Hybrid-Trefftz stress elements for plate bending

The hybrid-Trefftz stress element is used to emulate conventional finite elements for analysis of Kirchhoff and Mindlin-Reissner plate bending problems. The element is hybrid because it is based on the independent approximation of the stress-resultant and boundary displacement fields. The Trefftz variant is consequent on the use of the formal solutions of the governing Lagrange equation to approximate the stress-resultant field. In order to emulate conventional elements, nodal functions are used to approximate the displacements on the boundary of the element. Duality is used to set up the element solving system. The associated variational statements and conditions for existence and uniqueness of solutions are recovered. Triangular and quadrilateral elements are tested and characterized in terms of convergence, sensitivity to shear-locking, and shape distortion. Their relative performance is assessed using assumed strain Mixed Interpolation of Tensorial Components (MITC) elements and recently proposed Trefftz-based elements. This relative assessment is extended to a hypersingular problem to illustrate the effect of enriching the domain and boundary approximation bases.     

Hybrid-Trefftz plate bending elements with p-method capabilities

The p-method capabilities have thus far been used only in connection with the assumed displacement model elements. When such elements are used, singular points have to be isolated by one or two layers of small elements graded in a suitable geometrical progression towards singularity. This paper presents an alternative formulation which circumvents this drawback and enables excellent solution results to be obtained with uniform FE-grids. The formulation is based on the recently presented hybrid-Trefftz FE-model and makes use of optional expansion sets for various singularities or stress concentrations. Stress concentrations due to concentrated loads are also properly accounted for. It is shown that by augmenting the order of approximation over a fixed grid of such elements rapid convergence towards the accurate solution is obtained in the most efficient way. This paper summarizes the results of a preliminary study that had been undertaken to critically evaluate the potential of the new approach as a suitable basis for subsequent developments of a fully automated adaptive process.     

Automatic adaptive refinement for plate bending problems using Reissner–Mindlin plate bending elements

The influence of the presence of singular points and boundary layers associated with the edge effects in a Reissner–Mindlin (RM) plate in the design of an optimal mesh for a finite element solution is studied, and methods for controlling the discretization error of the solution are suggested. An effective adaptive refinement strategy for the solution of plate bending problems based on the RM plate bending model is developed. This two-stage adaptive strategy is designed to control both the total and the shear error norms of a plate in which both singular points and boundary layers are present. A series of three different order assumed strain RM plate bending elements has been used in the adaptive refinement procedure. The locations of optimal sampling points and the effect of element shape distortions on the theoretical convergence rate of these elements are given and discussed. Numerical experiments show that the suggested refinement procedure is effective and that optimally refined meshes can be generated. It is also found that all the plate bending elements used can attain their full convergence rates regardless of the presence of singular points and boundary layers inside the problem domain. Boundary layer effects are well captured in all the examples tested and the use of a second stage of refinement to control the shear error is justified. In addition, tests on the Zienkiewicz–Zhu error estimator show that their performances are satisfactory. Finally, tests of the relative effectiveness of the plate bending elements used have also been made and it is found that while the higher order cubic element is the most accurate element tested, the quadratic element tested is the most efficient one in terms of CPU time used. © 1998 John Wiley & Sons, Ltd.     

A study of three-node triangular plate bending elements

An assessment of flat triangular plate bending elements with displacement degrees-of-freedom at the three corner nodes only is presented, with the purpose of identifying the most effective for thin plate analysis. Based on a review of currently available elements, specific attention is given to the theoretical and numerical evaluation of three triangular 9 degrees-of-freedom elements; namely, a discrete Kirchhoff theory (DKT) element, a hybrid stress model (HSM) element and a selective reduced integration (SRI) element. New and efficient formulations of these elements are discussed in detail and the results of several example analyses are given. It is concluded that the most efficient and reliable three-node plate bending elements are the DKT and HSM elements.     

Trigonometric function representations for rectangular plate bending elements

A study of bending of plates for small deformations by applying the finite element method is presented in this paper. A new class of displacement functions has been developed for rectangular bending elements. For the first time a method is proposed for the derivation of displacement functions in which trigonometric expressions that had apparently failed to yield satisfactory results in earlier attempts can be used along with polynomial terms. The derivation leads to conforming type of displacement functions which satisfy the convergence criterion. The approach also demonstrates how many more new displacement functions can be found.     

Convergence of eigenvalue solution in conforming plate bending finite elements

The convergence proof of plate eigenvalue solution from conforming displacement finite elements is presented. The analysis is based on converting a thick plate free vibration problem in to a corresponding isoperimetric variational problem. A conforming thick, plate element is used to illustrate the mathematical development. On the basis of the derived asymptotic rate of convergence of the approximate eigenvalues, the authors propose a practical method of improving the numerical solutions. Extension of the mathematical proof to cover classical thin plate finite elements is briefly discussed.     

Mixed formulation finite elements for mindlin theory plate bending

Two new finite elements are developed for the Mindlin theory plate bending problem. The formulation is based on the modified Hellinger-Reissner principle with independent transverse shear strains. Numerical examples indicate that, with properly assumed transverse shear strains, these new elements designated as PLAT8 and PLAT8H do not exhibit locking effect even for very thin plates.     

Mixed plate bending elements based on least‐squares formulation

A finite element formulation for the bending of thin and thick plates based on least‐squares variational principles is presented. Finite element models for both the classical plate theory and the first‐order shear deformation plate theory (also known as the Kirchhoff and Mindlin plate theories, respectively) are considered. High‐order nodal expansions are used to construct the discrete finite element model based on the least‐squares formulation. Exponentially fast decay of the least‐squares functional, which is constructed using the L₂ norms of the equations residuals, is verified for increasing order of the nodal expansions. Numerical examples for the bending of circular, rectangular and skew plates with various boundary conditions and plate thickness are presented to demonstrate the predictive capability and robustness of the new plate bending elements. Plate bending elements based on this formulation are shown to be insensitive to both shear‐locking and geometric distortions. Copyright © 2004 John Wiley & Sons, Ltd.     

Convergence studies of eigenvalue solutions using two finite plate bending elements

The convergence rates of eigenvalue solutions using two finite plate bending elements are studied. The elements considered are the well-known 12 degree of freedom, non-conforming rectangular element and the 16 degree of freedom, conforming rectangular element. Three problems are analysed, a square plate simply supported on two opposite sides with the other two sides clamped, simply supported, or free. Closed form, finite element solutions for these problems are obtained by using shifting E-operators. With few exceptions, eigenvalue solutions found with the non-conforming element converge from below the exact answers at an asymptotic rate of n^?2, where n is the number of elements on a side. However, since the array size needed for such convergence is very large, little can be said about the convergence rates for practical arrays. The conforming element solutions converge from above at an asymptotic rate of n^?4. A comparison of the errors involved in using these two elements shows that the conforming element is far superior to the non-conforming element.     

On the use of singular displacement finite elements for cracked plate in bending

The objective of this work is to assess the performance, convergence and accuracy of four different displacement crack-tip elements used in modelling a cracked plate subjected to out-of-plane bending. A methodology is developed for calculating the singular field from the computed results and optimizing the mesh used in the numerical solution. It was found that using the quarter-node triangular elements surrounded by quadrilateral transition elements yields very accurate estimates of the singular fields of displacements and stresses as well as the stress-intensity factor for various materials and thicknesses of plates. It was also found that when the transition elements are incorporated, the optimum length of the singular element ranges from 0.5 to 1 per cent of half of the crack length.

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Résumé Le but de ce travail est d'établir les possibilités, la convergence et la sécurité de quatre modèles différents d'éléments finis pour extrémités de fissure, utilisés pour représenter une plaque fissurée soumise à flexion hors de son plan. On a développé une méthodologie pour le calcul du champ singulier qui résulte des calculs, et pour l'optimisation du réseau utilisé dans la solution numérique.On trouve qu'en utilisant des éléments triangulaires quart-onde entourés d'éléments quadrilatères de transition, on est conduit à des estimations très précises des champs singuliers de déplacements et de contraintes ainsi que du facteur d'intensité de contrainte, dans le cas de divers matériaux et de diverses épaisseurs de plaque.On trouve également que lorsque sont incorporés les éléments de transition, la longueur optimale d'un élément singulier varie entre 0,5 et 1% de la demi-longueur de la fissure.

     

On evaluating the inf–sup condition for plate bending elements

This paper addresses the evaluation of the inf–sup condition for Reissner–Mindlin plate bending elements. This fundamental condition for stability and optimality of a mixed finite element scheme is, in general, very difficult to evaluate analytically, considering for example distorted meshes. Therefore, we develop a numerical test methodology. To demonstrate the test methodology and to obtain specific results, we apply it to standard displacement-based elements and elements of the MITC family. Whereas the displacement-based elements fail to satisfy the inf–sup condition, we find that the MITC elements pass our numerical test for uniform meshes and a sequence of distorted meshes. © 1997 John Wiley & Sons, Ltd.     

Specifications of boundary conditions for Reissner/Mindlin plate bending finite elements

Plate bending finite elements based on the Reissner/Mindlin theory offer improved possibilities to pursue reliable finite element analyses. The physical behaviour near the boundary can be modelled in a realistic manner and inherent limitations in the Kirchhoff plate bending elements when modelling curved boundaries can easily be avoided. However, the boundary conditions used are crucial for the quality of the solution. We identify the part of the Reissner/Mindlin solution that controls the boundary layer and examine the behaviour near smooth edges and corners. The presence of boundary layers of different strengths for different sets of boundary conditions is noted. For a corner with soft simply supported edges the boundary layer removes the singular behaviour of Kirchhoff type from the stress resultants. We demonstrate the theoretical results by some numerical studies on simple plate structures which are discretized by an accurate, higher-order plate element. The results provide guidance in choosing efficient meshes and appropriate boundary conditions in finite element analyses.     

A simple algorithm for the stiffness matrix of triangular plate bending elements

For a hierarchy of polynomials on the triangle there is derived an algorithm for computing the stiffness matrix of the plate bending element. The algorithm is easy to program and means a considerable saving of the computing time. The same approach can be used for any elliptic equation with constant coefficients.     

Bicubic fundamental splines in plate bending

This paper deals with solving the two-dimensional variational problem for plate bending by the Ritz method using bicubic fundamental splines. It is a piecewise polynomial method, very adaptable to practical numerical computation, and can be an alternative for the well-known Finite Element Method¹.     

A study of quadrilateral plate bending elements with ‘reduced’ integration

The accuracy of certain thick plate elements when used in the context of thin plate problems can be improved by the use of reduced integration of the stiffness matrices. A series of numerical experiments on five different quadrilateral thick plate elements demonstrates the use of reduced integration and indicates the main reason for its success. This is the relaxation of a constraint on the shear strains. It is shown that the performance of a nine-noded Lagrangian element is near optimal.     

Kinked crack in a bending trapezoidal plate

The interaction problem of a kinked crack and the edges of a bending trapezoidal plate which takes the effects of transverse shear deformation into account is presented. The research method is based upon the complex potential technique of Muskhelishvili using conformal mapping. Furthermore, for the analysis of the moment intensities at the tips of the kinked crack, the concept of dislocation distribution is applied. The integral equations for the stress disturbance problem along the line that is the presumed location of the kinked crack are then obtained as a system of singular integral equations with simple Cauchy kernels. As a consequence, the variation of moment intensity factors at the crack-tips is also illustrated.     

A new method for derivation of locking-free plate bending finite elements via mixed/hybrid formulation

The shear-locking phenomenon in discrete bending analysis of Mindlin/Reissner plates is investigated. Mixed/hybrid variational principles are introduced which, unlike the rigorous displacement model, allow systematic derivation of locking-free finite elements. This is achieved by satisfaction of an auxiliary condition, having the clear physical interpretation of shear-force elimination on account of equilibrium. An example, using competitive techniques, demonstrates the applicability of the idea.     

An effective quadrilateral plate bending element

An effective arbitrary quadrilateral thin plate bending element with a quasi-conforming, QCQ element, is presented in this paper. The elements pass the patch test with constant strain and the patch test with linear strains approximately. When the element degenerates to a rectangle the patch test with linear strains is passed. The calculation of the element stiffness matrix is simple without numerical integration. The numerical examples show that the QCQ element has a higher accuracy and a faster convergence rate.     

A quintic conforming plate bending triangle

A new family of conforming triangular plate bending elements, based on the Displacement Method of formulation, is developed in detail in Reference 9. The elements can be of any order n?4, and do not present excess nodal continuties at the corner nodes. For a given element, the assumed vertical displacement w consists of a polynomial expansion supplemented by three rational function. Polynomial completeness is of order n for even n, and of order n-1 for odd values of n. This family is proposed as a desirable alternative to refined triangular elements which incorporate all three curvatures as corner node variables,⁸ or require constraint equations to enforce inter-element compatibility,¹⁰ The purpose of this paper is to describe the quintic member of the family. The traditional difficulties associated with the discontinuous curvatures of the rational functions are solved in a straightforward fashion. The element has the same good accuracy as the other, well-known, quintic triangle, ^3,4,8 and some additional advantages accruning from the simplicity of its nodal variables. A similar performance can be anticipated with the other members of the family.