Optimal design of laminated composite beam structures

This work deals with design sensitivity analysis and optimal design of composite structures modelled as thin-walled beams. The structures are treated as a torsion-bending resistant beams. The analysis problem is discretized by a finite element technique. A two-node Hermitean beam element is used. The beam sections are made from an assembly of elements that correspond to flat layered laminated composite panels. Optimal design is performed with respect to the lamina orientations and thickness of the laminates. The structural weight is considered as the objective function. Constraints are imposed on stresses, displacements, critical load and natural frequencies. Two failure criteria are used to limit the structural strength: Tsai-Hill and maximum stress. The Tsai-Hill criterion is also adopted to predict the first-ply-failure loads. The design sensitivity analysis is analytically formulated and implemented. An adjoint variable method is used to derive the response sensitivities with respect to the design. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional structures.     

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.     

Design of laminated composite structures for optimum fiber direction and layer thickness,using optimality criteria

In this study, two optimality criteria are presented for optimum design of composite laminates using finite element method. Thickness of the layers and fiber orientation angles in each finite element are considered as the design variables. It will be shown that the optimum design of composite laminates with varying fiber orientations and layers thicknesses may be found by using these optimality criteria in an efficient way, without performing the sensitivity analysis.     

Stacking sequence optimization with genetic algorithm using a two-level approximation

We propose a new method for laminate stacking sequence optimization based on a two-level approximation and genetic algorithm (GA), and establish an optimization model including continuous size variables (thicknesses of plies) and discrete variables (0/1 variables that represent the existence of each ply). To solve this problem, a first-level approximate problem is constructed using the branched multipoint approximate (BMA) function. Since mixed-variables are involved in the first-level approximate problem, a new optimization strategy is introduced. The discrete variables are optimized through the GA. When calculating the fitness of each member in the population of GA, a second-level approximate problem that can be solved by the dual method is established to obtain the optimal thicknesses corresponding to the each given ply orientation sequence. The two-level approximation genetic algorithm optimization is performed starting from a ground laminate structure, which could include relatively arbitrarily discrete set of angles. The method is first applied to cylindrical laminate design examples to demonstrate its efficiency and accuracy compared with known methods. The capacity of the optimization strategy to solve more complex problems is then demonstrated using a design example. With the presented method, the stacking sequence in analytical tools can be directly taken as design variables and no intermediate variables need be adopted.     

Topology and thickness optimization of laminated composites including manufacturing constraints

This paper presents a gradient based optimization method for large-scale topology and thickness optimization of fiber reinforced monolithic laminated composite structures including certain manufacturing constraints to attain industrial relevance. This facilitates application of predefined fiber mats and reduces the risk of failure such as delamination and matrix cracking problems. The method concerns simultaneous determination of the optimum thickness and fiber orientation throughout a laminated structure with fixed outer geometry. The laminate thickness may vary as an integer number of plies, and possible fiber orientations are limited to a finite set. The conceptual combinatorial problem is relaxed to a continuous problem and solved on basis of interpolation schemes with penalization through the so-called Discrete Material Optimization method, explicitly including manufacturing constraints as a large number of sparse linear constraints. The methodology is demonstrated on several numerical examples.     

A lagrange parametrization for the design of variable stiffness laminates

A Lagrange parameterization of lamination parameters which are used for optimization of variable stiffness laminates is presented. The advantages of the approach are: a) the design variables become independent of the finite element mesh and, b) the smoothness of the solution is inherently guaranteed. Due to independency of design variables of the finite element mesh, the reduction in the number of design variables is drastic, once variable stiffness laminates usually demand a fine mesh. The lamination parameters formulation allows a more precise and concise approach to laminate design, removing difficulties related to fiber direction and stacking sequence, increasing the chances to find a global optimal solution. In this paper, the Lagrange parameterization is used for the maximization of the buckling load of a variable stiffness composite plate.     

Cross-section optimal design of composite laminated thin-walled beams

This paper aims to perform optimal design of cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. The finite element method is used in discretizing the theory. For design sensitivity calculations, the cross-section is modelled throughout design elements. Geometrically, these elements may coincide with the laminates that constitute the cross-section. The developed formulation is based on the concept of adjoint structure. After a warping function is calculated for the cross-section, an adjoint problem may be formulated for each of the properties and a corresponding adjoint warping is determined. It can be applied in a unified way to open, closed or hybrid cross-sections. Design optimization is performed by nonlinear programming techniques. Laminate thickness and lamina orientations are considered as design variables.     

A repair operator for the preliminary design of a composite structure using genetic algorithms

This paper deals with the preliminary design of a composite structure where the design variables are the thicknesses and the percentages of fiber orientations in the zones of the structure. In this paper, we propose to include the design and manufacturing rules in the preliminary design. A stacking sequence generator is used to compute admissible stacking sequences with respect to these rules and which correspond to the design variables. Given that an admissible stacking sequence does not exist for every set of values of the design variables, a repair operator is proposed to cope with this problem. It aims at changing the values of the percentages of the fiber orientations in order to guarantee the existence of admissible stacking sequences with respect to the manufacturing rules. The repair operator is integrated into an optimization loop which uses a genetic algorithm to perform the preliminary design of a composite structure. Its efficiency is shown with a test case which involves a large number of fiber orientations and stacking sequences.     

基于假定应变法的协同转动四边形复合材料壳单元

为对复合材料层合板壳结构进行精确的大变形数值模拟,提出一种采用假定应变法的能分析层合结构大转动问题的协同转动四边形壳单元.该方法在建立有限元公式时引入假定应变法以克服膜闭锁和剪切闭锁的不利影响.与其他能分析大转动问题的复合材料壳单元相比,在新的协同转动框架中采用矢量型转动变量,可大大降低在非线性增量求解过程中更新转动变量的难度,且能得到对称的单元切线刚度矩阵,提高单元的计算效率.分析两个典型算例,并与其他学者的结果进行对比,结果表明在计算层合结构大转角问题时拥有较好的精度和收敛性.     

Weight optimization for flexural reinforced concrete beams with static nonlinear response

The weight optimization of reinforced concrete (RC) beams with material nonlinear response is formulated as a general nonlinear optimization problem. Incremental finite element procedures are used to integrate the structural response analysis and design sensitivity analysis in a consistent manner. In the finite element discretization, the concrete is modelled by plane stress elements and steel reinforcement is modelled by discrete truss elements. The cross-sectional areas of the steel and the thickness of the concrete are chosen as design variables, and design constraints can include the displacement, stress and sizing constraints. The objective function is the weight of the RC beams. The optimal design is performed by using the sequential linear programming algorithm for the changing process of design variables, and the gradient projection method for the calculations of the search direction. Three example problems are considered. The first two are demonstrated to show the stability and accuracy of the approaches by comparing previous results for truss and plane stress elements, separately. The last one is an example of an RC beam. Comparative cost objective functions are presented to prove the validity of the approach.     

Geometrically nonlinear analysis of thin-walled composite box beams

A general geometrically nonlinear model for thin-walled composite space beams with arbitrary lay-ups under various types of loadings has been presented by using variational formulation based on the classical lamination theory. The nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed. Numerical results are obtained for thin-walled composite box beam under vertical load to investigate the effect of geometric nonlinearity and address the effects of the fiber orientation, laminate stacking sequence, load parameter on axial–flexural–torsional response.     

Coupled thermal-structural problems in the optimization of laminated plates

The behaviour of a laminated plate with given boundary temperatures and displacement constraints may be tailored by varying the orientation of the reinforcement in the different layers. Because the material parameters in a thermal conductivity problem, as also in a structural problem, depend on the orientations of the layers, there is a coupled-field problem to be solved. FEM is applied here to the analysis of such problems, which now consists of two phases in each iteration cycle: first solution of the temperature distribution over the structure and then computation of the displacements, stresses and strains. Strain energy and the sum of selected displacements for the structure are minimized with respect to the fibre orientations in the layers. Only mid-plane symmetric laminates with constant temperature over the thickness are considered, i.e. the response of the laminate is restricted to in-plane behaviour. Mathematically, the problem is a nonlinear one, and thus the minimum point can be either a local or a global one. The gradients needed during minimization are computed analytically. Examples with different numbers of design variables are given.     

Variable-stiffness composite panels: Buckling and first-ply failure improvements over straight-fibre laminates

One of the primary advantages of using fibre-reinforced laminated composites in structural design is the ability to change the stiffness and strength properties of the laminate by designing the laminate stacking sequence in order to improve its performance. This procedure is typically referred to as laminate tailoring. Traditionally, tailoring is done by keeping the fibre orientation angle within each layer constant throughout a structural component. Allowing the fibres to follow curvilinear paths within the plane of the laminates constitutes an advanced tailoring option that can lead to modification of load paths within the laminate to result in more favourable stress distributions and improve the laminate performance.Based on numerical simulations, the present work demonstrates the advantages of variable-stiffness over straight-fibre laminates in terms of compressive buckling and first-ply failure. A physically based set of failure criteria, able to predict the various modes of failure of a composite laminated structure, is implemented in finite element models of straight and variable-stiffness panels under compression. Non-linear analyses are carried out to simulate first-ply failure in the postbuckling regime.     

Multi-objective optimization of fiber reinforced composite laminates for strength,stiffness and minimal mass

We present a methodology for the multi-objective optimization of laminated composite materials that is based on an integer-coded genetic algorithm. The fiber orientations and fiber volume fractions of the laminae are chosen as the primary optimization variables. Simplified micromechanics equations are used to estimate the stiffnesses and strength of each lamina using the fiber volume fraction and material properties of the matrix and fibers. The lamina stresses for thin composite coupons subjected to force and/or moment resultants are determined using the classical lamination theory and the first-ply failure strength is computed using the Tsai–Wu failure criterion. A multi-objective genetic algorithm is used to obtain Pareto-optimal designs for two model problems having multiple, conflicting, objectives. The objectives of the first model problem are to maximize the load carrying capacity and minimize the mass of a graphite/epoxy laminate that is subjected to biaxial moments. In the second model problem, the objectives are to maximize the axial and hoop rigidities and minimize the mass of a graphite/epoxy cylindrical pressure vessel subject to the constraint that the failure pressure be greater than a prescribed value.     

Three-dimensional finite-element analysis of laminated composites

A three-dimensional finite-element analysis treating the mechanical response of thick laminated composite plates in bending is presented. An isoparametric solid element with a cubic displacement expansion in planform and a linear variation through the thickness is used to model each layer of the laminate. The degrees-of-freedom of the element are retained at its boundaries so that interconnections between lamina with different fiber orientations can be made at their interfaces. An incore version of the conjugate gradient technique, which does not have bandwidth restrictions, is used to minimize the total potential energy of the system with the number of iterations to convergence being about one-fifth the total global degrees-of-freedom. Because a three-dimensional analysis is used, the effects of thickness-stretching, transverse shear, extension, and bending deformations are obtained. Comparisons with three-dimensional elasticity solutions are in excellent agreement and show the necessity of having individual elements for each layer when they have different fiber orientations and when the plates are thick.     

A modified ant colony algorithm for the stacking sequence optimisation of a rectangular laminate

This paper presents a modified Ant Colony Algorithm (ACA) called multi-city-layer ant colony algorithm (MCLACA). The research attention is focused on improving the computational efficiency in the stacking sequence optimisation of a laminated composite plate for maximum buckling load. A new operator, the so-called two point interchange, is introduced and proved to be effective for reducing the convergence time and enhancing the robustness in the MCLACA performance. The laminate optimisation is subject to balanced and symmetric layup with ply contiguous and strength constraints. In order to assess the MCLACA performance, a simply supported rectangular laminate plate, which was taken as numerical example in previous research using traditional ACA and genetic algorithm (GA) is chosen as a benchmark case study. Comparing with the ACA and GA results, it is shown that the presented MCLACA has better performance in terms of computational efficiency and robustness. To demonstrate the applicability of the MCLACA to a general case, an additional example of laminate optimisation has been taken with more design variables and five different boundary conditions by finite element analysis.     

Buckling load enhancement of damaged composite plates under hygrothermal environment using unified particle swarm optimization

Optimization with Unified Particle Swarm Optimization (UPSO) method is performed for the enhancement of buckling load capacity of composite plates having damage under hygrothermal environment which has received little or no attention in the literature. Numerical results are presented for effect of damage in buckling behavior of laminated composite plates using an anisotropic damage model. Optimized critical buckling temperature of laminated plates with internal flaw is computed with the fiber orientation as the design variable by employing a UPSO algorithm and results are compared with undamaged case for various aspect ratios, ply orientations, and boundary conditions. FEM formulation and programming in the MATLAB environment have been performed. The results of this work will assist designers to address some key issues concerning composite structures. It is observed that the degradation of buckling strength of a structural element in hygrothermal environment as a result of internal flaws can be avoided to a large extent if we use these optimized ply orientations at design phase of the composite structure. This specific application proves the contribution of present work to be of realistic nature.     

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.     

A finite element approach to global-local modeling in composite laminate analysis

A finite element model is described to study interlaminar stresses within polymer composite laminated materials. This model is based upon a global-local model proposed by Pagano and Soni in 1983. The development of solution procedures includes an out-of-core memory solving technique. The numerical results generated for simple plate problems with and without holes in the center under uniaxial loading are reported. Comments regarding the finite-element mesh-size, numerical stability, problem size and sensitivity of results to substructuring of the laminate into global and local regions have also been discussed.