A simple four-node quadrilateral finite element for plates

A simple and accurate four-node quadrilateral finite element based on the Mindlin plate theory and Kirchhoff constraints is presented for general thin plate bending applications. The derivation of the element stiffness properties is straightforward, starting with a specified eight-node interpolation; usual discrete Kirchhoff (DK) constraints are employed to constrain out the four midside nodes of the element. The present resulting DK element passes patch tests with elements of arbitrary and even highly distorted mesh types. Numerical studies of the element convergence behaviours are undertaken for various plate bending problems so far investigated. It is indicated from comparative examples that fairly good convergence characteristics have been achieved.     

A higher-order finite element scheme for incompressible lubrication calculations

A higher-order finite element scheme is formulated for incompressible lubrication calculations, based on the energy functional of the lubricating system (derived from the variational technique) and the hierarchical approximation concept. The current formulation ensures the pressure continuity across inter-element mating boundaries. Since this is a hierarchical formulation, it facilitates convergence studies of results. Numerical examples are provided to demonstrate the accuracy of the proposed method, the simplicity of modeling, applications, and the convergence characteristics of numerical solutions.     

A simple nonconforming quadrilateral finite element

We introduce and analyze a simple nonconforming quadrilateral finite element and then we derive optimal a priori error estimates for arbitrary regular quadrilaterals. The idea of extension to some non-conforming elements for three-dimensional hexahedrons is also presented.     

Stochastic multi-scale finite element based reliability analysis for laminated composite structures

This paper proposes a novel multi-scale approach for the reliability analysis of composite structures that accounts for both microscopic and macroscopic uncertainties, such as constituent material properties and ply angle. The stochastic structural responses, which establish the relationship between structural responses and random variables, are achieved using a stochastic multi-scale finite element method, which integrates computational homogenisation with the stochastic finite element method. This is further combined with the first- and second-order reliability methods to create a unique reliability analysis framework. To assess this approach, the deterministic computational homogenisation method is combined with the Monte Carlo method as an alternative reliability method. Numerical examples are used to demonstrate the capability of the proposed method in measuring the safety of composite structures. The paper shows that it provides estimates very close to those from Monte Carlo method, but is significantly more efficient in terms of computational time. It is advocated that this new method can be a fundamental element in the development of stochastic multi-scale design methods for composite structures.     

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

This paper presents a novel finite element formulation for static, free vibration and buckling analyses of laminated composite plates. The idea relies on a combination of node-based smoothing discrete shear gap method with the higher-order shear deformation plate theory (HSDT) to give a so-called NS-DSG3 element. The higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The numerical examples demonstrated that the present element is free of shear locking and shows high reliability and accuracy compared to other published solutions in the literature.     

An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates

This paper presents an alternative alpha finite element method using triangular meshes (AαFEM) for static, free vibration and buckling analyses of laminated composite plates. In the AαFEM, an assumed strain field is carefully constructed by combining compatible strains and additional strains with an adjustable parameter α which can produce an effectively softer stiffness formulation compared to the linear triangular element. The stiffness matrices are obtained based on the strain smoothing technique over the smoothing domains and the constant strains on triangular sub-domains associated with the nodes of the elements. The discrete shear gap (DSG) method is incorporated into the AαFEM to eliminate transverse shear locking and an improved triangular element termed as AαDSG3 is proposed. Several numerical examples are then given to demonstrate the effectiveness of the AαDSG3.     

Analysis of the static response of cross-ply simply supported plates and shells based on a higher-order theory

The boundary-discontinuous double Fourier series-based solution methodology is used to solve the problem of higher-order shear deformation of cross-ply plates and doubly curved panels, which are characterized by a system of five highly coupled linear partial differential equations with mixed-type simply supported boundary conditions prescribed at all four their edges. The present solution is related to a number of unsolved boundary-value problems and can serve as a tool in particular for early design stages and for benchmark comparisons and verifications of numerical results. The analytical results obtained are compared with finite-element calculations, and a good agreement is found to exist between them.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

A simple finite element method for the Stokes equations

The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate.     

A vibration analysis of composite laminated plates

A finite element formulation of the equations governing laminated anisotropic plates using Reddy's higher-order theory is presented. This simple higher-order shear deformable theory takes into account the parabolic distribution of the transverse shear deformation through the thickness of the plate and contains the same unknowns as in the first-order shear deformation theory. Finite element solutions are presented for rectangular plates of different layups, such as cross-ply, antisymmetric angle-ply, and sandwich plates with various material properties, boundaries, and plate aspect ratios. The numerical results are compared with the available closed-form results, the 3-D linear elasticity theory results, and the other available numerical results. A comparison is also made with test data from a laminated cantilever plate.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

Nonlinear vibration of moderately thick laminated beams using finite element method

A finite element model is developed to study the large-amplitude free vibrations of generally-layered laminated composite beams. The Poisson effect, which is often neglected, is included in the laminated beam constitutive equation. The large deformation is accounted for by using von Karman strains and the transverse shear deformation is incorporated using a higher order theory. The beam element has eight degrees of freedom with the inplane displacement, transverse displacement, bending slope and bending rotation as the variables at each node. The direct iteration method is used to solve the nonlinear equations which are evaluated at the point of reversal of motion. The influence of boundary conditions, beam geometries, Poisson effect, and ply orientations on the nonlinear frequencies and mode shapes are demonstrated.     

Buckling and free vibration analyses of laminated composite plates by using two new hyperbolic shear-deformation theories

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. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 217–230, March–April, 2008.     

Quasi-three-dimensional theory for solution of dynamic thermoelastic problem of laminated composite plates

An uncoupled dynamic thermoelastic problem for laminated composite plates has been considered. The hypotheses used take into account the nonlinear distribution of temperature and displacements over the thickness of a laminated plate. On the basis of these hypotheses a quasi-three-dimensional (layerwise) theory is constructed that makes it possible to investigate the internal thermal and stress-strain states, as well as the edge effects of the boundary layer type for laminated plates. Systems of the heat conduction and motion equations are derived using the variational method. The order of the equations depends on the number of layers and terms in expansions of temperature and displacements of each layer. An analytical solution of the dynamic thermoelastic problem is presented for a cross-ply laminated rectangular plate with simply supported edges. The reliability of the results is confirmed by a comparison with the known exact solutions. The results based on the proposed theory can be used for verifying various two-dimensional plate theories when solving the dynamic thermoelastic problems for laminated composite plates.     

A simple four-unknown refined theory for bending analysis of functionally graded plates

In the present paper, a refined trigonometric higher-order plate theory is simply derived, which satisfies the free surface conditions. Moreover, the number of unknowns of this theory is the least one comparing with other shear theories. The effects of transverse shear strains as well as the transverse normal strain are taken into account. The number of unknown functions involved in the present theory is only four as against six or more in case of other shear and normal deformation theories. The bending response of FG rectangular plates is presented. A comparison with the corresponding results is made to check the accuracy and efficiency of the present theory. Additional results for all displacements and stresses are investigated through-the-thickness of the FG rectangular plate.     

A p-version MITC finite element method for Reissner–Mindlin plates with curved boundaries

We consider the approximation of Reissner–Mindlin plates with curved boundaries, using a p-version MITC finite element method. We describe in detail the formulation and implementation of the method, and emphasize the need for a Piola-type map in order to handle the curved geometry of the elements. The results of our numerical computations demonstrate the robustness of the method and suggest that it gives near exponential convergence when the error is measured in the energy norm. For the robust computation of quantities of engineering interest, such as the shear force, the proposed method yields very satisfactory results without the need for any additional post-processing. Comparisons are made with the standard finite element formulation, with and without post-processing.     

On the mixed finite element approximation for the buckling of plates

We consider the mixed finite element method for the buckling problem of the thin plate by using piecewise linear polynomials. We give error estimates for the approximate eigenvalues and the eigenfunctions.