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General Higher-Order Equivalent Single Layer Theory for Free Vibrations of Doubly-Curved Laminated Composite Shells and Panels
... Among literature, a significative number of works has already assessed the enhancement of the structural response due to the application of composite materials, which are intended to be orthotropic [10][11][12][13][14]. However, classic composite materials can be optimized only from a variation of the material orientation angle, setting a discrete number of superimposed laminae with different orientations, as well as the Variable Angle Tow (VAT) technology [15][16][17][18][19]. ...
... In a more expanded form, Eqn. (12) can be written as: ...
... T is the second-order time de-rivative of the three-dimensional displacement field vector introduced in Eqn. (12), setting Ü (k) ...
... However, in this way such formulations were not reliable for the analysis of moderately thick structures with deep main curvatures. For this reason, refined models have been formulated accounting for the actual geometry of the shell [30][31]. ...
... are the three-dimensional stress and strain vectors, respectively, defined in each -th layer of the laminate; C (k) ij with i, j = 1, ..., 6 are the anisotropic elastic constants of the material, referred to the material reference system. Starting from Eqn. (31), it is possible to express the threedimensional linear elastic relations for each k-th lamina applying to C (k) an orthogonal transformation T (k) accounting for the rotation angle θ k . Thus, the rotated stiffness matrix C (k) comes out [30]: ...
... The generalized three-dimensional elastic stiffnesses, according to Eqn. (31) have been adopted for the identification of a Trigonal Material [125], whose density is equal to ρ (k) = 2649 kg/m 3 : ...
The present work investigates the modal response of doubly-curved shells characterized by unsymmetric lamination schemes embedding anisotropic materials and externally constrained with general boundary conditions along the edges. The Equivalent Single Layer (ESL) methodology has been adopted for the assessment of the fundamental governing equations, employing a generalized formulation for the setup of each component of the kinematic field variable, leading to a higher order through-the-thickness expression of the displacement field. A two-dimensional non uniform discrete grid has been sat up within the mapped physical domain. The fundamental equations have been derived from the Hamiltonian Principle, and the problem has been solved employing a weak formulation of the field variable based on a higher order Lagrange polynomials-based interpolation algorithm. General boundary conditions have been obtained starting from a series of translational springs for each principal direction of the shell, each of them distributed following a generalized analytical expression. After some validations with respect to refined three-dimensional models, a systematic analysis has been performed, where the mode frequencies and shapes of structures with different syngonies and curvatures have been investigated. It has been shown that the modal response of anisotropic doubly-curved shells can be significantly oriented if governing distribution parameters are correctly selected.
... For this reason, two-dimensional (2D) alternative approaches have been derived. The most famous strategy is Equivalent Single Layer (ESL) [15][16][17], which assesses the entire structural problem on a reference surface whose generic point is located in the middle thickness of the solid. In particular, in reference [16] an extensive study with higher order theories for doubly-curved shell structures is provided, and in reference [17] an ESL higher order model is developed for laminated composite curved structures. ...
... The most famous strategy is Equivalent Single Layer (ESL) [15][16][17], which assesses the entire structural problem on a reference surface whose generic point is located in the middle thickness of the solid. In particular, in reference [16] an extensive study with higher order theories for doubly-curved shell structures is provided, and in reference [17] an ESL higher order model is developed for laminated composite curved structures. In the same way, the Layer-Wise (LW) approach [18][19][20][21][22] introduces a 2manifold for each layer of the stacking sequence. ...
... If det (J) = 0, the inverse form J −1 of the Jacobian matrix occurring in Eq. (17) can be calculated, thus obtaining: ...
The article proposes an Equivalent Single Layer (ESL) formulation for the linear static analysis of arbitrarily-shaped shell structures subjected to general surface loads and boundary conditions. A parametrization of the physical domain is provided by employing a set of curvilinear principal coordinates. The generalized blending methodology accounts for a distortion of the structure so that disparate geometries can be considered. Each layer of the stacking sequence has an arbitrary orientation and is modelled as a generally anisotropic continuum. In addition, re-entrant auxetic three-dimensional honeycomb cells with soft-core behaviour are considered in the model. The unknown variables are described employing a generalized displacement field and pre-determined through-the-thickness functions assessed in a unified formulation. Then, a weak assessment of the structural problem accounts for shape functions defined with an isogeometric approach starting from the computational grid. A generalized methodology has been proposed to define two-dimensional distributions of static surface loads. In the same way, boundary conditions with three-dimensional features are implemented along the shell edges employing linear springs. The fundamental relations are obtained from the stationary configuration of the total potential energy, and they are numerically tackled by employing the Generalized Differential Quadrature (GDQ) method, accounting for non-uniform computational grids. In the post-processing stage, an equilibrium-based recovery procedure allows the determination of the three-dimensional dispersion of the kinematic and static quantities. Some case studies have been presented, and a successful benchmark of different structural responses has been performed with respect to various refined theories.
... where U is the vector for the 3D displacement components and u is the vector for the 2D τth generalized displacements defined at the middle surface of the shell structure [26]. ...
... where the explicit form of Ω D is detailed in [26]. The τth order stress resultants and the generalized strains are linked using the following relation ...
... The elastic coefficients included in the 6 × 6 matrix used for linear elastic constitutive equations are proposed in complete form in [26]. The governing equations of motion with related boundary conditions are calculated by using the Hamilton principle. ...
... Therefore, the materials used for the construction of the biosensor play an important role. Recently, laminated composites have become of great interest to various industries and applications [30][31][32][33][34][35][36][37][38][39][40][41][42]. The development of new composite materials is grabbing researchers' attention, as these materials are characterized by high surfaceto-volume ratios, high catalytic activity, good electrical conductivity, and good magnetic properties [43][44][45][46][47]. Yang et al. [47] extensively presented the synthesis of carbon nanotubes Compared to other types of biosensors, e.g., biomedical ones, biosensors for environmental monitoring have a nonaged phase due to the complexity of the analysis, such as the complex ecological matrix, which interferes with pollutant recognition. ...
... Therefore, the materials used for the construction of the biosensor play an important role. Recently, laminated composites have become of great interest to various industries and applications [30][31][32][33][34][35][36][37][38][39][40][41][42]. The development of new composite materials is grabbing researchers' attention, as these materials are characterized by high surface-tovolume ratios, high catalytic activity, good electrical conductivity, and good magnetic properties [43][44][45][46][47]. Yang et al. [47] extensively presented the synthesis of carbon nanotubes (CNT) (arc discharge, laser ablation, chemical vapor deposition (CVD), etc.) and the possibilities for their functionalization. ...
The monitoring of environmental pollution requires fast, reliable, cost-effective and small devices. This need explains the recent trends in the development of biosensing devices for pollutant detection. The present review aims to summarize the newest trends regarding the use of biosensors to detect environmental contaminants. Enzyme, whole cell, antibody, aptamer, and DNA-based biosensors and biomimetic sensors are discussed. We summarize their applicability to the detection of various pollutants and mention their constructive characteristics. Several detection principles are used in biosensor design: amperometry, conductometry, luminescence, etc. They differ in terms of rapidity, sensitivity, profitability, and design. Each one is characterized by specific selectivity and detection limits depending on the sensitive element. Mimetic biosensors are slowly gaining attention from researchers and users due to their advantages compared with classical ones. Further studies are necessary for the development of robust biosensing devices that can successfully be used for the detection of pollutants from complex matrices without prior sample preparation.
... If the kth lamina is reinforced by CNTs, the analytical model accounts for different possible agglomerations of the reinforcing particles, as visible in figure 4.1. The characteristic shape of CNTs, depicted in figure 4.1(a), is one of the main reasons which cause the agglomeration of these particles when scattered in a polymer matrix [94]. ...
... where the differential operator τη L ( ) assumes the following aspect whose explicit expressions can be found in [94]. A dynamic or a static analysis of shell structures in absence of an angular velocity, would represent different subcases of equation (4.87), when vanishing, conveniently, the terms related to ...
... In modern engineering applications, plate and shell components with different geometries can be produced via wide range of material properties such as isotropic, anisotropic, orthotropic composites, laminated composite, functionally graded composites and carbon nanotubes reinforced composites. Furthermore, different mechanical behaviors such as static or dynamic buckling and free vibrations of composite plates and shells have been largely studied till now [35][36][37][38][39][40][41]. Circular cylindrical shells, conical shells, shallow spherical shells are well known and frequently used shell components in engineering design. ...
... ,σ (k η)w jk (38) As similar the other numerical methods, performance of the DSC method is based on the different parameters such as some DSC parameters, kernels, and grid numbers. In fact, eight-nodded element can also be good alternative for skew plates in view of the accuracy [84]. ...
... Nevertheless, these papers have been focused only on the dynamic response of structures in which the angular velocity is applied about their axis of revolution. In addition, first-order or classical shell theories have been used to investigate this problem, whereas a theoretical framework based on Higher-order Shear Deformation Theories (HSDTs) is developed in this paper [101][102][103][104][105][106][107][108]. In fact, as highlighted in the papers [109][110][111][112][113][114][115], lower-order models could be inadequate to capture the effective mechanical behavior of shell structures made of advanced constituents. ...
... The meaning of each term in (64) can be found in the paper by Tornabene et al. [103]. It can be noticed that a proper expansion in all the three principal directions allows to consider complicated kinematical models which present non constant displacements through the thickness. ...
The paper presents a general theoretical framework to investigate the dynamic behavior of rotating doubly-curved shell structures made of Functionally Graded Materials (FGMs). A clear advancement of the present formulation is the possibility to apply a rotating speed (angular velocity) about a general axis of the global reference system. It is important to underline that this aspect is innovative with respect to the previous studies proposed in the literature, in which the angular velocity is only applied about the revolution axis the shell. Furthermore, several Higher-order Shear Deformation Theories (HSDTs) are used to investigate the problem at issue. The results of various numerical applications are presented to discuss the effect of the choice of the axis of rotation, as well as the geometric features of the shells, on the dynamic response of the structures. In particular, the analyses are performed to evaluate the critical value of rotating speed, which define the stiffness reduction of the structure. A five-parameter power law is developed to define the variation of the mechanical properties of the FGMs along the thickness of the structures. The accuracy of the current formulation, which includes the effects of both Coriolis and centripetal accelerations, is verified by means of the comparison with the results available in the literature. The solutions are carried out numerically by means of an efficient tool that allows to solve the strong formulation of the governing equations.
... Zhu et al. [44] investigated the nonlinear dynamic snap-through and vibrations of temperature-dependent FGM deep arch under sudden thermal shock based on the uncoupled thermoelastic theory by using Ritz method in conjunction with Newton-Raph-Thermodynamic response analysis of functionally graded [45][46][47] proposed a new meshfree method for analyzing the three-dimensional vibration characteristics of laminated doubly-shell subjected to various boundary conditions. Tornabene et al. [48][49][50][51] investigated the dynamic behaviors of doubly-curved shells and panels with various material types based on FSDT and HSDT by employing generalized differential quadrature method. Choe et al. [52] established a theoretical modeling for investigating the vibration characteristics of coupled FG doubly-curved revolution shell based on FSDT by employing unified Jacobi-Ritz method. ...
This paper presents thermodynamic characteristics of functionally graded (FG) doubly-curved panels with varying circumferential size. The circumferential size considered in this paper means the circumferential rotating angle and varies according to certain rules in the longitudinal direction. The theoretical formulation is derived by using Hamilton’s principle in conjunction with first-order shear deformation theory and the displacement and rotating components of the FG doubly-curved panels with varying circumferential size are described approximatively by employing meshfree Tchebychev point interpolation (TPIM) shape function. The thermal effects on the natural frequency of FG doubly-curved panels are studied by employing the thermo-elastic theory. For forced vibration analysis, rectangular and exponential pulse are considered. The convergence of the established theoretical modeling is verified by convergence studies, the validation of the established theoretical modeling is confirmed through comparison with the results of published literature and finite element method. In numerical example, the influences of geometry dimension, boundary conditions, thermal and external loads on the thermo-dynamic characteristics of FG doubly-curved panels with varying circumferential size are investigated systematically. The research results show that the established numerical model has converged basically when the number of node points is greater than Nx = 11 and the boundary spring stiffness increases by more than 1012; the maximum error of comparison results is less than 1%; the boundary conditions, power-law index p, starting angles φ0, temperature and starting circumference A have important influence on the thermodynamic response including free and forced vibration characteristics of FG doubly-curved panels with varying circumferential size.
... The ELW methodology differs from LW as a single reference middle surface is provided for the entire structure. Last but not least, the ESL framework accounts for the reference surface located in the middle thickness, whereas the displacement field variable is taken alongside the surface at issue [17][18][19][20]. As a matter of fact, parametrization of the reference surface of the structure is a key element for the correct theoretical development of the formulation. ...
... Sahoo and Singh (2013) proposed a HOZT for static responses of multi-layered and sandwich composite panels using a FE model. Tornabene, Viola, and Fantuzzi (2013) examined the deflection behavior of laminated curved panels by the Generalized Differential Quadrature (GDQ) technique. The bending and frequency responses of a functionally graded plate are investigated by Hebali et al. (2014) via the newly formulated shear deformation theory (hyperbolic). ...
The nonlinear deflection response of the sandwich shell panel with cutout is analyzed, numerically in the present article. The numerical model of the said curved panel is derived using higher-order shear deformation theory and finite element approximations. The geometrical nonlinearity is being considered via Green-Lagrange’s strain-displacement relation. The governing equation of the nonlinear bending analysis is derived using the energy approach and variational principle. Model accuracy is analyzed as a first step by matching current model responses with previously available data. Further, the model’s capability is explored by solving various new numerical examples and discussed in detail.
... Another interesting aspect regarding shell structural theories is the proper description of the mechanical behaviour for each layer. Due to relatively recent manufacturing issues, many theories have been proposed in the literature, accounting for isotropic, non-homogeneous and orthotropic media [47][48][49][50][51]. Most innovative materials, however, exhibit a completely anisotropic behaviour, and require the introduction of some coupling terms within the formulation, together with a series of homogenization methods for a proper definition of the equivalent mechanical properties. ...
In the present contribution a general formulation is proposed to account for general boundary conditions within the dynamic analysis of anisotropic laminated doubly-curved shell having arbitrary shape and variable thickness. Different analytical expressions are considered for the shell thickness variation along the geometrical principal directions, and the distortion of the physical domain is described by a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) curves. Mode frequencies and shapes are determined employing higher-order theories within an Equivalent Single Layer (ESL) framework. The related fundamental relations are tackled numerically by means of the Generalized Differential Quadrature (GDQ) method. The dynamic problem is derived from the Hamiltonian Principle, leading to a strong formulation of the governing equations. General external constraints are enforced along the edges of the shell employing a distribution of linear springs distributed on the faces of the three-dimensional solid and accounting for a spatial coordinate-dependent stiffness along both in-plane and out-of-plane directions. Moreover, a Winkler-type foundation with general distribution of linear springs is modelled on the top and bottom surfaces of the shell. A systematic set of numerical examples is carried out for the validation of the proposed theory by comparing mode frequencies with predictions from refined three-dimensional finite element analyses. Finally, we perform a sensitivity analysis of the dynamic response of mapped curved structures for different spring stiffnesses and general external constraints, according to various kinematic assumptions.
... Wang et al. [15][16][17] offered unified solutions for dynamic analysis of circular, annular and sector plate structures of orthotropic materials, laminated composites and functionally graded materials under general boundary conditions. Tornabene et al. [18] completed a general higher-order equivalent single layer theory to study free vibrations of doubly-curved laminated composite shells and panels. Li et al. [19,20] analyzed the vibration characteristics of rotating composite laminated cylindrical shells under subsonic air flow and axial load in humid and hot environments. ...
A vibration characteristic analysis model of a laminated composite double cylindrical shell system (LCDCSS) coupled with several annular plates under general boundary conditions is established. Artificial springs are used to simulate the coupling relationship between substructures to ensure the continuity of displacement both at ends of the shells and coupling boundaries. The variable number of annular plates can be distributed unevenly and coupled elastically. Displacement functions of LCDCSS are expressed with improved Fourier series. Based on the principle of energy, obtain the unknown coefficients of the displacement components by using the Rayleigh–Ritz method. The convergence and effectiveness of the proposed method are verified by comparing with the results with literature and FEM, and then carried out parametric investigation to study the free and steady-state response vibration characteristics of LCDCSS. Rapid prediction of free vibration and response vibration of a double-layer cylindrical shell system with various structures and scales is realized by exploiting the model, and some new results of double-layer cylindrical shell system are explored, which can provide reference for further research.
... Based on Carrera's unified formulation (CUF), these structural models are obtained using the N-order Taylor expansion to expand the unknown displacement variables. Tornabene et al. [24,25] derived a general formulation of 2D higher-order equivalent plate theory. The theoretical framework covers the static and dynamic analysis of shell structures by using a general displacement field based on CUF. ...
Thick wall structures are usually applied at a highly reduced frequency. It is crucial to study the refined dynamic modeling of a thick plate, as it is directly related to the dynamic mechanical characteristics of an engineering structure or device, elastic wave scattering and dynamic stress concentration, and motion stability and dynamic control of a distributed parameter system. In this paper, based on the partial differential operator theory, an exact elasto-dynamics theory without assumptions for bending vibrations is presented by using the formal solution proposed by Boussinesq–Galerkin, and its dynamic equations are obtained under appropriate gauge conditions. The exact plate theory is then compared with other theories of plates. Since the derivation of the dynamic equation is conducted without any prior assumption, the proposed dynamic equation of plates is more exact and can be applied to a wider frequency range and greater thickness.
... Since the research object of this work is related to the thin laminated composite shallow shells, the most basic thin shell theory is adopted to construct the theoretical formulation for vibration analysis of shell. For the numerical analysis method, most commonly used methods include the Rayleigh-Ritz method (RRM) [44][45][46][47][48][49][50][51][52][53][54][55][56][57]112], the finite element method (FEM) [58][59][60][61][62][63][64]111,115], and the generalized differential quadrature (GDQ) method [36][37][38][65][66][67][68]114] etc. Due to the standard evaluation procedure, fast convergence speed, and low computational cost as compare to FEM, the Rayleigh-Ritz method is commonly conducted for vibration analysis of shallow shells, where different admissible functions may be adopted, e.g., Legendre polynomials [69][70][71]110], orthogonal polynomials [44,72,73], (modified) Fourier series [74], or combination of them [45]. To facilitate the design optimization of laminated composite shallow shells, in this work, the Legendre polynomials is adopted in the Rayleigh-Ritz method to perform the vibration analysis. ...
A sequential permutation search (SPS) optimization algorithm is proposed for the stacking sequence design of doubly-curved laminated composite shallow shells. The SPS algorithm takes advantages of the inherent physical–mechanical features of composite laminates that the outer layers contribute more than the inner layers to bending rigidity and the convexity of lamination parameters. In SPS, the linear and nonlinear sensitive detection techniques are introduced, which detect the sensitive ply orientation at a proper stacking position. By assigning identical ply orientation at respective stacking position, and designing the stacking sequence from the inner to the outer positions sequentially and iteratively using the sensitive ply orientation, the solution will converge to the optimum when no sensitive ply orientation can be detected at any stacking position. In addition, the bending-twisting coupling effects of the laminates are regulated by means of a sign optimization algorithm (SOA) coupled in SPS. The stacking sequences of doubly-curved laminated composite shallow shells exhibit various boundaries and geometries are optimized to maximize the fundamental frequency, and the Rayleigh-Ritz method is developed for vibration analysis. The optimal results are compared with those of layerwise optimization approach and genetic algorithm, demonstrating the robustness and efficiency of the SPS algorithm.
... An FSDT based model was proposed by Kiani et al. [13] and panels. A general higher-order theory with stretching and zig-zag effects was proposed by Tornabene and Fantuzzi [22] to study the free vibration of doubly-curved shells and panels. Panda and Singh [23,24] studied the nonlinear free vibration behavior of single/doubly-curved shells within the post-buckled state under thermal loads. ...
In this research, a general model to study the vibration behavior of axially moving two-dimensional continuums in the presence of curvature along the moving axis is developed. To this end, an axially moving doubly-curved panel of variable radius of curvature is considered. The integral boundary value problem is obtained based on a higher-order shear deformation with first-order thickness stretching theory. Due to its high accuracy and computational performance, spectral Chebyshev approach is used to numerically solve the boundary value problem. Considering the geometry capabilities of the developed model, dynamics of various axially moving structures such as flat, singly- and doubly-curved plates/shells in different engineering applications with different boundary conditions can be investigated. The numerical results confirmed that the calculated natural frequencies for axially moving flat plates and circular cylindrical shells are in excellent agreement to those found in the literature and obtained via finite element approach. Furthermore, the effects of the axial velocity, thickness stretching, curvature ratio, and boundary conditions on the natural frequencies and stability behavior of the doubly-curved panels are investigated.
... Setoodeh et al. (2009) utilized the theory of 3D elasticity and FEM to evaluate the transient response of laminated composite plates under low velocity impact loads. Tornabene et al. (2013) presented a higher order theory to study the vibrational frequencies of doubly-curved panels made of laminated composite materials. Wang ...
This paper proposes the use of a porous core between layers of laminated composite plates to examine its effect on the natural frequencies of the resulted porous laminated composite sandwich plate (PLCSP) resting on a two-parameter elastic foundation. Moreover, it has been suggested that the dispersion of porosity has two different functionally graded (FG) patterns which are compared with a uniformly dispersed (UD) profile to find their best vibrational efficiency in the proposed PLCSPs. In FG patterns, two types of dispersions, including symmetric (FG-S) and asymmetric (FG-A) patterns have been considered. To derive the governing Eigen value equation of such structures, the first order shear deformation theory (FSDT) of plates has been employed. Accordingly, a finite element method (FEM) is developed to solve the derived Eigen value equation. Using the mentioned theory and method, the effects of porosity parameters, fiber orientation of laminated composite, geometrical dimensions, boundary conditions and elastic foundation on the natural frequencies of the proposed PLCSPs have been studied. It is observed that embedding porosity in core layer leads to a significant improvement in the natural frequencies of PLCSPs. Moreover, the natural frequencies of PLCSPs with FG porous core are higher than those with UD porous core. (PDF) The effect of embedding a porous core on the free vibration behavior of laminated composite plates.
... They investigated six groups of panels and shells such as spherical and cylindrical panels, rectangular and annular plates, in particular an elliptic paraboloid and a catenoidal shell. Tornabene et al. [21] proposed an equivalent single-layer model for laminated composite structures in order to study free vibrations of the modeled single-layer shell/panel. Their consideration also covered the zig-zag and stretching effects for the material. ...
A drawback to the material composition of thick functionally graded materials (FGM) beams is checked out in this research in conjunction with a novel hyperbolic-polynomial higher-order elasticity beam theory (HPET). The proposed beam model consists of a novel shape function for the distribution of shear stress deformation in the transverse coordinate. The beam theory also incorporates the stretching effect to present an indirect effect of thickness variations. As a result of compounding the proposed beam model in linear Lagrangian strains and variational of energy, the system of equations is obtained. The Galerkin method is here expanded for several edge conditions to obtain elastic critical buckling values. First, the importance of the higher-order beam theory, as well as stretching effect, is assessed in assorted tabulated comparisons. Next, with validations based on the existing and open literature, the proposed shape function is evaluated to consider the desired accuracy. Some comparative graphs by means of well-known shape functions are plotted. These comparisons reveal a very good compliance. In the final section of the paper, based on an inappropriate mixture of the SUS304 and Si3C4 as the first type of FGM beam (Beam-I) and, Al and Al2O3 as the second type (Beam-II), the results are pictured while the beam is kept in four states, clamped-clamped (C-C), pinned-pinned (S-S), clamped-pinned (C-S) and in particular cantilever (C-F). We found that the defect impresses markedly an FGM beam with boundary conditions with lower degrees of freedom.
... With the increase in the use of composite materials in engineering applications, many beam theories have been developed for predicting the response of laminated composite beams. In the open literature, the equivalent single layer theories such as classic, first-order and higherorder shear deformation theories has been developed for laminated composite beam analyses [32][33][34][35]. Literature reviews of equivalent single layer theories for isotropic and composite beams were presented Seyyid and Ghugal [36]. ...
Fiber reinforced polymer (FRP) composite materials produced by pultrusion technique attract the attention of researchers due to their superior properties. Limited studies are available in the literature where experimental and theoretical investigations on the FRP composites are carried out together. In this study, the flexure performances of pultruded glass FRP (P-GFRP) composite beams were investigated experimentally and theoretically. Theoretical approaches for flexural analysis of P-GFRP composite beams were developed with the help of variational methods. Kinematic relations of composite beams were defined based on high order shear deformation beam theory. Effective material properties of composite beams were obtained by using mixture rule model. The differential field equations were transformed to functional with Gâteaux differential method. The element matrix with a total of 10 degrees of freedom was obtained by using the mixed finite element method (MFEM). Moreover, classical finite element modeling (FEM) was performed with the help of ABAQUS program. Various beam specimens with different fiber orientations were extracted from the P-GFRP box profile. Tensile and burnout tests were performed to determine the mechanical properties of the obtained P-GFRP composite beam specimens. Three-point bending test was utilized to investigate flexure performance. Kinetic, macro and micro mechanical damage analyzes were performed to better understand the behavior of the P-GFRP composite beams. Experimental results, theoretical solution results and FEM simulation results were compared and numerical values were found to be very close to each other.
... A general higher-order equivalent single layer (GHESL) formulation based on Carrera unified formulation for the free vibrations of doubly-curved laminated composite shells and panels was illustrated by Tornabene et al. [18,19]. A comparison between classical two-dimensional (2D) finite elements (FEs) and an exact threedimensional (3D) solution for the free vibration analysis of onelayered and multilayered isotropic, composite and sandwich plates and cylinders was proposed by Brischetto and Torre [20]. ...
... since the radii of curvature are constant along the midsurface curvature coordinates " " and " ". The radii of curvature of a shell structure are considered as variables when they depends of a certain function of the midsurface curvilinear coordinates ( , ) an some references can be found in excellent works by Tornabene et al. [38][39][40], for further details on the calculations of radii of curvature the Ref. [41] is recommended. ...
This paper presents Best Theory Diagrams (BTDs) constructed from non-polynomial terms to identify best shell theories for bending analysis of cross-ply single skin and sandwich shell panels. This structure presents a constant radii of curvature. The shell theories are constructed using Axiomatic/Asymptotic Method (AAM). The different shell theories are described using the Carrera’s Unified Formulation. The governing equations are derived from the Principle of Virtual Displacement (PVD). Navier-Type closed form solution is used for solving the bending problem of simply supported doubly curved shell panels subjected to bi-sinusoidal transverse pressure. The BTDs built from non-polynomial functions are compared with Maclaurin expansions. Spherical shell panels with different layer-configurations are investigated. The results demonstrated that the shell models obtained from the BTD using non-polynomial terms can improve the accuracy obtained from Maclaurin expansion for a given number of unknown variables of a displacement field.
... So far, scholars mainly utilize numerical methods to solve the vibration problems of the arbitrary-shaped plate and shell, for instance, finite element approach [1], generalized differential quadrature finite element method [2], general higher-order equivalent single layer theory [3], differential quadrature method [4,5], complex variable methods [6], and differential volume method [7]. Ahmad et al. [8] proposed a method to overcome the disadvantage of former ways when approximate geometric structure and the influence of shear displacement were ignored, by using the curved thick shell finite element method. ...
In this investigation, an improved Rayleigh–Ritz method is put forward to analyze the free vibration characteristics of arbitrary-shaped plates for the traditional Rayleigh–Ritz method which is difficult to solve. By expanding the domain of admissible functions out of the structural domain to form a rectangular domain, the admissible functions of arbitrary-shaped plates can be described conveniently by selecting the appropriate admissible functions. Adopting the spring model to simulate the general boundary conditions, the problems of vibration of the arbitrary plate domain can be solved perfectly. Then, a numerical method is introduced to figure out the structure strain energy, kinetic energy, and elastic potential energy of the boundary. Finally, comparing the result with the simulation results and reference examples, the accuracy and convergence of this method are testified. Therefore, an effective new method is proposed for the guidance of the related research and practical engineering problems.
... Among them, the unified formulation, known as Unified Formulation (UF), should be mentioned [1,[68][69][70][71][72]. To the best of the authors' knowledge, the UF represents an efficient methodology to study the mechanical behavior of beams, plates and shells made of composite materials, as proven in the papers [73][74][75][76][77][78][79][80][81][82][83][84][85][86][87][88]. In particular, the researches just mentioned are based on an Equivalent Single Layer (ESL) approach, since the kinematic expansion of the displacement field is developed within the shell middle surface, which represents the reference domain of the structural problem at issue. ...
A theoretical framework based on an Equivalent Single Layer (ESL) approach is proposed in this chapter to develop several Higher-order Shear Deformation Theories (HSDTs) in a unified and compact manner. In particular, the maximum order of kinematic expansion can be arbitrarily chosen in order to define more refined displacement fields. The Murakami’s function can be also included in the model to take into account the so-called zig-zag effect. The proposed theory is employed to describe the mechanical behavior of doubly-curved shell structures made of composite materials. In particular, the differential geometry is used to define accurately the curved surfaces at issue. The strong formulation of the governing equation is solved by means of a numerical approach based on the Generalized Differential Quadrature (GDQ) method. The accuracy of both the theoretical model and the numerical method is shown through some applications, in which the solutions are compared with the results obtained by means of a three-dimensional finite element model.
... More general approaches based on this formulation have been also developed during the years, as it can be noted in the following papers [70][71][72][73][74][75]. It should be mentioned that in general HSDTs are classified as Equivalent Single Layer (ESL) theories if the governing equations are defined in the middle surface of the structure [76][77][78][79][80], but a kinematic expansion along the thickness can be also developed according to the so-called Layer-Wise (LW) approaches [81][82][83][84][85]. Hybrid formulations are also possible, such as an Equivalent Layer-Wise (ELW) model [86]. ...
The aim of this chapter is the development of an efficient and accurate higher-order formulation to solve the weak form of the governing equations that rule the mechanical behavior of doubly-curved shell structures made of composite materials, whose reference domain can be defined by arbitrary shapes. To this aim, a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) is introduced. It should be specified that the theoretical shell model is based on the Equivalent Single Layer (ESL) approach. In addition, the Generalized Integral Quadrature technique, that is a numerical tool which can guarantee high levels of accuracy with a low computational effort in the structural analysis of the considered shell elements, is introduced. The proposed technique is able to solve numerically the integrals by means of weighted sums of the values that a smooth function assumes in some discrete points placed within the reference domain.
... Kiani et al. [40] inspected dynamic behavior of a functionally graded (FG) doubly curved shell subjected to sensor piezoelectric considering FSDT. However, two-dimensional higher order theory was applied by Tornabene et al. [41] to analyze vibration of a doubly curved composite shell in different curvatures. They [42] also considered generalized displacements of the Carrera Unified Formulation to analyze dynamic of the FG doubly curved shells. ...
Multi-objective vibroacoustic optimization of the double-walled doubly curved composite shells having poroelastic lining in its core in a diffuse field is performed based on Non-dominated sorting Genetic Algorithm-II. To present an analytical model on the basis of multi-objective optimization, the summation of sound transmission loss and transverse displacement along with weight of the structure are considered as two cost functions, which should be optimized in a diffuse field. In fact, the significant achievement of this work is to design an optimization algorithm to improve vibroacoustic fitness and weight of the sandwich doubly curved shells. In the first part of the paper, a general formulation is prepared to analyze the dynamic of the poroelastic composite sandwich structures. Likewise, some validation configurations are presented to confirm the accuracy of the current formulation. Consequently, an optimization algorithm is provided on the basis of considering some appropriate design variables including material and porous types as well as stacking sequences. In this regard, a batch of 19 benchmarks of porous core is investigated. Furthermore, a configuration of optimized points in the Pareto front is plotted in which the simultaneous effects of optimizing the weight and vibroacoustic fitness can be observed. As a result, a new approach is made through optimization of the transverse displacement of the structure as a function of various incidence and azimuth angles in three dimensional configurations with respect to different frequencies.
... They also employed GDQ to present numerical solution for the two dimensional vibration problems. Fallowing the last works, Tornabene et al. [25,26] Galerkin methods were employed. These techniques were also applied in another work [33] to dynamic analyze of composite structure with negative Poisson's ratio. ...
Vibroacoustic performance of the doubly curved thick shell is explored based on the three dimensional sound propagation approach as well as state space solution. In fact, the main aim is particularly focused on inspecting the influence of using three-dimensional theory through sound transmission loss (STL)of the structure which includes more reliable and accurate results especially for relatively thick and thick shells even in high frequency domain in comparison with other theories. In order to achieve this end, firstly stress and strain components are developed to present the governing equations of thick shell. This procedure is carried out by dividing the shell into slayers. Then, a solution technique is provided on the basis of state vector methodology wherein approximate layer model along with local transfer matrix are performed. Moreover, this method is followed by global transfer matrix method for the all layers of structure. As an outcome, in results section, not only the accuracy of the offered results is proved but also the importance of employing the current theory in high frequencies is revealed. Another remarkable achievement of this work is related to nominate the dip points of STL diagram. On contrary to panels, doubly curved shells contain two dips, because of their both radii of curvatures. In this work, the first dip is nominated as curvature frequency. The second dip is similar to that of panels at high frequency zone. Thus, it is called as coincidence frequency. Finally, the behavior of the transmitted pressure and the effect of curvatures on the position of curvature frequency are discussed.
The main aim of this book is to analyze the mathematical
fundamentals and the main features of the Generalized
Differential Quadrature (GDQ) and Generalized Integral
Quadrature (GIQ) techniques. Furthermore, another interesting
aim of the present book is to shown that from the two numerical
techniques mentioned above it is possible to derive two different
approaches such as the Strong and Weak Finite Element
Methods (SFEM and WFEM), that will be used to solve various
structural problems and arbitrarily shaped structures.
A general approach to the Differential Quadrature is
proposed. The weighting coefficients for different basis
functions and grid distributions are determined. Furthermore,
the expressions of the principal approximating polynomials
and grid distributions, available in the literature, are shown.
Besides the classic orthogonal polynomials, a new class of basis
functions, which depend on the radial distance between the
discretization points, is presented. They are known as Radial
Basis Functions (or RBFs). The general expressions for the
derivative evaluation can be utilized in the local form to reduce
the computational cost. From this concept the Local Generalized
Differential Quadrature (LGDQ) method is derived.
The Generalized Integral Quadrature (GIQ) technique can be
used employing several basis functions, without any restriction
on the point distributions for the given definition domain.
To better underline these concepts some classical numerical
integration schemes are reported, such as the trapezoidal rule or
the Simpson method. An alternative approach based on Taylor
series is also illustrated to approximate integrals. This technique
is named as Generalized Taylor-based Integral Quadrature (GTIQ)
method.
The major structural theories for the analysis of the
mechanical behavior of various structures are presented in depth
in the book. In particular, the strong and weak formulations
of the corresponding governing equations are discussed and
illustrated. Generally speaking, two formulations of the same
system of governing equations can be developed, which are
respectively the strong and weak (or variational) formulations.
Once the governing equations that rule a generic structural
problem are obtained, together with the corresponding boundary
conditions, a differential system is written. In particular, the
Strong Formulation (SF) of the governing equations is obtained.
The differentiability requirement, instead, is reduced through
a weighted integral statement if the corresponding Weak
Formulation (WF) of the governing equations is developed. Thus,
an equivalent integral formulation is derived, starting directly
from the previous one. In particular, the formulation in hand is
obtained by introducing a Lagrangian approximation of the
degrees of freedom of the problem.
The need of studying arbitrarily shaped domains or
characterized by mechanical and geometrical discontinuities
leads to the development of new numerical approaches that
divide the structure in finite elements. Then, the strong form or
the weak form of the fundamental equations are solved inside
each element. The fundamental aspects of this technique,
which the author defined respectively Strong Formulation Finite
Element Method (SFEM) and Weak Formulation Finite Element
Method (WFEM), are presented in the book.
Modeling and analysis of composites with curvilinear fiber reinforcement is rather challenging in terms of accuracy and computational cost associated with variable material stiffness. In this study, to reduce the computational cost drastically without sacrificing the numerical accuracy, variable stiffness composite laminate (VSCL) is modelled as a single layer based on the refined zigzag theory (RZT). To this end, a three-node triangle RZT element formulation is adopted and effectively implemented for static analysis of multilayer composites and sandwich plates with curvilinear fiber paths. Moreover, to accurately model the strains in VSCL, the derivatives of the zigzag with respect to planar coordinates are considered for each ply within the laminate in the RZT kinematic-strain relations. Enhanced capability of the present model is verified by performing comprehensive numerical investigation on several benchmark cases. The obtained results are compared with those present in the literature and three-dimensional elasticity solutions. Hence, it is demonstrated that the triangular RZT element is a fast, robust, and accurate structural analysis platform that can potentially lend itself to the optimization of curvilinear fiber angles of VSCL.
The availability of limited literature on thickness-shear mode and thickness-twist mode frequencies of laminated composite plates and shells of double curvature motivated the authors to perform this research study. Cylindrical, spherical, hyperbolic paraboloid and elliptical paraboloid laminated composite shells are analyzed using a generalized higher-order shell theory. Governing equations of motion corresponding to the present theory are derived within the framework of Hamilton’s principle. Analytical solutions are obtained using Navier’s technique for the simply supported boundary conditions. Non-dimensional flexural mode, thickness-shear mode, and thickness-twist mode natural frequencies are presented. Natural frequencies obtained using the classical, first-order, and the present higher-order shell theories are compared with each other and previously published results wherever possible to verify the accuracy and efficiency of the present generalized shell theory. Thickness-shear mode and thickness-twist mode frequencies presented in this paper can be considered as one of the important contributions towards the research in dynamic analysis of laminated composite shells of double curvature.
In this paper, a unified method for free and forced vibration of functionally graded porous annular and circular plate is proposed, incorporating the Jacobi–Ritz method and the higher-order shear deformation theory. In order to improve the accuracy of calculation, the segment strategy is adopted. Meanwhile, the artificial spring is used to express the connection between each truncated plates and arbitrary boundary conditions. The convergency study and validation are implemented, and parametric study of both free and transient forced vibration are conducted. The present method shows a fast convergence property and high accuracy for thick plate. It has been found that boundary conditions are the dominant influencing factor of the central deflection of functionally graded porous annular plate, and the porous material with vertical gradient change in the plate leads to a better impedance effect than the uniformly distributed porous material.
The present study is concerned with an analytical solution for calculating sound transmission loss through an infinite double-walled circular cylindrical shell with two isotropic skins and a polymeric foam core. Accordingly, the two-walled cylindrical shell is stimulated applying an acoustic oblique plane wave. The equations of motion are derived according to Hamilton’s principle using the first-order shear deformation theory for every three layers of the construction. Additionally, by the aid of employing the Zener mathematical model for the core of polymeric foam, mechanical properties are determined. To authenticate the results of this study, the damping of the core layer goes to zero. Therefore, the numerical results in this special case are compared with those of isotropic shells. The results prove that the presented model has high accuracy. It is also designated that decreasing the power-law exponent of the core leads to improving the sound transmission loss through the thickness of the construction. Besides, in addition to probe some configurations versus alterations of frequencies and dimensions, the convergence algorithm is provided. Consequently, it is realized that by increasing the excitation frequency, the minimum number of modes to find the convergence conditions is enhanced. The results also contain a comparison between the sound transmission loss coefficient for four different models of a core of a sandwiched cylindrical shell. It is comprehended that the presented model has a transmission loss coefficient more than the other types of the core at high frequencies.
In this paper, a FSDT meshfree method based on the 3D continuous shell theory and moving-least squares approximation is proposed for the first time to investigate the free vibration behavior of arbitrary laminated composite shells and spatial structures. The novelty lies in that the resultant meshfree model has completely general geometric and kinematic descriptions, which are more readily applied to complex laminated shell geometries. In the formulation, the geometrical representation is based on the mapping technique without simplifying assumptions on the shell geometry, whereas the moving least-squares (MLS) approach and the first shear order theory (FSDT) are employed to build displacement fields. By using the mapping technique, a 3D arbitrary curved surface is expanded in a two-dimensional space, and the displacement approximation is established based on a set of nodes scattered in the Converted coordinate system. The simple Gauss integral are also formed in the Converted coordinate system. The essential boundary conditions are imposed by the full transformation method. According to Hamilton principle, the meshfree governing equations of free vibration of arbitrary laminated shells are obtained. The convergence, accuracy and applicability of the proposed method are demonstrated for the laminated shells of different geometrical shell shapes involving the square plates, shallow shells, open cylindrical shells, conical shells and corrugated plates with different boundary conditions and lamination schemes.
In this paper, an energy-spectral boundary element formulation is proposed to study the vibro-acoustic characteristics of elliptical caps immersed in acoustic medium. Based on the first-order shear deformation theory, the energy principle and segment technology are used to establish the structural vibration analysis model. The displacements are written by the Jacobian orthogonal polynomials and Fourier series. The one-dimensional Helmholtz integral formulation is adopted to describe the external acoustic field, where the formulation can be solved by discretization along the generator line of the elliptical cap. On this basis, the vibro-acoustic model of the elliptical cap can be established by considering the energy of the acoustic field acting on the elliptical surface. By the comparison of the results of the vibration and acoustic response with the coupled FEM/BEM method, the reliability and accuracy of the present analysis model are verified. Furthermore, the studies on effects of circumferential wave modes, boundary condition and acoustic medium on the vibro-acoustic behavior of the elliptical cap are made, which is helpful for the design of elliptical cap to some extent.
A beam finite element model is proposed for the static and free vibration analyses of FGM sandwich beams with viscoelastic nonlinear material behavior. In the analysis, zigzag theory is adopted for the displacement fields. The damping results from the shear properties of the viscoelastic layer and its low stiffness. Timoshenko 1st order and Reddy’s higher order shear models are implemented for static and vibration behaviors. Various viscoelastic frequency-dependent laws are considered. The resulting stiffness matrix is nonlinear and is frequency dependent. Solutions are possible according to a powerful asymptotic method combined with recent method for the power series terms. The efficiency of the present model is proven through simulations using 3D volume elements of Abaqus code, where new module for viscoelastic FGM materials is now available. The effects of power law index and boundary conditions on static, vibration and the damping properties of the viscoelastic sandwich FGM beam are investigated successfully. It is shown that the beam behavior is very sensitive on the loss factor. In vibration, the damping properties are nonlinearly power law index dependent. The boundary conditions have an incidence on the vibration modes. The cantilever case is particular and interesting that needs an optimization process.
This study presents, for the first time, a unified wavelet finite element formulation for static and free vibration analysis of laminated composite shells, combining the wavelet finite element method and general shell theory. The governing equations of the proposed unified wavelet finite element formulation are derived from general shell theory and first-order shear deformation theory (FSDT) using the principle of minimum total potential energy. The formulated kinematics of laminated shells can be applied to flat plates, cylindrical shells, and doubly-curved shells with the selection of appropriate Lamé coefficients with surface metric tensor and radii of curvature parameters. With the excellent approximation of B-spline wavelets on the interval (BSWI) for structure analysis, BSWI scaling functions are used as interpolating functions to construct wavelet finite elements for laminated shells. The accuracy and efficiency of the proposed BSWI method are assessed through numerical comparisons with analytical 3-D elasticity solutions and other reference solutions in the literature for static and free vibration analysis of laminated composite plates, cylindrical shells, and doubly-curved shells.
There is not enough mixed finite element method (MFEM) model developed for static and dynamic analysis of functionally graded material (FGM) beams in the literature. The main purpose of this study is to develop a reliable and efficient computational modeling using an efficient functional in MFEM for free vibration and static analysis of FGM composite beams subject to high order shear deformation effects. The modeling of material properties was performed using mixture rule and Mori-Tanaka scheme which are more realistic determination techniques. This method based on the assumption that a two phase composite material consisting of matrix reinforced by spherical particles, randomly distributed in the beam. To explain the displacement components of the shear deformation effects, it was accepted that the shear deformation effects change sinusoidal. Partial differential field equations were obtained with the help of variational methods and then these equations were transformed into a novel functional for FGM beams with the help of Gâteaux differential derivative operator. Thanks to the Gâteaux
differential method, the compatibility of the field equations was checked, and the field equations and boundary conditions were reflected to the function. A MFEM model was developed with a total of 10 degrees of freedom to apply the obtained functional. In the numerical applications section, free vibration and flexure problems solutions of FGM composite beams were compared with those predicted by other theories to show the effects of shear deformation, thickness changing and boundary conditions.
A semi-analytical method to conduct vibro-acoustic analysis of a composite laminated elliptical shell immersed in air is proposed. A variational method and multi-segment technique are used to formulate the dynamic model. The sound radiation of the exterior fluid field is calculated by a spectral Kirchhoff-Helmholtz integral formulation. The variables containing displacements and sound pressure are expanded by the combination of Fourier series and Chebyshev orthogonal polynomials. The collocation points are introduced to construct an algebraic system of acoustic integral equations, where these points are distributed on the roots of Chebyshev polynomials, and the non-uniqueness solution of system is eliminated by a combined Helmholtz integral. Numerical examples for sound radiation problems of composite laminated elliptical shells are presented and individual contributions of the circumferential modes to the acoustical results of composite laminated elliptical shells are also given. The effects of geometric and material parameters on sound radiation of composite laminated elliptical shells are also investigated.
As a first endeavor, this study scrutinizes the effects of general lamination schemes on the natural frequencies and loss factors of simply supported doubly-curved laminated composite panels. Transversely isotropic fibers obey linear elastic rule while isotropic polymeric matrix follows both standard solid and Kelvin-Voigt viscoelastic models. Frequency-dependent properties are obtained by multi-cell micromechanical approach and Alfrey correspondence principle. The governing equations of motions with frequency-dependent shear-stretching, bending-stretching and bending-twisting couplings are derived using Hamilton principle and third-order shear deformation theory. The coupled PDEs of motions are figured out via Galerkin method and nonlinear eigenvalue solver. To verify, vibrational characteristics are compared with available results of elastic laminated panels and viscoelastic sandwich plates. Then, the effects of constitutive relations, fiber volume fraction, different fibers, geometrical parameters, symmetric, anti-symmetric and asymmetric laminations schemes are investigated on vibrational characteristics of spherical, cylindrical and hyperboloid panels. Also, crossing phenomenon and curve veering of frequencies and loss factors are observed.
Free vibration of sandwich plates with temperature dependent functionally graded (FG) face sheets in various thermal environments is investigated. The material properties of FG face sheets are assumed to be temperature-dependent and vary continuously through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. Also, the material properties of the core are assumed to be temperature dependent. The governing equations of motion in polar system and in free natural vibration are derived usingHamilton’s principle and Galerkin method is used to solve the equations and obtain the natural frequency. In-plane stresses of the core that usually are ignored in the vibration characteristics of the sandwich structures are considered in this formulation. The results obtained by Galerkin method for symmetric circular sandwich plate with fixed support is compared with finite element method that obtained by ABAQUS and good agreement is found. The results show that varying the power-law index and temperature have important effects on natural frequency
Analysis of overhead power transmission lines (OPL) involves the simulation of statics and dynamics of conductors and cables together with spiral accessories, vibration dampers, and other devices accounting for internal conductors’ structures. As a typical conductor is formed by wire layers wound on each other at different angles, known issues arise in the estimates of stiffness, bearing capacity, and other properties of such structure. Indeed, the bending stiffness of the conductor depends considerably on its deformation and vary along the conductor axis as well as in time since the wire layers may slip relative to each other, and a separate wire is movable within the wire layer. On the other hand, each wire layer could be considered as an equivalent elastic anisotropic cylindrical shell on the basis of energy averaging, therefore a conductor or a spiral clamp could be modeled as a system of shells nested into each other and interacting by means of pressure and friction. This method allows one to obtain the formulae for the flexibility and stiffness of spiral structures. Some simulation results for two-layer connecting clamps with conductors including the estimates for the bearing clamp capacity limits are shown below.
The main content of this paper is to establish an analysis model for dynamic analysis of composite laminated doubly-curved revolution shell based on the Higher order Shear Deformation Theory (HSDT). Firstly, the energy functional of shell is established based on higher order shear theory. Then, the multi-segment partitioning technique is introduced to segment the shell along the generatrix direction, which is mainly to relax the boundary conditions of shell, and then reduce requirements for the displacement function. At the boundary position, the boundary spring parameters are used to obtain the corresponding boundary conditions. Similarly, the connecting spring parameters are introduced to simulate the continuity conditions between the segmented shells. The parameters of boundary spring and connecting spring can be regarded as the weight parameter. Thirdly, all displacement components of the composite laminated doubly-curved revolution shell are expressed by Jacobi orthogonal polynomials. Finally, the whole dynamic characteristics are obtained by Rayleigh-Ritz method with respect to unknown Jacobian expansion coefficient. The convergence, validity and dynamic characteristics of the analytical model established in this paper are given by a series of numerical examples.
This paper focuses on the dynamics of doubly-curved functionally graded and laminated composite structures with arbitrary geometries and boundary conditions. Integral boundary value problem is obtained following an energy-based approach where the strain energy of the structure is expressed using three-dimensional elasticity equations. The effective properties of functionally graded materials can be described based on Mori-Tanaka or theory of mixtures methods. To simplify the domain of the problem, coordinate transformations are applied to map the curved structure into a straight one; and furthermore, a one-to-one mapping technique is applied to map the (complex) curved geometry to a master geometry in the case of composites with arbitrary geometries. Then, the integral boundary value problem is discretized by means of Gauss-Lobatto sampling and solved using the three-dimensional spectral-Tchebychev approach. In this method, the system matrices are calculated through the exact evaluation of differentiation and integration operations using the derived Tchebychev matrix operators. Finally, if necessary, to impose the essential boundary conditions on the boundary value problem and to assemble multiple layers, the projection matrices approach is used. Various case studies including (i) doubly-curved structures, (ii) doubly-curved laminated composites and (iii) doubly-curved laminated composite structures with arbitrary geometries are analyzed. In each case study, to present the accuracy/precision of the developed solution technique, the predicted (non-dimensional) natural frequencies and mode shapes are compared to those obtained using either a commercial finite element software and/or to those found in literature. It is shown that the developed three-dimensional spectral-Tchebychev solution technique enables accurately and efficiently capturing the vibration behavior of doubly-curved laminated composite structures having arbitrary geometries under different boundary conditions.
The free vibration of different kinds of stepped coupled doubly-curved shell structures with elastically constrained edges are investigated by adopting the Jacobi-Ritz method for the first time. Coupled structures are comprised of substructures, where paraboloidal, hyperbolical, elliptical, and cylindrical stepped shells are typical ones. The Flügge's thin shell theory is utilized to construct the analytical model, together with the multi-segment partitioning strategy. For each shell segment, despite of various boundary conditions, the displacement components along the meridional directions are expressed by Jacobi polynomials and those along the circumferential directions are represented by Fourier series. Then the unknown coefficients of the displacements are obtained by introducing the Rayleigh-Ritz method. The solutions proposed here for coupled structures have two main advantages: first, there is no need to vary the displacement or the motion equations; and secondly, the efficiency of modeling can be notably enhanced. By comparison with (Finite Element method) FEM and others’ results, the reliability of current method can be validated. At last, the free vibrations of different kinds of coupled structures containing stepped shell are analyzed by presenting several numerical examples, the results of which may be served as reference data.
The thermo-mechanical vibration characteristics of doubly-curved nano-composite shells reinforced by graphene nanoplatelets are investigated by considering a uniform distribution of graphene and a first-order shear deformation theory. The mechanical properties of the nano-composite shells are estimated by using the modified Halpin–Tsai model. The governing equations are first derived by a variational formulation using Hamilton’s principle and are solved using the Galerkin technique. Numerical results are presented for various shell curvatures and compared with those available in the archival literature. Furthermore, parametric studies are offered to highlight the significant influence of graphene nanoplatelets’ weight fraction, dimensions of graphene nanoplatelets, and temperature variation, on the free vibration of the nano-composite shells.
In this paper, a comparative study of refined beam theories has been done for the free
vibration analysis of thick beams, taking into account transverse shear deformation effect.
The theories involves parabolic, sinusoidal, hyperbolic and exponential functions in-terms
of thickness coordinates to include transverse shear deformation effect. The numbers of
unknowns are same as that of first order shear deformation theory. The governing
differential equations and boundary conditions are obtained by using the principle of
virtual work. The results of bending and thickness shear mode frequencies for simply
supported beam are presented and discussed critically with those of other theories. The
results are found to agree well with the exact elasticity results wherever applicable.
Comparison of dynamic shear correction factor is carried out using various shear
deformation theories.
This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution struc-tures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present for-mulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to invest-tigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behav-ior of laminated shells of revolution.
This study deals with a mixed static and dynamic optimization of four-parameter functionally graded material (FGM) doubly curved shells and panels. The two constituent functionally graded shell consists of ceramic and metal, and the volume fraction profile of each lamina varies through the thickness of the shell according to a generalized power-law distribution. The Generalized Differential Quadrature (GDQ) method is applied to determine the static and dynamic responses for various FGM shell and panel structures. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT). Three different optimization schemes and methodologies are implemented. The Particle Swarm Optimization, Monte Carlo and Genetic Algorithm approaches have been applied to define the optimum volume fraction profile for optimizing the first natural frequency and the maximum static deflection of the considered shell structure. The optimization aim is in fact to reach the frequency and the static deflection targets defined by the designer of the structure: the complete four-dimensional search space is considered for the optimization process. The optimized material profile obtained with the three methodologies is presented as a result of the optimization problem solved for each shell or panel structure.
Two new hyperbolic displacement models, HPSDT1 and HPSDT2, are used for the buckling and free vibration analyses of simply
supported orthotropic laminated composite plates. The models contain hyperbolic expressions to account for the parabolic distributions
of transverse shear stresses and to satisfy the zero shear-stress conditions at the top and bottom surfaces of the plates.
The equation of motion for thick laminated rectangular plates subjected to in-plane loads is deduced through the use of Hamilton’s
principle. Closed-form solutions are obtained by using the Navier technique, and then the buckling loads and the fundamental
frequencies are found by solving eigenvalue problems. The accuracy of the models presented is demonstrated by comparing the
results obtained with solutions of other higher-order models given in the literature. It is found that the theories proposed
can predict the fundamental frequencies and buckling loads of cross-ply laminated composite plates rather accurately.
Two hyperbolic displacement models, HPSDT1 and HPSDT2, are developed for a bending analysis of orthotropic laminated composite
plates. These models take into account the parabolic distribution of transverse shear stresses and satisfy the condition of
zero shear stresses on the top and bottom surfaces of the plates. The accuracy of the analysis presented is demonstrated by
comparing the results with solutions derived from other higher-order models and with data found in the literature. It is established
that the HPSDT1 model is more accurate than some theories of laminates developed previously, and therefore the analysis can
be expanded to laminated composite shells.
A two-dimensional theory of flexural motions of isotropic, elastic plates is deduced from the three-dimensional equations of elasticity. The theory includes the effects of rotatory inertia and shear in the same manner as Timoshenko’s one-dimensional theory of bars. Velocities of straight-crested waves are computed and found to agree with those obtained from the three-dimensional theory. A uniqueness theorem reveals that three edge conditions are required.
This book by the late R D Mindlin is destined to become a classic introduction to the mathematical aspects of two-dimensional theories of elastic plates. It systematically derives the two-dimensional theories of anisotropic elastic plates from the variational formulation of the three-dimensional theory of elasticity by power series expansions. The uniqueness of two-dimensional problems is also examined from the variational viewpoint. The accuracy of the two-dimensional equations is judged by comparing the dispersion relations of the waves that the two-dimensional theories can describe with prediction from the three-dimensional theory. Discussing mainly high-frequency dynamic problems, it is also useful in traditional applications in structural engineering as well as provides the theoretical foundation for acoustic wave devices. © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
Composite materials consist of two or more materials which together produce desirable properties that may not be achieved with any of the constituents alone. Fiber-reinforced composite materials, for example, consist of high strength and high modulus fibers in a matrix material. Reinforced steel bars embedded in concrete provide an example of fiber-reinforced composites. In these composites, fibers are the principal loadcarrying members, and the matrix material keeps the fibers together, acts as a load-transfer medium between fibers, and protects fibers from being exposed to the environment (e.g., moisture, humidity, etc.).
This unique book explores both theoretical and experimental aspects of nonlinear vibrations and stability of shells and plates. It is ideal for researchers, professionals, students, and instructors. Expert researchers will find the most recent progresses in nonlinear vibrations and stability of shells and plates, including advanced problems of shells with fluid-structure interaction. Professionals will find many practical concepts, diagrams, and numerical results, useful for the design of shells and plates made of traditional and advanced materials. They will be able to understand complex phenomena such as dynamic instability, bifurcations, and chaos, without needing an extensive mathematical background. Graduate students will find (i) a complete text on nonlinear mechanics of shells and plates, collecting almost all the available theories in a simple form, (ii) an introduction to nonlinear dynamics, and (iii) the state of art on the nonlinear vibrations and stability of shells and plates, including fluid-structure interaction problems.
Vibrations drive many engineering designs in today's engineering environment. There has been an enormous amount of research into this area of research over the last decade. This book documents some of the latest research in the field of vibration of composite shells and plates filling a much-needed gap in the market. Laminated composite shells have many engineering applications including aerospace, mechanical, marine and automotive engineering. This book makes an ideal reference for researchers and practicing engineers alike.
Consider the element of the shell shown in figure 3-1(a), bounded by the normal sections α, α + dα, β, and β + dβ. The geometry of the middle surface of such an element was considered in chapter 2 (see figures 2-2 and 2-4). Here, we show the entire thickness h with the coordinate ζ defined in the direction of t n and depict a differential volume element dα dβ dζ with thickness dζ, parallel to and displaced from the middle surface a distance ζ.
Preface to the Second Edition. Preface to the First Edition. 1. Introduction to Composite Materials. 2. Anisotropic Elasticity and Composite Laminate Theory. 3. Plates and Panels of Composite Materials. 4. Beams, Columns and Rods of Composite Materials. 5. Composite Material Shells. 6. Energy Methods For Composite Material Structures. 7. Strength and Failure Theories. 8. Joining of Composite Material Structures. 9. Introduction to Composite Design. Appendices: A-1. Micromechanics. A-2. Test Standards for Polymer Matrix Composites. A-3. Properties of Various Polymer Composites. Author Index. Subject Index.
Elastic shells are pervasive in everyday life. Examples of these
thin-walled structures range from automobile hoods to basketballs, veins
and arteries, and soft drink cans. This book explains shell theory,
with numerous examples and applications. This second edition not only
brings all the material of the first edition entirely up to date; it
also adds two entirely new chapters on general shell theory and general
membrane theory. Aerospace, mechanical, and civil engineers, as well as
applied mathematicians, will find this book a clearly written and
thorough information source on shell theory.
The mixed layerwise shell theories that are presented in the companion article are evaluated here by solving several problems related to orthotropic cross-ply laminated, circular, cylindrical, and spherical shells subjected to static loadings for which closed-form solutions are given. Particular cases related to layerwise and equivalent single-layer models, based on classical displacement formulations, are evaluated for comparison purpose. A further comparison with three-dimensional elasticity exact solutions and to other higher-order shear deformations studies have been made. Results are given in the form of tables and diagrams. Approximations introduced by Donnell's shallow shell theories are evaluated for most of the problems. It has been concluded that the proposed mixed layerwise theories leads to a better description than the related analyses, which are based on displacement formulations. An excellent agreement, with respect to the exact solution, has been found for displacement and transverse stress components. These stresses have been herein calculated a priori. The importance of an adequate description of curvature terms related to the shell thickness to radii ratio h/R is also underlined. These effects have been contrasted by extensive use of fictitious interfaces in the conduced layerwise investigations.
Shell structures form key components in a very wide range of engineering enterprise. The theory of shell structures is an old and large subject, with a huge literature. However, this book is not a compilation of results from the past. Instead, it is an attempt to bring the essence of the subject within the grasp of engineers. It tackles the fundamental question of how bending and stretching effects combine and interact in shell structures from a physical point of view; and it shows that this approach leads to an understanding of the structural mechanics of shells in general, and to useful results in particular problems. The first half of the book is concerned mainly with the basic ideas and equations of equilibrium, geometry and elasticity, and their combination in various useful ways. In particular, it includes a simple treatment of the geometry of general curved surfaces. The second half of the book first investigates the behaviour of various practical shell structures under static loading. Then there are chapters on the buckling of shells, on vibration, and on the application of plastic theory to analysis and design.
This paper investigates the static analysis of doubly-curved laminated composite shells and panels. A theoretical
formulation of 2D Higher-order Shear Deformation Theory (HSDT) is developed. The middle surface
of shells and panels is described by means of the differential geometry tool. The adopted HSDT is
based on a generalized nine-parameter kinematic hypothesis suitable to represent, in a unified form,
most of the displacement fields already presented in literature. A three-dimensional stress recovery procedure
based on the equilibrium equations will be shown. Strains and stresses are corrected after the
recovery to satisfy the top and bottom boundary conditions of the laminated composite shell or panel.
The numerical problems connected with the static analysis of doubly-curved shells and panels are solved
using the Generalized Differential Quadrature (GDQ) technique. All displacements, strains and stresses
are worked out and plotted through the thickness of the following six types of laminated shell structures:
rectangular and annular plates, cylindrical and spherical panels as well as a catenoidal shell and an elliptic
paraboloid. Several lamination schemes, loadings and boundary conditions are considered. The GDQ
results are compared with those obtained in literature with semi-analytical methods and the ones computed
by using the finite element method.
The aim of this study is to clarify the discrepancy regarding the critical flow speed of straight pipes conveying fluids that appears to be present in the literature by using the Generalized Differential Quadrature method. It is well known that for a given “mass of the fluid” to the “mass of the pipe” ratio, straight pipes conveying fluid are unstable by a flutter mode via Hopf bifurcation for a certain value of the fluid speed, i.e. the critical flow speed. However, there seems to be lack of consensus if for a given mass ratio the system might lose stability for different values of the critical flow speed or only for a single speed value. In this paper an attempt to answer to this question is given by solving the governing equation following first the practical aspect related to the engineering problem and than by simply considering the mathematics of the problem. The
Generalized differential quadrature method is used as a numerical technique to resolve this problem. The differential governing equation is transformed into a discrete system of algebraic equations. The stability of the system is thus reduced to an eigenvalue problem. The relationship between the eigenvalue branches and the corresponding unstable flutter modes are shown via Argand diagram. The transfer of flutter-type instability from one eigenvalue branch to another is thoroughly investigated and discussed. The critical mass ratios, at which the transfer of the eigenvalue branches related to flutter take place, are determined.
This paper is focused on the Generalized Differential Quadrature (GDQ) Method to study the free vibration of conical shell structures. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion are expressed as functions of five kinematic parameters. Numerical solutions are obtained.
This paper presents an efficient shear deformation theory for vibration of functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded plate are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton’s principle. Analytical solutions of natural frequency are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates.
A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.
A new shear deformation theory for sandwich and composite plates is developed. The proposed displacement field, which is “m” parameter dependent, is assessed by performing several computations of the plate governing equations. Therefore, the present theory, which gives accurate results, is relatively close to 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature.
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.
In this paper, a nth-order shear deformation theory is proposed to analyze the free vibration of laminated composite plates. The present nth-order shear deformation theory satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Reddy’s third-order theory can be considered as a special case of present nth-order theory (n=3). Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties are computed by present nth-order theory and a meshless radial point collocation method based on the thin plate spline radial basis function. The results are compared with available published results which demonstrate the accuracy and efficiency of present nth-order theory.
A free vibration analysis of metal and ceramic functionally graded conical shell panels is presented using the element-free kp-Ritz method. The first-order shear deformation shell theory is used to account for the transverse shear strains and rotary inertia, and mesh-free kernel particle functions are employed to approximate the two-dimensional displacement fields. The material properties of the conical shell panels are assumed to vary continuously through their thickness in accordance with a power-law distribution of the volume fractions of their constituents. Convergence studies are performed in terms of the number of nodes, and comparisons of the current solutions and those reported in literature are provided to verify the accuracy of the proposed method. Two types of functionally graded conical shell panels, including Al/ZrO2 and Ti–6Al–4V/aluminum oxide, are chosen in the study, and the effects of the volume fraction, boundary condition, semi-vertex angle, and length-to-thickness ratio on their frequency characteristics are discussed in detail.
One of the current problems connected with multi-layer composite structures concerns the analysis of the distribution of the stresses around peculiarities (free edge and loaded edge) and at the interfaces of each layer. This work presents a new shear stress function in the form of the exponential function, to predict the mechanical behaviour of multi-layered laminated composite structures. As a case study, the mechanical behaviour of a laminated composite beam (90°/0°/0°/90°) is examined. The results are compared with the model “Sinus” and 2D finite element method studied. Results show that this new model is more precise than older ones as compared to the results obtained by the finite element analysis [Abaqus]. To introduce continuity on the interfaces of each layer, the kinematics defined by Ossadzow is used with the new exponential model. The equilibrium equations and natural boundary conditions are derived by the principle of virtual power.
In the present study, unified shear deformation theory which was proposed by Soldatos, K.P. and Timarci, T. (1993). A Unified Formulation of Laminated Composite, Shear Deformable Five-degrees-of-freedom Cylindrical Shells on the Basis of a Unified Shear Deformable Shell Theory, Compos. Struct., 25(1–4): 165–171 is used to analyze simply supported symmetric cross-ply rectangular plates for deflections, stresses, natural frequencies, and buckling loads. This theory enables the selection of different in-plane displacement components to represent shear deformation. Exponential shear deformation theory, which is proposed by Karama et al., 40(6) (2003). Mechanical Behavior of Laminated Composite Beam by New Multi-layered Laminated Composite Structures Model with Transverse Shear Stress Continuity, Int J Solids Struct., 40: 1525–1546, is used for the first time to analyze the problem considered. Results which are found by exponential theory are compared with those obtained by using the parabolic shear deformation theory of Reddy, J.N., 51(4) (1984). A Simple Higher-order Theory for Laminated Composite Plates, J Appl Mech., 51: 745–752, the trigonometric shear deformation theory of Touratier, M., 29(8) (1991). An Efficient Standard Plate Theory, Int. Jnl. Engng. Sci., 29(8): 901–916, the hyperbolic shear deformation theory of Soldatos, K.P., 94(3–4) (1992). A Transverse Shear Deformation Theory for Homogeneous Monoclinic Plates, Acta Mech., 94: 1995–220 and with the available three-dimensional elasticity solutions. The study shows that while the transverse displacement and the stresses are best predicted by the exponential shear deformation theory, the parabolic shear deformation and the hyperbolic shear deformation theories yield more accurate predictions for the natural frequencies and the buckling loads.