Stacking-sequence optimization using fractal branch-and-bound method for unsymmetrical laminates

The fractal branch-and-bound method has been developed by the authors for stacking-sequence optimizations of symmetric and balanced composite laminates comprise of two in-plane and two out-of-plane lamination parameters. Cylindrical structures such as tanks or pipes, however, are usually made from balanced laminates. In the present study, therefore, we focus on the stacking-sequence optimizations of unsymmetrical composite laminates. In the unsymmetrical laminates, nine lamination parameters including three coupling-lamination parameters exist, and its feasible design region of fractal pattern is unrevealed. The paper clarifies the feasible region in which the in-plane, out-of-plane and coupling lamination parameters create fractal patterns of tetrahedrons or tetradecahedrons. Using the fractal patterns of lamination parameters, the improved fractal branch-and-bound method is proposed for unsymmetrical laminates including coupling lamination parameters. This new method is applied to stacking-sequence optimization problems of maximization of buckling load of cylindrical laminated shells. As a result, the method is successfully applied, and a practical optimal stacking sequence is obtained with low computational cost.     

Stacking sequence optimization to maximize the buckling load of blade-stiffened panels with strength constraints using the iterative fractal branch and bound method

Laminated carbon fiber reinforced polymer (CFRP) composites have widespread applications in aerospace structures, and thus optimization of the stacking sequences in these composites is indispensable. Here, a fractal branch and bound method (FBB) is proposed for optimizing the stacking sequences. This method requires only low computational costs, and an optimal result can be obtained rapidly by means of the deterministic process. For practical stacking sequence optimizations, more than two laminates have to be optimized, because a practical aerospace structural component usually comprises a panel and stiffeners made from composite laminates. Since the stacking sequences of the skin panel and stiffeners affect the buckling load of the stiffened panel, the optimization of both laminates must be performed simultaneously. In the present study, a new method to implement a strength constraint for the FBB method is proposed for the simultaneous optimization of more than two laminates (such as a panel and stiffeners). Moreover, a quadratic polynomial objective function, which includes lamination parameter variables of the two laminates: the stiffeners and the panel, is adopted. The strength constraint is implemented by means of a response surface. The new method is applied to the buckling load maximization of a blade-stiffened composite panel, in which the strength constraint is demonstrated as a feasibility study. The method successfully obtained optimal stacking sequences with the strength constraint at low computational cost.     

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.     

Multiobjective stacking sequence optimization for laminated composite structures

A Pareto-based multiobjective evolutionary algorithm is presented for stacking sequence optimization of composite structural parts. Special attention has been paid to engineering design guidelines for stacking sequence design. These guidelines are included into the formulation of the optimization problem as constraints or additional objectives. A new initialization strategy is proposed following mechanical considerations. The method is applied to the optimal design of a composite plate for weight minimization and maximization of the buckling margins under three hundred load cases that make also the originality of this work. It is shown that the introduction of new ply orientations compared to the classical 0°, ±45° and 90° plies can lead to significantly improved optimal designs.     

铺设角度与铺层顺序对层合板稳定性的影响

以层合板结构的临界屈曲载荷系数最大化为优化目标,基于改进型模拟退火算法对层合板结构铺设角度和铺层顺序进行优化。由于层合板结构的铺层角度是离散变量,模拟退火算法适合求解离散变量的优化问题。利用模拟退火算法优化层合板铺层,在算法内采用并行计算、引入记忆功能同时设置双阈值终止准则,有效地提高了优化过程的收敛速度,同时避免优化过程中出现局部最优解。以临界屈曲载荷系数作为目标函数,选取复合材料层合板的铺设角度顺序为设计变量,采用改进的模拟退火算法得出复合材料层合板的最优铺设角度以及铺层顺序。     

Prediction of the postbuckling response of composite airframe panels including ply failure

Future design scenarios aim to allow buckling in composite airframe panels. Reliable simulation procedures should be able to capture the postbuckling elastic as well as the inelastic response associated with damage. Damage in composite laminates in terms of ply failure may primarily occur as fiber fracture or matrix cracking. This paper presents a model which is able to capture both geometrical and material nonlinearity. It bases on the finite element formulation of a layered, iso-parametric, quadrilateral shell element which allows for an arbitrary reference surface as well as an arbitrary stacking sequence. Geometrical nonlinearity is accounted for by utilizing Green strains and second Piola–Kirchhoff stresses. Material nonlinearity is considered via a layerwise ideally brittle damage model. The model is applied to a buckling test of a stringer-stiffened composite airframe panel. The numerical results are compared with an experiment proving the applicability of the proposed concept.     

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.     

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。      

考虑制造因素的变刚度层合板的抗屈曲铺层优化设计

与常规层合板相比,变刚度层合板的制造、有限元建模分析和铺层设计有其特殊之处。首先对设计时需考虑的制造因素进行了归纳,提出了变刚度层合板的铺层设计要求。然后给出了变刚度层合板的理想模型和考虑丝束宽度模型的建模方法。基于理想模型对ABAQUS的前处理模块进行二次开发,利用编制的参数化建模程序分析了不同铺放角的变刚度层合板的屈曲性能,并讨论了最小曲率半径对铺层的限制和变刚度设计提高屈曲载荷的机制。基于变刚度层合板的抗屈曲机制建立了一种铺层优化设计方法,使用遗传算法经两步优化得到最优铺层。对最优铺层建立考虑丝束宽度的模型以研究丝束宽度和铺层偏移对变刚度层合板抗屈曲铺层优化结果的影响。研究表明,在变刚度层合板的抗屈曲铺层优化中使用简化的理想模型通常来说是可行的。在考虑制造因素的情况下,优化后的变刚度层合板较常规层合板屈曲载荷有显著提高。     

Optimization of laminated composites subject to uncertain buckling loads

Optimal design of composite laminates under buckling load uncertainty is presented. The laminates are subjected to biaxial compressive loads and the buckling load is maximized under worst case in-plane loading which is computed using an anti-optimization approach. The magnitudes of the in-plane loads are not known a priori resulting in load uncertainty subject to the only constraint that the loads belong to a given uncertainty domain. Results are given for continuous and discrete fibre orientations which constitute the optimization problem coupled to load anti-optimization problem leading to a nested solution method. It is observed that the stacking sequence of a laminate designed for a deterministic load case only differs considerably from that of a robust laminate designed taking load uncertainties into account. Consequently the buckling load carried by a deterministic design is considerably less than the one carried by a robust design when both are subjected to uncertain loads.     

基于分级优化的复合材料带加强筋板屈曲优化

对工程中常用复合材料带加强筋板结构的优化方法进行研究。由于优化变量较多且变量类型涉及离散型和连续型两种类型,因此优化中引入了分级优化方法。在结构质量一定的前提下,确定各加强筋最优的位置、截面尺寸、铺层数及铺层顺序,以提高整体结构的抗屈曲性能。代理层的引入使优化问题可以在先不考虑铺层顺序的情况下对其余变量进行优化,通过析因实验设计方法确定剩余变量中影响结构屈曲载荷的主要变量和次要变量,并分别在第1、2级优化中独立对两类变量进行优化,在优化过程中根据变量的类型及取值范围,对部分变量建立径向基代理模型以提高优化效率。第3级优化采用遗传算法对铺层顺序进行优化。优化结果表明:优化后方案比原方案的抗屈曲能力提高了3.21倍。     

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.     

复合材料层合板屈曲稳定性优化设计及其灵敏度计算方法

笔者在有限元分析基础上研究了以屈曲稳定性作为约束条件或优化目标的复合材料层合板结构优化设计及其灵敏度分析方法,重点讨论了屈曲临界荷载灵敏度对内力场和载荷的依赖关系及其在铺层优化、尺寸优化和形状优化问题中的不同计算方法,并在JIFEX软件中实现了复杂结构复合材料层合板优化设计方法。数值算例验证了本文算法和程序的有效性。     

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.     

Optimum buckling design of composite stiffened panels using ant colony algorithm

Optimal design of laminated composite stiffened panels of symmetric and balanced layup with different number of T-shape stiffeners is investigated and presented. The stiffened panels are simply supported and subjected to uniform biaxial compressive load. In the optimization for the maximum buckling load without weight penalty, the panel skin and the stiffened laminate stacking sequence, thickness and the height of the stiffeners are chosen as design variables. The optimization is carried out by applying an ant colony algorithm (ACA) with the ply contiguous constraint taken into account. The finite strip method is employed in the buckling analysis of the stiffened panels. The results shows that the buckling load increases dramatically with the number of stiffeners at first, and then has only a small improvement after the number of stiffeners reaches a certain value. An optimal layup of the skin and stiffener laminate has also been obtained by using the ACA. The methods presented in this paper should be applicable to the design of stiffened composite panels in similar loading conditions.     

Multi-step blended stacking sequence design of panel assemblies with buckling constraints

In this paper, we propose a multi-step framework for design of composite panel assemblies and subsequent blending of the designs to ensure laminate continuity across multi-panel configurations. Multilevel optimisation is frequently used for solving complex optimisation problems. In composite design this approach leads to stacking sequence mismatch among adjacent structural components which is generally referred to as blending problem. To overcome stacking sequence mismatch, a guide-based genetic algorithm (GA) is used which in essence forces the design to be completely blended at any step in the design process. A serious drawback of guide based approach is that it necessitates repeated analysis of the entire structure within the GA iterations. A multi-step framework is proposed where the structure is first optimised using panel thickness and lamination parameters as continuous design variables. The continuous optimisation is performed using a successive convex approximation scheme. In the second step, discrete blended stacking sequences are obtained using a guide-based genetic algorithm. The fitness function in the guide-based GA is evaluated using convex approximations of the response. In this fashion, the cost of evaluating structural response within the GA optimisation is eliminated. The proposed framework is demonstrated via design of an eighteen panel horseshoe configuration, where each panel is optimised individually subject to a local buckling constraint. Numerical results indicate that the present algorithm is capable of producing near-optimal fully blended designs at a small fraction of the computational cost of traditional blending algorithms.     

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.     

Matrix cracking behavior of K3B/IM7 composite laminates subject to static and fatigue loading

The matrix cracking behavior of a new high-performance thermoplastic composite material, K3B/IM7, was systematically investigated. Laminates in various grouped thickness and ply stacking sequences, [0₂/90₂/0₂], [0₂/90₄/0₂], and a quasi-isotropic laminate [+45/0/−45/90]_s were tested under static and tension–tension fatigue loading. Depending on the stacking sequence of the laminates and the type of loading, various matrix cracking behavior were found. Under static loading, the matrix cracks were mainly close to the specimen edges. A few cracks were found to penetrate the specimen width, even when the load was large enough to break the specimen. However, under fatigue cyclic load, the edge initiated cracks propagated fully across the specimen width. Combined with the fatigue Paris Rule and considering the ply thickness and stacking sequence, the energy release rate method was applied to predict the relations between the loading strain amplitude and fatigue cycles for matrix cracking failure.     

复合材料层合板剪切稳定性试验及强度预测

对无损伤及含冲击损伤的复合材料层合板进行了剪切稳定性试验,基于数字图像相关方法 (Digital image correlation,DIC)对层合板屈曲后屈曲行为进行了实时测量。试验结果表明：引入冲击损伤后,复合材料层合板剪切屈曲波形、屈曲载荷无明显变化,失效模式转变,承载能力下降了9.69%。随后,基于断裂面失效理论,建立了考虑剪切非线性效应的复合材料渐进损伤失效模型,并对复合材料层合板剪切失效过程进行了模拟。模型采用软化夹杂法将冲击损伤等效简化,直接将损伤区的几何边界信息写入材料模型中,不需要对冲击损伤区进行切割,从而保证了整体网格质量。与试验结果对比发现：模型考虑剪切非线性对屈曲载荷预测无明显影响,对后屈曲承载能力的预测精度影响较大,不考虑剪切非线性效应时的误差可达20%以上;软化夹杂法可以有效地模拟冲击损伤,预测的含冲击损伤的复合材料层合板的屈曲载荷、破坏载荷误差分别为-3.17%、-1.27%。     

铺层混杂对复合材料层压板侵彻性能的影响

本文利用MTS和冲击侵彻测试装置，研究了由芳香族聚酰胺纤维、高强聚乙烯醇纤维制成的织物通过不同铺层方式与酚醛/PVB树脂复合的层压板的准静态和冲击侵彻性能。结果表明，芳香族聚酰胺织物层的加入能显著提高高强维纶织物树脂复合材料层压板的准静态侵彻刚度。随着芳纶混杂体积分数的提高，铺层混杂复合材料层压板的准静态侵彻阻力、穿孔能量（或单位面密度穿孔能量）将随之增加。从防护装具性能/重量比和性能/价格比的角度考虑，在芳香族聚酰胺与高强聚乙烯醇织物铺层混杂复合材料层压板中，高强聚乙烯醇纤维混杂体积分数可以确定为20%左右。