A case for Tsai’s Modulus, an invariant-based approach to stiffness

A11 angle-ply laminated plate is optimized with the objective of maximizing critical buckling load. The material properties of the plate are not known precisely but with sorrle scatt,er around their average values. Convex analysis approach is employed to determine the material properties producing the lowest buckling load. T h e laminate is optimized over the ply-angles for the worst case of elastic moduli. The results are given for uniaxial and biaxial loading cases as well as for various aspect ratios. 111 structural design problems it is generally the case tha t the material properties are assurned to be fixed a t some deterministic parameters. However, it is well established tha t t,he elastic and strength properties of structures, and composite struct,urrs in particular, exhibit a scatter, mainly due to manufacturing imperfections. Therefore it is necessary to take the material uncertainty into account in the design of mechanical components to improve the reliability and to avoid premature failure. Consideration of scatter in the material properties becomes even more important in design optimization as the optimal configurations depend heavily on material da ta . Previous studies of the optimal design of composite laminates have been based on the assumption that the propertirs of the composite materials are known in a deterministic Since the scatter in material properties does not,, as a rule, cornply with any predefined distribution function, the rrlethods of convex modeling8 are quite suitable to deal with uncertain material data. This approach was applied to study t,he effect of uncerta i r~t~y i n elastic moduli on the buckling of corrlposite plates and shells.g I11 the present study optimal buckling design of composite laminates is investligated taking into account scatter in the material properties of the c o n posite. Using the methods of convex anx!ysis the elastic properties producirlg the least favorable ( the lowest) critical buckling load are drterrrliiied arid the laminate is optimized over the ply angles for thv worst case of material properties. Buckling a n a l y s i s Consider a simply supported angleply laminated plate of length a , width b and tllickrless 11 i l r thc z, y and z directions, respe~t~ively. T h e plat,r is symrnetrically laminated and balanced with equal number of layers with positive autl negative ply anglcs. It corisists of 11 laycrs so that the tlist,ance fro111 th r r~lidplane, z = 0, to the upper surface of the k;th lamilia is equal to t k = k h l n . 'The ply angle is denoted b j 0 . and tjhe stiffnesscs QiJ are fuuctior~s of 8 , 1, and the vector of elastic moduli E = (El, E2, E3. E4). wherc El = E L , E2 = ET, E3 = VLT and E4 = GLT. The stacking sequence for a symmetric and balanced laminate is of the form ((+@/ B)j),y,r where j = n /4 is the number of +O and -Q combinations. The compression loads per unit length in the z arid y directions are denoted by N, = ,V and IVY = AIL ' , respectively. The equation governing the, hucklilig of the plate under these loading conditions is given b j Copyright c 1994 by the American Institute of Aeronautics and Astronautics, Inc. All right reserved.

Optimization of composite laminates requires the satisfaction of constraints where the design ply thicknesses and orientations can only take discrete values prescribed by the manufacturers. Heuristics such as particle swarm or genetic algorithms are inefficient in such cases because they provide suboptimal solutions when the number of design variables is large. They also are computationally expensive in handling the combinatorial nature of the problem. In contrast, with the help of binary decision variables, mixed integer programming can be adopted to optimize such laminates efficiently. This paper presents an approach to reformulate lamination parameters and failure constraints as functions of binary decision variables. The buckling load maximization for a simply supported laminated plate is initially demonstrated using integer linear programming. Next, the laminate weight is minimized by varying the number of plies for a given external bi-axial compressive load and subjected to buckling and material failure constraints. A variation of laminate weight minimization is demonstrated by fixing the number of plies and assuming discrete changes in ply thicknesses. This is achieved using a sequential two-level optimization for laminates having uniform ply thickness. Finally, a scalability study is performed to evaluate the performance of mixed integer programming for different problem sizes. It is demonstrated that all three formulations with integer programming achieve significant performance gain and robustness over standard heuristic solvers.

In response to the current challenges of low precision and efficiency in the optimization of composite material layups, the insufficient lightweight of automotive body floors, and the high cost of carbon fiber composites, this study introduces an optimized design method for carbon fiber composite flooring layups. Based on implicit parametric technology, a hybrid PSO-BFO (Particle Swarm Optimization-Bacterial Foraging Optimization) algorithm is employed. This approach achieves an integrated optimization of materials, processes, and structures, thereby balancing and reducing costs. The SFE-CONCEPT is utilized to establish an implicit parameterization model for the body floor, which is validated through experiments and finite element simulation analysis. Concept design and modeling of the carbon fiber composite material floor laminate are performed. Continuous variable optimization is employed to determine the thickness, tile shape, and number of layers for the front, middle, and rear floors. A continuous variable discretization rounding strategy is used to obtain the discrete layer numbers for each laminate orientation of the composite material floor. The continuous fiber lamination strategy is applied to create different shared lamination regions. The PSO-BFO hybrid optimization method is proposed to optimize the lamination sequence as a multi-objective optimization, addressing the challenges of discrete lamination sequence, explosive combinations, and multiple variables in the optimization design of carbon fiber composite material floor laminates. The optimization results demonstrate improvements of 34.4% in floor quality M, 6.0% in static bending stiffness BS, and 5.3% in lightweight coefficient QLX using the proposed PSO-BFO method. PSO-BFO and PSO-GA (Particle Swarm Optimization-Genetic Algorithm) methods are more capable of obtaining global optimal solutions for complex optimization problems than single optimization algorithms. Still, the results obtained by the PSO-BFO method are more balanced.

Rapid development of automotive, aviation, aerospace and defense industries are necessitated the use of lightweight and high strength materials. For this purpose, glass fiber reinforced plastic composite materials are began to be produced and widely used. These materials are in proportion to weight have higher strength and rigidity so that has found wide usage. In this study, glass fibre reinforced plastic composite materials were produced with hand lay-up as different number of layers of glass fibre. Mechanical properties of produced composite materials were investigated. Avarage strength values of produced composite materials are obtained with three point bending and tensile tests. In addition to these tests, impact strength of materials were investigated by low speed impact test. Macro and micro photographs of samples were taken before and after the tests. Samples were characterized by metallographic respect. The damage zones of resin and glass fibre materials were examined in detail in the test images. The variation of mechanical properties depending on the number of fibre layers were identified considering the experimental results. Mechanical properties of GFRP materials were investigated; mechanical properties of fibre composite materials had a good outcome, depending on the number of layers. When impact test results are examined, between the number of fibre layers and the values of the energy absorbed by GFRP were observed approximately linearity.

Concurrent multi-scale design optimization of fiber-reinforced composite material based on an adaptive normal distribution fiber optimization scheme for minimum structural compliance and additive manufacturing

The stacking sequence optimization of laminated composite plates while maximizing the structural performance or minimizing the weight is a subject investigated extensively in the literature. Meanwhile, research on the optimization of laminates with cut-outs has been relatively limited. Cut-outs being an indispensable feature of structural components, this paper concentrates on the stacking sequence optimization of composite laminates in the presence of circular cut-outs. The buckling load of a laminate is used as a metric to quantify the structural performance. Here the laminates are modeled as carbon fiber-reinforced composites using the finite element analysis software, ABAQUS. For the optimization, the widely used harmony search algorithm is applied. In terms of design variables, ply thickness, and fiber orientation angles of the plies are used as continuously changing variables. In addition to the stacking sequence, another geometric variable to consider is the aspect ratio (ratio of the length of the longer sides to the length of the shorter sides of the plate) of the rectangular laminates. The optimization is carried out for three different aspect ratios. It is shown that, by using dispersed stacking sequences instead of the commonly used 0°/±45°/±90° fiber angle stacks, significantly higher buckling loads can be achieved. Furthermore, changing the cut-out geometry is found to have a significant effect on the structural performance.

Benefits of directional properties of fiber reinforced composites could be fully utilized by proper placement of the fibers in their optimal spatial orientations. This paper investigates an application of a Cellular Automata (CA) based strategy for design of variable stiffness composite laminates for optimal bending stiffness. CA are iterative numerical techniques that use local rules to update both field and design variables to satisfy equilibrium and optimality conditions. In the present study, displacement update rules are derived using a cell level model governing the equilibrium of the CA neighborhood. Local fiber orientation angles are treated as continuous design variables, and their spatial distribution is determined based on an optimality criterion formulation for minimum compliance design. Numerical examples for simply supported and clamped square plates are used to demonstrate the improvement in bending stiffness.

The genetic algorithm BIANCA, developed for design and optimisation of composite laminates, is a multi-population genetic algorithm, capable to deal with unconstrained and constrained hard combinatorial optimisation problems in engineering. The effectiveness and robustness of BIANCA rely on the great generality and richness in the representation of the information, i.e. the structure of populations and individuals in BIANCA, and on the way the information is extensively exploited during genetic operations. Moreover, we developed proper and original strategies to treat constrained optimisation problems through the generalisation of penalisation methods. BIANCA can also treat constrained multi-objective problems based on the construction of the Pareto frontier. Therefore, BIANCA allows us to approach very general design problems for composite laminates, but also to make a step forward to the treatment of more general problems of optimisation of materials and structures. In this paper, we describe specifically the case of optimal design of composite laminates, concerning both the theoretical formulation and the numeric resolution.

Thermo-mechanical buckling analysis of symmetric VAT composite laminates with temperature-dependent material properties

The design of laminates with a minimum number of layers for obtaining the given elastic properties is addressed in the paper. The problem is treated and solved in the general case — no simplifying hypotheses are made about the type of their stacking sequence. The problem is stated as a nonlinear programming problem, where a unique objective function includes all the requirements to be satisfied by the solutions. The optimum solutions are found within the framework of the polar-genetic approach, since the objective function is written in terms of polar parameters of the laminates, while a nonclassical genetic algorithm is used as the optimization scheme. The optimization variables include the number of layers, layer orientations and layer thicknesses. In order to include the number of plies among the design variables, certain modifications of the genetic algorithm have been done, and new genetic operators have been developed. Some examples and numerical results concerning the design of laminates with a minimum number of layers for obtaining some prescribed elastic symmetries, bending-extension uncoupling, and quasi-homogeneity are shown in the paper.

A Geometric non-linear analysis of composite shells under thermomechanical loading has been discussed here. From the literature, it may be seen that the thermal stress analysis of structural elements has continued to remain a research topic for a couple of decades. No one computationally verified the geometric non-linear buckling of composite shells under thermomechanical loading using semiloof element. In this work, linear buckling analysis of Kari Thangaratnam (2) is extended to geometric non-linear analysis of composite shells under thermomechanical loading. A general shell element called the semiloof shell element has been extended to thermal stress analysis of laminated shells. The formulation is based on nonlinear theory and the finite element method using semiloof element. The validation checks on the program are carried out using results on homogeneous isotropic shells available in the literature. The parameters considered in analysis are (1) number of layers in the laminate, (2) Lay-up sequence (symmetry, antisymmetry, cross-ply etc.), (3) Fibre orientation angle, (4) Different aspect ratios, (5) Orthotrophy ratio, (6) Boundary conditions (simply supported, clamped and combination of boundary conditions).

Inspired by work on allocating risk between the different components of a system for a minimal cost, we explore the optimal allocation of uncertainty in a single component. The tradeoffs of uncertainty reduction measures on the weight of structures designed for reliability are explored. The uncertainties in the problem are broadly classified as error and variability. Probabilistic design is carried out to analyze the effect of reducing error and variability on the weight. As a demonstration problem, the design of composite laminates at cryogenic temperatures is chosen because the design is sensitive to uncertainties. For illustration, variability reduction takes the form of quality control, while error is reduced by including the effect of chemical shrinkage in the analysis. Tradeoff plots of uncertainty reduction measures, probability of failure and weight are generated that could allow choice of optimal uncertainty control measure combination to reach a target probability of failure with minimum cost. In addition, the paper also compares response surface approximations to direct approximation of a probability distribution for efficient estimation of reliability.

An innovative optimization technique is presented for the design of composite laminated plates subjected to in-plane loads. A list of quasi-homogeneous laminates that can be used as angle-ply materials is proposed as a comprehensive solution for optimum lay-up. Two optimization procedures are performed: Dimensioning of the flexural stiffness and the elastic modulus, which provides the optimal orientations for the layers and offer highest in-plane resistance to composite laminated structures. The polar formalism for plane anisotropy is used to represent the flexural stiffness and elastic modulus tensors. Numerical examples are resolved for two materials with different elastic moduli.

In this article, a ranking method is introduced to design laminated composites with stress concentrations. The ranking method can be approached in two ways: (1) by con sidering many different thickness ratios of a family or several families of laminates; (2) by rigid body rotation of a family or several families of laminates to obtain many combina tions of lamination. Numerical examples are given for the design of a slanted elliptical opening under a single load, combined loads, and multiple loads. The differences between the design of composite laminates with stress concentrations and those without stress con centrations are also investigated.

Multi-level intelligent design of variable angle tow laminates via image-driven method