: mechanical, thermal, electrical and magnetic. Constitutive equations, in terms of coupled mechanical–thermal–electrical–magnetic field variables, are obtained on the basis of a thermodynamic approach. The four-field principle of virtual displacements is employed to derive FE matrices. Three-fields, two-fields as well as pure mechanical problems have been discussed as relevant particular cases. A condensed notation, known as Carrera unified formulation, has been employed to establish a comprehensive two-dimensional modeling with variable kinematic features. Layer-wise/equivalent single layers plate elements have been developed according to linear up to fourth-order expansion in the layer/plate thickness directions. FE matrices have been obtained in terms of a few fundamental nuclei whose dimension is 6×6 for the full four fields case. Numerical results show the effectiveness of the proposed implementation by encompassing various static and dynamic multifield plate problems. [Copyright &y& Elsevier]

Abstract: This paper considers the Mixed Interpolation of Tensorial Components (MITC) technique, which was originally proposed for Reissner–Mindlin type plates to develop shear locking free refined multilayered plate elements. Refined elements are obtained by referring to variable kinematic modelling in the framework of the Carrera Unified Formulation (CUF): linear, parabolic, cubic and fourth-order displacement fields in the thickness direction of the plate are used; both equivalent single layer (the multilayered plate is considered as an equivalent one-layer plate) and layer-wise (each layer is considered as an independent plate) variable descriptions are accounted for. Four-node elements are considered and a number of applications are developed for isotropic and multilayered anisotropic plates. Results related to the mixed interpolation of tensorial components are compared to the reduced and selective integration technique in the static and dynamic linear analysis. The numerical results show that the MITC technique maintains its effectiveness in the case of variable kinematic plate elements, hence the obtained elements are free from shear locking mechanisms. The capability of MITC to reduce/remove spurious modes is confirmed for refined multilayered elements. [Copyright &y& Elsevier]

Abstract: The paper considers a fully coupled thermo-mechanical analysis of one-layered and multilayered isotropic and composite plates. In the proposed analysis, the temperature is considered a primary variable as the displacement; it is therefore directly obtained from the model and this feature permits the temperature field to be evaluated through the thickness direction in three different cases: – static analysis with imposed temperature on the external surfaces; – static analysis of structures subjected to a mechanical load, with the possibility of considering the temperature field effects; – a free vibration problem, with the evaluation of the temperature field effects. In the first case, imposing a temperature at the top and bottom of the plate, the static response is given in term of displacements, stresses and temperature field; the proposed method is very promising if compared to a partially coupled thermo-mechanical analysis, where the temperature is only considered as an external load, and the temperature profile must be a priori defined: considering it linear through the thickness direction or calculating it by solving the Fourier heat conduction equation. In the second case, a mechanical load is applied. The fully coupled thermo-mechanical analysis gives smaller displacement values than those obtained with the pure mechanical analysis; the temperature effect is not considered in this latter approach. The third case is the free vibration problem. The fully coupled thermo-mechanical analysis permits the effect of the temperature field to be evaluated: larger frequencies are obtained with respect to the pure mechanical analysis. Carrera’s Unified Formulation is applied to obtain several refined models with orders of expansion in the thickness direction, from linear to fourth-order, for displacements and temperature. Both equivalent single layer and layer wise approaches are considered for the multilayered plates. At present, no benchmarks are available within the framework of a fully coupled theory. This work aims to fill this gap. [Copyright &y& Elsevier]

Mixed plate elements for accurate evaluation of transverse mechanical stresses and electrical displacement are developed and compared in this work. Various forms of the Reissner mixed variational theorem are considered to a priori fulfill the interlaminar continuity of transverse electromechanical variables. Classical formulation based on the principle of virtual displacements are implemented for comparison purposes. 2D assumptions are introduced by means of the Carrera unified formulation (CUF). Layer-wise theories are developed by using Legendre type polynomials of various order in the plate-thickness expansion. The evaluation of the interlaminar continuity of transverse electric displacement and/or transverse stresses on the static response of piezoelectric actuated/sensing plate is provided and compared. Orthotropic and anisotropic simply supported plates are considered. Results are assessed by comparison with available 3D piezoelasticity solutions. The mixed formulations that a priori evaluate the transverse electric displacement appear of particular interest. These last, in fact, permit an easy, convenient and reliable calculation of the electric charge and make feasible the development of robust control algorithms. [ABSTRACT FROM AUTHOR]

Abstract: This paper shows the advantages of using the zig-zag function (−1) k ζ k (ZZF) in the bending analysis of sandwich flat panels. Higher order theories are developed by adding ZZF to displacement fields of known theories. The main advantage of ZZF lies in the fact that it introduces a discontinuity in the first derivative (so called zig-zag effect) of the displacements distribution with correspondence to the core–face interfaces. Results including and discarding ZZF are compared in the bending response of sandwich plates with soft core loaded by harmonic distribution of transverse pressure at the top surface. Different values of face-to-core-stiffness-ratio FCSR as well as length-to-thickness-ratio LTR have been analyzed. It is concluded that: (1) ZZF is highly recommended in the bending analysis of sandwich plates; (2) the use of ZZF makes the error almost independent by FCSR parameter; (3) ZZF is easy to implement and its use should be preferred with respect to other refined theories. [Copyright &y& Elsevier]

This article shows the advantages of using the zig-zag function (-1)kζk (ZZF) in the bending analysis of unsymmetrically laminated sandwich flat panels with a soft core. Higher order theories are developed by adding ZZF to displacement fields of known theories. From linear to seventh order cases in displacement are considered. The main advantage of ZZF lies in the fact that it introduces a discontinuity in the first derivative (so called zig-zag effect) of the displacement distribution corresponding to the core-face interfaces. Results including and discarding ZZF are compared in the bending response of sandwich plates loaded by an harmonic distribution of transverse pressure at the top surface. Different values of face-to-core stiffness ratio (FCSR) as well as length-to-thickness ratio (LTR) have been analyzed. It is concluded that: (1) ZZF is highly recommended in the bending analysis of unsymmetrically laminated sandwich plates; (2) the use of ZZF makes the error almost independent of the FCSR parameter; (3) ZZF is easy to implement and its use should be considered with respect to other theories. [ABSTRACT FROM AUTHOR]

In this work the deformations of a simply supported, functionally graded, rectangular plate subjected to thermo-mechanical loadings are analysed, extending Unified Formulation by Carrera. The governing equations are derived from the Principle of Virtual Displacements accounting for the temperature as an external load only. The required temperature field is not assumed a priori, but determined separately by solving Fourier's equation. Numerical results for temperature, displacement and stress distributions are provided for different volume fractions of the metallic and ceramic constituent as well as for different plate thickness ratios. They correlate very well with three-dimensional solutions given in the literature. [ABSTRACT FROM AUTHOR]

A study of the transverse normal strain effect on the static thermoelastic response of homogeneous and multilayered plates is presented. Numerical evaluations have been given for classical, refined, and advanced zig-zag plate theories. Constant, linear, and higher-order forms of temperature profile in the plate thickness direction have been accounted for. Basic assumptions of the considered theories are quoted. The related governing equations are not given because these were derived from a unified formulation that was described elsewhere. Closed-form solutions are discussed by addressing three plate problems: a homogeneous plate made of an isotropic layer; a two-layered plate consisting of two layers made of different isotropic materials; a multilayered composite plate made of three cross-ply layers. It has been confirmed that any refinements of classical models are generally meaningless, unless the effects of transverse shear and normal strains are both taken into account in a plate theory. Furthermore, it has been found that transverse normal strains cannot be discarded even though thin plates are considered and the accurate description of the temperature profile in the plate thickness direction could result to be meaningless, unless transverse normal strains are taken into account, and vice versa. [ABSTRACT FROM AUTHOR]

Two-dimensional theories and finite elements are assessed to analyze displacement and stress fields in sandwich, composites plates. Two benchmarks are used to conduct the assessment. The first benchmark is a sandwich plate loaded by harmonic distribution of transverse pressure for which a three-dimensional closed-form solution exists in the literature. The second benchmark is a rectangular sandwich plate loaded by a transverse pressure located at the center. More than 20 plate theories and finite elements were implemented in a unified formulation recently proposed by the authors. Classical theories based on displacement assumptions are compared to advanced mixed models formulated on the basis of Reissner's mixed variational theorem. Both equivalent single-layer models as well as layerwise models are considered. Analytical closed-form solutions and finite elements are given. The considered benchmarks highlight both the performance and limitations of the considered two-dimensional theories. The convenience of layerwise description and advanced mixed theories has been demonstrated. The second benchmark especially proved the need for layerwise models to capture the local effects. [ABSTRACT FROM AUTHOR]

Focuses on a study that analyzed the transverse normal stress in multilayered plate modeling. Information on the geometry and coordinate systems for the laminated plates; Governing equations at the multilayered plate level; Results and discussion.

Nonlinear dynamics and mode aberration of rotating plates and shells are discussed in this work. The mathematical formalism is based on the one-dimensional (1D) Carrera unified formulation (CUF), which enables to express the governing equations and related finite element arrays as independent of the theory approximation order. As a consequence, three-dimensional (3D) solutions accounting for couplings due to geometry, material, and inertia can be included with ease and with low computational costs. Geometric nonlinearities are incorporated in a total Lagrangian scenario and the full Green-Lagrange strains are employed to outline accurately the equilibrium path of structures subjected to inertia, centrifugal forces, and Coriolis effect. A number of representative numerical examples are discussed, including multisection blades and shells with different radii of curvature. Particular attention is focused on the capabilities of the present formulation to deal with nonlinear effects, and comparison with s simpler linearized approach shows evident differences, particularly in the case of deep shells. [ABSTRACT FROM AUTHOR]

This work addresses the static analysis of functionally graded material plates subjected to transverse mechanical loadings. The unified formulation and principle of virtual displacements are employed to obtain both closed-form and finite element solutions. The use of the unified formulation permits a large variety of plate models with variable kinematic assumptions to be compared in the same framework. These differ according to the order of the expansion in the thickness direction and the variable description: the order of the expansion ranges from 1 to 4, thus covering first-order as well as higher-order theories; the description of the unknowns can be equivalent single layer or layerwise. The dependence of the material data on the thickness direction was introduced by employing thickness functions that are derived from the Legendre polynomials that are used in the layerwise case. The proposed approach is independent of the used transition function, and as a result, any continuous variations of the material properties in the thickness direction may be easily implemented. The obtained solutions (closed form and finite element) are validated through comparison with three-dimensional exact solutions and other available solutions. These show the limitations of classical plate theories as well as the convenience of the use of the proposed variable kinematic models for the analysis of functionally graded material plates. [ABSTRACT FROM AUTHOR]

This paper presents a static analysis of functionally graded (FG) single and sandwich plates using Carrera Unified Formulation with five new displacement fields of the non-polynomial form. In particular, trigonometric, exponential and hyperbolic displacement fields are employed. The simply supported FG single and sandwich plates are subjected to a bi-sinusoidal load. The governing equations for the static bending analysis are obtained employing the Principal of Virtual Displacement (PVD) under CUF and solved using Navier type solutions. The results show that non-polynomial thickness functions are accurate although, in a few cases, the influence of some non-polynomial terms may be detrimental. [ABSTRACT FROM AUTHOR]

In this study, a nonlinear dynamic analysis of sandwich plate with flexible core and laminated composite face sheets embedded with shape memory alloy (SMA) wires is investigated using the mixed LW/ESL models in the framework of Carrera's Unified Formulation. The instantaneous phase transformation effects are considered for every point on the face sheets. Since, this research deals with the transient nonlinear problem and the structure is thick plate with flexible core, employing a model with high accuracy and low computational cost is vital in this work. For this aim, the new mixed LW/ESL models proposed are employed in this study. The constitutive equation proposed by Brinson is used for modeling the nonlinear behavior of SMA wires. The governing equations are derived employing the Reissner Mixed Variational Theorem (RMVT) in order to satisfy the interlaminar continuity of transverse stresses between the layers. The nonlinear governing equations are solved based on the transient finite element along with the iterative incremental method. Some parametric studies such as intensity of impulsive pressure, location of SMA wires, plate aspect ratio, ratio of face sheet's thickness to the total thickness, volume fraction of the SMA wires and also boundary conditions upon the loss factor are investigated. [ABSTRACT FROM AUTHOR]

Composite laminates are prone to delamination. Implementation of delamination in the Carrera Unified Formulation frame work using nine noded quadrilateral MITC9 element is discussed in this article. MITC9 element is devoid of shear locking and membrane locking. Delaminated as well as healthy structure is analyzed for free mode vibration. The results from the present work are compared with the available experimental or/and research article or/and the three dimensional finite element simulations. The effect of different kinds and different percentages of area of delamination on the first three natural frequencies of the structure is discussed. The presence of open-mode delamination mode shape for large delaminations within the first three natural frequencies is discussed. Also, the switching of places between the second bending mode, with that of the first torsional mode frequency is discussed. Results obtained from different ordered theories are compared in the presence of delamination. Advantage of layerwise theories as compared to equivalent single layer theories for very large delaminations is stated. The effect of different kinds of delamination and their effect on the second bending and first torsional mode shape is discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this study, a new nonlinear finite element model is presented in the frame work of Carrera’s Unified Formulation (CUF) for the dynamic analysis of SMA hybrid composite considering the instantaneous phase transformation and material nonlinearity effects, for every point on the plate. The CUF unify many theories in a unified form which can be differed by the order of expansion and definition of the variables in the thickness direction. The Brinson’s SMA constitutive equation is used to model the behavior of SMA wires. The governing equations are derived using the Reissner Mixed Variational Theorem (RMVT) in order to enforce the interlaminar continuity of transverse shear and normal stresses between two adjacent layers. A transient finite-element-based method beside an iterative incremental procedure is presented to study the dynamic response of multilayered composite plate embedded with SMA wires. A suppressed vibration of the plate is observed, which is due to the energy dissipation of SMA wires. The parametric effects like length-to-thickness ratio, plate aspect ratio and also the effect of different boundary conditions, upon the loss factors are investigated. Results show that as the length-to-thickness ratio and also the plate aspect ratio increases, the loss factor decreases. [Copyright &y& Elsevier]

Abstract: We present an isogeometric finite element formulation for static, free vibration and buckling analysis of laminated composite and sandwich plates. The idea behind this work is to associate an isogeometric analysis (IGA) with a layerwise theory [A.J.M. Ferreira. Analyis of composite plates using a layerwise deformation theory and multiquadrics discretization. Mech Adv Mater Struct 2005;12(2):99–112]. Isogeometric analysis based on non-uniform rational B-spline (NURBS) basic functions were recently proposed to preserve exact geometries and to enhance very significantly the accuracy of the traditional finite elements. B-splines basic function (or NURBS) is used to represent for both geometric and field variable approximations, which provide a flexible way to make refinement and degree elevation. They enable us to achieve easily the smoothness with arbitrary continuity order compared with the traditional FEM. The layerwise theory assumes a first-order shear deformation theory in each layer and the imposition of displacement continuity at the layers interfaces. This permits to remove shear correction factors and improves the accuracy of transverse shear stresses. Intensive numerical studies have been conducted to show the highly efficient performance of the proposed formulation. [Copyright &y& Elsevier]

Abstract: In the present work a non-linear dynamic response of a continuous sandwich beam with SMA hybrid composite face sheets and flexible core is analyzed taking into account the phase transformation and also the material non-linearity effects, for every point along the face sheets. The one-dimensional constitutive equation of SMA proposed by Brinson is employed. Equations of motion are derived using Hamilton’s principle and a new finite element is proposed based on a higher order sandwich panel theory. Due to the phase transformation, the equations of motion are coupled with the phase transformation’s kinetic equations of SMA wires. A new finite-element-based approach along with an iterative incremental method is developed to study the dynamic response of sandwich beam with SMA hybrid composite face sheets and flexible core. A damped response of the sandwich beam is observed, which is due to the hysteresis behavior of SMA wires. The influence of the SMA wires on the vibration suppression related to the resonance phenomena of a sandwich beam as well as the effect of the through thickness location of the SMA wires inside the composite face sheets and also the effect of different boundary conditions on the dynamic response are analyzed. [Copyright &y& Elsevier]

Abstract: In this paper the authors derive a higher-order shear deformation theory for modeling functionally graded plates accounting for extensibility in the thickness direction. The explicit governing equations and boundary conditions are obtained using the principle of virtual displacements under Carrera’s Unified Formulation. The static and eigenproblems are solved by collocation with radial basis functions. The efficiency of the present approach is assessed with numerical results including deflection, stresses, free vibration, and buckling of functionally graded isotropic plates and functionally graded sandwich plates. [Copyright &y& Elsevier]

Abstract: In this paper, a variation of Murakami’s Zig-Zag theory is proposed for the analysis of functionally graded plates. The new theory includes a hyperbolic sine term for the in-plane displacements expansion and accounts for through-the-thickness deformation, by considering a quadratic evolution of the transverse displacement with the thickness coordinate. The governing equations and the boundary conditions are obtained by a generalization of Carrera’s Unified Formulation, and further interpolated by collocation with radial basis functions. Numerical examples on the static analysis of functionally graded sandwich plates demonstrate the accuracy of the present approach. The thickness stretching effect on such problems is studied. [Copyright &y& Elsevier]

Abstract: This paper presents an original hyperbolic sine shear deformation theory for the bending and free vibration analysis of functionally graded plates. The theory accounts for through-the-thickness deformations. Equations of motion and boundary conditions are obtained using Carrera’s Unified Formulation and further interpolated by collocation with radial basis functions. The efficiency of the present approach combining the new theory with this meshless technique is demonstrated in several numerical examples, for the static and free vibration analysis of functionally graded plates. Excellent agreement for simply-supported plates with other literature results has been found. [Copyright &y& Elsevier]

Abstract: In this paper we present a new application for Carrera’s unified Formulation (CUF) to analyse functionally graded plates. In this paper the authors present explicit governing equations of a sinusoidal shear deformation theory for functionally graded plates. It addresses the bending and free vibration analysis and accounts for through-the-thickness deformations. The equations of motion are interpolated by collocation with radial basis functions. Numerical examples demonstrate the efficiency of the present approach. [Copyright &y& Elsevier]

Abstract: This paper addresses the static deformations analysis of functionally graded plates by collocation with radial basis functions, according to a sinusoidal shear deformation formulation for plates. The present plate theory approach accounts for through-the-thickness deformations. The equations of motion and the boundary conditions are obtained by the Carrera’s Unified Formulation, and further interpolated by collocation with radial basis functions. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a variation of the use of Murakami’s zig-zag theory for the analysis of laminated plates. The new theory accounts for through-the-thickness deformation, by considering a quadratic evolution of the transverse displacement with the thickness coordinate. The equations of motion and the boundary conditions are obtained by the Carrera’s Unified Formulation, and further interpolated by collocation with radial basis functions. This paper considers the analysis of static deformations, free vibrations and buckling loads on laminated composite plates. [Copyright &y& Elsevier]

Abstract: In this paper, we combine Carrera''s Unified Formulation and a radial basis function collocation technique for predicting the static deformations and free vibration behavior of thin and thick isotropic and cross-ply laminated plates. Through numerical experiments, the capability and efficiency of this collocation technique for static and vibration problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined. [Copyright &y& Elsevier]

Abstract In the present work, higher-order Legendre polynomials are adopted as shape functions of p -version plate finite elements (FEs) and used in combination with node-dependent kinematics (NDK) to construct computationally efficient global–local FE models for the analysis of multilayered plates. The use of higher-order Legendre polynomials enables the elements to accommodate the complex structural deformations with a fewer number of variables in the FE model. Derived from Carrera Unified Formulation (CUF), NDK can integrate plate kinematics based on Equivalent Single Layer (ESL) models and Layer-wise (LW) models to obtain global–local models using no ad hoc coupling. The combination of Legendre-type shape functions and NDK shows excellent rates of convergence, which can lead to FE models with high accuracy in the refined local area with a reduction in the computational efforts. The capabilities of the proposed approach are investigated through various numerical examples by comparing the solution accuracy versus the computational expenses. [ABSTRACT FROM AUTHOR]

This work presents a class of plate finite elements (FEs) formulated with node-dependent kinematics, which can be used to construct global-local models with high numerical efficiency. Taking advantage of Carrera Unified Formulation (CUF), plate theory kinematics can be individually defined on each FE node, realizing a variation of refinement levels within the in-plane domain of one element. When used in the bridging zone between a global model and a locally refined one, an efficient global-local model can be constructed. Elements with variable ESL/LW kinematics from node to node are developed and applied in the global-local analysis of laminated structures. This work includes numerical examples in which LW models with refined kinematics are employed in local regions while ESL models are adopted in the less critical area, and modeling domains are connected by transition zone composed of elements with node-dependent kinematics. The obtained results are compared with solutions from literature and 3D FE modeling. For laminated plates with local effects to be considered, the proposed plate models can reduce the computational costs significantly while guaranteeing numerical accuracy without using special global-local coupling methods. [ABSTRACT FROM AUTHOR]

Static bending and free vibration of cross-ply laminated plates with simply supported boundary conditions are studied using layerwise description for field variables. The layerwise approach accounts for the through-the-thickness deformations. The equations of motion and the boundary conditions are obtained by the Carrera's unified formulation. The stiffness matrix is computed by using the Reissner mixed variational theorem (RMVT), in which the transverse stresses are also treated as independent variables apart from the displacements. To this end, a mixed form of Hooke's law is defined. A cell-based smoothed finite element method is employed to compute the terms in the stiffness matrix. The influence of various parameters on the static bending and free vibration are numerically studied. [ABSTRACT FROM AUTHOR]

This paper presents Best Theory Diagrams (BTDs) employing combinations of Maclaurin, trigonometric and exponential terms to build two-dimensional theories for laminated cross-ply plates. The BTD is a curve in which the least number of unknown variables to meet a given accuracy requirement is read. The used refined models are Equivalent Single Layer and are obtained using the Unified Formulation developed by Carrera. The governing equations are derived from the Principle of Virtual Displacement (PVD), and Navier-type closed form solutions have been obtained in the case of simply supported plates loaded by a bisinuisoidal transverse pressure. BTDs have been constructed using the Axiomatic/Asymptotic Method (AAM) and genetic algorithms (GA). The influence of trigonometric and exponential terms in the BTDs has been studied for different layer configurations, length-to-thickness ratios and stresses. It is shown that the addition of trigonometric and exponential expansion terms to Maclaurin ones may improve the accuracy and computational cost of refined plate theories. The combined use of CUF, AAM and GA is a powerful tool to evaluate the accuracy of any structural theory. [ABSTRACT FROM AUTHOR]

This paper proposes variable kinematic, mixed theories for laminated plates built via the asymptotic/axiomatic method (AAM). This method has been recently developed and successfully applied to develop refined theories for multilayered plates and shells. The AAM evaluates the accuracy of each unknown variables of a structural model. The present paper extends the AAM to mixed theories based on the Reissner Mixed Variational Theorem (RMVT). The displacement and transverse stress fields are modeled by means of the Carrera Unified Formulation (CUF), and expansions up to the fourth-order are employed. Equivalent Single Layer (ESL) and Layer Wise (LW) schemes are adopted, and closed-form Navier-type solutions are considered. The AAM is exploited to determine the set of active terms of a refined plate model. The inactive terms are then discarded. The effectiveness of each variable is evaluated with respect to an LW, fourth-order mixed model. Reduced models are built for different thickness ratios, stacking sequences and displacement/stress variables. The results suggest that reduced models with significantly less unknown variables than full models can be built with no accuracies penalties. Such models are problem dependent, and full models should be preferred in the case of thick, asymmetric plates. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive review of the various methods employed to study the static, dynamic and stability behavior of Functionally Graded Material (FGM) plates. Both analytical and numerical methods are considered. The review is carried out with an emphasis to present stress, vibration and buckling characteristics of FGM plates predicted using different theories proposed by several researchers without considering the detailed mathematical implication of various methodologies. The effect of variation of material properties through the thickness, type of load case, boundary conditions, edge ratio, side-to-thickness ratio and the effect of nonlinearity on the behavior of FGM plates are discussed. The main objective of this paper is to serve the interests of researchers and engineers already involved in the analysis and design of FGM structures. [ABSTRACT FROM AUTHOR]