Buckling of shear deformable laminated composite plates

A shear deformable finite element is developed for the buckling analysis of laminated composite plates. The finite element formulation is based on Mindlin's theory in which shear correction factors are derived from the exact expressions for orthotropic materials. A variety of problems on uniaxial and shear bucklings of laminated composite plates are solved. The effects of material properties, plate aspect ratio, length-to-thickness ratio, number of layers and lamination angle on the buckling loads of symmetrically and antisymmetrically laminated composite plates are investigated. Optimal lamination arrangements of layers for maximizing the buckling loads of the plates are determined.     

Linear buckling analysis of orthotropic inhomogeneous rectangular plates under uniform in-plane compression

Summary A linear buckling analysis is carried out for orthotropic inhomogeneous rectangular plates under uniform in-plane compression. It is assumed that material inhomogeneities of Young's modulus and shear modulus of elasticity are continuously changed in the thickness direction with the power law of the coordinate variable, while Poisson's ratio is assumed to be constant. The buckling equation can be successfully constructed as the linearized von Kármán plate model by introducing the newly defined position of the reference plane which enables us to easily deal with the problem. The critical buckling loads of the simply supported rectangular plate are presented using the derived fundamental relations. Effects of material inhomogeneity, material orthotropy, aspect ratio, width-to-thickness ratio and load ratio are discussed.     

Single-mode solution for post-buckling analysis of composite panels with elastic restraints loaded in compression

A closed-form solution is obtained to determine the buckling and post-buckling behavior of elastically restrained composite panels under compressive loading. The approach allows to study the response of stiffened panels undergoing local buckling modes, taking into account the restraints provided by the stiffeners to the rotation of the skin edges. The panels are modeled as thin plates referring to Marguerre type equations together with classical lamination theory. The equations are written in non-dimensional form, allowing for the study of a wide class of orthotropic laminates. The problem is formulated in terms of out of plane displacement, represented with a single-mode approximation, and Airy stress function. The compatibility equation is solved exactly, while the method of Galerkin is applied to impose the equilibrium. The buckling load, the out of plane displacement at different load levels, and the post-buckling stiffness are derived and compared with finite element analyses, revealing good accuracy. Sensitivity analyses are also performed obtaining design charts.     

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.     

考虑屈曲的复合材料加筋壁板铺层顺序优化

提出了一种考虑屈曲的复合材料加筋壁板铺层顺序优化方法。基于复合材料加筋壁板屈曲载荷求解的能量法,系统推导了轴压载荷作用下复合材料加筋壁板蒙皮、筋条局部屈曲载荷的显示表达式,考虑了加筋壁板各板元之间的弹性支持作用及筋条下缘条的影响,引入工程法求解了加筋壁板整体屈曲载荷。基于国产自主结构分析软件HAJIF中的复合材料铺层工程数据库,以铺层参数为中间变量,利用本文提出的复合材料加筋壁板屈曲载荷求解方法,构建了考虑屈曲的复合材料加筋壁板铺层顺序优化设计流程并完成程序实现,将最小二乘法用于最优铺层顺序与工程铺层数据库的匹配。相比于传统有限元计算方法,本文提出的复合材料加筋壁板屈曲载荷求解方法具备较好的求解精度及求解效率。复合材料加筋壁板优化算例表明,采用本文提出的加筋壁板屈曲载荷分析及其优化方法,在结构重量不变的前提下,屈曲载荷提高约17%,且铺层顺序优化结果可直接从铺层工程数据库中提取并用于工程实际。      

On refined computational models of composite laminates

Finite element models of the continuum-based theories and two-dimensional plate/shell theories used in the analysis of composite laminates are reviewed. The classical and shear deformation theories up to the third-order are presented in a single theory. Results of linear and non-linear bending, natural vibration and stability of composite laminates are presented for various boundary conditions and lamination schemes. Computational modelling issues related to composite laminates, such as locking, symmetry considerations, boundary conditions, and geometric non-linearity effects on displacements, buckling loads and frequencies are discussed. It is shown that the use of quarter plate models can introduce significant errors into the solution of certain laminates, the non-linear effects are important even at small ratio of the transverse deflection to the thickness of antisymmetric laminates with pinned edges, and that the conventional eigenvalue approach for the determination of buckling loads of composite laminates can be overly conservative.     

A design of laminated composite plates using graded orthotropic fiber-reinforced composite plies

A new lamination scheme is proposed through the design of a graded orthotropic fiber-reinforced composite ply for achieving continuous variations of material properties along the thickness direction of laminated composite plates. First, a micro-structure of graded unidirectional fiber-reinforced composite ply is designed and its effective graded elastic properties are estimated using finite element procedure. Next, the new lamination scheme is demonstrated through the conversion of a conventional laminated composite plate (CLCP) into a conventional-graded laminated composite plate (CGLCP) utilizing presently designed graded orthotropic composite ply. The suitability of this conversion/proposed lamination scheme is substantiated through the bending analysis of both the plates (CLCP and CGLCP).     

A generalized higher-order theory for buckling of thick multi-layered composite plates with normal and transverse shear strains

The onset of buckling in square laminated multi-layered composite plates, subject to unidirectional in-plane loads, is investigated within the framework of a generalized higher-order shear deformation theory suitable to capture significant transverse shear and thickness-wise deformation effects. The displacement field is expanded in a Taylor series of the thickness coordinate with arbitrary polynomial degree; in turn, the series coefficients, expressed as a superposition of admissible functions, are determined according to the Rayleigh–Ritz method. Truly higher-order polynomial terms, along with a sufficient number of in-plane admissible functions, are shown to be necessary for convergence towards the fundamental buckling load multiplier. As a by-product, reduced-order models are identified for various plate geometries and lamination schemes. The sensitivity of the lowest buckling load with respect to the nondimensional parameters (the thickness ratio, the ratio between the elastic moduli, the ply angle) is investigated. In particular, the attention is focused on the cross-over phenomenon between the lowest two buckling eigenvalues in multi-layered composite square plates with different lamination schemes. The presented results shed light onto the buckling behavior of thick shear-deformable multi-layered plates.     

Buckling of the CCFF orthotropic rectangular plates under in-plane pure bending

Solution of the buckling problem for the CCFF orthotropic plate subjected to in-plane pure bending is presented. The two parallel clamped edges of the plate are loaded by linearly distributed in-plane loads statically equivalent to the in-plane bending moments. The problem is solved using method of lines for partial differential equations and Galerkin’s method. The buckling problems are solved for isotropic, orthotropic and multilayered CFRP composite plates with various aspect ratios. Results of calculations of critical loads are compared with those based on finite-element modelling and analyses. The comparisons demonstrate efficiency of the proposed approach to the buckling analysis of composite CCFF plates with various dimensional and stiffness parameters.     

Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates

Accurate free-vibrations and linearized buckling analysis of anisotropic laminated plates with different lamination schemes and simply supported boundary condition are addressed in this paper. Approximation methods, such as Rayleigh-Ritz, Galerkin and Generalized Galerkin, based on Principle of Virtual Displacement are derived in the framework of Carrera’s Unified Formulation (CUF). CUF widely used in the analysis of composite laminate beams, plates and shells, have been here formulated both for the same and different expansion orders, for the displacement components, in the thickness layer-plate direction. An extensive assessment of advanced and refined plate theories, which include Equivalent single Layer (ESL), Zig-Zag (ZZ) and Layer-wise (LW) models, with increasing number of displacement variables is provided. Accuracy of the results is shown to increase by refining the theories. Convergence studies are made in order to demonstrate that accurate results are obtained examining thin and thick plates using trigonometric approximation functions. The effects of boundary terms, upon frequency parameters and critical loads are evaluated. The effects of the various parameters (material, number of layers, fiber orientation, thickness ratio, orthotropic ratio) upon the frequencies and critical loads are discussed as well. Numerical results are compared with 3D exact solution when available from the open literature.     

Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates

Accurate free-vibrations and linearized buckling analysis of anisotropic laminated plates with different lamination schemes and simply supported boundary condition are addressed in this paper. Approximation methods, such as Rayleigh-Ritz, Galerkin and Generalized Galerkin, based on Principle of Virtual Displacement are derived in the framework of Carrera’s Unified Formulation (CUF). CUF widely used in the analysis of composite laminate beams, plates and shells, have been here formulated both for the same and different expansion orders, for the displacement components, in the thickness layer-plate direction. An extensive assessment of advanced and refined plate theories, which include Equivalent single Layer (ESL), Zig-Zag (ZZ) and Layer-wise (LW) models, with increasing number of displacement variables is provided. Accuracy of the results is shown to increase by refining the theories. Convergence studies are made in order to demonstrate that accurate results are obtained examining thin and thick plates using trigonometric approximation functions. The effects of boundary terms, upon frequency parameters and critical loads are evaluated. The effects of the various parameters (material, number of layers, fiber orientation, thickness ratio, orthotropic ratio) upon the frequencies and critical loads are discussed as well. Numerical results are compared with 3D exact solution when available from the open literature.     

Buckling of the SSCF rectangular orthotropic plate subjected to linearly varying in-plane loading

The paper presents the solution of the buckling problem for an orthotropic rectangular plate having two parallel edges simply supported, one edge clamped and the remaining edge free (the SSCF plate). The plate considered is subjected to a linearly varying in-plane load that can take the form of uniform compression, combination of in-plane bending and uniform compression, or pure in-plane bending. The solution technique involves reduction of the relevant variational buckling equation to a one-dimensional form using the Kantorovich procedure and subsequent application of the generalised Galerkin method. The buckling problems are solved for isotropic and orthotropic plates with various aspect ratios. The analytical solution is verified using the finite-element analysis. The comparisons of computational results demonstrate the appropriateness and efficiency of the approach developed in this work for the calculation of critical loads of composite SSCF plates with various dimensional and stiffness parameters.     

复合材料帽形加筋壁板轴压屈曲与后屈曲性能

针对2种边界条件和3种板元宽度情况,采用几种屈曲方法对轴压载荷作用下的复合材料帽形加筋壁板进行了计算,得到了不同情况下的屈曲载荷,通过对复合材料加筋壁板几种计算方法的计算结果与试验结果的对比,得到一种更简单有效计算复合材料帽形加筋壁板轴压屈曲载荷的方法;依据帽形加筋壁板的结构特点和试验后的破坏模式,提出一种估算复合材料帽形加筋壁板轴压破坏载荷的方法,用该方法在几个项目上的计算结果与试验结果进行比较,发现两者较吻合,验证了复合材料帽形加筋壁板轴压承载能力估算方法的合理性,为结构设计人员在初始设计阶段对复合材料帽形加筋壁板轴压强度评估提供了一种简洁的途径。     

Damage Tolerance of Pre-Stressed Composite Panels Under Impact Loads

An experimental test campaign studied the structural integrity of carbon fibre/epoxy panels preloaded in tension or compression then subjected to gas gun impact tests causing significant damage. The test programme used representative composite aircraft fuselage panels composed of aerospace carbon fibre toughened epoxy prepreg laminates. Preload levels in tension were representative of design limit loads for fuselage panels of this size, and maximum compression preloads were in the post-buckle region. Two main impact scenarios were considered: notch damage from a 12 mm steel cube projectile, at velocities in the range 93–136 m/s; blunt impact damage from 25 mm diameter glass balls, at velocities 64–86 m/s. The combined influence of preload and impact damage on panel residual strengths was measured and results analysed in the context of damage tolerance requirements for composite aircraft panels. The tests showed structural integrity well above design limit loads for composite panels preloaded in tension and compression with visible notch impact damage from hard body impact tests. However, blunt impact tests on buckled compression loaded panels caused large delamination damage regions which lowered plate bending stiffness and reduced significantly compression strengths in buckling.     

复合材料帽型加筋板轴压试验及承载能力预测

随着复合材料的广泛使用,复合材料帽型加筋板在飞机结构上的使用也越来越多。为研究复合材料帽型加筋板承受轴向压缩的能力,首先对不同蒙皮半径、蒙皮厚度及长桁间距的复合材料帽型加筋板进行了轴压试验,得到了局部屈曲载荷、破坏载荷与加筋板曲率系数、长桁间距的关系,然后,通过引入曲率修正系数,修正了现有加筋板屈曲载荷的工程估算公式;最后,利用分段处理法结合有效宽度概念改进了加筋板轴压极限承载的工程算法。结果表明:帽型复合材料加筋板局部屈曲载荷及最终破坏载荷与曲率系数正相关;改进的方法能对复合材料加筋板的极限承载进行准确预测。所得结果表明该方法为复合材料加筋板结构设计及载荷估算提供了一种新方法,具有一定的工程应用价值。      

Buckling of open-section bead-stiffened composite panels

Stiffened panels are structures that can be designed to efficiently support in-plane compression, bending, and shear loads. Although the stiffeners are usually discrete elements which are fastened or bonded to a flat or continuously curved plate, manufacturing methods such as thermoforming allow integral formation of the stiffeners in a panel. Such a configuration offers potential advantages in terms of a reduced number of parts and manufacturing operations. For thermoplastic composite panels stiffened by integrally formed open-section beads, the effects of bead spacing and bend cross-section geometry on the initiation of buckling under uniaxial compression and uniform shear loading were investigated. Finite elements results for a range of stiffened panel sizes and bead geometries are presented and compared with approximate closed-form solutions based on an effective flat plate size. Experimental verification of analytical predictions for one of the shear panels and one of the compression panels is described. Compensation of the forming tool to reduce the degree of initial curvature of the panels was found to be necessary.     

Fundamental frequency and design of the CFCF composite sandwich plate

The design efficiency of sandwich panels is often associated with the value of fundamental frequency. This paper investigates the free vibrations of rectangular sandwich plates having two adjacent edges fully clamped and the remaining two edges free (CFCF). The vibration analysis is performed by applying Hamilton’s principle in conjunction with the first-order shear deformation theory. The analytical solution determining the fundamental frequency of the plate is obtained using the generalised Galerkin method and verified by comparison with the results of finite element modal analysis. The approach developed in the paper and equations obtained are applied to the design of sandwich plates having composite facings and orthotropic core. Design charts representing the effects of the thickness of the facings and core on the mass of composite sandwich panel for a given value of the fundamental frequency are obtained.     

Buckling of functionally graded conical panels under mechanical loads

This paper presents an analytical approach to investigate the linear buckling of truncated conical panels made of functionally graded materials and subjected to axial compression, external pressure and the combination of these loads. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and linear stability equations in terms of displacement components for conical panels are derived by using the classical thin shell theory. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain closed-form relations of bifurcation type buckling loads. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the linear stability of conical panels.     

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。