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).     

Dynamic Stability Optimization of Laminated Composite Plates under Combined Boundary Loading

Dynamic stability and design optimization of laminated simply supported plates under planar conservative boundary loads are investigated in current study. Examples can be found in internal connecting elements of spacecraft and aerospace structures subjected to edge axial and shear loads. Designation of such elements is function of layup configuration, plate aspect ratio, loading combinations, and layup thickness. An optimum design aims maximum stability load satisfying a predefined stable vibration frequency. The interaction between compound loading and layup angle parameter affects the order of merging vibration modes and may stabilize the dynamic response. Laminated plates are assumed to be angle-plies symmetric to mid-plane surface. Dynamic equilibrium PDE has been solved using kernel integral transformation for modal frequency values and eigenvalue-based orthogonal functions for critical stability loads. The dictating dynamic stability mode is shown to be controlled by geometric stiffness distributions of composite plates. Solution of presented design optimization problem has been done using analytical approach combined with interior penalty multiplier algorithm. The results are verified by FEA approach and stability zones of original and optimized plates are stated as final data. Presented method can help designers to stabilize the dynamic response of composite plates by selecting an optimized layup orientation and thickness for prescribed design circumstances.     

Optimal lay-up design of variable stiffness laminated composite plates by a layer-wise optimization technique

The optimal lay-up design for the maximum fundamental frequency of variable stiffness laminated composite plates is investigated using a layer-wise optimization technique. The design variables are two fibre orientation angles per ply. Thin plate theory is used in conjunction with a p-element to calculate the fundamental frequencies of symmetrically and antisymmetrically laminated composite plates. Comparisons with existing optimal solutions for constant stiffness symmetrically laminated composite plates show excellent agreement. It is observed that the maximum fundamental frequency can be increased considerably using variable stiffness design as compared to constant stiffness design. In addition, optimal lay-ups for the maximum fundamental frequency of variable stiffness symmetrically and antisymmetrically laminated composite plates with different aspect ratios and various combinations of free, simply supported and clamped edge conditions are presented. These should prove a useful benchmark for optimal lay-ups of variable stiffness laminated composite plates.     

Maximum stiffness design of laminated composite plates via a constrained global optimization approach

The optimal lamination arrangements of laminated composite plates with maximum stiffness subject to side constraints are investigated via a constrained multi-start global optimization approach. In the optimal design process, the deformation analysis of laminated composite plates is accomplished by utilizing a shear deformable laminated composite finite element and the optimal design problem, which has been converted into an unconstrained minimization problem via the general augmented Lagrangian method, is solved by utilizing the proposed unconstrained multi-start global optimization technique to determine the optimal fiber angles and layer group thicknesses of the laminated composite plates for attaining maximum stiffness and simultaneously satisfying the imposed side constraints. The feasibility of the proposed constrained multi-start global optimization algorithm is validated by means of a simple but representative example and its applications are demonstrated by means of a number of examples on the maximum stiffness design of symmetrically laminated composite plates. The effects of length-to-thickness ratio, aspect ratio, and number of layer groups upon the optimum fiber angles and layer group thicknesses of the plates are investigated.     

Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters

The buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads of axial compression and torsion are examined on the basis of Flügge’s theory. In the buckling analysis of long laminated composite cylindrical shells, 12 lamination parameters are introduced and used as design variables for layup optimization. Applying a variational approach, the feasible region in the design space of the 12 lamination parameters is numerically obtained. The buckling characteristics are discussed in the design space of the 12 lamination parameters. In the layup optimization, the optimum lamination parameters for maximizing the buckling loads and the laminate configurations for realizing the optimum lamination parameters are determined by mathematical programming methods. It is found that in case of combined loads of axial compression and torsion, the optimum laminate configurations are unsymmetric.     

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。     

Use of material grading for enhanced buckling design of thin-walled composite rings/long cylinders under external pressure

This paper presents a mathematical model for enhancing the buckling stability of composite, thin-walled rings/long cylinders under external pressure using radial material grading concept. The main structure to be analyzed is built of multi-angle fibrous laminated lay-ups having different volume fractions of the constituent materials within the individual plies. This leads to a piecewise grading of the material in the radial direction. The objective is to maximize the critical buckling pressure while preserving the total structural mass at a constant value equal to that of a baseline design. The fiber volume fractions are included among the standard design variables such as fiber orientation angles and ply thicknesses, which are used by many investigators in the field. The model employs the classical lamination theory, where an analytical solution that accounts for the effective axial and flexural stiffness separately is given. The critical buckling pressure contours subject to the mass equality constraint are given for several types of anisotropic rings/long cylinders showing the functional dependence of the constrained objective function on the selected design variables. It is shown that material grading can have significant contribution to the whole optimization process in achieving the required structural designs with enhanced stability limits.     

复合材料层合板力学性质分析及角铺设层优化设计

基于Kirchhoff经典理论,用样条有限元法以三次B样条函数构成的样条基对反对称多层角铺设层合板的三个独立位移进行插值,推导了复合材料层合板刚度阵,质量阵列式,阻尼阵列式,并由Lagrange方程导出了层合板的动力学方程,通过瑞利一李兹法建立了特征方程。分析了层合板的固有频率及不同层数和不同约束条件下的基频变化等力学特性,在Kirchhoff假设的基础上,对层合板的非线性弯曲的力学特性进行了探讨。基于样条有限元法和遗传算法进行复合材料层合板的角铺设层的优化设计,数值算列验证了算法的有效性。     

基于几何因子的复合材料气动弹性剪裁设计

实现了基于几何因子的复合材料层合板建模,解决了几何因子与Natran的参数输入问题,并根据工艺约束中的最小铺层比例对几何因子可行空间进行了推导补充。在此基础上,提出了一种基于几何因子和Nastran的复合材料气动弹性剪裁优化设计方法。首先以总厚度和几何因子作为设计变量以及以Nastran作为求解器,以强度、刚度、颤振和发散速度以及几何因子相关性约束作为约束条件进行结构寻优,得到最优的铺层总厚度和几何因子。其次,以最优几何因子作为目标,进行铺层结构逆问题求解,约束条件为复合材料铺层工艺约束。因几何因子为铺层厚度和铺层顺序的表达式,与传统的多级优化相比,以几何因子作为设计变量可以避免铺层厚度和铺层顺序的解耦,进而获得更大的设计空间,且得到的铺层结构可以满足工艺约束。最后,对一矩形悬臂复合材料层合板进行剪裁设计,使得铺层结构满足气动弹性约束且质量最小。结果显示,运用该优化方法可以得到质量更小且满足工艺约束的铺层结构。     

基于分层设计变量的复合材料层合板优化设计及灵敏度分析

在复合材料层合板的结构优化设计中,提出分层设计变量的优化方法以满足实际工程需要。在结构位移、自振频率和屈曲临界荷载灵敏度分析中,给出了刚度矩阵对分层厚度和分层角度设计变量的灵敏度计算公式,考虑了分层厚度变化引起层合板对称中心的改变,保证了计算准确性。数值算例验证了灵敏度算法的精度,应用实例显示了分层设计变量方法的实用性。     

Optimal design of laminated composite plates using a global optimization technique

A multi-start global optimization technique is used to investigate the lamination arrangements of laminated composite plates designed for maximum stiffness. The multi-start global optimization technique which originated from the concept of minimizing the potential energy of a moving particle in a conservative force field is extended to the optimal design of laminated composite plates in which the strain energies of the plates are minimized. The optimization algorithm has been proved to be efficient and effective in producing the global optima. Numerical examples of the selection of optimal lamination arrangements of symmetrically laminated composite plates with different aspect ratios subject to different loading conditions are given. The results show that aspect ratio, loading condition and material property can affect the optimal lamination arrangement.     

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.     

Reliability-based optimization of fibrous laminated composites

This paper is concerned with the optimum design of multiaxial fiber reinforced laminate systems under probabilistic conditions of loads and material properties. A multiaxially laminated composite is treated as a structural system with each ply contained in the composite as one element. The Tsai-Wu failure criterion is adopted as the limit state function of a unidirectional ply. It is assumed that the system failure occurs when any one of the plies in a laminate system fails. The multiple-check-point method is successfully applied to evaluate the system reliabilities of multiaxial laminates under probabilistic in-plane stresses. An optimization problem is defined to find the optimal number of fiber orientation axes, optimum orientation angles, and optimum ply ratios which yield the highest system reliability.     

基于遗传算法的大型风机复合材料叶片根部强度优化设计

为了提高复合材料叶片承担载荷的能力, 尤其是承受最大弯矩的叶片根部的承载能力, 研究了遗传算法的优化原理并将遗传算法应用到复合材料叶片根部铺层的优化设计中。针对复合材料层压结构遗传算法优化设计中, 层压结构参数具有离散型的特点, 提出了适合复合材料层压结构遗传算法优化设计的整数编码策略, 以整数来表征层压结构参数。在分析层压结构强度的基础上, 针对结构强度优化的目标构造了可用于遗传算法的适应度函数。同时参考了一定的铺层规则, 在铺层角度限制为工程中常用的四种角度的前提下, 应用遗传算法对叶片根部进行了铺层优化设计。结果表明, 由于遗传算法特有的处理离散型问题的优势, 在叶片根部的铺层优化设计中应用遗传算法是可行和可信的。     

Damping optimization of symmetrically laminated plates with transverse shear deformation using lamination parameters

This paper deals with the damping characteristics of symmetrically laminated plates with transverse shear deformation. First, the effect of laminate configuration on the damping characteristics is investigated for cantilevered laminated plates based on the Reissner–Mindlin’s first-order shear deformation theory. To examine the effect of laminate configuration, the concept of specific damping capacity is introduced and the damping characteristics are represented on the lamination parameter plane, where the damped stiffness invariants in transverse shear are newly proposed in this paper. Next, the optimal laminate configurations for the cantilevered laminated plates with maximal damping are determined taking into account the transverse shear effect by using differential evolution in which lamination parameters are used as intermediate design variables. The relation between the laminate configurations and the damping characteristics is discussed based on the concept of lamination parameters.     

A design study into the delamination behaviour of tapered composites

A fracture mechanics based analysis has been used to predict the tensile delamination load of tapered laminated plates. Simple laminate examples are used to show the effect of dropped ply thickness, number of delaminating surfaces, and dropped ply axial stiffness on the delamination load. Using these trends and acknowledged guidelines, a design is presented for a complex tapered plate with a view to maximising the onset of delamination.     

Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM

In this paper, the nonlinear behavior of symmetric and antisymmetric cross ply, thin to moderately thick, elastic rectangular laminated plates resting on nonlinear elastic foundations are studied using differential quadrature method (DQM). The first-order shear deformation theory (FSDT) in conjunction with the Green’s strain and von Karman hypothesis are assumed for modeling the nonlinear behavior. Elastic foundation is modeled as shear deformable with cubic nonlinearity. The differential quadrature (DQ) discretized form of the governing equations with the various types of boundary conditions are derived. The Newton–Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Comparisons are made and the convergence studies are performed to show the accuracy of the results even with a few number of grid points. The effects of thickness-to-length ratio, aspect ratio, number of plies, fiber orientation and staking sequence on the nonlinear behavior of cross ply laminated plates with different boundary conditions resting on elastic foundations are studied.     

Generating realistic laminate fiber angle distributions for optimal variable stiffness laminates

The advent of advanced fiber placement technology has made it possible, through the use of fiber steering, to exploit the anisotropic properties of composite materials to a larger extent than was previously possible. Spatial variation of stiffness can be induced by steering composite fibers in curvilinear paths to give beneficial load and stiffness distribution patterns. Buckling of composite panels is one area where fiber steering has been proven to be very effective. Fiber angles and predefined fiber angle variations are used in most of the research on fiber steered composites reported in the literature, however, from an optimization point of view it is attractive to design such variable stiffness (VS) structures in terms of lamination parameters (LPs). This results in a two-step design approach. In the first step a VS composite is designed in terms of LPs, and in the second step the LPs are converted into fiber angle distributions for each layer in the laminate. A methodology is proposed to convert a known LP distribution for a VS composite laminate into a realistic design in terms of fiber angles, with minimum loss of structural performance, whilst satisfying a constraint on in-plane fiber angle curvature. The proposed conversion process is formulated as an optimization problem and can be used for any number of equi-thickness plies. The methodology was tested by converting a known optimal LP design for a sample structure, a square plate under bi-axial compression into a fiber angle design. The effect of the in-plane curvature constraint, the number of layers in the laminate, and the choice of objective function for the conversion process were studied for a balanced symmetric lay-up.     

Impact surface fractures of glass-fiber/epoxy lamina-coated glass plates by small steel-ball

Fiber orientation effects on the impact surface fracture of glass plates coated with a glass-fiber/epoxy lamina layer were investigated using a small-diameter steel-ball impact experiment. Four kinds of materials were used: soda-lime glass plates, unidirectional glass-fiber/epoxy layer (one ply, two plies) coated glass plates, crossed glass-fiber/epoxy layer (only two plies) coated glass plates. The maximum stress and absorbed fracture energy of these plates were measured by a single-grid strain gage bonded to the back surface of the glass plates during the impact of the steel ball. With increasing impact velocity, various surface cracks, such as ring, cone, radial and lateral cracks, occurred near the impact sites of the uncoated glass plates. Plates with glass-fiber coating had a plastic deformation zone between the fiber layer and the glass plate that formed around the impact site while the surface cracks in the plates drastically diminished. The principal direction of this plastic deformation and delamination followed the fiber orientation. The impact surface-fracture index expressed in terms of the maximum stress and the absorbed energy could be used as an effective evaluation parameter for surface resistance.     

考虑纤维面外起伏的变刚度层合板优化策略研究

针对变刚度层合板在自动铺放制造过程中因间隙/重叠缺陷产生大量纤维面外起伏缺陷的问题,提出采用铺层偏移法与断送纱策略两种铺层优化策略来进行变刚度层合板的铺层设计,在研究过程中同时引入考虑间隙/重叠缺陷建模的方法。根据变刚度层合板铺层的特点提出缺陷重复单元的概念,通过对缺陷重复单元的分析来反映纤维面外起伏的影响,并提出通过纤维面外系数来表征变刚度层合板的纤维面外起伏尺度,最后对不同优化策略的变刚度层合板的屈曲性能进行分析。研究表明:基准设计方案、铺层偏移法与断送纱策略所对应的纤维面外起伏系数为0.83、0.95、0.93,所提出的优化策略对变刚度层合板的纤维面外起伏尺度有着明显的抑制作用。铺层偏移法优化后的[±<50/65>]_6s变刚度层合板最大厚度超差为33%,所对应的屈曲载荷为9117.1 N,屈曲载荷提升17.6%;断送纱策略优化后的[±<50/65>]_6s变刚度层合板最大厚度超差为50%,所对应的屈曲载荷为9716.3N,屈曲载荷提升25.3%。