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A higher-order Reissner-Mindlin plate element method is presented based on the framework of assumed stress quasi-conforming method and Hellinger-Reissner variational principle. A novel six-node triangular plate element is proposed by utilizing this method for the static and free vibration analysis of Reissner-Mindlin plates.

First, the initial assumed stress field is derived by using the fundamental analytical solutions which satisfy all governing equations. Then the stress matrix is treated as the weighted function to weaken the strain-displacement equations after the strains are derived by using the constitutive equations. Finally, the arbitrary order Timoshenko beam function is adopted as the string-net functions along each side of the element for strain integration.

The proposed element can pass patch test and is free from shear locking and spurious zero energy modes. Numerical tests show that the element can give high-accurate solutions, good convergence and is a good competitor to other models.

This work gives new formulations to develop high-order Reissner-Mindlin plate element, and the new strategy exhibits advantages of both analytical and discrete methods.

This paper aims to describe the formulation of a new finite element by assuming the strain field rather than the displacement field and by using the Reissner–Mindlin plate theory for the free vibration analysis of bending plates. This quadrilateral element consists of four-nodes and twelve degrees of freedom. The suggested element is based on assumed functions of the strain field that satisfy the compatibility equation.

After the proposition of the new element, several numerical tests for plates with regular and distorted meshes are presented to assess the performance of the new element. In addition, a parametric study is carried out to analyze the effects of biaxial loads on the natural frequencies of square plates with various boundary conditions. Detailed discussions are proposed after each benchmark problem.

The formulated element has verified the shear locking test and passes the patch test. The obtained results from the developed element show an excellent accuracy and fast convergence, and the natural frequencies are in excellent agreement when compared with analytical and other available numerical solutions.

The present element is simple in its formulation and has been proven to be applicable to thin or thick plate situations with sufficient accuracy. This element with full integration is free from shear locking, however, the numerical results provided by the standard four-node plate element R4 element show locking phenomena in thin plates. In addition to these features, the imposition of the compatibility conditions and the rigid body modes allow obtaining a finite element with higher-order terms for displacements field, which can increase the performance of the finite elements.

We briefly summarize the theoretical formulations of our MITC plate bending elements and then present numerical convergence results. The elements are based on Reissner‐Mindlin plate theory and a mixed‐interpolation of the transverse displacement, section rotations and transverse shear strain components. We consider our 4, 9 and 16‐node quadrilateral elements and our 7 and 12‐node triangular elements. The theoretical and numerical results indicate the high reliability and effectiveness of our elements.

The purpose of this paper is to address error‐controlled adaptive finite element (FE) method for thin and thick plates. A procedure is presented for determining the most suitable plate model (among available hierarchical plate models) for each particular FE of the selected mesh, that is provided as the final output of the mesh adaptivity procedure.

The model adaptivity procedure can be seen as an appropriate extension to model adaptivity for linear elastic plates of so‐called equilibrated boundary traction approach error estimates, previously proposed for 2D/3D linear elasticity. Model error indicator is based on a posteriori element‐wise computation of improved (continuous) equilibrated boundary stress resultants, and on a set of hierarchical plate models. The paper illustrates the details of proposed model adaptivity procedure for choosing between two most frequently used plate models: the one of Kirchhoff and the other of Reissner‐Mindlin. The implementation details are provided for a particular case of the discrete Kirchhoff quadrilateral four‐node plate FE and the corresponding Reissner‐Mindlin quadrilateral with the same number of nodes. The key feature for those elements that they both provide the same quality of the discretization space (and thus the same discretization error) is the one which justifies uncoupling of the proposed model adaptivity from the mesh adaptivity.

Several numerical examples are presented in order to illustrate a very satisfying performance of the proposed methodology in guiding the final choice of the optimal model and mesh in analysis of complex plate structures.

The paper confirms that one can make an automatic selection of the most appropriate plate model for thin and thick plates on the basis of proposed model adaptivity procedure.

An adaptive mesh refinement procedure is used in static plate bending finite element analysis to study the edge effects in Mindlin—Reissner plates.

The aim of this paper is to extend the Element Free Galerkin method (EFGM) in order to perform the elasto‐plastic analysis of isotropic plates.

The EFGM shape‐function construction is briefly presented. The Newton‐Raphson method and the elasto‐plastic algorithm adapted to the EFGM, are described. Several plate bending non‐linear material problems are solved and the obtained solutions are compared with available finite element method (FEM) solutions.

The paper finds that the developed EFGM approach is a good alternative to the FEM for the solution of non‐linear problems, once the obtained results with the EFGM show a high similarity with the obtained FEM results.

Comparing the FEM and the EFGM there are some drawbacks for the EFGM. The computational cost of the EFGM is higher, the imposition of the essential boundary conditions is more complex and there is a high sensitivity of the method in what concerns the choice of the influence domain and the choice of the weight function.

The knowledge that the EFGM formulation can be treated almost as the FEM formulation once the EFGM parameters are calibrated and optimized.

The extension of the EFGM to the elasto‐plastic analysis of isotropic plates.

The purpose of this paper is to extend the natural neighbour radial point interpolation method (NNRPIM) to the dynamic analysis (free vibrations and forced vibrations) of two‐dimensional, three‐dimensional and bending plate problems.

The NNRPIM shape‐function construction is briefly presented, as are the dynamic equations and the mode superposition method is used in the forced vibration analysis. Several benchmark examples of two‐dimensional and plate bending problems are solved and compared with the three‐dimensional NNRPIM formulation. The obtained results are compared with the available exact solutions and the finite element method (FEM) solutions.

The developed NNRPIM approach is a good alternative to the FEM for the solution of dynamic problems, once the obtained results with the EFGM shows a high similarity with the obtained FEM results and for the majority of the studied examples the NNRPIM results are more close to the exact solution results.

Comparing the FEM and the NNRPIM, the computational cost of the NNRPIM is higher.

The paper demonstrates extension of the NNRPIM to the dynamic analysis of two‐dimensional, three‐dimensional and bending plate problems. The elimination of the shear‐locking phenomenon in the NNRPIM plate bending formulation. The various solved examples prove a high convergence rate and accuracy of the NNRPIM.

A simple shape-free high-order hybrid displacement function element method is presented for precise bending analyses of Mindlin–Reissner plates. Three distortion-resistant and locking-free eight-node plate elements are proposed by utilizing this method.

This method is based on the principle of minimum complementary energy, in which the trial functions for resultant fields are derived from two displacement functions, F and f, and satisfy all governing equations. Meanwhile, the element boundary displacements are determined by the locking-free arbitrary order Timoshenko’s beam functions. Then, three locking-free eight-node, 24-DOF quadrilateral plate-bending elements are formulated: HDF-P8-23β for general cases, HDF-P8-SS1 for edge effects along soft simply supported (SS1) boundary and HDF-P8-FREE for edge effects along free boundary.

The proposed elements can pass all patch tests, exhibit excellent convergence and possess superior precision when compared to all other existing eight-node models, and can still provide good and stable results even when extremely coarse and distorted meshes are used. They can also effectively solve the edge effect by accurately capturing the peak value and the dramatical variations of resultants near the SS1 and free boundaries. The proposed eight-node models possess potential in engineering applications and can be easily integrated into commercial software.

This work presents a new scheme, which can take the advantages of both analytical and discrete methods, to develop high-order mesh distortion-resistant Mindlin–Reissner plate-bending elements.

A numerical method is presented in this paper for the free vibration analysis of circular and elliptical first order shear deformable plates. In this method, the ellipse is mapped into a circle and then the circular geometry of the plate is mapped using parabolic interpolation function of natural coordinates and eight nodal points of prescribed coordinates. The displacement fields are defined by a set of relatively very high order interpolation functions and for the displacement degrees of freedom a set of nodal points are defined separate from those of the geometric interpolation. Numerical results for the fully clamped elliptical plate are obtained and compared with the available data from the literature. Additional results for the simply 1‐supported complete elliptical plate and the annular elliptical plates subjected to various boundary conditions are presented and discussed.

Numerous finite elements are proposed based on analytical solutions. However, it is difficult to find the solutions for complicated governing equations. This paper aims to present a novel formulation in the framework of assumed stress quasi-conforming method for the static and free vibration analysis of anisotropic and symmetric laminated plates.

Firstly, an initial stress approximation ruled by 17 parameters, which satisfies the equilibrium equations is derived to improve the performance of the constructed element. Then the stress matrix is treated as the weighted function to weaken the strain-displacement equations. Finally, the Timoshenko’s laminated composite beam functions are adopted as boundary string-net functions for strain integration.

Several numerical examples are presented to show the performance of the new element, and the results obtained are compared with other available ones. Numerical results have proved that the new element is free from shear locking and possesses high accuracy for the analysis of anisotropic and symmetric laminated plates.

This paper proposes a new QC element for the static and free vibration analysis of anisotropic and symmetric laminated plates. In contrast with the complicated analytical solutions of the equilibrium equations, an initial stress approximation ruled by 17 parameters is adopted here. The Timoshenkos laminated composite beam functions are introduced as boundary string-net functions for strain integration. Numerical results demonstrate the new element is free from shear locking and possesses high accuracy for the analysis of anisotropic and symmetric laminated plates.