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.     

Optimum stacking sequence design of laminated composite circular plates with curvilinear fibres by a layer-wise optimization method

The optimum stacking sequence design for the maximum fundamental frequency of symmetrically laminated composite circular plates with curvilinear fibres is investigated for the first time using a layer-wise optimization method. The design variables are two fibre orientation angles per layer. The fibre paths are constructed using the method of shifted paths. The first-order shear deformation plate theory and a curved square p-element are used to calculate the objective function. The blending function method is used to model accurately the geometry of the circular plate. The equations of motion are derived using Lagrange’s method. The numerical results are validated by means of a convergence test and comparison with published values for symmetrically laminated composite circular plates with rectilinear fibres. The material parameters, boundary conditions, number of layers and thickness are shown to influence the optimum solutions to different extents. The results should serve as a benchmark for optimum stacking sequences of symmetrically laminated composite circular plates with curvilinear fibres.     

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.     

Thermomechanical postbuckling analysis of symmetric and antisymmetric composite plates with imperfections

In the present paper the postbuckling response of symmetrically and antisymmetrically laminated composite plates subjected to a combination of uniform temperature distribution through the thickness and in-plane compressive edge loading is presented. The structural model is based on a higher-order shear deformation theory incorporating von Karman nonlinear strain displacement relations. Adopting the Galerkin procedure, the governing nonlinear partial differential equations are converted into a set of nonlinear algebraic equations, which are solved using the Newton–Raphson iterative procedure. The critical buckling temperature is obtained from the solution of the linear eigenvalue problem. Postbuckled equilibrium paths are obtained for symmetrically laminated cross-ply, quasi-isotropic and antisymmetric angle-ply composite plates with and without initial geometric imperfections. Modal participation of each mode in the postbuckling deflection is reported using a multi-term Galerkin procedure.     

Post-buckling analysis of imperfect laminated plates under combined axial and thermal loads

A post-buckling analysis of composite laminated plates subjected to combined axial compression and uniform temperature loading is presented. The two cases of thermal post-buckling of initially compressed plates and of compression post-buckling of initially heated plates are considered. The initial geometrical imperfection of the plates is taken into account. The analysis uses a perturbation technique to determine buckling loads and post-buckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. Typical results are presented in dimensionless graphical form.     

基于直接搜索方法的层合板多目标优化设计

直接多目标搜索方法(DMS)是一种不需要计算梯度信息并且能实现全局收敛的多目标优化方法。基于直接多目标搜索方法,以简支层合板铺设角度为设计变量,基频和声功率为目标函数进行层合板结构振动与声多目标优化。分别以4层、8层复合材料层合板为例,用DMS方法对其优化设计,并与传统的遗传算法(GA)对比。结果表明,对于4层复合材料层合板,DMS方法比GA方法优化速度快,且能得到全局最优解;对于8层复合材料层合板,用DMS方法比4层板优化所需时间多,但相比GA方法,DMS方法优化更快。     

The Artificial Bee Colony algorithm in layer optimization for the maximum fundamental frequency of symmetrical laminated composite plates

In this study the layer optimization was carried out for maximizing the lowest (first) fundamental frequency of symmetrical laminated composite plates subjected to any combination of the three classical boundary conditions, and the applicability of the Artificial Bee Colony (ABC) algorithm to the layer optimization was investigated. The finite element method was used for calculating the first natural frequencies of the laminated composite plates with various stacking sequences. The ABC algorithm maximizes the first natural frequency of the laminated composite plate defined as an objective function. The optimal stacking sequences were determined for two layer numbers, twenty boundary conditions and two plate length/width ratios. The outer layers of the composite plate had a stiffness increasing effect, and as the number of clamped plate edges was increased both he stiffness and natural frequency of the plate increased. The optimal stacking sequences were in good agreement with those determined by the Ritz-based layerwise optimization method (Narita 2003: J. Sound Vibration 263 (5), 1005–1016) as well as by the genetic algorithm method combined with the finite element method.     

Normal stress distribution adjacent to optimum holes in composite plates

An approximate solution in the form of a polynomial is presented for the normal stress distribution adjacent to a class of optimum holes in symmetrically laminated infinite composite plates under uniaxial loading. Comparison of the present approximate solution with that of the finite element method shows good agreement. The approximate solution is useful in assessing the strength of the laminated composite plates with holes. The present simple solution will be of immense use to design engineers who are interested in obtaining quick and reliable solutions for circular and optimized holes in symmetrically laminated composite plates.     

复合材料对称层合板特征值问题的灵敏度分析

由于复合材料层合板的横向剪切模量较低,因此基于克希霍夫理论而计算的层合板特征值及其灵敏度与实际情况相差较大。本文采用层合板的一阶剪切变形理论,将纤维铺设角、材料参数和从层合板的中面到第K层板上表面的距离Z_K视为设计变量,研究了对称层合板特征值的灵敏度计算问题。     

Reliability predictions of laminated composite plates with random system parameters

Reliability predictions of laminated composite plates with random system parameters subjected to transverse loads are performed using different methods. System parameters such as material properties, layer thicknesses, and lamina strengths of a laminated composite plate are treated as base-line random variables and an appropriate failure criterion is used to construct the limit state equation of the plate in the reliability analysis. Based on the statistics of the base-line random variables obtained from experiments, different methods, namely, Monte Carlo method, β method, and first-order second moment method, are used to calculate the reliability of the laminated composite plates. In the first-order second moment method, the stochastic finite element method is used to derive for the statistics of the first-ply failure load of the laminated composite plates from those of the base-line random variables. The reliability of the laminated plate is then computed using the theoretically determined statistics together with an assumed probability distribution function of the first-ply failure load. The feasibility and accuracy of the different methods are studied by means of the experimental data of centrally loaded laminated composite plates with different lay-ups. The suitability of several commonly used failure criteria for reliability analysis of laminated composite plates is also investigated by means of several examples.     

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.     

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.     

A two-step optimization scheme for maximum stiffness design of laminated plates based on lamination parameters

The purpose of this paper is to propose an effective solution scheme of simultaneous optimization design of layup configuration and fiber distribution for maximum stiffness design of laminated plates. Firstly, a numerical analysis of the lamination parameters feasible region for a laminated plate consisting of various given number of ply groups (each ply group may have different thickness and all the fibers in one ply group are orientated in an identical direction) is carried out, and it is found that the feasible region based on only a few ply groups is very close to the overall one determined by infinite plies. Therefore, it is suggested that the feasible region of lamination parameters of a laminated plate could be approximately determined by the layup configuration of least ply groups. Secondly, a two-step simultaneous optimization scheme of layup configuration and fiber distribution for maximum stiffness design of laminated plates is proposed. Accordingly, by using ply thickness, fiber orientation angle and fiber volume fraction in a laminated plate of least ply groups as design variables, the optimal lamination parameters for maximum stiffness is obtained. Then, taking the optimal lamination parameters as the design objective, a detailed layup design optimization is implemented by considering some limitations on manufacturing, such as preset ply thickness, and specific fiber orientation angle and a limited maximum number of consecutive plies in the same fiber orientation. Numerical examples are also presented to validate the proposed two-step optimization scheme.     

层合阻尼结构各向异性设计之阻尼特性分析

将各向异性设计引入层合阻尼结构中,从理论上分析了各向异性层合阻尼结构的阻尼特性及其控制机理,从而验证了建立约束阻尼层合结构各向异性优化设计新体系的可行性。文中衡量结构阻尼减振效果的重要指标是结构系统的损耗因子,采用半无限简支板为例,对各种铺层形式作了大量的计算;并以最大损耗因子为目标函数,以铺层角度、频率、厚度等为约束条件进行结构优化设计。分析研究表明,该结构内部柔性层对阻尼的影响要比应力耦合对其影响大得多;在高于基本模式的固有频率下,能显著地提高损耗因子。     

基于复合材料层合箱梁改进解析模型计算等效刚度

提出了基于复合材料层合箱梁改进解析模型的等效刚度计算方法。在考虑三维应变效应的同时用复合材料单层的二维折算模量分量来表示三维折算模量分量,简化了复合材料层合箱梁等效刚度系数的计算,得到了由梁横截面几何尺寸和层合板刚度系数表达的等效抗弯刚度和等效抗扭刚度的解析式。该解析式适用于环向刚度一致的复合材料层合箱梁,并充分考虑了弯曲-剪切耦合和扭转-拉伸耦合效应对等效刚度的影响。通过三点弯试验和扭转试验,验证了解析式的正确性;通过与分层等效叠加法、有限元法进行对比,分析了解析式的计算精度。结合经典层合板理论,研究了铺层方式对等效刚度产生的影响及原因,预测了不同铺层复合材料层合箱梁等效刚度的变化规律。      

Vibration analysis of variable stiffness laminated composite sandwich plates

Abstract

This paper presents the free vibration analysis of a variable stiffness laminated composite sandwich plates. The fiber orientation angle of the face sheets (Skin) is assumed to vary linearly with the x-axis. The problem formulation is based on the higher-order shear deformation plate theory HDST C⁰ coupled with p-version of finite element method. The elements of the stiffness and mass matrices are calculated analytically. The sandwich plate is presented with a uniform mesh of four p-elements and the convergence properties are achieved by increasing the degree p of the hierarchical shape functions. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New comparison study of vibration response of laminated sandwich plate between the straight and curvilinear fibers is presented.     

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.     

复合材料浅球壳模态分析与结构优化设计

复合材料层合壳体在航空、航天、工程结构中得到了广泛应用.结合复合材料层合浅球壳的结构特点,基于有限元分析软件ANSYS,采用层合壳单元建立有限元模型,分析了几种壳体参数对浅球壳自振频率的影响,并采用函数逼近法和梯度寻优法相结合的方法对壳体参数进行优化设计,给出了壳体参数的最优组合,使壳体的一阶自振频率为最大,改善了壳体的动态特性,为复合材料层合浅球壳的结构设计提供了有价值的理论依据,也为进一步进行结构动力学分析奠定基础.     

各向异性复合材料对称角铺设层合板非线性弯曲强迫振动

笔者曾经研究讨论了各向异性复合材料反对称角铺设层合板的非线性弯曲强迫振动问题[1]。本文是这一研究领域的继续和扩展。在这里,笔者详细研究了各向异性复合材料对称角铺设层合板在简谐激振力作用下的非线性弯曲强迫振动问题,做了包括算例的全解析过程分析,得到了各向异性对称角铺设层合板在各种量级载荷作用下的幅一频关系,分析了对称角铺设层合板结构的铺设方向角对非线性弯曲强迫振动的幅一频关系影响,所求得的解析形式结果保证了结果的可靠性。本文无疑是对复合材料层合板非线性强迫振动问题研究的一个有意义的深入扩展,它们对于探讨其它复合材料结构的非线性动力问题具有一定的意义。      

Design of shear-deformable antisymmetric angle-ply laminates to maximize the fundamental frequency and frequency separation

An antisymmetrically laminated angle-ply plate is optimized with the objectives of maximizing the fundamental eigenfrequency and the distance between two consecutive natural frequencies. The formulation includes the contribution of the shear deformation, but neglects the in-plane and rotary inertias. The design variables are the fiber orientations and the thicknesses of individual layers. The design problems are cast into a mathematical programming format and solved by using a quasi-Newton function maximization algorithm. A penalty function method is employed to maximize the fundamental frequency, subject to lower bound constraints on higher-order frequencies. Numerical results are presented for laminates constructed of high-modulus fiber-reinforced materials, and the effects of various problem parameters on the efficiency of the designs are investigated. It is shown that the design variables may not be determined optimally if the effect of shear deformation is neglected. Moreover, it was observed that the classical plate theory leads to erroneous results in optimal material selection problems.