STRUCTURAL COMPOSITE DESIGN: CONCEPTS AND CONSIDERATIONS
Nicolais weoc238.tex V1 - 09/06/2011 5:11 A.M.STRUCTURAL COMPOSITE DESIGN: CONCEPTSAND CONSIDERATIONSCAE of composite structures is described, and severalways to parameterize the laminates are explained. Theseparameterizations form the basis for the introduction offormulations for the structural composite optimal design,which is the main topic addressed here. Since the aeronautics industry has driven the innovation in structuralcomposite design, most of the concepts described here arebased on the developments carried out for large compositeaircrafts.MICHAL BRUYNEELSAMTECH Headquarters,Analysis and OptimizationGroups, Angleur, BelgiumCEZAR DIACONUSAMTECH Japan, Nagoya,Aichi, JapanTHE STRUCTURAL COMPOSITE DESIGN PROCESSGeneral Layout of Composite StructuresINTRODUCTIONA composite structure is made up of several plies of different orientations and shapes. The plies are stacked togetherand define zones. In each zone, a laminate with a givenstacking sequence (i.e., the order of the plies in the laminate) is obtained. An example is given in Fig. 1. In thiscase, the stiffeners and ribs of the wing naturally definethe zones of constant stacking sequence.The structures and materials considered in this articleare thin-walled structures made up of plies with continuous unidirectional fibers or woven fabrics, embedded ina polymer matrix. Nowadays, such composite materialsare used extensively in the primary structures of aircrafts [1] and their design for advanced applications isaccomplished using computers and numerical tools. Thistypically involves two disciplines. The first one, calledcomputer-aided design (CAD), aims to define the overallgeometry of the part and the zones of laminates with theirstacking sequence. It is linked to computer-aided manufacturing (CAM), which provides specific capabilities forthe manufacturing processes simulation. These tools areused to determine the accurate fiber orientations and thedeformation of the plies during draping. The second discipline, called computer-aided engineering (CAE), is used toanalyze the structural integrity of the composite structurewhen subjected to the expected loads. Since the mechanicalproperties of composite structures are highly dependenton the stacking sequence, it is important that the regionsof laminates with their detailed stacking sequence shouldbe determined in the analysis phase and further reportedby the designer in the CAD model for a draping simulation. In most cases, the finite element method (FEM)[2] is used, especially for complex geometries. Compositestructures exhibiting nonlinear material behaviors, largedisplacements, and instabilities under the in-service loadscan be analyzed nowadays. One particularity of structuralcomposite components, when compared with metallic components, is the large number of parameters needed todescribe their mechanical properties, for example, thedimension and the location of plies, their thickness, theirorientation, and the definition of the stacking sequences.In addition, a very large number of analysis results shouldbe considered, since failure indices are typically computedin each ply. As a consequence, the use of optimizationtechniques becomes essential in the design and analysis phases, especially if the fiber-reinforced materialsare to be tailored to the specific needs and the benefit of their anisotropy is to be maximized. This is theprice to be paid for the design of competitive compositestructures.This article presents the structural composite designprocess, including CAD–CAE–CAM capabilities. TheThe Design Phase: CAD and Link toward CAMThe design process uses these zones as a basis for the preliminary design of the composite part. This is a zone-baseddesign, in which the CAD software assigns a given number of laminates simply defined by the total number ofplies and their orientations (usually conventional angles:0 , 45 , 90 , and 45 ) in each zone. Typically, these laminates are the results of a CAE step. Specific CAD capabilities are then used in order to define more accuratelya first trial of the stacking sequences (order of the plies),while satisfying specific design rules and ply continuityconstraints across the zones. At this stage, it is possible toestimate the deviation of fiber orientations resulting fromthe draping or the misalignments of fibers. The ply-baseddesign is then generated, with its ply drops (i.e., the gradual thickness changes at the boundary of the laminates).A link toward CAM can be provided: in this case, a specific software is used to conduct a simulation of the plydeposition on a virtual machine (Fig. 2).The Analysis Phase: CAE ToolsThe structural analysis of complex composite parts is carried out with the finite elements method [2]. Only forsimple geometries and approximated boundary conditionsare analytical solutions possible. During the CAE phase,the design provided by the designer in the previous stepis validated and possibly modified by the analyst. Structural integrity is checked, and design improvements areprovided, the ultimate goal being to provide a correct (optimal) stacking sequence in each region of the structure.The methods used to estimate the integrity of the composite component are described in the section titled ‘‘TheStructural Composite Design Process.’’ Different ways toidentify the optimal stacking sequences are presented inthe section titled ‘‘The Analysis of Composite Structures:CAE.’’Wiley Encyclopedia of Composites, Second Edition. Edited by Luigi Nicolais and Assunta Borzacchiello. 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.1P. 1
Nicolais weoc238.tex V1 - 09/06/2011 5:11 A.M.2STRUCTURAL COMPOSITE DESIGN: CONCEPTS AND CONSIDERATIONSZone of given stacking sequenceZone with a smaller thicknesse.g. laminate with 6 pliese.g. laminate with 4 pliesA ply in the structure,defined across zonesFigure 1. A wing made of composite materials.Another ply, definedacross other zonesAnalystFigure 2. Illustration of the CAD and CAM capabilities for the design of a fuselage [3,4].SizinglaminateZone ssignmentsFigure 3. Illustration of the compositedesign process [3,5]: analysis (CAE) anddesign (CAD).The Complete Structural Composite Design Process:CAD–CAE–CAM InteractionsIn practice, CAD and CAE are alternatively used, asillustrated in Fig. 3. The analyst conducts structural analyses to check the structural integrity, while the designertranslates the modifications into an improved design,and determines the accurate fiber orientations, which areessential for the next CAE step. The final design is transferred to CAM, for the simulation of the manufacturingprocess and the generation of the numerical commandprograms needed to produce the part.THE ANALYSIS OF COMPOSITE STRUCTURES: CAEIntroductionComposite structures are laminated, thin-walled structures. These specific aspects must be considered in theDetailedanalysisRegion onsRegion laminatespecificationsUpdatesPer cificationanalysis. Since the structural composites are usually submitted to compression and shear, they must thereforewithstand buckling [6]. Moreover, specific failure modes,at the inter- and intralaminar levels, must be includedin the design to assess damage tolerance. In most aircraft applications, the structure is usually made of flator curved stiffened panels, divided into base cells calledsuper-stiffeners, which are composed of a portion of paneland the corresponding stiffener (Fig. 4).Analytical SolutionsOn the basis of the classical lamination and plates theories,it is possible to obtain analytical solutions for compositestructures with simple geometries, for static, dynamic,buckling, and failure analyses. Nonlinearities can also betaken into account to some extent. That is, the case forthe super-stiffeners in aircraft applications. It is thereforepossible to carry out local analyses on such representative isolated single structural elements to obtain resultsP. 2
Nicolais weoc238.tex V1 - 09/06/2011 5:11 A.M.STRUCTURAL COMPOSITE DESIGN: CONCEPTS AND CONSIDERATIONS3Figure 4. A stiffened composite panel made of superstiffeners.Local buckling modeGlobal buckling modeFigure 5. Buckling modes in a stiffened compositestructure obtained with FEM.quickly [7]. While such analyses are useful in the conceptual and preliminary design phases, such analyticalsolutions are, in most cases, approximated solutions withonly limited accuracy.Finite Element AnalysisQ1Linear Analysis. Using the FEM, a linear analysis yieldsestimates of the stiffness of the structure and the plyresistance, with classical first ply failure criteria such asTsai–Hill [8]. Buckling can also be studied enabling thebifurcation points, which are the intersections of different(stable or unstable) equilibrium paths [6], to be identified.Figure 5 presents the local and global buckling modesappearing in a curved composite structure made of sixsuper-stiffeners (hat stiffeners) submitted to shear andcompression. These structural responses can be computedat the component (local) level (Fig. 4) or at the wholestructural level (global level).Geometric Nonlinearities. The reliability of a linearbuckling analysis is questionable for structures capableof withstanding large displacements observed beyond abifurcation point in the postbuckling range or those assuming a limit point in the equilibrium path. To simulatesuch behaviors more realistically, a nonlinear analysis isneeded to identify the collapse (limit) load of the structure.Figure 6 illustrates the equilibrium path of the compositestructure shown in Fig. 5 when it sustains buckling andpostbuckling, up to the final collapse. The inclusion of geometrical nonlinearities in the analysis enables the designof lighter structures by allowing them to operate in thepostbuckling state [9].Material Nonlinearities and Damage Tolerance. Delamination and fiber and matrix breaks are typical compositefailure modes, which must be taken into account in thedesign of structural composites [1,10,11]. In most cases,simplified criteria are applied in the preliminary designphase to obtain a damage tolerant structure. Examples ofsuch criteria can be found in Refs 8 and 12.For more information on the FEM, and especially thediscretized equations of the physical phenomena describedabove, refer to Ref. 2.Integrated Analysis of Large Composite StructuresThe design of large stiffened composite structures usedin aeronautics is carried out by the combination of aglobal analysis on the whole structural model and localFigure 6. Equilibrium path of a composite structure sustaining buckling and postbuckling.P. 3
Nicolais weoc238.tex V1 - 09/06/2011 5:11 A.M.4STRUCTURAL COMPOSITE DESIGN: CONCEPTS AND CONSIDERATIONS1. Global model used tocompute the forces and fluxesPx3. Update the global modelwith new values of the localparameters2. Local model for the optimaldesign of super stiffenerNyNxyFigure 7. Global/local analysis of acomplex composite structure.Q2computations on the super-stiffeners. The forces actingon each super-stiffener are obtained from the global FEManalysis. They form input for the local analyses, whichare conducted as described previously. At this local level,the values of the parameters (e.g., ply thickness, fractionof 90 plies) can be modified in order to provide a safedesign. Since local modifications alter the global structuralbehavior, the global model is updated and a new globalanalysis is carried out, in order to re-evaluate the localforces modified by the new values of the local parameters(Fig. 7).NxPxThe Mathematical Optimization ProblemThe optimization problem is written in Equation (1). Itincludes one objective function g0 (x) to be minimized andm constraints gj (x). These functions depend on n designvariables x {xi i 1, . . . , n}, which are the parameterswhose values are varied to find an optimal solution. Atthe optimum, each constraint must be lower than or equalto a given value gj . Side constraints are added to theproblem, providing lower and upper limits on the designvariable values, representing physical or manufacturingrestrictions.min g0 (x) submitted to gj (x) gj , j 1, . . . , m andxTHE STRUCTURAL COMPOSITE OPTIMAL DESIGNIntroductionThe use of optimization algorithms is essential in orderto assign optimal values to the numerous parametersdefining the mechanical properties of structures made offiber-reinforced composite materials. These parametersinfluencing the composite design, called the design variables, can be the number of plies, their thickness and fiberorientation, the stacking sequence, and the shape andtopology of the structure. The structural responses of composites, for example, the buckling load, the strain energydensity, the ply strains, and failure indices, can typicallypresent highly nonlinear and nonmonotonous behaviorswith respect to the design variables. As a result, the optimization problem is extremely difficult to solve, since itis not convex and therefore characterized by several localoptima and many infeasible solutions. Moreover, when theoptimization of the stacking sequence is addressed, theproblem is combinatorial and specific solution proceduresare required. The optimal design of composite structureshas been studied for more than 30 years, and althoughseveral interesting solutions have been proposed, a general solution procedure still has not been obtained. Thekey point of these optimization procedures relies on howthe plies and laminates are parameterized. In this section,after a brief review on how a structural optimization problem is formulated, the main innovative concepts developedover the past years for the structural composite optimaldesign are described and discussed.x i xi xi , i 1, . . . n.(1)The functions gj in the problem (1) are generally nonlinear. They can be global and impact the whole structure,as is the case for the structural stiffness, the buckling loads, or the vibration frequencies. They can alsobe local, and so defined in each ply, examples of whichare the Tsai–Wu and Tsai–Hill criteria. The solution ofproblem (1) is obtained iteratively. At each iteration, astructural analysis is carried out, the results of which feedthe optimizer which provides new values for the designvariables. Several optimization methods exist and canbe used to solve the problem (1). Gradient-based methods, such as the mathematical programming methods[13] and the sequential convex programming methods [14]use the first-order derivatives of the structural functions.Zero-order methods, such as the genetic algorithms [15] orthe surrogate-based optimization methods [16], use onlythe function values. The latter requires a large numberof structural analyses but can be used directly when thegradients are not available.The Optimal Design of Structural Composites: Aspects andConsiderationsThe ultimate goal of composite design is to definezones of optimal stacking sequences in the structure(Fig. 1). This is clearly a complicated task, since this isa combinatorial problem including a very large numberof parameters and restrictions. In aircraft applications,conventional laminates made of stacking sequences of0 , 45 , 90 , and 45 fiber angles have been used. WithQ3P. 4
Nicolais weoc238.tex V1 - 09/06/2011 5:11 A.M.STRUCTURAL COMPOSITE DESIGN: CONCEPTS AND CONSIDERATIONS5Figure 8. Draping of six plies, with continuity constraints over the regions of different thickness.Figure 9. Summary of the methods usedfor the optimal design of structural composites.the improvement of manufacturing capabilities andespecially the developments of advanced fiber and tapeplacement machines, it is now possible to design structures with curved fibers. Allowing fiber orientations thatchange continuously over the structure with respect to thelocal internal forces improves the design, as reported inRef. 17. For conventional laminates, specific and complicated design rules related to mechanical or manufacturingrequirements should be taken into account. Some of themimpact the ply continuity of the draping, as illustratedin Fig. 8. Others stipulate that, for damage toleranceconsiderations, no more than four consecutive superposedplies should have the same orientation. Such constraintsare clearly difficult to take into account in the formulationof the optimization problem.Confronted with such difficulties, it is thus understandable that a global and general solution for theoptimal structural composite design is not yet available.An overview of the many approaches studied over the past30 years is shown in Fig. 9 and reviewed in the followingsections.Parameterizations of Composites in View of Their OptimalDesignDepending on the choice of the design variables in (1),different optimization problems can be addressed. Various parameterization formulations have been proposed forcomposite structures, each of them with their own advantages and disadvantages, as discussed in the followingsections. In order to introduce the notations, the constitutive relations in a ply with unidirectional fiber orientationand the classical lamination theory are briefly reviewed.Constitutive Relations for the Orthotropic Ply and theLaminate [8]. The constitutive relations of an orthotropiccomposite ply in its orthotropic axes are given in (2) for aplane stress state, where Ex and Ey are the Young modulusin the longitudinal and transverse directions, Gxy is theshear modulus, and νxy is the Poisson ratio: mExmνyx Ex0 σx εx σymEy0 εy mνxy Ey 00Gxyσxyγxy Qxx Qxy0 εx 10 εy Qyx Qyy.m 1 νxy νyxγxy00Qss(2)The constitutive relations of an orthotropic compositeply having an orientation θ with respect to the structural(or laminate) axes is given in (3), where the three indices1, 2, and 6 stand for the global longitudinal direction, thetransversal direction, and the shear direction, respectively(Fig. 10). Q11 (θ ) Q12 (θ ) Q16 (θ ) ε1 σ1 σ2 Q12 (θ ) Q22 (θ ) Q26 (θ ) ε2σ Q(θ )ε. σ6ε6Q16 (θ ) Q26 (θ ) Q66 (θ )(3)The stiffness coefficients Qij (θ ) in (3) are related tothe orthotropic coefficients Qxx , Qyy , Qxy , and Qss in (2) bytrigonometric functions either of the fourth power (e.g.,sin4 θ , cos4 θ , sin2 θ cos2 θ , sin3 θ cos θ , . . .) or depending onmultiples of the angle θ (sin 2θ , sin 4θ , cos 2θ , and cos 4θ ).In this case, it can be shown thatQ(θ ) γ0 γ1 cos 2θ γ2 cos 4θ γ3 sin 2θ γ4 sin 4θ ,(4)where the 3 3 matrices γ are functions of the orthotropiccoefficients Qxx , Qyy , Qxy , and QssIn the classical lamination theory, the constitutive relations of the laminate are expressed in (5), where A, B,and D are the in-plane (membrane), coupling, and bending stiffness matrices, N and M are the local forces andmoments by unit length, and ε0 and κ are the in-planeP. 5
Nicolais weoc238.tex V1 - 09/06/2011 5:11 A.M.6STRUCTURAL COMPOSITE DESIGN: CONCEPTS AND CONSIDERATIONSFibreyMaterial axes(orthotropy)2xpli 4x2xpli 33z,3θ4θθ311Structural axes(laminate)MatrixFigure 10. Laminate including four plies: structural axes (1,2,3) and material axes (x, y, z).3nhtkzkkh2Figure 11. Definition of the plies location throughthe laminate’s thickness.21strain and curvatures, respectively. The stiffness matricesare obtained by considering the superposition of n plieswith their own properties given by (3), using the notationsof Fig. 11. N1 N2 ½ 0 ¾¾ ·½ N6A BNε MB DMκ 1 M2 M6 0 A11 A12 A16 B11 B12 B16 ε10 A12 A22 A26 B12 B22 B26 ε 2 A16 A26 A66 B16 B26 B66 ε0 6. B11 B12 B16 D11 D12 D16 κ1 κ2 B12 B22 B26 D12 D22 D26 B16 B26 B66 D16 D26 D66κ6(5)Direct Parameterization. When the thickness tk and thefiber orientation θk of each ply k are defined to describethe laminate, the coefficients of the stiffness matrices in(5)
Integrated Analysis of Large Composite Structures The design of large stiffened composite structures used in aeronautics is carried out by the combination of a global analysis on the whole structural model and local Figure 6. Equilibrium path of a composite structure sustaining buckling and postbuckling.
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