Closed Form Solutions for The Stability and Free Vibration Analysis of Laminated Composite Plates

Analytical formulations and solutions to fundamental frequency and buckling loads analysis of simply supported isotropic and layered anisotropic cross-ply composite plates are presented. The displacement model based on Generalized Laminate Plate Theory (GLPT) assumes piece-wise linear variation of in-plane displacement components and constant transverse displacement through thickness of the plate. It also includes the quadratic variation of transverse shear stresses within each layer of the plate. The large deflection theory (in Von Karman sense) is incorporated into the buckling analysis. The equations of motion are obtained using Hamilton’s principle. Closed form solution is derived following the Navier’s technique and by solving the eigenvalue problem. The effects of side-to-thickness ratio, aspect ratio, coupling between bending and stretching and number of layers on the fundamental frequencies and critical buckling loads are investigated. The results of the presented theory (GLPT) are compared with the exact 3D elasticity theory, Higher-order Shear Deformation Theory (HSDT) and Classical Laminated Plate Theory (CLPT) solutions. The study concludes that the present model accurately predicts fundamental frequencies and buckling loads of composite plates, while the CLPT is inadequate for the analysis of non-homogeneous laminated plates

Free vibrations of delaminated composite and sandwich plates

The aim of this paper is comparison of fundamental dynamic characteristics of intact and damaged laminated composite and sandwich plates. Generalized Layerwise Plate Theory is used for analysis of different plates with presence of delaminations. Jumps in displacement field are incorporated using Heaviside step functions. Different layerwise FE are derived. Mathematical model is coded in MATLAB® for free vibrations analysis of laminated composite and sandwich plates. Effects of plate aspect ratio, lamination scheme, degree of orthotropy and delamination position on fundamental dynamic characteristics are analyzed. All numerical solutions are compared with existing dat

Linear Transient Analysis of Laminated Composite Plates using GLPT

The objective of this work is to study the transient response of laminated composite plates under different types of dynamic loading. For this purpose, laminated composite plate is modeled using Reddy’s generalized layerwise plate theory (GLPT). This theory assumes layerwise linear variation of displacements components. Transverse displacement is constant through the thickness of the plate. Using the assumed displacement field, linear kinematic relations, as well as Hooke’s constitutive law, equations of motion are derived using Hamilton’s principle. Analytical solution for cross-ply laminates is derived using the Navier method. Numerical solution is obtained using FEM. Governing partial differential equations in both solutions are reduced to a set of ordinary differential equations in time using Newmark integration scheme. The equations of motion are solved using constant-average acceleration method. Effects of time step, mesh refinement and lamination scheme on accuracy of transient response are considered. Illustrative comments are given about the influence of shear deformation on transient response. Finally, different schemes of dynamic loading are investigated. Good agreement is obtained with results from the literature.Obiectivul acestei lucrări este de a studia răspunsul tranzitoriu al plăcilor compozite laminate sub diferite tipuri de încărcare dinamică. În acest scop, placa compozită laminată este modelată folo-sind teoria generalizată a plăcilor propusă de Reddy (GLPT). Această teorie presupune variația liniară a componentelor deplasării în raport cu straturile plăcii. Deplasarea transversală este constantă în grosimea plăcii. Ecuațiile de mișcare sunt derivate folosind principiul lui Hamilton, utilizând câmpul de deplasare asumat, relațiile liniare cinematice, precum și legea constitutivă a lui Hooke. Soluția analitică pentru plăci laminate din fibre din lemn încrucișate este derivată folosind formula lui Navier. Soluția numerică este obținută cu ajutorul metodei elementului finit. Ecuațiile cu derivate parțiale în ambele soluții sunt reduse la un set de ecuații diferențiale ordinare în timp, utilizând Metoda Newmark. Ecuațiile de mișcare sunt rezolvate folosind metoda de inte-grare implicită Newmark β. Sunt luate în considerare efectele integrării numerice în timp, pas cu pas, ale rafinării discretizării și ale sistemului de laminare asupra acurateței răspunsului tranzitoriu. Sunt prezentate comentarii ilustrative despre influența deformării cauzate de forfecare asupra răspunsului tranzitoriu. În cele din urmă, sunt investigate diferite scheme de încărcare dinamică. Rezultatele obținute sunt în concordanță cu cele din literatura de specialitate

Termička analiza laminatnih kompozitnih i sendvič ploča primenom slojevitog konačnog elementa

In this paper the quasi static response of laminated composite and sandwich plates subjected to lineary varaying through the thickness and sinusoidall distributed in plane temperature field, is analyzed. Mathematical model, based on layer-wise displacement field of Reddy [15], is formulated using small deflection linear-elasticity theory. The principle of virtual displacements (PVD) is used to obtain the weak form of the mathematical model. The weak form is discretized utilizing isoparametric finite element approximation. The originally coded MATLAB program is used to investigate the influence of plate thickness on thermo-elastic response of laminate composite and sandwich plates. The accuracy of the numerical model is verified by comparison with the available results from the literature.U ovom radu analiziran je kvazi statički odgovor laminatnih kompozitnih i sendvič ploča ploča izloženih temperaturnom polju linearno promenlivom po debljini i sinusoiudalno promenlijvom u ravni. Matematički model, zasnovan na slojevitom polju pomeranja koje je predložio Reddy [15], formulisan je koristeći pretpostavke o geometrijskoj, materijalnoj i statičkoj linearnosti problema. Princip virtulanih pomeranja je primenjen za dobijanje slabe forme matematičkog modela. Slaba forma je diskretizovana koristeći izoparametarsku aproksimaciju konačnim elementom. Originalan MATLAB računski program je korišćen za analizu uticaja razlicitih debljina ploče na termo-elastičan odgovor laminatnih kompozitnih i sendvič ploča. Tačnost numeričkog modela je potvrđena poređenjem sa rešenjima iz literature

Geometrically Nonlinear Analysis of Composite Plates Using Layerwise Displacement Model

The low mass density and the high tensile strength, usually expressed through the specific modulus of elasticity and the specific strength, have made composite materials lighter and stronger compared with most traditional materials (such as steel, concrete, wood, etc.) and have increased their application not only for secondary, but during the last two decades also for primarily structural members in aerospace and automotive industry, ship building industry and bridge design. Although weight saving has eliminated constrain of slenderness and thickness and has made possible use of very thin plate elements, they have become susceptible to large deflections. In such cases, the geometry of structures is continually changing during the deformation and geometrically nonlinear analysis should be adopted. In this paper the geometrically nonlinear continuum plate finite element model, hitherto not reported in the literature, is developed using the total Lagrange formulation. With the layerwise displacement field of Reddy [Reddy JN, Barbero EJ, Teply JL. A plate bending element based on a generalized laminated plate theory. International Journal for Numerical Methods in Engineering 1989; 28: 2275-2292], nonlinear Green-Lagrange small strain large displacements relations (in the von Karman sense) and linear elastic orthotropic material properties for each lamina, the 3D elasticity equations are reduced to 2D problem and the nonlinear equilibrium integral form is obtained. By performing the linearization on nonlinear integral form and then the discretization on linearized integral form, tangent stiffness matrix is obtained with less manipulations and in more consistent form, compared to the one obtained using laminated element approach of Reddy [E. J. Barbero, J. N. Reddy, Nonlinear Analysis of Composite Laminates Using a Generalized Laminated Plate Theory, AIAAL Journal, Vol. 28(11), 1990, 1987-1994]. Symmetric tangent stiffness matrixes, together with internal force vector are then utilized in Newton Raphson’s method for the numerical solution of nonlinear incremental finite element equilibrium equations. Despite of its complex layer dependent numerical nature, the present model has no shear locking problems, compared to ESL (EquivalenSingleLayer) models, or aspect ratio problems, as the 3D finite element may have when analyzing thin plate behavior. The originally coded MATLAB computer program for the finite element solution is used to verify the accuracy of the numerical model, by calculating nonlinear response of plates with different mechanical properties, which are isotropic, orthotropic and anisotropic (cross ply and angle ply), different plate thickness, different boundary conditions and different load direction (unloading/loading). The obtained results are compared with available results from the literature and the linear solutions from the previous paper [Ćetković M, Vuksanović Dj, Bending, Free Vibrations and Buckling of Laminated Composite and Sandwich Plates Using a Layerwise Displacement Model. Composite Structures 2009; 88(2): 219-227].Link ka sajtu na kome je objavljen rad na konferenciji (on CD) https://paginas.fe.up.pt/~iccs16/CD/1-99.htm

Free vibrations of laminated composite plates using layerwise displacement model

In this paper Generalized Layer Wise Plate Theory of Reddy (GLPT) is used to formulate an isoparametric finite element model for free vibration of laminated composite plates. With the assumed displacement field, linear strain displacement relations and linear elastic orthotropic material properties for each lamina, virtual work statement is utilized in order to formulate isoparametric finite element model. The original MATLAB computer program is coded for finite element solution. Some new results using GLPT finite element model for soft core sandwich plate is presented, which may be used as the guideline for their optimal design in the laboratory

Geometrically Nonlinear Analysis of Composite Plates Using Layerwise Displacement Model

The low mass density and the high tensile strength, usually expressed through the specific modulus of elasticity and the specific strength, have made composite materials lighter and stronger compared with most traditional materials (such as steel, concrete, wood, etc.) and have increased their application not only for secondary, but during the last two decades also for primarily structural members in aerospace and automotive industry, ship building industry and bridge design. Although weight saving has eliminated constrain of slenderness and thickness and has made possible use of very thin plate elements, they have become susceptible to large deflections. In such cases, the geometry of structures is continually changing during the deformation and geometrically nonlinear analysis should be adopted. In this paper the geometrically nonlinear continuum plate finite element model, hitherto not reported in the literature, is developed using the total Lagrange formulation. With the layerwise displacement field of Reddy [Reddy JN, Barbero EJ, Teply JL. A plate bending element based on a generalized laminated plate theory. International Journal for Numerical Methods in Engineering 1989; 28: 2275-2292], nonlinear Green-Lagrange small strain large displacements relations (in the von Karman sense) and linear elastic orthotropic material properties for each lamina, the 3D elasticity equations are reduced to 2D problem and the nonlinear equilibrium integral form is obtained. By performing the linearization on nonlinear integral form and then the discretization on linearized integral form, tangent stiffness matrix is obtained with less manipulations and in more consistent form, compared to the one obtained using laminated element approach of Reddy [E. J. Barbero, J. N. Reddy, Nonlinear Analysis of Composite Laminates Using a Generalized Laminated Plate Theory, AIAAL Journal, Vol. 28(11), 1990, 1987-1994]. Symmetric tangent stiffness matrixes, together with internal force vector are then utilized in Newton Raphson’s method for the numerical solution of nonlinear incremental finite element equilibrium equations. Despite of its complex layer dependent numerical nature, the present model has no shear locking problems, compared to ESL (EquivalenSingleLayer) models, or aspect ratio problems, as the 3D finite element may have when analyzing thin plate behavior. The originally coded MATLAB computer program for the finite element solution is used to verify the accuracy of the numerical model, by calculating nonlinear response of plates with different mechanical properties, which are isotropic, orthotropic and anisotropic (cross ply and angle ply), different plate thickness, different boundary conditions and different load direction (unloading/loading). The obtained results are compared with available results from the literature and the linear solutions from the previous paper [Ćetković M, Vuksanović Dj, Bending, Free Vibrations and Buckling of Laminated Composite and Sandwich Plates Using a Layerwise Displacement Model. Composite Structures 2009; 88(2): 219-227].Link ka sajtu na kome je objavljen rad na konferenciji (on CD) https://paginas.fe.up.pt/~iccs16/CD/1-99.htm

Closed Form Solutions for The Stability and Free Vibration Analysis of Laminated Composite Plates

Analytical formulations and solutions to fundamental frequency and buckling loads analysis of simply supported isotropic and layered anisotropic cross-ply composite plates are presented. The displacement model based on Generalized Laminate Plate Theory (GLPT) assumes piece-wise linear variation of in-plane displacement components and constant transverse displacement through thickness of the plate. It also includes the quadratic variation of transverse shear stresses within each layer of the plate. The large deflection theory (in Von Karman sense) is incorporated into the buckling analysis. The equations of motion are obtained using Hamilton’s principle. Closed form solution is derived following the Navier’s technique and by solving the eigenvalue problem. The effects of side-to-thickness ratio, aspect ratio, coupling between bending and stretching and number of layers on the fundamental frequencies and critical buckling loads are investigated. The results of the presented theory (GLPT) are compared with the exact 3D elasticity theory, Higher-order Shear Deformation Theory (HSDT) and Classical Laminated Plate Theory (CLPT) solutions. The study concludes that the present model accurately predicts fundamental frequencies and buckling loads of composite plates, while the CLPT is inadequate for the analysis of non-homogeneous laminated plates

Uticaj graničnih uslova na nelinearan odgovor laminatnih kompozitnih ploča

In this paper the influence of different boundary conditions on geometrically nonlinear response of laminated composite plates is analyzed. Mathematical model, based on layer-wise displacement field of Reddy [1], is formulated using the von Karman, small strain large deflection theory. The principle if virtual displacements (PVD) is used to obtain the weak form of the problem. The weak form is discretized utilizing isoparametric finite element approximation. The originally coded MATLAB program is used to investigate the influence of different boundary conditions on geometrically nonlinear response of laminate composite plates. The accuracy of the numerical model is verified by comparison with the available results from the literature.U ovom radu analiziran je uticaj različitih graničnih uslova na geometrijski nelinearan odgovor laminatnih kompozitnih ploča. Matematički model, zasnovan na slojevitom polju pomeranja koje je predložio Reddy [1], formulisan je koristeći von Karman’ovu teoriju velikih pomeranja i malih deformacija. Princip virtulanih pomeranja je primenjen za dobijanje slabe forme problema. Slaba forma je diskretizovana koristeći izoparametarsku aproksimaciju konačnim elementom. Originalan MATLAB računski program je korišćen za analizu uticaja razlicitih graničnih uslova na nelinearna odgovor laminatnih kompozitnih ploča. Tačnost numeričkog modela je potvrđena poređenjem sa rešenjima iz literature

Free vibrations of laminated composite plates using layerwise displacement model

In this paper Generalized Layer Wise Plate Theory of Reddy (GLPT) is used to formulate an isoparametric finite element model for free vibration of laminated composite plates. With the assumed displacement field, linear strain displacement relations and linear elastic orthotropic material properties for each lamina, virtual work statement is utilized in order to formulate isoparametric finite element model. The original MATLAB computer program is coded for finite element solution. Some new results using GLPT finite element model for soft core sandwich plate is presented, which may be used as the guideline for their optimal design in the laboratory