Optimal design of hybrid laminated composite plates with dynamic and static constraints

Optimization procedures are presented that consider the static and dynamic characteristic constraints for laminated composite plates and hybrid laminated composite plates subject to a concentrated load on the center of the plate. The design variables adopted are ply angle or ply thickness. Considered constraints are deflection, natural frequency and specific damping capacity. Using a recursive linear programming method, nonlinear optimization problems are solved, and by introducing the design scaling factor, the number of iterations is reduced significantly. Relating interactive optimization procedures with the finite element method analysis, various hybrid composite plates with arbitrary boundary conditions can be designed optimally. In the optimization procedure, verification of analysis and design of the laminated composite plates are compared with a previous paper. Various design results are presented on laminated composite plates and hybrid laminated composite plates.     

An analytical solution for static stability of multi-scale hybrid nanocomposite plates

An analytical answer to the buckling problem of a composite plate consisted of multi-scale hybrid nanocomposites is presented here for the first time. In other words, the constituent material of the structure is made of an epoxy matrix which is reinforced by both macro- and nanosize reinforcements, namely, carbon fiber (CF) and carbon nanotube (CNT). The effective material properties such as Young’s modulus or density are derived utilizing a micromechanical scheme incorporated with the Halpin–Tsai model. To present a more realistic problem, the plate is placed on a two-parameter elastic substrate. Then, on the basis of an energy-based Hamiltonian approach, the equations of motion are derived using the classical theory of plates. Finally, the governing equations are solved analytically to obtain the critical buckling load of the system. Afterward, the normalized form of the results is presented to emphasize the impact of each parameter on the dimensionless buckling load of composite plates. It is worth mentioning that the effects of various boundary conditions are covered, too. To show the efficiency of presented modeling, the results of this article are compared to those of former attempts.

     

On choice of optimal anisotropy of composite plates against buckling, with special attention to bending-twisting coupling

The lay-up optimization problems of composite plates against buckling are studied. The plate has symmetric lay-up and is loaded by an in-plane loading. The Classical Lamination Plate Theory is used. The necessary optimality conditions for the plate orthotropy/anisotropy optimization problems are considered, with special attention to bending-twisting coupling. The purpose of the optimization is to maximize the (lowest) buckling eigen value. The varied lay-up/layer orientation angles are considered both as smooth functions of the location coordinates and as having the same values at every point. It is shown that the twisting moment (calculated in the principal curvature axes) plays an important role in the conditions. Two example problems for a thin composite plate, loaded by shear, are considered. The first one corresponds to the case of one shear loading direction. The details of some known numerical solutions are studied. The obtained numerical results are in agreement with the theoretical results. The maximal lowest eigen values are 2-folded ones. A schematic model for treatment of the known optimal numerical solution (60° unidirectional lay-up for a long plate loaded by shear) is proposed. The second example problem corresponds to the case of shear loading acting in two opposite directions. The ways of equalizing the lowest buckling values corresponding to both loading directions are proposed. Numerical results for various aspect ratio and plate thickness values are presented. The potential of weight saving is demonstrated.     

Maximization of buckling load of laminated composite plates with central circular holes using MFD method

This paper addresses optimal design of simply supported symmetrically laminated composite plates with central circular holes. The design objective is the maximization of the buckling load, and the design variable is considered as the fiber orientation. The first-order shear deformation theory is used for the finite element analysis. The study is complicated because the effects of bending–twisting coupling are also included for the buckling optimization. The modified feasible direction method is used to solve the optimization problems. Finally, the effect of different number of layers, boundary conditions, width-to-thickness ratio, plate aspect ratios, hole daimeter-to-width ratio, and load ratios on the results is investigated.     

Optimum stacking sequence design of composite laminates for maximum buckling load capacity using parameter-less optimization algorithms

This paper presents a comparative study of the application of parameter-less meta-heuristic algorithms in optimum stacking sequence design of com of composite laminates for maximum buckling load capacity. Here, JAYA algorithm, along with Salp Swarm Algorithm, Colliding Bodies Optimization, Grey Wolf Optimizer, and Genetic Algorithm with standard setting and self-adaptive version are implemented to the problem of composite laminates with 64 graphite/epoxy plies with conventional ply angles, under several bi-axial cases and panel aspect ratios. Optimization objective is to maximize the buckling load of symmetric and balanced laminated plate. Statistical analysis are performed for six cases and the results are compared in terms of mean, standard deviation, the coefficient of variation, best and worst solutions, accompanied by the percentage of the independent runs that found the global optimum \(\left( {{R_{{\text{op}}}}} \right)\) and near global optimum \(\left( {{R_{{\text{no}}}}} \right)\). The Kruskal–Wallis nonparametric test is also utilized to make further confidence in the examinations. Numerical results show the high capability of the JAYA algorithm for maximizing the buckling capacity of composite plates.

     

Buckling optimization of laminated composite plates using genetic algorithm and generalized pattern search algorithm

In this paper, genetic algorithm and generalized pattern search algorithm are used for optimal stacking sequence of a composite panel, which is simply supported on four sides and is subject to biaxial in-plane compressive loads. The problem has several global optimum configurations in the vicinity of local optima. The composite plate under consideration is 64-ply laminate made of graphite/epoxy. The laminate is taken to be symmetric and balanced, comprised of two-ply stacks with discrete fiber angles of 0₂, ± 45, 90₂ in the laminate sequence. The critical buckling loads are maximized for several combinations of load case and plate aspect ratio, and are compared with published results. Performance of both algorithms is compared in terms of capability of identifying global optima. It is found that genetic algorithm is efficient for problems with global optima.     

An ant colony optimization approach to multi-objective optimal design of symmetric hybrid laminates for maximum fundamental frequency and minimum cost

An ant colony optimization algorithm for optimum design of symmetric hybrid laminates is described. The objective is simultaneous maximization of fundamental frequency and minimization of cost. Number of surface and core layers made of high-stiffness and low-stiffness materials, respectively, and fiber orientations are the design variables. Optimal stacking sequences are given for hybrid graphite/epoxy-glass/epoxy laminated plates with different aspect ratios and number of plies. The results obtained by ant colony optimization are compared to results obtained by a genetic algorithm and simulated annealing. The effectiveness of the hybridization concept for reducing the weight and keeping the fundamental frequency at a reasonable level is demonstrated. Furthermore, it is shown that the proposed ant colony algorithm outperforms the two other heuristics.     

Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM

In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)₂]_s and angle-ply [(15°/−75°)₂]_s, [(30°/−60°)₂]_s, [(45°/−45°)₂]_s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates.     

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

     

Optimal orientation of orthotropic materials for plates designed against buckling

The paper deals with the optimization of anisotropic plates loaded by in-plane forces and designed against buckling. The internal structure of the plate is seen as a twodimensional locally orthotropic solid and the orientations of the axes of orthotropy are taken as the design variables. The problem of optimization consists in determining the best orientation of the axes of orthotropy from the condition of the maximum behaviour of the critical buckling loads. General properties of the problem, such as multiplicity of the critical parameters and splitting of multiple eigenvalues, are studied. Optimization algorithms are developed to improve the anisotropic properties of the plate. The results of numerically finding the optimal orientation of orthotropic properties are compared with conventional layouts for square and rectangular plates loaded by normal in-plane forces.     

Design of laminated composite plates for optimal dynamic characteristics using a constrained global optimization technique

The lamination arrangements of moderately thick laminated composite plates for optimal dynamic characteristics are studied via a constrained multi-start global optimization technique. In the optimization process, the dynamical analysis of laminated composite plates is accomplished by utilizing a shear deformable laminated composite finite element, in which the exact expressions for determining shear correction factors were adopted and the modal damping model constructed based on an energy concept. The optimal layups of laminated composite plates with maximum fundamental frequency or modal damping are then designed by maximizing the frequency or modal damping capacity of the plate via the multi-start global optimization technique. The effects of length-to-thickness ratio, aspect ratio and number of layer groups upon the optimum fiber orientations or layer group thicknesses are investigated by means of a number of examples of the design of symmetrically laminated composite plates.     

Optimization formulations for the maximum nonlinear buckling load of composite structures

This paper focuses on criterion functions for gradient based optimization of the buckling load of laminated composite structures considering different types of buckling behaviour. A local criterion is developed, and is, together with a range of local and global criterion functions from literature, benchmarked on a number of numerical examples of laminated composite structures for the maximization of the buckling load considering fiber angle design variables. The optimization formulations are based on either linear or geometrically nonlinear analysis and formulated as mathematical programming problems solved using gradient based techniques. The developed local criterion is formulated such it captures nonlinear effects upon loading and proves useful for both analysis purposes and as a criterion for use in nonlinear buckling optimization.     

Response of hybrid laminated composite plates under low-velocity impact

The response of hybrid laminated composite plates subjected to low velocity impact was investigated using shear deformation theory. As a result, the fractional energy loss of two hybrid composite plates with the same component ratio has different values according to the stacking sequence. A Graphite-Kevlar-Graphite plate has low energy loss and a Kevlar-Graphite-Kevlar plate much higher energy loss. Contact forces between the impactor and plates, center deflections of the plates and velocity changes of the impactor to time have different values according to the material properties of the impacted surface. Various composite plates with the same material in the impacted surface behaved with a similar response.     

Analysis and optimum design of fibre-reinforced composite structures

The optimal design of a carbon-fibre-reinforced plastic (CFRP) sandwich-like structure with aluminium (Al) webs is addressed. The material parameters are determined using tensile tests, whereafter the results of an analytical model, a numerical model and an experimental setup are compared. The analytical and numerical approximations are then used to optimize the structure in a multi-algorithm approach for minimum cost and maximum stiffness. The selected algorithm and approximation are motivated by their accuracy and computational efficiency. The CFRP plates are optimized with respect to ply arrangement, while the complete sandwich-like structure is optimized with respect to the combination of manufacturing and material cost. Design constraints on maximum deflection of the total structure, buckling of the CFRP composite plates, buckling of the Al webs, stress in the composite plates and stress in the Al stiffeners are included in the formulation. For the different phases in the optimization process, we use the recently proposed particle swarm optimization algorithm, a dynamic search technique and a continuous-discrete optimization technique .     

Thermal buckling analysis of temperature-dependent FG-CNTRC quadrilateral plates

The main objective of the present study is to analyze the thermal buckling of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) quadrilateral plates. Functionally graded patterns are introduced for the distribution of the carbon nanotubes (CNTs) through the thickness direction of the plate. The effective material properties of nanocomposite plate reinforced by CNTs are considered to be temperature-dependent (TD) and estimated using the micromechanical model. By the use of minimum total potential energy principle and based on the first-order shear deformation theory of plates, the stability equations are obtained. In order to use the generalized differential quadrature (GDQ) method and solve the stability equations, the irregular domain of quadrilateral plate is transformed into regular computational domain employing the mapping technique. The efficiency and accuracy of the proposed approach are first validated. Then, a comprehensive parametric study is presented to examine the effects of model parameters on the thermal buckling of FG-CNTRC quadrilateral plates. The results indicate that considering temperature dependency of the material properties plays an important role in the stability of the FG-CNTRC quadrilateral plates subjected to thermal loading.     

复合材料层合板非线性热振动分析

采用有限元方法研究复合材料层合板结构在线性温度场作用下非线性热振动特性．采用特征值屈曲分析方法,判断了结构在线性温度场作用下的临界屈曲分歧点,计算了结构的一阶弯曲固有频率,分析了铺层角度及铺层层数对结构临界屈曲温度分布和结构固有频率的影响,总结了其对复合材料层合板结构热振动特性影响的一般规律．这些结论对复合材料结构设计、抗热设计有一定的指导意义．     

Detection of delamination in laminated composites with limited measurements combining PCA and dynamic QPSO

This paper presents an output only damage diagnostic algorithm based on frequency response functions and the principal components for health monitoring of laminated composite structures. The principal components evaluated from frequency response data, are employed as dynamical invariants to handle the effects of operational/environmental variability on the dynamic response of the structure. Finite element models of a laminated composite beam and plate are used to generate vibration data for healthy and damaged structures. Three numerical examples include a laminated composite beam, cantilever plate made of carbon–epoxy and a laminated composite simply supported plate. Varied levels of delamination of laminated composite plies and matrix cracking at varied locations in the plies are simulated at different spatial locations of the structure. Numerical investigations have been carried out to identify the spatial location of damage using the proposed principal component analysis (PCA) based algorithm. In order to limit the number of sensors on the structure, an optimal sensor placement algorithm based on PCA is employed in the present work and the effectiveness of the proposed algorithm with a limited number of sensors is also investigated. Finally, the inverse problem associated with the detection of delamination and matrix cracking is formulated as an optimization problem and is solved using the newly developed dynamic quantum particle swarm optimization (DQPSO) algorithm. Studies carried out and presented in this paper clearly indicate that the proposed SHM scheme can robustly identify the instant of damage, spatial location, the extent of delamination and matrix cracking even with limited sensor measurements and also with noisy data.     

Buckling and postbuckling of circular plates containing concentric penny-shaped delaminations

Buckling and postbuckling analyses of circular laminated composite plates with delaminations are presented. An axisymmetric finite element model based on a layer-wise laminated composite plate theory is developed to formulate the problem. Geometric nonlinearity in the sense of von Kármán and imperfections in the form of initial global deflection and initial delamination openings are included. A simple contact algorithm which precludes the physically inadmissible overlapping between delaminated surfaces is proposed and incorporated into the analysis.

Numerical results are obtained addressing the effects of the initial imperfections, the number of delaminations and their sizes on the critical buckling load and buckling mode shapes as well as postbuckling responses.     

Genetic algorithms with local improvement for composite laminate design

This paper describes the application of a genetic algorithm to the stacking sequence optimization of a laminated composite plate for buckling load maximization. Two approaches for reducing the number of analyses required by the genetic algorithm are described. First, a binary tree is used to store designs, affording an efficient way to retrieve them and thereby avoid repeated analyses of designs that appeared in previous generations. Second, a local improvement scheme based on approximations in terms of lamination parameters is introduced. Two lamination parameters are sufficient to define the flexural stiffness and hence the buckling load of a balanced, symmetrically laminated plate. Results were obtained for rectangular graphite-epoxy plates under biaxial in-plane loading. The proposed improvements are shown to reduce significantly the number of analyses required for the genetic optimization.Presented at the ASME Winter Annual Meeting Structures and Controls Optimization, pp. 13–28. Printed with permission from ASME.     

Asymmetric thermal buckling of temperature dependent annular FGM plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behaviour of FGM annular plates resting on partial Winkler-type elastic foundation under uniform temperature elevation is investigated. Material properties of the plate are assumed to be temperature dependent. Each property of the plate is graded across the thickness direction using a power law function. First order shear deformation plate theory and von Kármán type of geometrical nonlinearity are used to obtain the equilibrium equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained considering the deflection-less conditions. Only plates which are clamped on both inner and outer edges are considered. Applying the adjacent equilibrium criterion, the linearised stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region and the second set which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalised differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved iteratively to obtain the critical conditions of the plate, the associated circumferential mode number and buckled shape of the plate. Benchmark results are given in tabular and graphical presentations dealing with critical buckling temperature and buckled shape of the plate. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and power law index of the graded plate. It is shown that, stiffness foundation, and radius of foundation may change the buckled shape of the plate in both circumferential and radial directions. Furthermore, as the stiffness of the foundation or radius of foundation increases, critical buckling temperature of the plate enhances.