Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

Hierarchic finite elements with selective and uniform reduced integration for Reissner-Mindlin plates

It is well known that the numerical solution of Reissner-Mindlin plates degenerates very rapidly for small thickness (locking phenomenon) when standard finite elements are used for the approximation. We have introduced a family of hierarchic high order finite elements in order to assess reliability and robustness with respect to the locking behavior. In a previous note we have given numerical results obtained with exact numerical integration. In this paper we present the results obtained with selective and uniform reduced integration. The results show that, compared with exact integration, selective reduced techniques improve the quality of the numerical performance and are preferable since computational cost is made smaller.     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.     

Physical shape functions in finite element analysis of moderately thick plates

A new rectangular finite element for moderately thick plates is presented. The element is based on the concept of physical shape functions, i.e. functions which contain in themselves physical and geometrical properties of the element. The analytic formulae of stiffness, geometric stiffness and mass matrices are presented for isotropic material. The element is free from locking and zero energy modes not corresponding to rigid body motions. Several examples are presented for static, initial stability and free vibration problems.     

Two hybrid elements for analysis of thick,thin and sandwich plates

Two general quadrilateral elements for plate bending are developed. Each has 12 degrees of freedom and accounts for transverse shear deformation effects. Each is built from four triangular elements whose properties are derived by the assumed-stress hybrid approach. Matrices needed to generate the stiffness properties of the triangular elements are explicitly stated so as to facilitate their use. Numerical test cases, in which shear deformation effects range from negligible to very important, are used to illustrate the behaviour of the elements. It is concluded that the simpler of the two quadrilaterals is one of the best plate elements currently available.     

Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells

An elasto-plastic analysis of anisotropic plates and shells is undertaken by means of the finite element displacement method. A thick shell formulation accounting for shear deformation is considered, which is based on a degenerate three-dimensional continuum element. The accommodation of variable material properties, not only along the surface of the structure but also through the thickness, is made possible by a discrete layered approach. Although isoparametric elements of the Serendipity family give satisfactory solutions for thick and moderately thin shells the results exhibit ‘locking’ for an increasing ratio of span to thickness. To develop a numerical model which is applicable to thick or thin plates and shells, the nine-node Lagrangian element and the Heterosis element are also introduced into the present model. Plastic yielding is based on the Huber-Mises yield surface extended by Hill for anisotropic materials. The yield function is generalized by introducing anisotropic parameters of plasticity which are updated during the material strain hardening history. Numerical examples are presented and compared with available solutions. The effects of anisotropy on these solutions are also discussed.     

Strain gradient differential quadrature finite element for moderately thick micro-plates

In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C¹-type four-node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro-plates. This element is free of shape functions and shear locking. The C¹-continuity requirements of deflection and rotation functions are accomplished by a fourth-order differential quadrature (DQ)-based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss-Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non-rectangular domains is proceeded by the natural-to-Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro-plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro-plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro-plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner-Fox-Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.     

Robust C0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mechanical analysis

This paper presents a new C⁰ eight-node quadrilateral finite element (FE) for geometrically linear elastic plates. This finite element aims at modeling both thin and thick plates without any pathologies of the classical plate finite elements (shear and Poisson or thickness locking, spurious modes, etc). A C¹ FE was previously developed by the first author based on the kinematics proposed by Touratier. This new FE can be viewed as an evolution towards three directions: (1) use of only C⁰ FE approximations; (2) modeling of thick to thin structures; and (3) capability in multifield problems. The transverse normal stress is included allowing use of the three-dimensional constitutive law. The element performances are evaluated on some standard plate tests, and comparisons are given with exact three-dimensional solutions for plates under mechanical and thermal loads. Comparisons are made with other plate models using C¹ and semi-C¹ FE approximations as well as with an eight node C⁰ FE based on the Reissner–Mindlin model. All results indicate that the present element is highly insensitive to mesh distortion, has very fast convergence properties and gives accurate results for displacements and stresses. Copyright © 2012 John Wiley & Sons, Ltd.     

MITC finite elements for laminated composite plates

Within the framework of the first‐order shear deformation theory, 4‐ and 9‐node elements for the analysis of laminated composite plates are derived from the MITC family developed by Bathe and coworkers. To this end the bases of the MITC formulation are illustrated and suitably extended to incorporate the laminate theory. The proposed elements are locking‐free, they do not have zero‐energy modes and provide accurate in‐plane deformations. Two consecutive regularizations of the extensional and flexural strain fields and the correction of the resulting out‐of‐plane stress profiles necessary to enforce exact fulfillment of the boundary conditions are shown to yield very satisfactory results in terms of transverse and normal stresses. The features of the proposed elements are assessed through several numerical examples, either for regular and highly distorted meshes. Comparisons with analytical solutions are also shown. Copyright © 2001 John Wiley & Sons, Ltd.     

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

This work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner–Mindlin thick plates in shear‐bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner–Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second‐order cone programming problem and is solved using the industrial software package MOSEK . A comprehensive study of the shear‐locking phenomenon is also conducted, and it is shown that discontinuous elements avoid such a phenomenon quite naturally whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so‐obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates. Copyright © 2015 John Wiley & Sons, Ltd.     

Triangular finite element for analysis of thick laminated plates

The development of a general triangular C⁰ element, based on an assumed quadratic displacement potential energy approach, is presented for the analysis of arbitrarily laminated thick plates. The element formulation assumes transverse inextensibility and layerwise constant shear-angle. Convergence of transverse displacement, moments and stresses, the effects of two different Gauss quadrature schemes and comparison of the present solutions with the available analytical/finite-element results also form a part of the investigation. Furthermore, numerical results indicate close agreement between the LCST (layerwise constant shear-angle theory) and the three-dimensional elasticity theory with the length (or width) to thickness ratio as low as 4. Detailed comparison of the LCST-based finite-element solutions with those based on the CST (constant shear-angle theory) and the CLT (classical lamination theory) clearly demonstrates the superiority of the former over the latter two, especially in the prediction of the distribution of the in-plane displacements and stresses through the laminate thickness. This paper also introduces a new non-dimensionalized parameter, Δθ*, which is shown to be a very useful measure for classification of the laminated plates and the suitability of different plate theories over various ranges of length-to-thickness ratio.     

A discrete shear triangular nine D.O.F. element for the analysis of thick to very thin plates

This paper deals with the formulation and the evaluation of a new three node, nine d.o.f. triangular plate bending element valid for the analysis of thick to thin plates. The formulation is based on a generalization of the discrete Kirchhoff technique to include the transverse shear effects. The element, called DST (Discrete Shear Triangle), has a proper rank and is free of shear locking. It coincides with the DKT (Discrete Kirchhoff Triangle) element if the transverse shear effects are not significant. However, an incompatibility of the rotation of the normal appears due to shear effects. A detailed numerical evaluation of the characteristics and of the behaviour of the element has been performed including patch tests for thin and thick plates, convergence tests for clamped and simply supported plates under uniform loading and evaluation of stress resultants. The overall performance of the DST element is found to be very satisfactory.     

Benchmark solution for free vibration of functionally graded moderately thick annular sector plates

In the present article, an exact analytical solution for free vibration analysis of a moderately thick functionally graded (FG) annular sector plate is presented. Based on the first-order shear deformation plate theory, five coupled partial differential equations of motion are obtained without any simplification. Doing some mathematical manipulations, these highly coupled equations are converted into a sixth-order and a fourth-order decoupled partial differential equation. The decoupled equation are solved analytically for an FG annular sector plate with simply supported radial edges. The accurate natural frequencies of the FG annular sector plates with nine different boundary conditions are presented for several aspect ratios, some thickness/length ratios, different sector angles, and various power law indices. The results show that variations of the thickness, aspect ratio, sector angle, and boundary condition of the FG annular sector plates can change the vibration wave number. Also for an FG annular sector plate with one free edge, in opposite to the other boundary conditions, the natural frequency decreases with increasing the aspect ratio for small aspect ratios. Moreover, the mode shape contour plots are depicted for an FG annular sector plate with various boundary conditions. The accurate natural frequencies of FG annular sector plates are presented for the first time and can serve as a benchmark solution.     

A mixed 3D finite element for modelling thick plates

Based on the Hellinger-Reissner variational principle, we formulate a mixed 3-d finite element for plate bending. This element is used to model thick plates and alleviates the problem of shear-locking in plates with large length/thickness ratios. The computer code which was used here, is available.     

Hybrid stress reduced-Mindlin elements for thin multilayer plates

A hybrid-stress formulation of isoparametric elements for the analysis of thin multilayer laminated composite plates is presented. The element displacement behaviour is characterized by laminate reference surface inplane and transverse displacements and laminate non-normal cross-section rotations; as a result, simple C^o interpolation of displacement and rotation can be used, and the number of degrees-of-freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. Attention is restricted here to thin laminates, for which it is shown that the contributions of transverse shear stress and transverse normal stress to the internal complementary energy can be neglected. As a result of this reduction, a modified stiffness-formation algorithm can be used which provides a significant improvement in computational efficiency. The formulation presented is used to develop an 8-node isoparametric thin multilayer plate element. The resulting element is naturally invariant, of correct rank, and non-locking in the thin plate limit.     

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

Hybrid finite elements for the computation of sandwich plates

In this article, new hybrid finite elements are developed on the basis of the displacements and the Pian and Tong functionals using Lagrange multipliers in order to compute correctly and efficiently interlaminar stresses in sandwich structures. These elements represent the mechanical behaviour of sandwich structures in an accurate way, especially at interfaces, where the force equilibrium state must be ensured. They permit to obtain the values of interlaminar stresses using a coarse mesh through the thickness of the sandwich structure. These hybrid elements are assessed and compared through several examples of static linear problems with solutions found in the literature. Copyright © 2001 John Wiley & Sons, Ltd.     

A triangular finite element for thin plates and shells

A finite element modelling technique which utilizes a triangular element with 45 degrees-of-freedom and seven-point integration has been tested for analysis of thin plate and shell structures. The element is based on the degenerate solid shell concept and the mixed formulation with assumed independent inplane and transverse shear strains. Numerical result indicates effectiveness of the present modelling technique which features combined use of elements with kinematic modes and those without kinematic modes in an attempt to eliminate both locking and spurious kinematic modes at global structural level.     

Nonlinear vibrations of thick plates using mindlin plate elements

Large amplitude vibrations of Mindlin plates are studies using Lagrangian, isoparametric, quadrilateral elements with selective integration. Square, rectangular, circular and elliptical plates with clamped and simply supported boundary conditions are considered.     

Improved numerical integration of thick shell finite elements

A quadratic thick shell element derived from a three-dimensional isoparametric element was first introduced by Ahmad and co-workers in 1968. This element was noted, however, to be relatively inefficient in representing bending deformations in thin shell or thin plate applications. The present paper outlines a selective integration scheme for evaluating the stiffness matrix of the element, in which each component of the strain energy is evaluated separately using a different Gaussian integration grid for each contribution. By this procedure, the excessive bending stiffness of the element, which results from the use of me quadratic interpolation functions, is avoided. The improved performance of this element, as compared with the original thick shell element, is demonstrated by analyses of a variety of thin and thick shell problems.

1 Editors' note: A similar development was outlined by O. C. Zienkiewicz and co-workers in lnt. J. num. Meth. Engng, 3 , 275–290 (1971). Some important details differ between the two papers which are thus complementary.