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Gives a bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view. The range of applications of FEMs in this area is wide and cannot be presented in a single paper; therefore aims to give the reader an encyclopaedic view on the subject. The bibliography at the end of the paper contains 2,025 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1992‐1995.

A finite element formulation for flexure of isotropic plates based on a recent refined theory is developed. The refined theory incorporates effects of transverse shear, transverse normal stress and transverse normal strain. The Galerkin finite element method was used to develop the finite element equations for both plate bending and inplane problems. The performance of the proposed finite element model was evaluated by solving problems of uniformly loaded thick plates with different support conditions. The results of the present formulation are compared with Mindlin/Reissner and elasticity solutions.

A formulation has been developed for thermo‐elastoviscoplastic finite element analyses of continuous fibre‐reinforced composite plates subject to bending loading using a generalized continuum mechanics approach. Such an approach is used to model the non‐homogeneity in a composite, which is constituted by fibres embedded in a matrix material. The present formulation computes the respective stresses occurring in each constituent so that the respective yield criterion and flow rule of each constituent may be used if there is a material yielding in any constituent. Thermo‐elastic deformation of fibre and thermo‐elastoviscoplastic deformation of matrix are considered in the present study because the yield strength of fibre is substantially higher than that of matrix in many cases. Both constituents are assumed to be isotropic so that the von‐Mises yield criterion may be used for viscoplastic yielding of matrix. As numerical examples, a parametric study is performed for thermo‐elastoviscoplastic deformations of laminated composite plates subject to thermal bending loads.

The purpose of this study is to propose a generalized integral transform technique (GITT) to investigate the bending behavior of rectangular thin plates with linearly varying thickness resting on a double-parameter foundation.

The bending of plates with linearly varying thickness resting on a double-parameter foundation is analyzed by using the GITT for six combinations of clamped, simply-supported and free boundary conditions under linearly varying loads. The governing equation of plate bending is integral transformed in the uniform-thickness direction, resulting in a linear system of ordinary differential equations in the varying thickness direction that is solved by a fourth-order finite difference method. Parametric studies are performed to investigate the effects of boundary conditions, foundation coefficients and geometric parameters of variable thickness plates on the bending behavior.

The proposed hybrid analytical-numerical solution is validated against a fourth-order finite difference solution of the original partial differential equation, as well as available results in the literature for some particular cases. The results show that the foundation coefficients and the aspect ratio b/a (width in the y direction to height of plate in the x direction) have significant effects on the deflection of rectangular plates.

The present GITT method can be applied for bending problems of rectangular thin plates with arbitrary thickness variation along one direction under different combinations of loading and boundary conditions.

The purpose of this study is to propose generalised integral transform technique (GITT) to obtain the exact solutions for bending of clamped parallelogram plate resting on elastic foundation.

The GITT is used to solve the bending problem of the full clamped parallelogram plate under an elastic foundation. The auxiliary problem was developed and the corresponding eigenfunction and eigenvalue were calculated simultaneously. The original partial differential governed equation has been represented by the transformed ordinary differential equation system and solved by the subroutine DBVPFD from International Mathematics and Statistics Library.

The GITT has been proven to be an efficient approach to solve the bending problem of the plate with different loads, boundary conditions and elastic foundations. The parametric study indicates that the elastic foundation modulus has significant contribution in reducing the vertical deflections and moments for both rectangular and parallelogram plates. With the increasing of aspect ratio (a/b) and the elastic foundation modulus, the trends of the deflection and moment reduction decreased significantly.

The present hybrid analytical-numerical methodology was first used to solve the mechanics problem of the clamped parallelogram plate resting on elastic foundation. Excellent convergence and high accuracy was observed by comparing with the published results. It exhibits potential application to investigate the mechanics problem of the composite plate with different boundary conditions in the shipbuilding and civil engineering.

“V-bending” is the most commonly used bending process in which the sheet metal is pressed into a “V-shaped” die using a “V-shaped” punch to form a required angular bend. When the punch is removed after the operation, because of elastic recovery, the bent angle varies. This shape discrepancy is known as spring back which causes problems in the assembly of the component in the modern aerospace industry. Regarding the optimization of spring-back accuracy, this research will illustrate the laws of the transition area (TA) of the nondeformation area (NDA) during the 90° “V-shape” bending process.

According to the traditional “V-bending” process to optimize the spring-back accuracy, the bent sheets are divided into deformation area (DA) and NDA. For this reason, the traditional “V-bending” process may prolong error to optimize the spring-back accuracy because NDA has a certain amount of deformation, which the researcher always avoids. Firstly, bent sheets are divided into three parts in this research: DA, TA and NDA to avoid the distortion error in TA that are not considered in the NDA in traditional theory. Then, the stress and strain in the DA and TA were discussed during theoretical derivation and some hypotheses were proposed. In this research, the interval, position and distortion degree of the TA of the bending sheet are used by finite element analysis. Finally, V-shape bending tests for aluminum alloy at room temperature are used and labeled all the work pieces' TAs to realize the deformation amount in the TA.

The bending radius does not affect the range of the TA, it only changes the position of TA in the bending sheet. It is evident that the laws of TA were explored in the width direction and gradually changed from the inner layer to the outer layer based on the ratio of width and thickness of the bending plate/sheet.

In the modern aerospace industry, aircraft manufacturing technology must maintain high accuracy. This research has practical value in the 90° “V-shape” bending of metal sheets and the development of its spring-back accuracy.

FAILURE of panels under static compression, or for that matter under any loads, involves a vast array of problems ranging from properties of material to initial instability and post‐buckling phenomena as occurring in various types of panels. It is not intended here to do justice to all these aspects of the subject but to select a single—but at the same time very important—topic, develop its analysis as fully as possible, and present the results in a readily applicable form. The structure investigated is the single skin stiffened panel under compression and the mode of failure considered, denoted by flexural cum torsional failure, involves predominantly flexure and torsion of the stringer with a wavelength of greater order of magnitude than stringer height and pitch. By torsional deformation of the stringer we understand a rotation of its undistorted cross‐section about a longitudinal axis R in the plane of the plate, the position of which will be selected later on (see FIG. 1b). The panel may, of course, also fail in a local mode of stringer and plate with a short wave‐length of the order of magnitude of stringer height and pitch, but the analysis of this case is not included here (see, however, Argyris and Dunne). Note that a local mode of deformation of a stringer formed by straight walls is commonly defined as a distortion of the cross‐section in which the longitudinal edges where two adjacent walls meet remain straight (see FIG. 1c).

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 unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).

The original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.

The bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.

The symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.