Zafer Gürdal

Cellular Automata for design of truss structures with linear and nonlinear response

The feasibility of the use of Cellular Automata (CA) techniques for the design of two-dimensional structural problems, such as trusses and continuum structures under static loading, is investigated. The study implements an integrated analysis and design approach using the methods of CA to achieve an optimal structural configuration. The paper summarizes the basic features of the CA and demonstrates a formulation for the design of twodimensional truss structures that exhibit linear and geometrically nonlinear response characteristics. The solution variables chosen for the approach include the in-plane displacements of the cells, which represent nodal points of the truss, and the cross sectional areas of the members that connect the neighboring cells. Equations derived from local equilibrium are used as the rules that govern the cell displacements, and simple rules that are based on fully stressed design conditions are used for sizing of the members. In addition to the theoretical formulation, an investigation into the numerical implementation of the CA was performed. Initially the Mathematica programming language was used for demonstration purposes. For large-scale problems, an efficient and flexible Fortran 90 computer code capable of modeling complex two-dimensional geometries was developed. Examples demonstrating the capabilities of the design tools are included.

Tow-Placement Technology and Fabrication Issues for Laminated Composite Structures

‡A review of fabrication issues encountered during the manufacture of tow-steered laminates is presented. Tow-steering is a novel laminate design concept that attempts to tailor the stiffness of a traditional composite laminate by using non-traditional curvilinear fiber paths within the plane of a ply. Manufacture of such tow-steered plies has been demonstrated with advanced tow-placement machines, and a representative machine is used to provide an example of the fabrication process. Stiffness tailoring of the composite laminate is achieved through spatial variation of the fiber orientation angle. The resulting curvilinear tow paths are based on constant curvature arcs, which provide simple numerical solutions as well as direct correspondence with the limiting turning radius constraint of the tow-placement machine. A basic framework for a general constant curvature path is presented, along with additional concepts to extend the fiber angle definition throughout the entire domain of the ply. These construction techniques generate gaps and/or overlaps between neighboring tow courses, which must be adequately considered during both the structural analysis and fabrication phases. The major concerns within the analysis portion due to the presence of tow-steered plies involve the balancing and symmetry of the laminate. Balancing is an issue due to the variable stiffness nature of each ply, and proper configurations are introduced which are required to obtain a globally balanced laminate. Symmetry about the middle surface of the laminate ensures that unintentional curvature does not arise during the curing process. For laminates using plies with overlaps, both the lay-down order of the tow courses and the alignment of the layers on the mandrel surface play an important role for this symmetry issue. Appropriate modifications to the stiffness measures of Classical Lamination Theory are discussed so that better correlation between the modeling and the actual part can be achieved. Relevant information concerning the fabrication of such laminates is based on previous production of several prototype flat panels which were built to validate the tow-steering design concept. The fabrication of these panels provided many opportunities to improve several details of the tow-steering concepts, as well as assess and correct unforeseen complications that the manufacturing phase brought into focus. The most important of these issues involves the minimization of the gaps that exist due to the curvilinear nature of the tow paths and the tow-dropping between neighboring courses. The presence of these lamina irregularities can often lead to degraded performance of the structure, therefore a coverage parameter and an interweaving technique are introduced which aim to diminish the effect of the gaps. Examples from actual parts are employed to demonstrate the usefulness of the techniques to produce a more reliable structure. Similarly, the interweaving technique applied to plies that permit overlaps instead of tow-dropping is shown to smooth out the thickness variation that occurs due to the extra plies within the overlap regions.

Approximate Feasible Regions for Lamination Parameters

Lamination parameters represent the laminate lay-up configuration in a compact form. According to their definitions, lamination parameters are related and cannot be arbitrarily selected. The region which defines physically meaningful lamination parameters is called the feasible lamination parameters region. In order to formulate a typical structural design problem using lamination parameters, the feasible domain for the particular set of lamination parameters is required. Such a domain is known to be convex and is available in closed form only for a limited number of combinations of the lamination parameters. In the present study, we exploit the convexity of these feasible regions and outline a numerical approach using successive convex hull approximations. In the successive convex hull approximation we refine the feasible region by increasing number of layers as well as number of possible fiber orientation angles. In this process, the volume of the feasible domain is monitored to measure the convergence of the approach. The final approximation is presented in the form of linear inequality constraints (hyperplanes) that can be included in typical structural optimization problems as side constraints. The major advantage of such a methodology is that this approximation needs to be done only once and the resulting set of constraints can be stored and reused for any other structural optimization problem. First, numerical results are compared for the cases where exact definitions exist for feasible domains. Next we apply the present approach to sets of lamination parameters for which exact definitions for feasible regions do not exist and compare our results with approximate results using other methods already available in the literature.

Generating Curvilinear Fiber Paths from Lamination Parameters Distribution

Contrary to the classical stacking sequence design of composite laminates with straight fibers, each ply can be designed with curvilinear fiber paths resulting in variation of the stiffness properties over the structure. Such laminates are often denoted as “variablestiffness panels” in the literature. Instead of treating fiber orientation angles as spatial design variables, lamination parameters can be used. However, retrieving the actual stacking sequence requires additional efforts at a post processing level. In this paper, we presume that the optimal distribution of lamination parameters is already obtained for a particular design problem. Then we use curve fitting techniques to obtain continuous fiber paths that result in a distribution of the lamination parameters close to the optimal distribution in a least square sense while satisfying the manufacturing curvature constraint. The fiber orientation angle is expanded using a set of basis functions and unknown coefficients. The unknown coefficients are then computed such t hat the assumed form represents the optimal distribution of the lamination parameters in a least square sense. The key feature of such approach is that at this post-processing step, no more expensive finite element analyses are needed and the curve fitting is performed using simple polynomial and trigonometric function evaluations. The curve fitting problem is then solved using a constrained nonlinear least square solver where maximum curvature is controlled using a side constraint. Numerical results demonstrate the efficiency of the proposed formulation for minimum compliance design problems. For the cantilever plate problem investigated, the compliance of the approximate design is only than 2.5% larger than than the compliance of optimal lamination parameters design. A methodology is also proposed to estimate the thickness buildup due to the curved fiber paths.

Stacking Sequence Dispersion and Tow-Placement for Improved Damage Tolerance

C. Lopes∗, Delft University of Technology, The Netherlands and Universidade do Porto, Portugal O. Seresta†, M. Abdalla‡, Z. Gurdal§ Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands B. Thuis ¶ National Aerospace Laboratory NLR, Voorsterweg 31, 8316 PR Marknesse, The Netherlands and P.P. Camanho‖ DEMEGI, Faculdade de Engenharia, Universidade do Porto, R. Dr. Roberto Frias, 4200-465 Porto, Portugal

A Genetic Algorithm Based Blending Scheme for Design of Multiple Composite Laminates

Genetic algorithms (GA) are widely used for stacking sequence design of composite laminates. Design of large composite structure is done by subdividing it into a number of local panel design problems. Independent design of local panels leads to stacking sequence mismatch (referred to as blending issue) between adjacent panels. Blending is important for both structural integrity and manufacturability of final designs. In this paper, we present a robust, simple, and efficient way to impose blending in a GA framework for multiple composite laminates design. A multi-chromosomal GA is proposed to implement a novel parametrization of composite laminate stacking sequence that seamlessly blends stacking sequence across multiple laminates. The advantages and the numerical issues of the present approach are addressed.

Retrieving Variable Stiffness Laminates from Lamination Parameters Distribution

Throughout literature variable sti!ness composite structures have been reported to perform significantly better than constant sti!ness baseline designs. In order to take full benefit of the advantages tailored composites o!er, several design and optimisation methods have been investigated. By using lamination parameters (LP) the number of design variables of such optimisation can be successfully reduced to a finite set of continuous, dimensionless variables. In order to construct a manufacturable laminate, the optimal LP distribution found for a structure must be converted into a stacking sequence distribution. Which requires a procedure that is able to retrieve a stacking sequence from a given set of LPs. This paper investigates stacking sequence retrieval from a given set of LPs, and the importance of the relation between in-plane and flexural LPs in this process. Based on this study a new analytical retrieval procedure is presented. The new procedure is compared to two existing procedures, and a method to incorporate manufacturing constraints using the new procedure is suggested. The ability of the proposed framework is demonstrated for a square plate, optimised for buckling performance using LPs. Not only is the new procedure found to perform better than existing methods, but it does appear possible to convert an LP distribution into a stacking sequence distribution satisfying production constraints with a negligible loss in performance.

Bending Test of a Variable-Stiffness Fiber-Reinforced Composite Cylinder

Two carbon-fiber-reinforced composite cylinders were tested in bending. One cylinder, the baseline cylinder, consisted of 0o, 90o and ±45o plies, whereas the other cylinder, called the variable-stiffness cylinder, contained plies with fiber orientations that varied in the circumferential direction, which caused a variation in laminate stiffness. The cylinders were optimized for maximum buckling load carrying capability under bending. Simulations showed that the variable-stiffness cylinder was able to redistribute the applied loads around the circumference, resulting in lower strain values at both the tension and the compression side of the cylinder and an improvement of the buckling load by 18 percent compared to the baseline cylinder. The purpose of the bending test was to show that the improvements obtained in the analytical results could also be achieved experimentally. The baseline cylinder was tested first to serve as a benchmark for the variable-stiffness cylinder. The finite element model was adjusted based on the baseline cylinder tests to represent the experimental conditions correctly. The model took into account the flexible connection between the cylinder and the test fixture, the test mechanism and geometric imperfections present in the cylinder and showed good agreement with the experimental results. The variable-stiffness cylinder was tested twice: first oriented in the direction it was designed for and later rotated 180 degrees about the cylinder axis, such that the loading direction on the cylinder was reversed. The predicted global response and strain distributions for both configurations corresponded well with the experimental data. The flexible boundary conditions and the geometric imperfections affected the load and strain distributions of the baseline and the variable-stiffness cylinders, but the relative improvements of the variablestiffness cylinder in the preferred orientation with respect to the baseline cylinder were not affected. Follow-up tests of the cylinders including cutouts or induced damage are planned in the future.

Design of anisotropic composite shells using an isogeometric approach

Thin-walled composite structures, typically modeled as Cosserat continua during the design phase, are of particular importance in aerospace and automotive applications. At the dawn of industrial scale adoption of advanced fibre placement technology, it became viable to better exploit the directional properties of composite materials. In the recent past, numerous researches were devoted to the design of shells with optimal anisotropy. In the present work, combined stiffness tailoring and shape optimal design is proposed that is naturally facilitated in a non-uniform rational B-spline based isogeometric approach. Spatial variation of stiffness properties is parameterised by means of lamination parameters and the thickness of the shell. Shape changes are easily achieved by modifying selected control point co-ordinates and weights. The method of successive approximations has been employed to solve the optimisation problem. The formulation is separable in terms of sizing variables, however, separability of the shape design problem is not enforced. The design framework is verified through selected compliance minimisation problems.

On the Effect of Geometric Nonlinearities on Static Load Alleviation

This paper investigates the importance of nonlinear aeroelastic static analysis in the structural design of high aspect ratio wings. As an indicator of the importance of static nonlinearity, the root bending moment is used. A relatively accurate prediction of the bending moment of a flexible wing is important in early design stages, since it largely determines the wing box weight. A linear Timoshenko beam element in a corotational framework is used as a nonlinear structural model. For the aerodynamics, Weissinger’s method is used. The nonlinearly coupled structural and aerodynamical equations are solved using the Newton-Raphson method. The load distribution is obtained for the aircraft trimmed at a range of load factors. Results for root bending moment, trimmed angle-ofattack, and wing tip deformation are presented for a high aspect ratio composite wing of a high performance open class sail plane. It is shown that the root bending moment obtained using nonlinear analysis diers significantly from the results of a linear analysis at high load factors. The reasons for the dierence are found to be (i) end shortening, (ii) stiening of the wing, (iii) change in trimmed angle-of-attack, and (iv) load redistribution. Therefore, it is concluded that nonlinear static aeroelastic eects are significant for the structural design of flexible high aspect ratio wings and ought to be included in the preliminary design stage.