BENDING RESPONSES OF AN EXPONENTIALLY GRADED SIMPLY-SUPPORTED ELASTIC/VISCOELASTIC/ELASTIC SANDWICH PLATE

The bending response for exponentially graded composite (EGC) sandwich plates is investigated.The three-layer elastic/viscoelastic/elastic sandwich plate is studied by using the sinusoidal shear deformation plate theory as well as other familiar theories.Four types of sandwich plates are considered taking into account the symmetry of the plate and the thickness of each layer.The effective moduli and Illyushin’s approximation methods are used to solve the equations governing the bending of simply-supported EGC fiber-reinforced viscoelastic sandwich plates.Then numerical results for deflections and stresses are presented and the effects due to time parameter,aspect ratio,side-to-thickness ratio and constitutive parameter are investigated.     

SEMI-ANALYTICAL SOLUTION FOR FREE VIBRATION OF THICK FUNCTIONALLY GRADED PLATES RESTED ON ELASTIC FOUNDATION WITH ELASTICALLY RESTRAINED EDGE

This paper deals with free vibration analysis of functionally graded thick circular plates resting on the Pasternak elastic foundation with edges elastically restrained against translation and rotation.Governing equations are obtained based on the first order shear deformation theory(FSDT) with the assumption that the mechanical properties of plate materials vary continuously in the thickness direction.A semi-analytical approach named differential transform method is adopted to transform the differential governing equations into algebraic recurrence equations.And eigenvalue equation for free vibration analysis is solved for arbitrary boundary conditions.Comparison between the obtained results and the results from analytical method confirms an excellent accuracy of the present approach.Afterwards,comprehensive studies on the FG plates rested on elastic foundation are presented.The effects of parameters,such as thickness-to-radius,material distribution,foundation stiffness parameters,different combinations of constraints at edges on the frequency,mode shape and modal stress are also investigated.     

A two variable refined plate theory for the bending analysis of functionally graded plates

Bending analysis of functionally graded plates using the two variable refined plate theory is presented in this paper.The number of unknown functions involved is reduced to merely four,as against five in other shear deformation theories. The variationally consistent theory presented here has, in many respects,strong similarity to the classical plate theory. It does not require shear correction factors,and gives rise to such transverse shear stress variation that the transverse shear stresses vary parabolically across the thickness and satisfy shear stress free surface conditions.Material properties of the plate are assumed to be graded in the thickness direction with their distributions following a simple power-law in terms of the volume fractions of the constituents.Governing equations are derived from the principle of virtual displacements, and a closed-form solution is found for a simply supported rectangular plate subjected to sinusoidal loading by using the Navier method.Numerical results obtained by the present theory are compared with available solutions,from which it can be concluded that the proposed theory is accurate and simple in analyzing the static bending behavior of functionally graded plates.     

Dynamic buckling of stiffened plates under fluid-solid impact load

A simple solution of the dynamic buckling of stiffened plates under fluid-solid impact loading is presented. Based on large deflection theory, a discretely stiffened plate model has been used. The tangential stresses of stiffeners and in-plane displacement are neglected. Applying the Hamilton‘ s principle, the motion equations of stiffened plates are obtained. The deflection of the plate is taken as Fourier series, and using Galerkin method, the discrete equations can be deduced, which can be solved easily by Runge-Kutta method. The dynamic buckling loads of the stiffened plates are obtained from Budiansky-Roth ( B-R ) curves.     

ANALYTICAL RELATIONS BETWEEN EIGENVALUES OF CIRCULAR PLATE BASED ON VARIOUS PLATE THEORIES

Based on the mathematical similarity of the axisymmetric eigenvalue problems of a circular plate between the classical plate theory(CPT), the first-order shear deformation plate theory(FPT) and the Reddy's third-order shear deformation plate theory (RPT), analytical relations between the eigenvalues of circular plate based on various plate theories are investigated. In the present paper, the eigenvalue problem is transformed to solve an algebra equation. Analytical relationships that are expressed explicitly between various theories are presented. Therefore, from these relationships one can easily obtain the exact RPT and FPT solutions of critical buckling load and natural frequency for a circular plate with CPT solutions. The relationships are useful for engineering application, and can be used to check the validity, convergence and accuracy of numerical results for the eigenvalue problem of plates.     

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material(FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-ofvariables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.     

A free rectangular plate on the two-parameter elastic foundation

This paper provides a rigorous solution of a free rectangular plate on the V.Z. Vlazov two-parameter elastic foundation by the method of superposition[1]. In this paper we derive basic solutions under the various boundary conditions. To superpose these basic solutions the most generally rigorous solution of a free rectangular plate on the two-parameter elastic foundation can be obtained. The solution strictly satisfies the differential equation of a plate on the two-parameter elastic model foundation, the boundary conditions of the free edges and the free corner conditions. Some numerical examples are presented The calculated results show that when the plane dimension of plate is given and the ratio between the laver depth and the plate thick is equal to 15, the two-parameter elastic model is near the Winkler’s. It shows that the Winkler model can be applied to the thinner layer.     

ANALYSIS OF INTER-LAMINAR STRESSES FOR COMPOSITE LAMINATED PLATE WITH INTERFACIAL DAMAGE

A constitutive model for composite laminated plates with the damage effect of the intra-layers and inter-laminar interface is presented. The model is based on the general six-degrees-of-freedom plate theory, the discontinuity of displacement on the interfaces are depicted by three shape functions, which are formulated according to solutions satisfying three equilibrium equations, By using the variation principle, the three-dimensional non-linear equilibrium differential equations of the laminated plates with two different damage models are derived. Then, considering a simply supported laminated plate with damage, an analytical solution is presented using finite difference method to obtain the inter-laminar stresses.     

Differential quadrature method for bending of orthotropic plates with finite deformation and transverse shear effects

Based on the Reddy ‘s theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadrature ( DQ ) method of nonlinear analysis to the problem was presented. New DQ approach, presented by Wang and Bert ( DQWB), is extended to handle the multiple boundary conditions of plates. The techniques were also further extended to simplify nonlinear computations. The numerical convergence and comparison of solutions were studied. The results show that the DQ method presented is very reliable and valid. Moreover, the influences of geometric and material parameters as well as the transverse shear deformations on nonlinear bending were investigated. Numerical results show the influence of the shear deformation on the static bending of orthotropic moderately thick plate is significant.     

The Scheme to Determine the Convergence Term of the Galerkin Method for Dynamic Analysis of Sandwich Plates on Nonlinear Foundations

The vibration of a plate resting on elastic foundations under a moving load is of great significance in the design of many engineering fields,such as the vehicle-pavement system and the aircraft-runway system.Pavements or runways are always laminated structures.The Galerkin truncation method is widely used in the research of vibration.The number of truncation terms directly affects the convergence and accuracy of the response results.However,the selection of the number of truncation terms has not been clearly stated.A nonlinear viscoelastic foundation model under a moving load is established.Based on the natural frequency of linear undisturbed derivative systems,the truncation terms are used to determine the convergence of vibration response.The criterion for the convergence of the Galerkin truncation term is presented.The scheme is related to the natural frequency with high efficiency and practicability.Through the dynamic response of the sandwich beam under a moving load,the feasibility of the scheme is verified.The effects of different system parameters on the scheme and the truncation convergence of dynamic response are presented.The research in this paper can be used as a reference for the study of the vibration of elastic foundation plates.Especially,the model established and the truncation analysis method proposed are helpful for studying the vibration of vehicle-pavement system and related systems.     

Effect of partial elastic foundation on free vibration of fluid-filled functionally graded cylindrical shells

The free vibration characteristics of fluid-filled functionally graded cylindrical shells buried partially in elastic foundations are investigated by an analytical method.The elastic foundation of partial axial and angular dimensions is represented by the Pasternak model. The motion of the shells is represented by the first-order shear deformation theory to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shells are composed of stainless steel and silicon nitride. Material properties vary continuously through the thickness according to a power law distribution in terms of the volume fraction of the constituents. The governing equation is obtained using the Rayleigh–Ritz method and a variation approach. The fluid is described by the classical potential flow theory. Numerical examples are presented and compared with existing available results to validate the present method.     

Shear deformable finite beam elements for composite box beams

The shear deformable thin-walled composite beams with closed cross-sections have been developed for coupled flexural, torsional, and buckling analyses. A theoretical model applicable to the thin-walled laminated composite box beams is presented by taking into account all the structural couplings coming from the material anisotropy and the shear deformation effects. The current composite beam includes the transverse shear and the restrained warping induced shear deformation by using the first-order shear deformation beam theory. Seven governing equations are derived for the coupled axial-flexural-torsional-shearing buckling based on the principle of minimum total potential energy. Based on the present analytical model, three different types of finite composite beam elements, namely, linear, quadratic and cubic elements are developed to analyze the flexural, torsional, and buckling problems. In order to demonstrate the accuracy and superiority of the beam theory and the finite beam elements developed by this study,numerical solutions are presented and compared with the results obtained by other researchers and the detailed threedimensional analysis results using the shell elements of ABAQUS. Especially, the influences of the modulus ratio and the simplified assumptions in stress–strain relations on the deflection, twisting angle, and critical buckling loads of composite box beams are investigated.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

ELASTIC-PLASTIC DYNAMIC RESPONSE OF FULLY BACKED SANDWICH PLATES UNDER LOCALIZED IMPULSIVE LOADING

An analytical model is developed to assess the elastic-plastic dynamic response of fully backed sandwich plates under localized impulse load.The core is modeled as an elastic-perfectly plastic foundation.The top face sheet is treated as an individual plate resting on the foundation.The elastic-plastic analysis for the top face sheet is based on a minimum principle in dynamic plasticity associated with the finite difference technique.The effects of spatial and temporal distributions of the impulsive loading on the dynamic response of sandwich plates are discussed.The model can be used to predict the impulse-induced local effect on fully backed sandwich plates.     

Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis

Kármán-type nonlinear large deflection equations are derived occnrding to the Reddy’s higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.     

BUCKLING AND POSTBUCKLING OF MODERATELY THICK PLATES

This paper gives the basic differential equations for finite deflections of elastic platesaccording to Reissner’s approximate stress distributions.The buckling and postbucklingproblems of elastic rectangular plates.including the effect of transverse shear deformation.are solved and discussed by using perturbation method suggested in ref.[8].Thepostbuckling equilibrium paths of perfect and imperfect moderately thick rectangularplates are presented and compared with the results based on thin plate theory.     

ADOMIAN POLYNOMIALS FOR NONLINEAR RESPONSE OF SUPPORTED TIMOSHENKO BEAMS SUBJECTED TO A MOVING HARMONIC LOAD

This paper investigates the steady-state responses of a Timoshenko beam of infinite length supported by a nonlinear viscoelastic Pasternak foundation subjected to a moving harmonic load. The nonlinear viscoelastic foundation is assumed to be a Pasternak foundation with linear-plus-cubic stiffness and viscous damping. Based on Timoshenko beam theory, the nonlinear equations of motion are derived by considering the effects of the shear deformable beams and the shear modulus of foundations at the same time. For the first time, the modified Adomian decomposition method（ADM） is used for solving the response of the beam resting on a nonlinear foundation. By employing the standard ADM and the modified ADM, the nonlinear term is decomposed, respectively. Based on the Green＇s function and the theorem of residues presented,the closed form solutions for those linear iterative equations have been determined via complex Fourier transform. Numerical results indicate that two kinds of ADM predict qualitatively identical tendencies of the dynamic response with variable parameters, but the deflection of beam predicted by the modified ADM is smaller than that by the standard ADM. The influence of the shear modulus of beams and foundation is investigated. The numerical results show that the deflection of Timoshenko beams decrease with an increase of the shear modulus of beams and that of foundations.     

A THREE-DIMENSIONAL ELASTICITY SOLUTION FOR TWO-DIRECTIONAL FGM ANNULAR PLATES WITH NON-UNIFORM ELASTIC FOUNDATIONS SUBJECTED TO NORMAL AND SHEAR TRACTIONS

In the present paper, bending and stress analyses of two-directional functionally graded （FG） annular plates resting on non-uniform two-parameter Winkler-Pasternak founda- tions and subjected to normal and in-plane-shear tractions is investigated using the exact three- dimensional theory of elasticity. Neither the in-plane shear loading nor the influence of the two- directional material heterogeneity has been investigated by the researchers before. The solution is obtained by employing the state space and differential quadrature methods. The material proper- ties are assumed to vary in both transverse and radial directions. Three different types of variations of the stiffness of the foundation are considered in the radial direction： linear, parabolic, and sinu- soidal. The convergence analysis and the comparative studies demonstrate the high accuracy and high convergence rate of the present approach. A parametric study consisting of evaluating effects of different parameters （e.g., exponents of the material properties laws, the thickness to radius ratio, trends of variations of the foundation stiffness, and different edge conditions） is carried out. The results are reported for the first time and are discussed in detail.     

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.     

A FREE RECTANGULAR PLATE ON ELASTIC FOUNDATION

In the theory of elastic thin plates, the bending of a rectangular plate on the elastic foundation is also a difficult problem. This paper provides a rigorous solution by the method of superposition. It satisfies the differential equation, the boundary conditions of the edges and the free corners. Thus we are led to a system of infinite simultaneous equations. The problem solved is for a plate with a concentrated load at its center. The reactive forces from the foundation should be made to be in equilibrium with the concentrated force to see whether our calculation is correct or not.