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The aim of this work is the study of the adhesion integrity of metallic Single Lap Joints (SLJs) through the assessment of the MUL2 CODE, software developed by the MUL2 Research Group - Department of Mechanical and Aerospace Engineering of Politecnico di Torino. The MUL2 CODE is implemented through the Carrera Unified Formulation (CUF) for 2D structures based on Hierarchical Legendre Expansion (HLE) polynomials. An efficient method for the Structural Health Monitoring (SHM) of bonded joints is simulated and verified by CUF approach, in order to reduce the computational cost of analyses: by using transient excitations (toneburst signals), the structural health of damaged SLJ can be numerically evaluated. The interaction mechanism between the waves traveling through the investigated specimens is numerically modeled with a simple Finite Elements (FE) model and it is solved via MUL2 CODE and commercial software Ansys Workbench, respectively. Experimental campaigns data are compared with CUF and Ansys results demonstrating the consistence of the MUL2 formulation that is computationally simpler, but very efficient for the joint analysis. The presented and discussed CUF application is able to quantify with a high accuracy the debonding extension in the damaged SLJ, simply tuning the excitation frequency of the SHM technique.

This work focuses on the assessment of a novel so-called “homogenization method” allowing to transform a heterogeneous material with inclusions or holes into an equivalent homogeneous material with...

This work wants to investigate the soundproofing level of passive acoustic metamaterials made of Melamine Foam and cylindrical Aluminum inclusions. Latest research shows promising acoustical possibilities on controlling certain frequencies, varying their geometry or material configuration. Typically, acoustic metamaterials are plates with inclusions that resonate in low frequencies and block basses. Investigation is carried to use them for the improvement of aircraft cabin quietness. An homogenization method is adopted to study the metamaterial: this is modelled as viscoelastic material with inclusions, using a frequency-dependent approach, and the effective properties of the homogenized metamaterial are derived using a micromechanics-based method that allow us to prescribe the simplest mesh for periodical geometries (in this case, cylindrical) by reducing drastically the computation time. MSC Actran is used for vibro-acoustic simulations, in particular for the evaluation of Sound Transmission Loss in metamaterial panels with different volume fractions and in a sandwich panel with metamaterial core. This last has been compared with a classical sandwich panel used in aeronautics and it has been demonstrated that metamaterial gets higher transmission loss over all the frequency range.

Towards improving the numerical efficiency in the analysis of multi-layered shell structures with the finite element (FE) method, an adaptable two-level mathematical refinement approach is proposed for refined curvilinear shell elements. Based on Carrera Unified Formulation (CUF), the approximation of displacement functions of shell elements can be improved by refining the through-the-thickness assumptions and enriching the shape functions. By using the hierarchical Legendre polynomial expansions (HLE) as shape functions, the element capabilities can be enhanced conveniently without re-meshing. To further increase the numerical efficiency of shell FE models, Node-Dependent Kinematics (NDK) is utilized to implement local kinematic refinements on the selected FE nodes within the domain of interest. The conjunction of NDK with the two-level refinements of the shell FE models leads to an adaptable refinement approach in the analysis of shell structures, which can be used to build FE models with optimal efficiency and high fidelity. The competence of the proposed approach is investigated through numerical studies on laminated shells.

A new layerwise mixed model is developed for the coupled hygro-thermo-mechanical static analysis of multilayered plates, addressing hybrid composite laminates, fibre metal laminates and sandwich plates under hygro-thermo-mechanical loadings. This model is based on a mixed least-squares formulation, i.e. a strong form model, with a layerwise variable description for displacements, transverse stresses and in-plane strains, along with temperature, transverse heat flux and in-plane components of the thermal gradient, as well as moisture, transverse moisture flux and in-plane components of the moisture gradient, all taken as independent variables. This mixed formulation ensures that the interlaminar C 0 continuity requirements, where the material properties change, are fully fulfilled a priori by all independent variables. The effects of hygrothermal environments in the behaviour of multilayered plates are here demonstrated considering hybrid composite laminates, fibre metal laminates and sandwich plates with different side-to-thickness ratios under a series of hygro-thermo-mechanical loadings. Three-dimensional (3D) exact solutions are used to assess the results by the present model and some further by an alternative weak form model in the framework of Carrera Unified Formulation (CUF). The present model is shown to predict a highly accurate quasi-3D hygro-thermo-mechanical description of the through-thickness distributions of displacements and stresses, temperature and heat flux, moisture and moisture flux, all together.

In this paper, we present some advanced shell models for the analysis of orthotropic multilayered structures in which the mechanical and physical properties may change in the thickness direction. The finite element method showed successful performances to approximate the solutions of the advanced structures. In this regard, two variational formulations are available to reach the stiffness matrices , the principle of virtual displacement (PVD) and the Reissner mixed variational theorem (RMVT). Here we introduce a strategy similar to MITC (Mixed Interpolated of Tensorial Components) approach, in the RMVT formulation, in order to construct an advanced locking-free finite element. Moreover, assuming the transverse stresses as independent variables, the continuity at the interfaces between layers is easily imposed. We show that in the RMVT context, the element exhibits both properties of convergence and robustness when comparing the numerical results with benchmark solutions from literature, even for higher span to thickness ratios , and both interlayer continuity conditions and boundary conditions are fully satisfied.

This chapter presents refined shell finite element models with variable kinematics for the analysis of multi-layered structures involved in four physical fields: mechanical, electric, thermal, and hygroscopic. Variable kinematic models in the framework of Carrera Unified Formulation (CUF) with various kinematic assumptions are discussed. An efficient tool to realize adaptable refinement in finite element models, Node-Dependent Kinematics approach, is introduced. Refined doubly curved shell finite element formulations derived from the principle of virtual displacements accounting for multi-field coupling effects are presented.

This paper focuses on developing and exploiting the potential of miscellaneous through-the-thickness approximating functions for FEM analysis of laminated composite plates/shells. Considering the theory of series expansion, Taylor series, trigonometric series, exponential functions, and miscellaneous expansions are implemented in the equivalent single layer models of Carrera Unified Formulation (CUF). Their performances in obtaining a good approximation of stress distribution through the thickness of the plate/shell are investigated by performing several static mechanical studies, and the inclusion of Murakami’s zig-zag function is also evaluated. The results are compared with layer-wise theories in the framework of CUF by adopting as thickness functions both Legendre polynomials and Lagrange interpolations on Chebyshev nodes (Sampling-Surfaces method, SaS). The governing equations are derived from Principle of Virtual Displacement (PVD) and Finite Element Method (FEM) is adopted to get the numerical solutions. Nine-node 2D elements for plates and shells are employed, using Mixed Interpolation of Tonsorial Components (MITC) method to contrast the membrane and shear locking phenomenon. Simply-supported cross-ply plate and shell structures with various lay-ups and span-to-thickness ratios subjected to transverse bi-sinusoidal pressure load are analyzed. The results show that all the refined kinematic theories are able to capture the exact solution if a sufficient number of expansion (number of terms in the expansion of the displacement field) is taken, but the maximum computational cost can change for the different types of models. In some cases, combinations of different expansion theories (miscellaneous expansions) can show a significant reduction of computational costs.

The present article considers the linear static analysis of composite shell structures with double-curvature geometry by means of a shell finite element with variable through-the-thickness kinematic. The refined models used are grouped in the Unified Formulation by Carrera (CUF) and they permit the distribution of displacements and stresses along the thickness of the multilayered shell to be accurately described. The shell element has nine nodes and the mixed interpolation of tensorial components (MITC) method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement (PVD) and the finite element method (FEM) is employed to solve them. Cross-ply spherical shells with simply-supported edges and subjected to bi-sinusoidal pressure are analyzed. Various laminations, thickness ratios, and curvature ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elast...

This paper deals with the analysis of layered structures under thermal and electro-mechanical loads. Constitutive equations for multifield are considered and the Principle of Virtual Displacements (PVD) is employed to derive the governing equations. The MITC9 shell finite element based on the Carrera's Unified Formulation (CUF) has been applied for the analysis. The models grouped in the CUF have variable through-the-thickness kinematic and they provide an accurate distribution of displacements and stresses along the thickness of the laminate. The shell element has nine nodes and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomenon. The finite element analysis of multilayered plates and shells has been addressed. Variable kinematics, as well as layer-wise and equivalent single layer descriptions, have been considered for the presented FEs, according to CUF. A few problems are analyzed to show the effectiveness of the proposed approach. Various laminations, thickness ratios and curvature ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in literature and the analytical solutions obtained using the CUF and the Navier's method.

This paper discusses the robustness of plate elements based on Mixed Interpolation of Tensorial Components (MITC) technique and the variable kinematics approach of Carrera Unified Formulation (CUF) with respect to the problem of distorted meshes. MITC was originally proposed for Reissner-Mindlin type plates to develop shear locking free plate elements. In the present framework, refined plate elements are obtained by referring to high-order Equivalent Single Layer as well as Layer-Wise models expressed in CUF for the analysis of multilayered anisotropic structures. Four-node and nine-node elements are considered and some applications are developed for both isotropic and multilayered composite plates. Results related to the MITC approach are compared to the isoparametric elements, including selectively reduced quadrature schemes, for both, the static and free-vibration analysis. They show that CUF-MITC elements maintain their effectiveness also in the case of distorted meshes, for all the materials studied and kinematic models considered: the obtained elements are robust as free from shear locking and spurious zero-energy modes.

Node-dependent Kinematic (NDK) shell finite element (FE) formulations are presented for the steady-state thermo-mechanical analysis of laminated structures. The displacements and temperature change are treated as primary variables in the FE models and are directly solved through the coupled thermo-mechanical models. The enforcement of distributed temperature boundary conditions on the top or the bottom surface of hierarchical shell elements is conducted through the Linear Least Squares. The effectiveness of the proposed FE approaches is verified by comparing the results against those from the literature. The application of adaptive refinement approach based on the hierarchical elements and NDK to build FE models with optimal efficiency is demonstrated through numerical examples.

Carrera Unified Formulation (CUF) is taken a step further to render node-dependent kinematics (NDK) capabilities to new layerwise finite elements based on Reissner’s Mixed Variational Theorem (RMVT), especially suited for global–local stress analysis of multilayered plates, ensuring high numerical accuracy and computational efficiency, all together. In the framework of CUF, as introduced originally for multilayered structures, any degree of kinematic refinement can be considered in agreement with Equivalent Single-Layer (ESL) or Layer-Wise (LW) theories to develop advanced finite element models, whether based on the Principle of Virtual Displacements (PVD) or RMVT. The degree of kinematic refinement, which usually holds equally for the entire element, can be taken a step further, by being assigned locally to each of its nodes, making full use of CUF to render NDK capabilities to the elements. Besides, even though the elements can adopt any type of nodal shape functions, high p -order hierarchical Legendre expansions (HLE) can also be combined with NDK, achieving excellent convergence rates. These capabilities combined, explored first under the PVD, are for once integrated in the proposed elements under RMVT to further benefit accurate stress analysis. These elements can be applied throughout the entire mesh, adapting to local, transitional and global regions straightforwardly, providing high numerical accuracy, locally, with minimal computational efforts, globally. The numerical results focus on stress analysis of multilayered composite plates, including local effects, to demonstrate the predictive capabilities of the proposed RMVT-based LW elements with NDK and HLE combined, considering well-known benchmark three-dimensional exact solutions for assessment.

The present paper considers the modal analysis of delaminated composite shell structures with double-curvature geometry. The finite element for shell with variable through-the-thickness kinematic is adopted for the analysis. The refined models are grouped in the Unified Formulation by Carrera (CUF) and they permit the distribution of displacements along the thickness of the multilayered shell to be accurately described. The shell element has nine nodes and the Mixed Interpolation of Tensorial Components (MITC) method is used to alleviate the membrane and shear locking phenomenon. The governing equations are derived from the Principle of Virtual Displacement (PVD) and the Finite Element Method (FEM) is employed to solve them. From the analysis, one can conclude that the shell element based on the CUF is very efficient and the results obtained match closely with three-dimensional finite element simulations. The effect of delamination size, curvature, stacking sequence, and boundary conditions is stu...

This paper presents some solutions for mechanical responses of angle-ply laminated plates under transverse distributed loads, which are obtained by using refined finite element models adopting variable kinematics based on Carrera’s Unified Formulation (CUF). Plates with several types of stacking sequence under different boundary conditions are considered. Layer-wise (LW) models based on Chebyshev polynomials (first kind) and Equivalent Single Layer (ESL) models based on Trigonometric series are used in the analysis. To compare the performances of different displacement-based kinematic models, a set of simply supported boundary conditions and mixed clamped-free boundaries are adopted in the numerical study. A nine-node MITC (Mixed Interpolated of Tensorial Components) plate element is employed to contrast the shear locking phenomenon of thin plates. CUF-based variable kinematic models are used in the numerical study and the number of expansion terms in the thickness direction is increased until the requisite numerical accuracy is achieved. By comparing the numerical results obtained with CUF-based refined models and ABAQUS 3D models as well as reference solutions from literature, the effectiveness of the adopted models is verified. The newly studied numerical cases can be taken as benchmarks for future research.

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.

Advanced plate models with variable kinematics for steady state hygrothermal analysis of composite laminates are proposed. The refined models discussed include both layer-wise (LW) and equivalent single layer (ESL) models, and the Carrera Unified Formulation (CUF) is used. The mixed interpolation of tensorial component (MITC) method is applied to a nine-node element to contrast the shear locking phenomena. The governing equations are derived from the principle of virtual displacement (PVD) taking into account elastic mechanical, thermal and hygroscopic effects. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. Cross-ply plates with symmetrical lamination and simply supported edges subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are compared. Variable kinemat...

In this article various sinusoidal shear deformation theories are used for the buckling analysis of functionally graded sandwich plates. The theories may account for through-the-thickness deformations and/or zig-zag effect. The governing equations and boundary conditions are derived using the Principle of Virtual Work under a generalization of Carrera’s Unified Formulation and further interpolated by collocation with radial basis functions. A numerical investigation has been conducted on the buckling analysis of sandwich plates with functionally graded skins. The influence of the thickness stretching and the zig-zag effects on these problems is investigated. Numerical results demonstrate the accuracy of the present approach.

This chapter provides various constitutive laws for the most widely used materials in the analysis of thermal structures. In particular, the stress-strain relationships of isotropic, orthotropic, anisotropic and functionally graded materials are proposed. In the latter case, temperature-dependent materials for different types of FGMs are presented. The constitutive law for the RMVT case is also derived. Finally, after defining the Gibbs free-energy and thermo-mechanical enthalpy density, the constitutive equations for thermoelastic problems are obtained.

This article presents advanced shell models for the steady-state hygrothermal analysis of composite laminates. The Carrera Unified Formulation is used to derive refined models that include both lay...

This chapter is dedicated to the solution of various sample problems in the classical theory of thermoelasticity. Classical rectangular Cartesian, cylindrical and spherical coordinate systems are employed. More specifically, the first part of the chapter deals with thermoelastic problems involving bars, beams, plates, cylinders, spheres and disks. The second part focuses on simple and complex 1D, 2D and 3D steady-state and transient heat conduction problems.

This chapter deals with the fundamentals of mechanics of beam, plate and shell structures. The asymptotic and axiomatic approaches in structural mechanics are thoroughly discussed. The main strengths and weaknesses, of the above mentioned methodologies, are highlighted. The most known classical models for beam, plate and shell structures are reviewed. Various examples involving different structural elements are proposed.

The main purpose of the thermoelasticity is the determination of strain and stress states of a continuum deformable body when subjected to a prescribed temperature distribution. Similarly to the classical theory of elasticity, the variables involved in the thermoelasticity are uniquely determined by the solution of the thermoelastic problem including the governing differential equations of motion, the compatibility equations and the constitutive equations. This chapter is mainly devoted to the derivation of the basic equations of the 3D and 2D thermoelasticity. In particular, first the three-dimensional thermoelasticity is introduced and the two classical displacement and stress formulations are proposed. Then the problem is reduced to a two-dimensional one and the concepts of plane strain and plane stress are thoroughly examined.

The present chapter proposes a deep investigation of the static response of uncoupled thermoelastic problems for composite plate and shell structures. Advanced theories based on PVD and RMVT are employed. Both the exact Navier-type closed-form solution and the advanced MITC9 shell element are used in the analysis. The temperature profile through-the-thickness direction is both assumed considering a linear variation and calculated by solving Fourier's heat conduction equation.

This chapter studies the effect of thermomechanical load interactions on the panel flutter phenomena. The classical panel flutter of laminated composite and FGM flat panels is investigated. Aero-thermo-elastic stability diagrams for various panel configurations, different temperatures and dynamic pressure have been obtained.

This chapter presents free vibration and thermal stability analyses of functionally graded (FG) sandwich plates under temperature gradients. In particular, various through-the-thickness temperature distributions such as uniform, linear and non-linear are taken into account. The latter is given in different forms: i) Power law variation; ii) Solution of the one-dimensional Fourier's heat conduction equation; iii) Sinusoidal. The influence of initial thermal stresses on the free vibration characteristics of FG sandwich plates is investigated. The effect of volume fraction index, aspect ratio, boundary conditions, length-to-thickness ratio, sandwich plate type and temperature distributions through-the-thickness plate direction, on both natural frequencies and critical temperatures is thoroughly investigated.

This chapter presents classical and unconventional variational principles. The Principle of Virtual Displacements (PVD), derived from Hamilton's principle, and Reissner's Mixed Variational Theorem (RMVT) are proposed and discussed in detail. Two alternative versions of RMVT in the pure mechanical case are also given. Finally, thermoelastic variational formulations within the framework of the PVD and RMVT, for both coupled and uncoupled thermoelasticity, are provided in various forms.

This chapter proposes a thorough thermal stress investigation of layered structures. The temperature profiles and their influence on the static response are evaluated. Stress and displacement fields are computed by accounting for several temperature fields as well as various laminate stacking sequences and geometry. In particular, four sample problems are proposed and the results, obtained by using advanced plate theories, are compared with those available in literature.

The present chapter proposes the development of classical approximate solution methods used to solve various thermal stress problems given in the third part of the book. Refined variable-kinematics classical and mixed formulations derived in a general form exploiting the method of power series expansion of the displacements/stresses components, based on the PVD and RMVT respectively, are proposed. In the framework of meshless methods three advanced Hierarchical Ritz, Galerkin and Generalized Galerkin Formulations (HRF, HGF, HGGF) are used. For all of the aforementioned approximate methods a comprehensive mathematical description for beam, plate and shell structures, is provided. Both weak- and strong-form of the governing equations is derived. A full mathematical description for both coupled and uncoupled thermoelastic formulations, derived by using the PVD, is also given.

This chapter deals with the study of free vibration of uncoupled thermoelastic problems. The effect of the thermal environment on the modal characteristics of sandwich plate structures is thoroughly investigated. Two different sandwich structure configurations, accounting for cross-ply and angle-ply face sheets, respectively, are analyzed. Results obtained by using the proposed advanced and refined plate models are assessed against those present in literature and/or obtained by using commercial FE software.

This work presents the best theory diagrams (BTDs) for multilayered plates involved in multifield problems (mechanical, thermal and electrical). A BTD is a curve that reports the minimum number of terms of a refined model for a given accuracy. The axiomatic/asymptotic technique is employed in order to detect the relevant terms, and the error is computed with respect to an exact or quasi-exact solution. The models that belong to the BTDs are constructed by means of a genetic algorithm and the Carrera Unified Formulation (CUF). The CUF defines the displacement field as an expansion of the thickness coordinate. The governing equations are obtained in terms of few fundamental nuclei, whose form does not depend on the particular expansion order that is employed. The Navier closed-form solution has been adopted to solve the equilibrium equations. The analyses herein reported are related to plates subjected to multifield loads: mechanical, thermal and electrical. The aim of this study is to evaluate the influence of the type of the load in the definition of the BTDs. In addition, the influence of geometry, material parameters and displacement/stress components are considered. The results suggest that the BTD and the CUF can be considered as tools to evaluate any structural theory against a reference solution. In addition, it has been found that the BTD definition is influenced to a great extent by the type of load.

The present chapter provides a comprehensive thermal buckling analysis of laminated composite and sandwich plate structures. Analytical as well as approximate solution methods are used in the investigation. In particular, the Ritz, Galerkin and Generalized Galerkin method are compared. The influence of lamination angle, length-to-thickness ratio and orthotropic ratio on the critical temperature parameter is examined. Finally, the modal characteristics of thermo-mechanically pre/post-buckled anisotropic laminated composite plates are shown.

This chapter discusses the static and dynamic responses of composite structures by using coupled thermoelastic formulations. Advanced Q9 FEs based on refined models generated by means of the UF are used. First the effect of the thermoelastic coupling for the static response of isotropic plates subjected to a static instantaneous thermomechanical load is proposed. Then the investigation continues evaluating the effect of the higher order models on the stress and displacement fields due to thermal loading. An assessment of the temperature profile in the case of steady-state solution is also provided. Finally, thermomechanical dynamic analysis are proposed.

The present chapter deals with the development of advanced PVD and RMVT-based beam, plate and shell theories. The latter are featured by hierarchical capabilities and are generated by using the Unified Formulation (UF). In the first part of the chapter, the UF notation is introduced and carefully described. The concept of fundamental nucleus within the framework of the UF is proposed. Advanced Finite Element (FE) formulations based on refined UF models are presented. Several examples involving the derivation of the FE matrices as well as the assembly procedure are shown. In the second part, more advanced theories, such as those that involve 1×1 1 × 1 secondary nuclei and those based on the mixed axiomatic-asymptotic approach (MAAA) are proposed. Finally, the chapter ends with the construction of the best theory diagram (BTD).

This work deals with refined models for the thermal stress analysis of multilayered plates. The Carrera Unified Formulation has been used in order to generate refined models of any order, and both equivalent single layer (ESL) and layer-wise (LW) schemes have been adopted. A Navier-type solution has been employed and, as a result, only simply supported orthotropic plates have been considered. A linear temperature distribution along the thickness direction is considered. The purpose of this work is to establish the relevance of the displacement variables and to discard the irrelevant terms in order to obtain refined models with the same accuracy as full theories, but with a lower computational cost. The axiomatic/asymptotic technique has been employed. The effectiveness of each displacement variable has been measured considering the influence of several parameters, such as the length-to-thickness ratio (a/h), the stacking sequence and the kind of material (isotropic and orthotropic). The “best” reduced mod...

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. (C) 2014 Elsevier Ltd. All rights reserved.

In this paper, we present some advanced shell models for the analysis of multilayered structures in which the mechanical and physical properties may change in the thickness direction. The finite element method showed successful performances to approximate the solutions of the advanced structures. In this regard, two variational formulations are available to reach the stiffness matrices, the Principle of Virtual Displacement (PVD) and the Reissner Mixed Variational Theorem (RMVT). Here we introduce a strategy similar to MITC (Mixed Interpolated of Tensorial Components) approach, in the RMVT formulation, in order to construct an advanced locking-free finite element. Moreover, assuming the transverse stresses as independent variables, the continuity at the interfaces between layers is easily imposed. We show that in the RMVT context, the element exhibits both properties of convergence and robustness when comparing the numerical results with benchmark solutions from literature.

This paper compares 1D and 2D assumptions for the analysis of isotropic thin-walled structures. Refined 1D (beam) and 2D (shell) models were exploited. These models are based on a variable kinematic approach which means that the displacement field assumed can be refined to any extent since the orders of the polynomial expansions are taken as free parameters. The Finite Element (FE) approach was used to provide numerical results. FEM is, in fact, the most popular technique used in the analysis of structures in engineering. The Carrera Unified Formulation (CUF) was adopted in order to write finite element matrices of 1D and 2D models in a concise form based on the so-called fundamental nuclei. In this work, 1D models are used to assessing well-established benchmarks which are often used to solve shell problems, such as the pinched shell problem. 1D and 2D models were compared in terms of accuracy and computational costs. Furthermore, locking phenomena are discussed. It is concluded that some of the well-known shell problems, including those that are usually used as 'benchmarks' in FE shell developments, can be dealt with 1D refined models. The use of 1D refined models results in minor numerical locking, much lower computational efforts and comparable accuracy with shell models

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.

In this paper, we analyze a two dimensional model of multilayered plates for which the main interest is to study the mechanical and physical properties, that may change in the thickness direction. The finite element method showed successful performances to approximate the solutions of the advanced structures. In this regard, two variational formulations are available to reach the stiffness matrices, the principle of virtual displacement (PVD) and the Reissner mixed variational theorem (RMVT). Here we introduce a strategy similar to Mixed Interpolated of Tensorial Components (MITC) approach, in the RMVT formulation, in order to construct an advanced locking-free finite element. Assuming the transverse stresses as independent variables, the continuity at the interfaces between layers is easily imposed. It is known that unless the combination of finite element spaces for displacement and stresses is chosen carefully, the problem of locking is likely to occur. Following this suggestion, we propose a finite element scheme that it is known to be robust with respect to the locking phenomenon in the classical PVD approach. We show that in the RMVT context, the element exhibits both properties of convergence and robustness when comparing the numerical results with benchmark solutions from literature.

In this paper we present a layerwise finite element model for the analysis of sandwich laminated plates with a viscoelastic core and laminated anisotropic face layers. The stiffness and mass matrices of the element are obtained by Carrera’s Unified Formulation (CUF). The dynamic problem is solved in the frequency domain with viscoelastic frequency-dependent material properties for the core. The dynamic behaviour of the model is compared with solutions found in the literature, including experimental data.

In this work a nine-nodes shell finite element, formulated in the framework of Carrera’s Unified Formulation (CUF), is presented. The exact geometry of cylindrical shells is considered. The Mixed Interpolation of Tensorial Components (MITC) technique is applied to the element in order to overcome shear and membrane locking phenomenon. High-order equivalent single layer theories contained in the CUF are used to perform the analysis of shell structures. Benchmark solutions from the open literature are taken to validate the obtained results. The mixed-interpolated shell finite element shows good properties of convergence and robustness by increasing the number of used elements and the order of expansion of displacements in the thickness direction.

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.