Delamination buckling growth in laminated composites using layerwise-interface element

In this paper, a numerical investigation on the buckling of composite laminates containing delamination, under in-plane compressive loads, is presented. For this purpose, delamination propagation is modeled using the softening behavior of interface elements. The full layerwise plate theory is applied for approximating the displacement field of laminates and the interface elements are considered as a numerical layer between any two adjacent layers where the delamination is expected to propagate. A non-linear computer code was developed to handle the numerical procedure of delamination buckling growth in composite laminates using layerwise-interface elements. The load/displacement behavior and the contours of embedded and through-the-width delamination propagation for composite laminates are presented. It is shown that delamination growth can be well predicted using this layerwise-interface elements with decohesive law.     

Buckling and free vibration of non-homogeneous composite cross-ply laminated plates with various plate theories

Various theories of homogeneous laminated plates are extended to study the buckling and free vibration behavior of non-homogeneous rectangular composite laminates. The equations governing the dynamic response of non-homogeneous composite laminates are deduced. Numerical results for the natural frequencies and critical buckling loads of symmetric cross-ply laminates are presented. The influences of the degree of non-homogeneity, aspect ratio, thickness ratio and in-plane orthotropy ratio on the natural frequencies and critical buckling loads are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature. The study concludes that the classical plate theory is inadequate for predicting the structural response of non-homogeneous laminates, and that the free vibration and the state of the stability are affected strongly by the degree of nonhomogeneity.     

层合板在压剪载荷下屈曲分析的三角有限差分法

本文利用J.M.Housner和M.Stein采用过的、具有以较少的自由度获得较精确结果优点的三角有限差分方法,对复合材料加筋层合板的总位能表达式进行单元划分后的差分变换,再利用能量变分原理导出以差分节点挠度为未知特征向量、外加均布载荷为未知特征值的矩阵形式的屈曲控制方程。对在简支、固支或用挠曲弹簧和弯曲弹簧模拟的弹性支持情况下,承受面内单向压力、剪切或压、剪联合载荷作用的复合材料加筋层合板进行了有效的屈曲分析。文中对以上几种情况进行了数值计算,计算结果同实验结果吻合良好。     

Lateral buckling analysis of thin-walled laminated channel-section beams

The lateral buckling of a laminated composite beam with channel section is studied. A general analytical model applicable to the lateral buckling of a channel-section composite beam subjected to various types of loadings is derived. This model is based on the classical lamination theory, and accounts for the material coupling for arbitrary laminate stacking sequence configuration and various boundary conditions. The effects of the location of applied loading on the buckling capacity are also included in the analysis. A displacement-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite beam with arbitrary boundary conditions. Numerical results are obtained for thin-walled composites under central point load, uniformly distributed load, and pure bending with angle-ply and laminates. The effects of fiber orientation, location of applied load, and types of loads on the critical buckling loads are parametrically studied.     

Thermomechanical buckling of hybrid cross-ply laminated rectangular plates

A thermomechanical buckling analysis is presented for simply supported rectangular symmetric cross-ply laminated composite plates that are integrated with surface-mounted piezoelectric actuators and are subjected to the combined action of in-plane compressive edge loads, two types of thermal loads, and constant applied actuator voltage. The formulation of equations is based on the classical laminated plate theory and the von-Karman non-linear kinematic relations. The analysis uses an exact method to obtain closed-form solutions for the buckling load. The effects of applied actuator voltage, thermal and mechanical loads, plate geometry, and lay-up configuration of the laminated plates are investigated. The novelty of the present work is to obtain closed-form solutions for electro-thermomechanical buckling of hybrid composite plates, and to cover non-uniform temperature distribution loading. The results for various states are verified with known data in the literature.     

Explicit local buckling analysis of rotationally-restrained orthotropic plates under uniform shear

The explicit closed-form local buckling solution of in-plane shear-loaded orthotropic plates with two opposite edges simply supported and other two opposite edges either both rotationally restrained or one rotationally restrained and the other free is presented. Based on the boundary condition of the other two opposite edges, two types of plates are considered: the RR (both the edges rotationally restrained) and RF (one edge restrained and the other free) plate elements. Different plate buckled shape functions are proposed, and the approximate explicit expressions for the buckling loads are derived using the Rayleigh–Ritz method for the plate with the generic rotationally-restrained (R) boundary conditions which can be reduced to two extreme cases, i.e., simply supported (S) and clamped (C). The accuracy of the derived explicit solutions is verified by comparing the predictions with the existing solutions and numerical finite element analysis, and excellent agreements are obtained. The effects of material and boundary restraining parameters on the local shear buckling behavior of the plate elements are discussed. The derived explicit formulas for the shear buckling loads are straightforward, efficient and reliable for preliminary engineering design and analysis of composite structures under primarily shear-dominant loading conditions.     

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.     

Exact buckling solutions for composite laminates: proper free edge conditions under in-plane loadings

Summary This paper considers the elastic buckling of symmetric cross-ply laminated rectangular plates with two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The first-order shear deformation plate theory is used in the analysis. An error apparently made by previous researchers for boundary conditions at free edges subjected to in-plane loads is corrected. Closed-form buckling factors are obtained using a generalised Levy-type solution method to solve the differential equations which govern the buckling behaviour of the laminates. Comparisons are made with previously published results, and the differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are examined. The variation of buckling factors with plate aspect ratio, thickness ratio and the number of layers is investigated. Sets of first-known buckling solutions for cross-ply laminates are reported in design charts and tables.     

Buckling analysis of rectangular composite plates with rectangular cutout subjected to linearly varying in-plane loading using fem

A numerical study is carried out using finite element method, to examine the effects of square and rectangular cutout on the buckling behavior of a sixteen ply quasi-isotropic graphite/epoxy symmetrically laminated rectangular composite plate [0°/+45°/-45°/90°]_2s, subjected to various linearly varying in-plane compressive loads. Further, this paper addresses the effects of size of square/rectangular cutout, orientation of square/rectangular cutout, plate aspect ratio(a/b), plate length/thickness ratio(a/t), boundary conditions on the buckling bahaviour of symmetrically laminated rectangular composite plates subjected to various linearly varying in-plane compressive loading. It is observed that the various linearly varying in-plane loads and boundary conditions have a substantial influence on buckling strength of rectangular composite plate with square/rectangular cutout.     

Stability and vibration analyses of carbon nanotube-reinforced composite beams with elastic boundary conditions: Chebyshev collocation method

This article aims to investigate stability and vibration behavior of carbon nanotube-reinforced composite beams supported by classical and nonclassical boundary conditions. To include significant effects of shear deformation and rotary inertia, Timoshenko beam theory is used to formulate the coupled equations of motion governing buckling and vibration analyses of the beams. An effective mathematical technique, namely Chebyshev collocation method, is employed to solve the coupled equations of motion for determining critical buckling loads and natural frequencies of the beams with different boundary conditions. The accuracy and reliability of the proposed mathematical models are verified numerically by comparing with the existing results in the literature for the cases of classical boundary conditions. New results of critical buckling loads and natural frequencies of the beams with nonclassical boundary conditions including translational and rotational springs are presented and discussed in detail associated with many important parametric studies.     

纤维增强复合材料层合板屈曲性态分析的边界元法

本文用边界元法分析了纤维增强复合材料正交各向异性层合板的屈曲性态。为了克服在用边界元法求解正交各向异性层合板屈曲时寻求相应的基本解的困维,本文采用了双重傅立叶级数和引用等效荷载的概念,建立了层合板屈曲临界荷载的特征方程。算例说明了本文方法的可行性和有效性。      

The vibration and stability of symmetrically-laminated composite rectangular plates subjected to in-plane stresses

Consideration is given to the twin problems of the elastic buckling of rectangular, symmetrically-laminated composite plates and of the vibration in the presence of applied in-plane stress of such laminates. First-order shear deformation plate theory provides the mathematical model of plate behaviour and the Rayleigh-Ritz and finite strip methods are used to generate numerical results for laminates of thin and moderately thick geometry, with various combinations of standard plate edge conditions. The applied stresses include uniform shear stress as well as direct stresses, and anisotropic material properties can be included. The presented results demonstrate the accuracy of the numerical methods and highlight the very significant influence that transverse shear and related thickness effects can have in the subject problems.     

Effect of in-plane boundary restraints on optimal design of skew laminates under buckling loads

Symmetrically laminated cross-ply and angle-ply skew plates subject to uniaxial buckling loads and various combinations of in-plane boundary restraints are studied using a shear deformable theory. For this purpose a finite element code is developed and applied to a couple of verification problems. The formulation of the parabolic iso-parametric plate element is briefly given and numerical results obtained for the verification problems related to stability analysis and stress diffusion are presented. The effect of in-plane restraints on the non-uniform distribution of in-plane stresses is studied by means of contour graphs. Next the buckling loads are maximized with respect to layer thicknesses in the case of cross-ply laminates and with respect to fiber orientations in the case of angle-ply laminates. The optimization results show that the exclusion of the in-plane restraints, which arise in several engineering applications, may lead to errors in the stability analysis and consequently in the design of laminated plates against buckling.     

Postbuckling Analysis of Shear-Deformable Composite Laminated Plates on Two-Parameter Elastic Foundations

Postbuckling analysis is presented for a simply supported, shear-deformable, composite laminated plate subjected to uniaxial compression and resting on a two-parameter (Pasternak-type) elastic foundation. The initial geometric imperfection of the plate is taken into account. Two cases of in-plane boundary conditions are considered. The formulations are based on Reddy’s higher-order shear-deformation plate theory, including plate–foundation interaction. The analysis uses a deflection-type perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performance of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated plates resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, transverse shear deformation, the character of the in-plane boundary conditions, plate aspect ratio, total number of plies, fiber orientation, and initial geometric imperfections are studied.     

Postbuckling response and failure of symmetric laminates under in-plane shear

This paper summarizes a study aimed at understanding the postbuckling behaviour and progressive failure of thin, simply supported symmetric rectangular laminates with various possible in-plane boundary conditions and under the action of in-plane shear loads. First-order shear deformation theory and geometric non-linearity, in the von-Karman sense, is used with a finite-element procedure. The 3D Tsai–Hill criterion is used to predict failure of lamina and the maximum stress criterion is used to predict the onset of delamination at the interface of two adjacent layers. The effect of in-plane boundary conditions, plate lay-ups, plate aspect ratio, fiber orientations and lamina material properties on the load deflection response, buckling load, first-ply failure load, ultimate load and the maximum transverse displacement associated with failure loads is presented.     

非对称复合材料层板在面内载荷作用下的非线性横向变形或屈曲

本文利用广义富里叶级数法对非对称复合材料层板在面内载荷作用下的非线性横向变形或屈曲问题进行了分析。文中主要研究了四边简支边界条件下,非对称层板的铺层方式和面内载荷形式对横向变形的影响。通过求解非线性控制方程得到了层板的载荷—挠度曲线或特殊情况下的后屈曲平衡曲线。计算结果表明,对于大多数非对称复合材料层板在面内载荷作用下所产生的是横向变形问题,而不是分叉屈曲问题。     

Study of critical buckling loads and modes of cross-ply laminated annular plates

Buckling of composite annular plates under uniform internal and external radial edge loads have been investigated using energy method. Trefftez rule is used in the stability equations. The symmetric buckling of symmetric cross-ply laminates is considered. In this paper, buckling behavior for the three laminates (90/0)_2s, (90/0₂/90)_s and (90₂/0₂)_s are studied. Influence of some parameters such as thickness, stacking sequence, type of supports and the ratio of hole to sheet radius on buckling loads and modes are investigated. The results of the energy method are compared with the results of numerical method. Based on the results, in the plates with clamped boundary conditions the symmetric buckling assumption is not accurate, contrary to other boundary conditions.     

Thermal buckling of various types of FGM sandwich plates

The sinusoidal shear deformation plate theory is used to study the thermal buckling of functionally graded material (FGM) sandwich plates. This theory includes the shear deformation and contains the higher- and first-order shear deformation theories and classical plate theory as special cases. Material properties and thermal expansion coefficient of the sandwich plate faces are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. Several kinds of symmetric sandwich plates are presented. Stability equations of FGM sandwich plates include the thermal effects. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. Numerical examples cover the effects of the gradient index, plate aspect ratio, side-to-thickness ratio, loading type and sandwich plate type on the critical buckling for sandwich plates.     

The effect of extension/twist anisotropy on compression buckling in cylindrical shells

The study of axial compression buckling of isotropic cylinders has received much attention by various researchers over the years. It is commonly acknowledged that the presence of minute imperfections significantly reduces potential buckling loads in comparison with classical linear predictions. This approach, of including geometric imperfections, has been extended by a significant, yet fewer, number of researchers to composite cylindrical shells. As the current study shows, imperfections may not be the only major factor for the discrepancy between experimental and linear buckling loads. Flexural/twist anisotropy, present in most balanced, symmetric laminates with angle-ply layers, is shown to play a significant role in reducing buckling loads from those predicted by classical analysis. Indeed, the assumption of deflections in the form of a double sine series appears to be questionable for such laminates. A recently reported classically linear analysis that included the effects of flexural/twist coupling on buckling loads has been further developed to include the effects of extension/twist coupling. It is shown that buckling loads can be improved by making the laminate antisymmetric rather than symmetric, for the class of quasi-isotropic laminates, whilst retaining a spiral mode shape.