A finite element formulation for multilayered and thick plates

A bilinear isoparametric finite element concept is used for the numerical analysis of multilayered plates. The underlying theory used allows for transverse shear and normal strains in each layer, thus extending the analysis to very thick plates and laminates. To illustrate the versatility of the multilayered element, three examples are presented and the results are compared with available exact solutions.     

A three-dimensional nonlinear analysis of cross-ply rectangular composite plates

The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropie composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement     

Bending analysis of rectangular moderately thick plates using spline finite element method

A bending analysis of rectangular, moderately thick plates with general boundary conditions is presented using the spline element method. The cubic B spline interpolate functions are used to construct the field function of generalized displacements w, φ_itxand φ_ity. The spline finite element equations are derived based on the potential energy principle. For simplicity, the boundary conditions, which consist of three local spline points, are amended to fit specified boundary conditions. The shear effect is considered in the formulations. A number of numerical examples are described for rectangular, moderately thick plates. Since the cubic B spline interpolate functions have sufficient continuity and are piecewise polynomial, so the present numerical solutions show not only that the method gives accurate results, but also that the unified solutions of thick and thin plates can be directly obtained; the trouble with the so-called shear locking phenomenon does not occur here.     

Asymmetric buckling of bimodulus thick annular plates

An annular element with Lagrangian polynomials and trigonometric functions as shape functions is developed for asymmetric finite element stability analysis. The annular element is based on the Mindlin plate theory so that the effect of transverse shear deformation is included. Using the asymmetric finite element model, the asymmetric static buckling of bimodulus thick annular plates subjected to a combination of a pure bending stress and compressive normal stress is investigated. The obtained results of non-dimensional critical buckling coefficients are shown to be very accurate when compared with the exact solutions. The effects of various parameters on the buckling coefficients are studied. The bimodulus properties are shown to have significant influences on the buckling coefficients.     

A mixed FSDT finite element for monoclinic laminated plates

A 4-node finite element for the analysis of laminated composite plates with monoclinic layers, as it occurs for example in piezoelectric applications, is developed. The element is built through the linked interpolation scheme proposed by Taylor and Auricchio [Int J Numer Meth Eng 1993;36:3057–66] and is a generalization of the element presented in [Auricchio F, Sacco E. A mixed-enhanced finite-element for the analysis of laminated composite plates. Int J Numer Meth Eng 1999;44:1481–1504]. Starting from a first-order shear deformation theory (FSDT), a mixed-enhanced variational formulation is considered. It includes as primary variables the resultant shear stresses as well as enhanced incompatible modes, which are introduced to improve in-plane deformations. Bubble functions for rotation degrees of freedom and functions linking transversal displacement to rotations are employed. The solvability of the variational formulation is proved whereas effectiveness and convergence of the proposed finite element are confirmed through several numerical applications. Finally, numerical results are compared with the corresponding analytical solutions as well as to other finite-element solutions.     

On optimal stabilized MITC4 plate bending elements for accurate frequency response analysis

The mixed interpolation technique of the well-established MITC4 quadrilateral plate finite element is combined with shear and generalized least-squares stabilization methods for accurate frequency response analysis. Dispersion analysis is used to determine optimal combinations of stabilization parameters, which, for a given mesh, provide for a three-fold increase in the frequency range over which accurate solutions are obtained, thus allowing for accurate solutions at significantly lower cost. Numerical results for the forced vibration of Reissner–Mindlin plates validate the observations made from the dispersion analysis.     

Natural vibrations of shear deformable cantilevered skewed trapezoidal and triangular thick plates

In a recent companion paper, the efficacy of higher-order shear deformable, C⁰ continuous, Lagrangian isoparametric plate finite element analyses has been demonstrated on cantilevered skewed (parallelogram) thick plates. The present work extends the method to include skewed thick plates having trapezoidal and triangular planforms, and is the first known vibrational study of such plates. Extensive and accurate nondimensional frequency tables and graphical charts are presented for a series of trapezoidal plates showing the effect of aspect ratio, chord ratio, thickness ratio, and skew angle. The need for the present higher-order shear deformable plate finite element method for skewed trapezoidal plate vibrations increases as the skew angle increases and as the aspect ratio, chord ratio, and thickness ratio decreases. Some theoretical and experimental data hitherto published for delta and skewed triangular cantilevered plates are compared with results obtained using the present finite element method. No published theoretical results for cantilevered skewed trapezoidal and triangular thick plates are known to exist.     

A generalised technique for fracture analysis of cracked plates under combined tensile,bending and shear loads

The objective of this paper is to propose a generalized technique called numerically integrated modified virtual crack closure integral (NI-MVCCI) technique for fracture analysis of cracked plates under combined tensile, bending and shear loads. NI-MVCCI technique is used for post-processing the results of finite element analysis (FEA) for computation of strain energy release rate (SERR) components and the corresponding stress intensity factor (SIF) for cracked plates. NI-MVCCI technique has been demonstrated for 4-noded, 8-noded (regular and quarter-point) and 9-noded (regular and quarter-point) isoparametric plate finite elements. These elements are based on Mindlin’s plate theory that considers shear deformation. For all the elements, reduced integration/selective reduced integration techniques have been employed in the studies. In addition, for 9-noded element assumed shear interpolation functions have been used to overcome the shear locking problem. Numerical studies on fracture analysis of plates subjected to tension–moment and tension–shear loads have been conducted employing these elements. It is observed that among these elements, the 9-noded Lagrangian plate element with assumed shear interpolation functions exhibits better performance for fracture analysis of cracked plates.     

Development of an anisotropic,multilayered, shear-deformable rectangular plate element

A multilayered anisotropic flat plate element which includes the effects of the transverse shear deformation is developed by making use of the displacement formulation. The discrete element is a rectangle with 32 degrees of freedom which include extension, bending, and transverse shear deformation states. The formulation is based on an improved bi-dimensional transverse shear deformation plate theory which accounts for piecewise linear distribution across the thickness of the inplane displacements u and v, and allows the contact conditions at the interfaces between the layers to be satisfied. In order to demonstrate the accuracy and efficiency of the developed finite element, the solutions for two sample problems are given. Problems, for which analytical solutions are available, were selected so that a comparison between the analytical and numerical solutions could be made. The first example is the well-established cylindrical bending problem of cross-ply and angle-ply laminates; the second one is the bending of a symmetric cross-ply square plate simply-supported on all edges. The tests carried out show that the element is very efficient in predicting the responses of thick and thin laminated plates. Also predicting, at very low span-to-thickness ratios and models, the warpage of the cross-section, which has been found to be unfeasible by other conventional finite-element displacement approaches.     

A generalized unconstrained theory and isogeometric finite element analysis based on Bézier extraction for laminated composite plates

This work presents an isogeometric finite element formulation based on Bézier extraction of the non-uniform rational B-splines (NURBS) in combination with a generalized unconstrained higher-order shear deformation theory (UHSDT) for laminated composite plates. The proposed approach relaxes zero-shear stresses at the top and bottom surfaces of the plates and no shear correction factors are required. A weak form of static, free vibration and transient response analyses for laminated composite plates is then established and is numerically solved using isogeometric Bézier finite elements. NURBS can be written in terms of Bernstein polynomials and the Bézier extraction operator. IGA is implemented with the presence of C°-continuous Bézier elements which allow to easily incorporate into existing finite element codes without adding many changes as the former IGA. As a result, all computations can be performed based on the basis functions defined previously as the same way in finite element method (FEM). Numerical results performed over static, vibration and transient analysis show high efficiency of the present method.     

An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates

An edge-based smoothed finite element method (ES-FEM) for static, free vibration and buckling analyses of Reissner–Mindlin plates using 3-node triangular elements is studied in this paper. The calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements. In order to avoid the transverse shear locking and to improve the accuracy of the present formulation, the ES-FEM is incorporated with the discrete shear gap (DSG) method together with a stabilization technique to give a so-called edge-based smoothed stabilized discrete shear gap method (ES-DSG). The numerical examples demonstrated that the present ES-DSG method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature.     

Design of laminated composite plates for optimal dynamic characteristics using a constrained global optimization technique

The lamination arrangements of moderately thick laminated composite plates for optimal dynamic characteristics are studied via a constrained multi-start global optimization technique. In the optimization process, the dynamical analysis of laminated composite plates is accomplished by utilizing a shear deformable laminated composite finite element, in which the exact expressions for determining shear correction factors were adopted and the modal damping model constructed based on an energy concept. The optimal layups of laminated composite plates with maximum fundamental frequency or modal damping are then designed by maximizing the frequency or modal damping capacity of the plate via the multi-start global optimization technique. The effects of length-to-thickness ratio, aspect ratio and number of layer groups upon the optimum fiber orientations or layer group thicknesses are investigated by means of a number of examples of the design of symmetrically laminated composite plates.     

基于边光滑有限元法的剪切变形板几何非线性分析

为改善在计算板的几何非线性问题时有限元法系统过硬的数值缺陷,提高计算精度,在考虑剪切变形的yon Karman假设下,基于全拉格朗日描述方法,将边光滑有限元法应用于板的几何非线性分析.计算公式基于1阶剪切变形理论,并采用离散剪切间隙有效地消除剪切自锁.在三角形单元的基础上进一步形成边界光滑域,在每个光滑域内对应变进行光...     

Analysis of surface cracks in plates and shells using the line-spring model and ADINA

The line-spring model of Rice and Levy is used in conjunction with thin shell finite elements to analyze surface cracks in plates and cylindrical shells. Implementation of the approach in the finite element code ADINA is discussed in detail. Both linear and nonlinear cases are considered. The stress intensity factor and the J-integral results are obtained for several example problems concerning surface cracked plates and axial and circumferential flaws in cylinders. Comparisons are made with the available literature solutions. Sensitivity of the results relative to the mesh size, integration scheme and other relevant parameters is also investigated.     

Electric potential approximations for an eight node plate finite element

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C⁰ finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387–411] and C¹ finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305–24]. In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations.Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.     

Formulation of a nonlinear compatible finite element for the analysis of laminated composites

Increased use of laminated composites has pointed out the need for better analytic tools. These tools must be able to correctly account for normal shear in the laminates and should be able to solve nonlinear problems. The finite element method is applied in this paper to analyze laminated composites using an approach which includes both nonlinear and normal shear effects. A finite element is formulated from basic elasticity equations. The most unique characteristic of the element is the manner in which normal shear is handled. The normal to a reference surface is allowed to not only rotate but also to change shape. Not only is displacement continuity imposed at lamina interfaces, but also slope continuity is guaranteed. After a complete general formulation of the finite element is presented, the method is specialized to plates so that the convergence characteristics, the accuracy and the applicability of the element can be studied. The finite element does an excellent job of analyzing laminated composites, and gives the analyst the ability to do nonlinear analysis of laminated composites under a variety of loading and boundary conditions.     

Postbuckling behaviour of plates and shells using a mindlin shallow shell formulation

The paper presents a finite element Mindlin shallow shell formulation. Compared to a previous flat plate formulation it is shown that the addition of a shallow shell capability adds very little extra computational effort. Results are given for the postbuckling behaviour of square and circular plates subject to direct inplane loading and a square plate subject to inplane shear loading. Examples are also presented of the analyses of a shallow truss and cylindrical and spherical shells, all exhibiting snap through behaviour. Agreement with existing solutions is generally good and where possible the results are presented numerically.     

宽翼T形梁桥动力学理论与特性分析

考虑了剪滞翘曲应力自平衡条件、剪切变形和剪力滞后效应等因素的影响,本文提出了一种对宽翼薄壁T形梁动力学特性的分析方法.分析中为了准确反应T形梁翼板的动位移变化,三个广义动位移被引入,且以能量变分原理为基础建立了T形梁动力反应的控制微分方程和自然边界条件,据此对T形梁的动力反应特性进行了分析,揭示了T形梁桥动力反应的规律.算例中,对比了考虑和不考虑剪滞翘曲应力自平衡条件对T形梁动力反应的影响,结果显示考虑剪滞翘曲应力自平衡条件的计算方法与有限元数值解吻合更好.