Three-dimensional finite element analysis of thick laminated plates

A displacement-based, three-dimensional finite element scheme is proposed for analyzing thick laminated plates. In the present formulation, a thick laminated plate is treated as a three-dimensional inhomogeneous anisotropic elastic body. Particular attention is focused on the prediction of transverse shear stresses. The plane of a laminated plate is first discretized into conventional eight-node elements. Various through-thickness interpolation is then denned for different regions of the plate; layerwise local shape functions are used in the regions where transverse shear stresses are of interest, while an ad hoc global-local interpolation is used in the region where only the general deformation pattern is concerned. For satisfying the displacement compatibility between these two regions, a transition zone is introduced. The model incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Details of formulation will be presented together with several numerical examples for demonstrating the proposed scheme.     

Hybrid-interface element for thick laminated composite plates

A novel 27-node three-dimensional hexahedral hybrid-interface finite element (FE) model has been presented to analyze laminated composite plates and sandwich plates using the minimum potential energy principle. Fundamental elasticity relationship between components of stress, strain and displacement fields are maintained throughout the elastic continuum as the transverse stress components have been invoked as nodal degrees of freedom. Continuity of the transverse stresses at lamina interface has been maintained. Each lamina is modeled by using hybrid-interface elements at the top and the bottom interfaces and conventional displacement based elements sandwiched between these interfaces. Results obtained from the present formulation have found to be in excellent agreement with the elasticity solutions for thin and thick composite cross-ply, angle-ply laminates, as well as sandwich plates. Additional results have also been presented on the variation of the transverse strains to highlight magnitude of discontinuity in these quantities due to difference in properties of face and core materials of sandwich plates. Present formulation can be used effectively to interface hybrid formulation that uses transverse stresses and displacements as degrees of freedom with conventional purely displacement based formulation for realistic estimates of the transverse stresses.     

Mixed least-squares finite element model for the static analysis of laminated composite plates

This paper presents a mixed finite element model for the static analysis of laminated composite plates. The formulation is based on the least-squares variational principle, which is an alternative approach to the mixed weak form finite element models. The mixed least-squares finite element model considers the first-order shear deformation theory with generalized displacements and stress resultants as independent variables. Specifically, the mixed model is developed using equal-order C⁰ Lagrange interpolation functions of high p-levels along with full integration. This mixed least-squares-based discrete model yields a symmetric and positive-definite system of algebraic equations. The predictive capability of the proposed model is demonstrated by numerical examples of the static analysis of four laminated composite plates, with different boundary conditions and various side-to-thickness ratios. Particularly, the mixed least-squares model with high-order interpolation functions is shown to be insensitive to shear-locking.     

A hybrid/mixed model finite element analysis for buckling of moderately thick plates

Two modified complementary energy principles which only require C⁰ fields are derived in this paper. It is pointed out that if the fields of moment and deflection satisfy the respective second order homogeneous differential equation then they can both be considered as independent of each other and satisfy the equilibrium equation. The linear fields of moment and deflection are used in finite element analysis. The calculated results show that the method used in this paper is simple and reliable.     

Free vibration of laminated plates using a high-order element

Free vibration analysis of laminated composite plates using the finite element method has been presented. A high-order quadratic isoparametric element has been employed in the analysis. Both the eight-node serendipity and the nine-node Lagrangian shape functions have been used and their performances have been compared. Various schemes for the generation of the mass matrix have been discussed and a comparative study of these schemes has been presented. The results from the present method have been compared with the closed form solutions and experimental observations of the previous investigators.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

复合材料层合板非线性热振动分析

采用有限元方法研究复合材料层合板结构在线性温度场作用下非线性热振动特性．采用特征值屈曲分析方法,判断了结构在线性温度场作用下的临界屈曲分歧点,计算了结构的一阶弯曲固有频率,分析了铺层角度及铺层层数对结构临界屈曲温度分布和结构固有频率的影响,总结了其对复合材料层合板结构热振动特性影响的一般规律．这些结论对复合材料结构设计、抗热设计有一定的指导意义．     

Static,vibration and buckling analysis of axisymmetric circular plates using finite elements

The static, vibration, and buckling analysis of axisymmetric circular plates using the finite element method is discussed. For the static analysis, the stiffness matrix of a typical annular plate element is derived from the given displacement function and the appropriate constitutive relations. By assuming that the static displacement function, which is an exact solution of the circular plate equation ?²?²W = 0, closely represents the vibration and buckling modes, the mass and stability coefficient matrices for an annular element are also constructed. In addition to the annular element, the stiffness, mass, and stability coefficient matrices for a closure element are also included for the analysis of complete circular plates (no center hole). As an extension of the analysis, the exact displacement function for the symmetrical bending of circular plates having polar orthotropy is also given.     

A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis

This and a companion paper (Computers and Structures 26, 915–923, 1987) present a local finite element model based on a refined approximate theory for thick anisotropic laminated plates. The three-dimensional problem is reduced to a two-dimensional case by assuming piecewise linear variation of the in-plane displacements u and ρ and a constant value of the lateral displacement w across the thickness. By using a substructuring technique the present model is demonstrated to be practical and economical. The static bending stresses, transverse shearing stresses and in-plane displacements are predicted in the present paper. The vibration and buckling analyses will be presented in the second paper. Comparison with both exact three-dimensional analysis and a high-order plate bending theory shows that this model provides results which are accurate and acceptable for all ranges of thickness and modular ratio.     

Axisymmetric buckling of bimodulus thick circular plates

The static buckling of bimodulus thick circular and annular plates subjected to a combination of a pure bending stress and compressive stress is investigated. The thick finite element model, which includes the effect of transverse shear deformation, are created for axisymmetric buckling problems. The obtained results of buckling coefficient are compared with the exact solutions for ordinary thin plates. The accuracy of the finite element solutions are shown to be very good. The effects of various parameters on the buckling coefficients and neutral surfce locations are studied. The bimodulus properties are shown to have significant influences on the buckling coefficient.     

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.     

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

     

An efficient facet shell element for corotational nonlinear analysis of thin and moderately thick laminated composite structures

In the present work, an efficient facet shell element for the geometrically nonlinear analysis of laminated composite structures using the corotational approach is developed. The facet element is developed by combining the discrete Kirchhoff-Mindlin triangular bending element (DKMT), and the optimal membrane triangular element (OPT). The membrane-bending coupling effect of composite laminates is incorporated in the formulation, and inconsistent stress stiffness matrix is formulated. Using corotational formulation and the proposed facet element, some example laminated composite structures with geometric nonlinearity are analyzed, and the results are compared with those found using other facet elements.     

A new heterosis plate element for geometrically non-linear finite element analysis of laminated plates

In this study, a new nine-node quadrilateral, shear-deformable heterosis element is developed. In order to model this element, Kirchhoff constraints are modified using Reissner-Mindlin theory assumptions. All of the modifications are performed for first-order shear-deformation theory (FSDT). This new heterosis element is developed by modifying 8-node serendipity and twelve-node cubic polynomials. The new heterosis element is used with nine-node Lagrangian elements in finite element analysis of composite plates. A modified element is used in finite element analysis of linear and non-linear analysis considering the advantages of free of ‘shear locking’. Numerical results are presented by comparing Navier's series solution.     

A mixed shear flexible finite element for the analysis of laminated plates

A mixed shear flexible finite element based on the Hencky-Mindlin type shear deformation theory of laminated plates is presented and their behavior in bending is investigated. The element consists of three displacements, two rotations, and three moments as the generalized degrees of freedom per node. The numerical convergence and accuracy characteristics of the element are investigated by comparing the finite element solutions with the exact solutions. The present study shows that reduced-order integration of the stiffness coefficients due to shear is necessary to obtain accurate results for thin plates.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates

In view of the increasing interest in using composite materials for aerospace structures, the analysis of laminated composite plates becomes essential. A three-dimensional eight-node hybrid stress finite element method is developed for the analysis of laminated plates. The hybrid stress model is based on the modified complementary energy principle and takes into account the transverse shear deformation effects. The displacement field is interpolated through shape functions and nodal displacements. All three displacement components are assumed to vary linearly through the thickness of each lamina. The stress field is interpolated through assumed stress polynomials with 55 stress parameters for each lamina. All six stresses are included and satisfy the homogeneous equilibrium equations. The validity of the hybrid stress finite element model is determined by comparing the predicted numerical results with the existing three-dimensional elasticity solutions. Excellent accuracy and fast convergence are observed in the numerical results.     

Finite element free vibration analysis of eccentrically stiffened plates

A new finite element model is proposed for free vibration analysis of eccentrically stiffened plates. The formulation allows the placement of any number of arbitrarily oriented stiffeners within a plate element without disturbing their individual properties. A plate-bending element consistent with the Reissner-Mindlin thick plate theory is employed to model the behaviour of the plating. A stiffener element, consistent with the plate element, is introduced to model the contributions of the stiffeners. The applied plate-bending and stiffener elements are based on mixed interpolation of tensorial components (MITC), to avoid spurious shear locking and to guarantee good convergence behaviour. Several numerical examples using both uniform and distorted meshes are given to demonstrate the excellent predictive capability of this approach.     

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.