Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C¹ continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.     

Three-dimensional semi-analytical model for the static response and sensitivity analysis of the composite stiffened laminated plate with interfacial imperfections

For the stiffened composite laminated plates with interfacial imperfections, the problem of static response and sensitivity analysis was investigated in Hamilton system. Firstly, the meshfree formulation of Hamilton canonical equation for the composite laminated plate with interfacial imperfections was deduced by the linear spring-layer and the state-vector equation theory. And then, based on the equation of plates and stiffeners, governing equation of the composite stiffened laminated plate was assembled by using the spring-layer model again to ensure the compatibility of stresses and the discontinuity of displacements at the interface between plate and stiffeners. At last, a three-dimensional hybrid governing equation was developed for the static response analysis and sensitivity analysis.     

Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory

Natural frequencies and buckling stresses of cross-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the lowest natural frequency and buckling stress are examined in detail. Numerical results are compared with those of the published existing theories and FEM solutions. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. It is noticed that the present global higher-order approximate theories can predict the natural frequencies, buckling stresses and stresses of thick multilayered composite laminates as accurately as three-dimensional solutions.     

Postbuckling Analysis of Laminated Composite Plates Using a Higher-Order Zig-Zag Theory

This study examines the effects of incorporating zig-zag kinematics in the postbuckling analysis of laminated composite plates. A higher-order zig-zag plate element for nonlinear analysis was developed based on works of Averill and Yip. Their zig-zag element is especially suitable for a nonlinear structural laminate analysis due to its high accuracy and a low, constant number of degrees of freedom regardless of the number of layers. The article examines global postbuckling response as well as local displacement and stress fields of various laminated plates. The results derived from higher-order zig-zag theory are compared with predictions of first-order shear deformation theory (FSDT). Significant differences between these two theories are obtained for laminated plates with drastically different transverse stiffness properties with length-to-thickness aspect ratios L / t = 30 and 50. FSDT leads to good predictions of global and local behavior only for L / t = 50 and 100 with a typical layup in which the adjacent plies do not have very different transverse stiffness properties. Results presented in this article indicate that the zig-zag theory is required to predict accurately stresses and in-plane displacements through the thickness in moderately thick plates in the postbuckled state.     

A new finite element for buckling analysis of laminated stiffened plates

A new stiffened plate element for stability analysis of laminated stiffened plates has been presented. The basic plate element is a combination of Allman's plane stress triangular element and a Discrete Kirchhoff–Mindlin plate bending element. The element includes transverse shear effects. The model accommodates any number of arbitrarily oriented stiffeners within the plate element and eliminates constraints on the mesh division of the plate. The element has no problem associated with shear locking – a phenomenon usually encountered in isoparametric elements. The stability analysis of laminated stiffened plates has been carried out under different loading conditions with the present element.     

Finite element analysis of stiffened laminated plates under transverse loading

A finite element model is developed to study the behavior of stiffened laminated plates under transverse loadings. Transverse shear flexibility is incorporated in both beam and plate displacement fields. A laminated plate element with 45 degrees of freedom is used in conjunction with a laminated beam element having 12 degrees of freedom for the bending analysis of eccentrically-stiffened laminated plates. The validity of the formulation is demonstrated by comparing with the available solutions in the literature. The numerical results are presented for eccentrically-stiffened layered plates having various boundary conditions and with stiffeners varying in number.     

A quadrilateral hybrid-Trefftz plate bending element for the inclusion of warping based on a three-dimensional plate formulation

A plate formulation, for the inclusion of warping and transverse shear deformations, is considered. From a complete thick and thin plate formulation, which was derived without ad hoc assumptions from the three-dimensional equations of elasticity for isotropic materials, the bending solution, involving powers of the thickness co-ordinate z, is used for constructing a quadrilateral finite plate bending element. The constructed element trial functions, for the displacements and stresses, satisfy, a priori, the three-dimensional Navier equations and equilibrium equations, respectively. For the coupling of the elements, independently assumed functions on the boundary are used. High accuracy for both displacements and stresses (including transverse shear stresses) can be achieved with rather coarse meshes for thick and thin plates.     

An improved in-plane displacement model for the stability analysis of laminated composites with general lamination configurations

This paper describes how the derivatives of lateral displacement are eliminated in the general displacement field of the global–local higher-order theory for stability analysis of laminated composite and sandwich plates with general lamination configurations. In contrast to previous models, the present model is applicable not only to the cross-ply but also to the angle-ply laminated composite and sandwich plates. Based on known traction forces on the surface boundaries, the derivatives of transverse displacement have been taken out from the general displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. To verify the proposed theory, the classical quadratic six-node C⁰ triangular element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate. Numerical results show that following the proposed theory simple C⁰ finite elements could accurately predict the critical loads of sandwich plate with soft core, which has long been a difficult case for the other global higher-order theories.     

具有初始几何缺陷加劲板的动态屈曲

针对工程中常用的加劲板,研究了动态屈曲的求解方法。将加劲板分为母板与加劲肋两个部分考虑,其中母板按经典薄板理论计算,加劲肋视为Euler梁。假定加劲板的位移,利用Hamilton原理结合系统能量和振型叠加法建立了加劲板的动态屈曲特征方程。最后,选择四边简支加劲板进行数值分析,分析中考虑初始几何缺陷的影响,并讨论了初始几何缺陷、加劲肋的数量及其刚度的变化对动态屈曲临界荷载的影响。结果表明：一阶模态的初始几何缺陷对加劲板的临界荷载影响很大,而增加加劲肋的数量及其刚度可以提高加劲板的抗动态屈曲能力。研究结果也为加劲板的结构设计方法提供一定的参考。     

Higher-order finite element modelling of laminated composite plates

P-version finite elements based on higher-order theory are developed for the two-dimensional modelling of general bending and cylindrical bending of thin-to-thick laminated composite plates. In the case of general laminated plate elements, three displacement fields are used. In the special case of cylindrically bent laminated plate elements, two displacement fields are needed. In each case the displacement is expressed as the product of two functions—one in terms of out-of-plane co-ordinates alone and the other in terms of in-plane co-ordinates. The shape functions used to build the displacement fields are based on integral of Legendre' polynomials. The quality and performance of the elements are evaluated in terms of convergence characteristics of displacements and stresses. The predicted response quantities are compared with those available in the published literature based on analytical as well as conventional finite element models.     

A Simple Locking-Alleviated 3D 8-Node Mixed-Collocation C0 Finite Element with Over-Integration,for Functionally-Graded and Laminated Thick-Section Plates and Shells,with & without Z-Pins

Following previous work of [Dong, El-Gizawy, Juhany, Atluri (2014)], a simple locking-alleviated 3D 8-node mixed-collocation C0 finite element (denoted as CEH8) is developed in this study, for the modeling of functionally-graded or laminated thick-section composite plates and shells, without using higher-order or layer-wise zig-zag plate and shell theories which are widely popularized in the current literature. The present C0 element independently assumes an 18-parameter linearly-varying Cartesian strain field. The independently assumed Cartesian strains are related to the Cartesian strains derived from mesh-based Cartesian displacement interpolations, by exactly enforcing 18 pre-defined constraints at 18 pre-selected collocation points. The constraints are rationally defined to capture the basic kinematics of the 3D 8-node C0 element, and to accurately model each basic deformation mode of tension, bending, shear, and torsion. A 2x2x2 Gauss quadrature is sufficient for evaluating the stiffness matrix of CEH8 C0 3D elements for homogeneous materials, but over-integration (with a higher-order Gauss Quadrature, a layer-wise Gauss Quadrature, or a simple Trapezoidal Rule in the thickness direction) is used for functionally-graded materials or thick-section laminated composite structures with an arbitrary number of laminae. Through several numerical examples, it is clearly shown that the present CEH8 3D C0 element can accurately capture the stress distribution of FG and thick laminated structures with an arbitrary number of laminae even when only one element is used in the thickness direction. In stark contrast to the higher-order or layer-wise zig-zag plate and shell theories, with assumptions for displacement or stress fields in the thickness direction, which may require complicated C1 finite element, the present C0 element can accurately compute the jumps in bending stresses at the interfaces of layers, while the out-of plane normal and shear stresses can be accurately recovered by exploring the equilibrium equations of 3D linear elasticity. By adding the contributing stiffness of z-pins into the stiffness matrix of CEH8, it is also demonstrated that the presently developed method can be used to study the effect of using z-pin reinforcements to reduce the inter-laminar stresses of composite structures, in a very simple and computationally-efficient manner.     

Free vibration and stability of angle-ply laminated composite and sandwich plates under thermal loading

A two-dimensional global higher-order deformation theory is presented for the free vibration and stability problems of angle-ply laminated composite and sandwich plates subjected to thermal loading. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal stresses is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of a simply supported angle-ply multilayered plate. Natural frequencies and critical temperatures of angle-ply laminated composite and sandwich plates subjected to thermal loading are obtained. Critical temperatures are obtained by increasing the temperature until the natural frequency vanishes. The effects of prebuckling displacements on the natural frequencies and critical temperatures are taken into account. Modal displacement distributions through the transverse direction of the laminates are plotted for the specific temperature parameter. Numerical results are compared with those of the published existing theories. The present global higher-order approximate theories can predict the natural frequencies and critical temperatures of angle-ply laminated composite and sandwich plates subjected to thermal loading accurately within small number of unknowns.     

Enhanced modeling of laminated and sandwich plates via strain energy transformation

An enhanced first-order shear deformation theory has been developed for the deformation and stress recovery of laminated and sandwich plates. Based on the definition of Reissner–Mindlin’s plate theory, the relationships between three-dimensional and first-order shear deformation theories have been derived. It is assumed that the displacements, in-plane strains and stresses of Reissner–Mindlin’s plate theory can approximate those of three-dimensional theory, in the least-square sense. Their relationships have been systematically established and verified through the strain energy transformation. These relationships provide the closed-form recovering relations for three-dimensional variables expressed in terms of the variables of Reissner–Mindlin’s plate theory. An efficient higher-order plate theory is utilized to obtain the in-plane warping functions. Comparisons of deflection, stresses and shear correction factors of both laminated and sandwich plates using the present theory are made with the original first-order shear deformation theory and three-dimensional exact solutions.     

爆炸载荷作用下双向加筋方板的大挠度塑性动力响应

从分别列出加筋板面板以及加强筋的运动方程出发,分析了爆炸载荷作用下双向加筋固支方板的大挠度塑性动力响应。分析表明:取决于加强筋的相对刚度以及爆炸载荷峰值的大小,加筋板的运动将呈现3种不同的模式。该文限于讨论加筋板的总体变形模式,具体讨论了十字加筋以及双十字加筋固支方板在忽略弯矩影响下的薄膜解法。理论结果与已有的试验结果在多数情况下符合良好,表明该文提出的简化理论分析方法能对爆炸载荷下双向加筋方板的永久变形做出较为合理的预报。     

厚度不连续悬臂梁板的自由振动分析

将厚度不连续梁板视为层合板, 分别应用Hamilton 正则方程半解析法建立每一层的线性方程。考虑到每两层连接界面上应力和位移的连续性, 联立各层的方程得到整个结构的特征方程, 其主要的优越性表现为: 控制方程不限制不连续梁板的厚度, 并能适合处理厚度不对称且不连续的层合板。本文中的方法可修改或扩展用来分析加筋压电材料层合板或带有压电材料传感器和驱动器块的板壳等问题。      

Flexural analysis of cross-ply laminated plates subjected to nonlinear thermal and mechanical loadings

The objective of this paper is to present an equivalent single-layer shear deformation theory for evaluation of displacements and stresses of cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load. A trigonometric shear deformation theory is used. The in-plane displacement field uses a sinusoidal function in terms of the thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of a shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Stresses and displacements for orthotropic, two-layer antisymmetric, and three-layer symmetric square cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load are obtained. Numerical results of the present theory for displacement and thermal stresses are compared with those of classical, first-order and higher-order shear deformation plate theories.     

Vibration and stability of an initially stressed laminated plate based on a higher-order deformation theory

Equations of motion for a laminated plate in a general state of nonuniform initial stress where the higher-order shear deformation terms are included are derived. The equations are derived by using the average stress method. The natural frequencies and buckling loads of cross-ply laminated plates subjected to initial stress are investigated. The initial stress is taken to be a combination of uniaxial pure bending and extensional stresses in the plane of the plate. The results obtained from the present higher-order theory are compared with first-order theory and other higher-order theory results. In addition, the effects of various parameters on the natural frequencies and buckling loads are studied.     

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect sandwich plates based on a generalized three-dimensional high-order global–local plate theory

The available plate theories either have not considered the interlaminar stress continuity condition or have been calibrated based on linear strain–displacement relations. Moreover, almost all buckling analyses performed so far employing the global–local plate theories, were restricted to linear, static buckling analyses of the perfect plates, neglecting the transverse normal strain and stress. Researches available in literature for dynamic buckling analyses of the sandwich plates are very rare. In the present paper, a generalized high-order global–local theory that satisfies all the kinematic and transverse stress continuity conditions at the interfaces of the layers, is proposed to investigate dynamic buckling of imperfect sandwich plates subjected to thermo-mechanical loads. In comparison to the layerwise, mixed, and available global–local theories, the present theory has the advantages of: (1) less required computational time due to using the global–local technique and matrix formulations, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions and considering the transverse flexibility, (3) suitability for non-linear analyses, (4) capability of investigating the local phenomena, such as the wrinkling. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.     

复合材料加筋板热变形和热应力分析

复合材料及其结构的热响应是近年来复合材料力学研究的热点课题之一。本文用有限元法研究了复合材料加筋板的热变形和热应力。基于Mindlin假定导出了一阶剪切变形层合板梁理论。这两种理论也适用于中厚板和深梁。考虑的热载可以是瞬态的也可以是稳志的,实际工程应用表明,根据本丈模型研制的计算机程序具有较高的计算精度和效率。      

The development of laminated composite plate theories: a review

This study investigates and reviews approaches to modelling laminated composite plates. It explores theories that have been proposed and developed and assesses their suitability and functionality. The particular focus in this study has been on normal stresses and the through-thickness distributions of transverse shear. These are important for composite plates as stress-induced failures can occur in three different ways. Therefore, it is essential to understand and calculate transverse shear and normal stress through the thickness of the plate accurately. In this study, previous laminated composite plate theories are categorised and reviewed in a general sense, i.e. not problem specific, and the advantages and disadvantages of each model are discussed. This research mainly focuses on how accurate and efficient the models can predict the transverse shear. It starts with displacement-based theories from very basic models such as Classical laminate plate theory to more complicated and higher-order shear deformation theory. Models are furthermore categorised by how the models consider the overall laminate. In this article, the theories are divided into two parts: Single layer theory, where the whole plate is considered as one layer; and Layerwise theory, where each layer is treated separately. The models based on zig-zag and Discrete Theories are then reviewed, and finally the mixed (hybrid) plate theories are studied.