Curvature effects in the buckling of symmetrically-laminated rectangular plates with transverse shear deformation

It is shown that the effect of stress discontinuities from ply-to-ply must be taken into account when curvature terms are included along with shear deformation in the buckling analysis of rectangular, symmetrically-laminated plates. Such ply-stress discontinuities lead to curvature terms in the governing equations which differ considerably from those derived for homogeneous plates. Critical buckling loads are determined for orthotropic laminates subjected to biaxial inplane loading and for cylindrical bending of anisotropic plates subjected to uniaxial compression loading. Simply-supported boundary conditions are considered in conjunction with the rectangular, orthotropic laminate, while simply-supported and clamped boundaries are considered for the case of cylindrical bending of anisotropic plates. Numerical results indicate that the curvature terms have little effect on critical buckling loads for the laminates investigated. The effect of transverse shear deformation is shown to depend on the degree of boundary constraint, laminate stacking geometry, and inplane load ratio.     

面内线性变化载荷作用复合材料层合板的振动与屈曲优化

采用广义微分求积（GDQ）法开展了不同边界条件下承受面内线性变化载荷作用复合材料层合板振动与屈曲的分析与优化。针对GDQ法求解面内线性变化载荷工况复合材料层合板屈曲问题存在计算振荡、不收敛现象,提出载荷扰动策略实现了GDQ法对复合材料层合板屈曲问题的稳定高效求解。基于基础圆频率和临界屈曲载荷系数的归一化指标,分析了铺层角度对复合材料层合板综合性能的影响,并结合直接搜索模拟退火算法开展了复合材料层合板的铺层顺序优化。结果表明:铺层角度变化对屈曲性能的影响明显强于频率特性;面内线性变化载荷中,以弯曲载荷作用下复合材料层合板的优化综合性能受边界条件变化的影响最小,而优化铺层角度受边界条件变化的影响最大。研究结果为复杂载荷作用下复合材料层合板的设计提供了参考。      

Optimal design of hybrid laminates with discrete ply angles for maximum buckling load and minimum cost

The design of hybrid symmetric laminated plates consisting of high-stiffness surface and low-stiffness core layers is presented. In the first problem the maximization of buckling load is carried out over a discrete set of ply angles. In the second problem the minimum number of high-stiffness plies is determined for a given buckling load to minimize the material cost. Boolean variables are introduced to specify stacking sequence. Solution of the linear optimization problem yields an optimal stacking sequence. The effect of hybridization is investigated for various problem parameters such as the aspect ratio of the laminate and the number of plies. The optimal designs are obtained with upper bound constraints on the effect of bending-twisting coupling stiffnesses. Results are given for hybrid graphite-epoxy/glass-epoxy laminates under both uniaxial and biaxial loadings.     

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.     

Optimization of shear-deformable laminated plates under buckling and strength criteria

A shear deformable laminated theory is used to study the optimal design of rectangular plates under biaxial compressive loads. Such loads lead to plate failure by buckling or material failure which corresponds to the violation of the selected strength criterion. The minimum of the two loads (buckling load or material failure load) determines the critical failure load for a given set of problem parameters. At the optimum values of the ply angles, buckling or both failure criteria may be operational depending on the laminate thickness. The present study investigates the effect of laminate thickness on the optimal design and gives numerical results for symmetrically laminated angle-ply plates.     

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.     

Influence of the prebuckling stress-field on the critical loads of inhomogeneous composite laminates

The paper studies the uniaxial buckling behavior of composite laminates in which preselected variations of fiber spacing in the constituent laminae are adopted. Such laminates are referred to as inhomogeneous laminates because of the variable elastic stiffness along the coordinate axes. A non-uniform prebuckling stress state observed even under constant uniaxial compression has a pronounced influence on the buckling behaviour of an inhomogeneous laminate. A procedure is summarized for computing the critical load of a laminate using the Ritz method which exploits an analogy between the bending and stretching formulations and utilizes Gram-Schmidt orthogonal polynomials. The paper illustrates that the variation in fiber spacing is an innovative way of increasing the critical load for a prescribed amount of fiber and highlights its remarkable sensitivity to the nature of fiber spacing, in-plane and out-of-plane boundary conditions, fiber type and the aspect ratio of the laminate.     

Stability and failure of composite laminates with various shaped cutouts under combined in-plane loads

The objective of this paper is to study stability and failure of a composite laminate with a centrally placed cutout of various shapes (i.e., circular, square, diamond, elliptical-vertical and elliptical-horizontal) under combined action of uni-axial compression and in-plane shear loads. The FEM formulation based on the first order shear deformation theory and von Karman’s assumptions has been utilized. Newton–Raphson method is used to solve nonlinear algebraic equations. Failure of a lamina is predicted by the 3-D Tsai–Hill criterion whereas the onset of delamination is predicted by the interlaminar failure criterion. The effects of cutout shape, direction of shear load and composite lay-up on buckling and postbuckling responses, failure loads and failure characteristics of the laminate has been discussed. An efficient utilization of material strength is observed in the case of laminate with circular cutout as compared to the laminate with other shaped cutouts. In addition, it is also concluded that although the buckling strength of the (0/90)_4s laminate is lower than that of the (+45/?45/0/90)_2s and (45/?45)_4s laminates, but its strength is increased in the advanced stage of postbuckling deformation.     

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。     

Stress and Failure Analysis of Composite Laminates with an Inclusion under Multiaxial Compression-Tension Loading

The present paper investigates the failure of orthotropic laminates with a filled hole subjected to biaxial compression-tension loading. This is an idealised case that simulates a situation where the impact damaged laminate has been repaired by drilling a hole and then plugging the hole with a perfect-fit core made of a dissimilar material. The exact stress distribution in the laminate has been determined using the complex variable mapping method. These stresses are then employed in a fracture mechanics failure model to predict the failure load of a laminate with a filled hole under in-plane compression-dominated multi-axial loading. Failure strength predictions are compared to open hole results and experimental data.     

Analytical and experimental studies on the low-velocity impact response and damage of composite laminates under in-plane loads with structural damping effects

In this paper, low-velocity impact response and damage of composite laminates under in-plane loads are analytically and experimentally investigated. The authors recently proposed a modified displacement field of plate theory, considering the effect of initially loaded in-plane strain, and used a finite element program to analyze the structural behavior of the composite laminate. In this study, the program is upgraded to account for the structural damping effect of the laminate. A pendulum type impact test system and an in-plane loading fixture are constructed for the experimental study. The analytical and experimental impact behaviors are compared at different impact energy levels for cases with an initial in-plane tensile load and a compressive load, as well as cases without the initial in-plane load. The results show good correspondence between the analytical and experimental impact force histories. The effect of the initial in-plane load reduces for higher impact energies. The numerical estimation of the damaged area is in good agreement with the results from C-scanning experiments.     

Low-velocity impact damage on dispersed stacking sequence laminates. Part I: Experiments

The stacking sequence design of composite laminates is often limited to combinations of 0°, 90°, and ±45° fibre angle plies. Furthermore, in order to comply to certain stiffness requirements, clustering of plies becomes unavoidable. Although such laminates might have the desired stiffness properties, they may show poor impact and/or compression-after-impact behaviour.A method to redesign the traditional stacking sequences such that the alternative laminates have improved damage resistance whilst keeping similar in-plane and bending stiffness properties as their original traditional stacking sequences is proposed. This method makes use of optimisation tools based on genetic algorithms. In the alternative laminates, the difference between fibre angles of two consecutive plies is maximised and allowed to vary in the 0–90° fibre angle range at intervals of 5°. Manufacturing of such laminates is practical nowadays as the industry is changing its production techniques into accurate automated fibre-placement and tape-laying technologies. A two-step approach is proposed for the design of laminates. In the first step, the optimal laminate is designed in the traditional fashion to cope with the expected quasi-static loads on the structure. The second step consists of redesigning this laminate to better withstand impact loads by dispersing its stacking sequence while keeping similar stiffness properties as in the first step.A traditional laminate and two dispersed stacking sequence alternative layups were tested under low-velocity impact and compression-after-impact loads in order to compare their impact resistance and damage tolerance characteristics. The evaluation of these laminates will also be carried out by the innovative numerical tools proposed in the follow-up of the present paper.     

含脱层纤维增强复合材料层合板在湿热环境下的屈曲分析

依据复合材料细观力学研究了含脱层的正交各向异性复合材料层合板的力学性能与温度和湿度的关系,在宏-细观力学模型框架下建立了考虑湿热效应的屈曲控制方程,利用空间上分离变量的方法求解了控制方程,得到满足连续性条件和边界条件的控制方程的解。通过含脱层的复合材料层合板的屈曲数值算例,综合讨论了几何参数、材料参数和湿热环境等多方面的参数变化对含脱层的正交对称铺设复合材料层合板的临界屈曲载荷的 影响。     

Local buckling loads of sandwich panels made with laminated faces

The paper is devoted to assessing the optimal arrangements of hybrid laminated faces of sandwich panels in order to maximize local buckling loads corresponding to the wrinkling of compressed faces. The analysis is carried out by modelling compressed faces as thin unsymmetric laminates resting on elastic two-parameter foundations. The First-order Shear Deformation Theory, in conjunction with the Rayleigh-Ritz method, has been used to evaluate buckling loads of simply supported flat laminates subjected to in-plane biaxial compression and shear forces. A numerical investigation is intended to support evidence for the influence of laminate parameters (fibre orientation, geometrical dimensions) and foundation parameters (modulus of subgrade reaction and shear modulus); obtained results are reported and discussed in the paper.     

A scatter search algorithm for stacking sequence optimisation of laminate composites

This paper explores the metaheuristic approach called scatter search for lay-up sequence optimisation of laminate composite panels. Scatter search is an evolutionary method that has recently been found to be promising for solving combinatorial optimisation problems. The scatter search framework is flexible and allows the development of alternative implementations with varying degree of sophistication. The main objective of this paper is to demonstrate the effectiveness of the proposed scatter search algorithm for the combinatorial problem like stacking sequence optimisation of laminate composite panels. Preliminary investigations have been carried out to compare the optimal stacking sequences obtained using scatter search algorithm for buckling load maximisation with the best known published results. Studies indicate that the optimal buckling load factors obtained using the proposed scatter search algorithm found to be either superior or comparable to the best known published results.

Later, two case studies have been considered in this paper. Thermal buckling optimisation of laminated composite plates subjected to temperature rise is considered as the first case study. The results obtained are compared with an exact enumerative study conducted on the problem to demonstrate the effectiveness and performance of the proposed scatter search algorithm. The second case study is optimisation of hybrid laminate composite panels for weight and cost with frequency and buckling constraints. The two objectives are considered individually and also collectively to solve as multi-objective optimisation problem. Finally the computational efficiency of the proposed scatter search algorithm has been investigated by comparing the results with various implementations of genetic algorithm customised for laminate composites. It was shown in this paper through numerical experiments that the scatter search is capable of finding practical solutions for optimal lay-up sequence optimisation of composite laminates and results are comparable and sometimes even superior to genetic algorithms.     

Optimization of composite structures under multiple load cases using a discrete approach based on lamination parameters

This paper presents a new optimization technique applicable to optimization of composite structures subjected to multiple objectives. The composite structures may be composed of an arbitrary number of laminates. The technique is especially suited for the case where the layers of the laminates may assume a discrete number of orientations. However, given the efficiency of the technique, it is readily extendable to situations where the ply orientations vary quasi‐continuously, for instance, by one degree in one degree. The high efficiency is obtained through application of lamination parameters, which, in the case of symmetric laminates, consist of only 10 parameters per laminate. Three traditional structures, a rectangular composite plate, a cantilever composite beam, and a stiffened composite panel, are optimized against buckling when subjected to multiple load cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Battle damage repair of a helicopter composite frame-to-skin junction – A sole external repair approach

Research was conducted to examine the effectiveness of a rapid repair to a helicopter composite frame-to-skin junction subjected to battlefield damage. The repair design consists of a laminate patch and aluminium angle bracket adhesively bonded and riveted, respectively, to the helicopter external surface. The assessment involved a relative comparison of three models, representing pristine, damaged and repaired configurations. Computational analyses were conducted to examine the stiffness and buckling onset load of the overall structure and the strengths of individual components (laminates, adhesive bondlines and rivets) under three typical load conditions, namely in-plane shear, axial compression and transverse compression. The results showed that the damage would cause significant stiffness and strength reduction. The repair could sufficiently restore the stiffness and static strength for the load cases considered. However, for the specimen without support from its adjacent helicopter structure, it is predicted that the failure mode under the transverse compression loading would be via buckling under a relatively low load. A compression test was conducted to further validate the repair design. The result agreed well with the prediction. It showed that compared with an un-repaired damaged specimen, the external repair increased the strength by 83%. The equivalent far field failure strain exceeded 3300 με which is considered satisfactory for a rapid field battle damage repair (BDR).     

Nonlinear dynamic behavior of a piezothermoelastic laminated plate with anisotropic material properties

Summary. In this paper, we analyze the nonlinear dynamic behavior of a piezothermoelastic laminated plate with anisotropic material properties. The analytical model is a rectangular laminate composed of fiber-reinforced laminae and piezoelectric layers. The model is assumed to be a symmetric cross-ply laminate with all egdes simply-supported and to be subjected to mechanical, thermal and electrical loads as intended control procedures or as disturbances. The von Kármán strains are introduced to treat non-linear deformation. The behavior of the laminate is analyzed by using the Galerkin Method. We discuss the following quantities: (i) the buckling temperature due to in-plane thermal load; (ii) the large static deflection due to combined in-plane and anti-plane loads; (iii) the natural frequency of infinitesimal oscillation around the static equilibrium state; (iv) the natural frequency of the oscillation with finite amplitude around the static equilibrium state. Moreover, numerical examples are shown to investigate the methods to rise the buckling temperature, to linearize the thermal deflection and the natural frequencies by applying the electrical voltage to the piezoelectric actuators.     

Optimum design of laminated composite under axial compressive load

In the present study optimal design of composite laminates, with and without rectangular cut-out, is carried out for maximizing the buckling load. Optimization study is carried out for obtaining the maximum buckling load with design variables as ply thickness, cut-out size and orientation of cut-out with respect to laminate. Buckling load is evaluated using a ‘simple higher order shear deformation theory’ based on four unknown displacements u, v, w _b and w _s . A C¹ continuous shear flexible finite element based on HSDT model is developed using Hermite cubic polynomial. It is observed that for thick anti-symmetric laminates, the non-dimensional buckling load decreases with increase in aspect ratio and increase in fibre orientation angle. There is a decrease in the non-dimensional buckling load of symmetric laminate in the presence of cut-out.     

Optimum in situ strength design of laminates under combined mechanical and thermal loads

In this paper, optimum laminate configurations are sought for multidirectional fibre-reinforced composite laminates under combined in-plane mechanical and thermal loads. The design objective is to enhance the value of the loads over and above the first-ply-failure loads which are judged by a transverse failure criterion and the Tsai-Hill criterion, respectively. The in situ strength parameters previously obtained are incorporated in these criteria. It is found that the optimum designs under combined mechanical and thermal loads are not the same as those under pure mechanical loads for three of the four loading cases studied. For all cases the optimum loads are significantly larger than those for a quasi-isotropic design.