Recentresearchonvariablestiﬀnesslaminateshaveshownbothnumericallyandexperimentallythat further improvement on the buckling behaviour is possible by incorporating overlaps that result in variable thickness proﬁles, with the thickness non-linearly coupled with the local steering angle. We present the concept of overlap-stiﬀened panels, developing a design method that allows for incorpo- rating higher-stiﬀness regions into individual plies of a variable-angle tow (VAT) laminate, taking advantage of the non-linear coupling between the tow steering angles and the local thickness. The proposed method naturally copes with minimum steering radius constraints of diﬀerent manufactur- ingprocesses,andthepresentstudyconsiderstwotowsteeringprocesses: automatedﬁberplacement (AFP)andcontinuoustowshearing(CTS).Theminimumradiusconstraintissatisﬁedbyconnecting twotransitionregionsofthicknessspeciﬁedoneachplybymeansofcircularﬁbertowarcs,ofwhich the radius of curvature always exceed the minimum manufacturing constraint. Each individual ply exploring the overlap-stiﬀened design is described using 5 design variables. Laminates made up of theseoverlap-stiﬀenedpliesareoptimizedforamaximumvolume-normalizedbucklingperformance underbi-axialcompression,measuredthroughFEM,byageneticalgorithmandbenchmarkedagainst astraightﬁberpaneloptimizedforthesameloadcase. TheconclusioncanbedrawnthatbothAFPand CTSoverlap-stiﬀenedVATpanelscanatleastachievethedoubleofthevolume-normalizedbuckling performance of an optimized straight ﬁber panel, demonstrating the potential of the proposed design method. 1. Introduction Novel automated manufacturing techniques have the ca- pability to steer the ﬁbers of each layer towards curvilin- earpaths,producingvaryingﬁberorientationthatreﬂectsin variablestiﬀness,whichultimatelyenablesahighertailoring potential of composite materials. This opens even more the development and application of advanced lightweight com- posite structural designs. For instance, substantial improve- ments in buckling loads were achieved by varying the lam- inate stiﬀness in the direction perpendicular to the applied load due to a redistribution of the primary loads from the center section of the panel to the simply supported sides of the panel [1, 2, 3]. The variable angle tows utilized in the present study can berealizedduringmanufacturingmainlybymeansofbend- ing and shearing [4]. When the tows are sheared, as in the continuous tow shearing process [5], the tow width mea- sured perpendicularly to the tow path changes according to wtowcos; wherewtowis the nominal tow thickness and  the tow steering angle. Note that when shearing occurs the tow thickness must change due to conservation of mass [6]. Variable angle tows can also be achieved by means of in- planebending[1,7],keepingthetowwidthconstantifmea- suredperpendicularlytothetowpath,whichcreatesresidual in-plane stresses on the tows to accommodate the variable- angle[4]. Ultimately,thisresidualin-planestresseswillde- terminetheminimumradiusofcurvatureachievableinVAT <Corresponding Authors r.ummels@student.tudelft.nl (R. Ummels); S.G.P.Castro@tudelft.nl (S.G.P. Castro) ORCID(s): 0000-0001-9711-0991 (S.G.P. Castro)designs [7]. In automated ﬁber placement (AFP), it is cus- tomary to avoid thickness variation during ﬁber steering by means of cut-and-restart [3]. In the present study, advanced compositedesignsareexploredbyallowingthicknessbuild- up created by the nonlinear steering-thickness coupling in- herent of the CTS process, and also achievable using an overlap-based design in the AFP process. An approximate relationship between the local thickness of an individual ply and the local ﬁber angle plies where tow overlaps are allowed, was derived by Castro et al. [6], whereby a ply can thus be designed for a speciﬁc thickness distribution, by tailoring the ﬁber directions within the ply to create that thickness distribution. Using this, stiﬀener- like structures could essentially be woven into the plies of a composite laminate structure during manufacturing, osten- sibly resulting in a higher buckling failure load of the ﬁnal product. The present study sets out to complete two main goals. First,developadesignmethodthatcanreliablygeneratethe aforementioned overlap-stiﬀened plies by taking advantage of the nonlinear steering-thickness coupling. These plies shouldbephysicallypossibletomanufacturewithoutirregu- larities,byhavingtheminimumradiusoftowcurvaturecon- strained according to the selected manufacturing process. Moreover, the design should preferably be described with as little variables as possible. Second,developamethodfortestingandoptimizingsuch overlap-stiﬀenedpliesforagivenstructuralperformance. In thepresentstudy,thelinearbucklingperformanceisinvesti- gatedforpanelsmanufacturedbymeansofAFPandaCTS, and compared with the performance of a well-known opti- R Ummels, SGP Castro, preprint submitted to Elsevier Page 1 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints mized straight-ﬁber laminate plate from Haftka [8]. By fulﬁlling these two goals, the authors can prove the concept for the new overlap-stiﬀened design method herein proposed,openingupawholenewworldofnewdesignpos- sibilities. 2. Proposed design concept The production of variable-angle tow (VAT) composite laminatedstructuresviaautomatedﬁberplacement(AFP)is achievedbymeansofrobotsthatarecapableoflayingtapes ortowsofpre-impregnated(prepreg)ﬁbersalongpredeﬁned computerizedpaths[2]. MoremodernAFPprocessesmake useofthermoplasticprepregs,wherebythetapesareheated byalasersource[7]. Bylayingthetowsnexttooneanother, full pliescovering the region ofinterest are created, andthe processisrepeateduntilthedesiredfullVATlaminateisde- posited. An arbitrary tow within a VAT ply is depicted in Figure 1, where represents the local ﬁber angle between the local tow path and the x-axis of the plate domain. xya b �+ TowTow Path Figure 1: Arbitrary tow path of one tow within a ply. Theta is counter clockwise positive, originating from the positive x axis. 2.1. Thickness and ﬁber angle distribution Uninterrupted curved tows cannot be laid neatly next to each other to produce a ply without creating gaps or over- laps [2, 3], or without using alternative production meth- ods of variable angle tow such as continuous tow shearing (CTS) [5]. In general, one would like to avoid irregularities like gaps or localized thickness increases due to overlaps. However,the thicknessincreasecreated byoverlapsinﬁber reinforced composite laminates can actually be used to sig- niﬁcantly increase the load carrying capacity and buckling performance [9]. Blometal. [3]derivedanapproximateformulaforarela- tionship between the width of a tow and the eﬀective width asafunctionofthelocaltowanglewasproposed. Basedon thisrelationship,Castroetal. [6]derivedEquation1,which gives an approximate relationship between the eﬀective lo- calplythickness, t.x;y/andthelocalﬁberangle, .x;y/within anoverlapply,basedonconservationofmass. Inthepresent study,tprepresentsthenominalthicknessofatowand t.x;y/the local varying thickness. t.x;y/ùtp cos..x;y//(1) BasedonEq. 1,notethatalocalﬁbersteeringangleof60 ° wouldthusresultinanapproximatedoublingoftheeﬀective local ply thickness. Theoriginofthiseﬀectivethicknessincreaseisillustrated in Figure 2 and Figure 3. In Figure 2 adjacent tows fol- low a path from 0 °to 70 °and back again from left to right. Moreandmoreregionsofdoubledandtripledplythickness created by tow overlaps crop up as the local ﬁber angle in- creases. When these localized overlaps are spread out, or smeared out, over the surrounding area, it creates the thick- nessdistributionshowninFigure3. Thissmearedoutthick- ness distribution was the basis for Equation 1. Lander and Castro [10, 11] compared the diﬀerences in modelling the exactthicknessdistributionvs. thesmearedthicknessdistri- bution, demonstrating the good correlation and equivalence in buckling response of the smeared approach compared to a discrete representation of the overlaps. Figure 2: The exact thickness distribution created by overlap- ping adjacent tows through curvature. [6] Figure 3: The approximate smeared out thickness distribution created by overlapping adjacent tows through curvature. [6] 2.2. Overlap-stiﬀened plies and laminates The relationship given in Equation 1 gives rise to the idea that one can design VAT overlap plies and laminates for a given thickness distribution by tailoring the ﬁber an- gle distribution to create the wanted thickness distribution. R Ummels, SGP Castro, preprint submitted to Elsevier Page 2 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints x b xsφswsθ0 θ1 t1 t0 xzStiffener Stiffener Centerline Ideal Tow Pathya Figure 4: The geometric description of an ideal stiﬀener within an overlap-stiﬀened ply. Stiﬀener-like thickness distributions could essentially be woven into the plies of the laminates during production. The geometry of an idealized arbitrary overlap-stiﬀener within a ply is pictured in Figure 4. The darker grey region istheregionofincreasedthickness,whichmimicsastiﬀener. The ﬁber angle is set to 1in the dark grey region, with the rest of the ply having a ﬁber angle of 0. If a tow laying machine were to follow these angles it would lay paths like theexampletowpathgiveninred. Thetowpathsfollowthe direction of the speciﬁed ﬁber angle distribution from edge to edge. Angles 1and0respectively translate to t1andt0 using Equation 1. Here 1>0, and thust1>t0. Thewidthofthestiﬀenerisgivenby ws. Theangleofthe stiﬀener with respect to the x-axis is deﬁned as s. Finally, the position of the stiﬀener along the horizontal axis of the plyisgivenby xs. Aplyhasalengthof ainx-direction,and a width ofbin the y-direction. The thickness of the ply is measured in z-direction. As a result, an idealized overlap-stiﬀened ply within an abyblaminate made with a tow thickness of tpcan be de- scribed by a vector, P, of only 5 variables, given in Equa- tion 2. Pi;n=[0i;n;1i;n;wsi;n; si;n;xsi;n] (2) Afulllaminateof npliescanthenbedescribedbyavector of plies, L, as per Equation 3. Li=[Pi;1;Pi;2;§;Pi;n] (3)The previously mentioned geometric variables have a number of constraints. First oﬀ, to ensure that plies can re- liablybeautomaticallygeneratedwithouthavingthechange ofthetowpathenteringthestiﬀenerandneverleavingagain, sis always set to be of opposite sign compared to the sign of0, and1. It is theoretically possible to have them be all the same sign in certain ranges of angle combinations, but the current constraint ensures physically possible angle distributionswithminimaladdedcomplexitytotheplygen- eration algorithm in all cases. The next constraint is given in Equation 4. A slarger than 45 °or smaller than -45 °combined with a 0 °ﬁber an- gle along the x-axis would result in the overlap eﬀect being ﬂipped,withthetowsnowactuallydivergingawayfromone another if they enter the stiﬀener, thus resulting in gaps in- stead of overlaps. ð sðg45° (4) Stiﬀenerswith-45 °< s<45 °canstillbecreatedthrough arotationtrickwiththeplygenerationalgorithm. If scor- respondstothisrangeitisﬁrstconvertedto s;ausingEqua- tion 5. s;a=90+ s sf0 s;a=*90+ s s>0(5) All of the other ply variables are fed into the ply gener- ation algorithm as normal together with s;a, but with the widthaandtheheight bvaluesswapped. Thisresultsinthe R Ummels, SGP Castro, preprint submitted to Elsevier Page 3 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints ab θ1,a φs, aθ0, a y xφs, ba bClockwise Rotation y xa) b) θ1,b θ0,b Figure 5: a) Starting orientation of a ply with -45 °< s< 45 °. Note that the width is denoted by b, and the height by a. b) Orientation of the ply with -45 °< s< 45 °after 90 °clockwise rotation to align with the other plies in the axblaminate. Note that the general ﬁber direction is in the y direction for such a ply, not the x direction. plyshowninFigure5–a),where 0;a=0and1;a=1. The localthicknessvaluesaresubsequentlyderivedfromtheval- ues of0;aand1;a. Iftheplyisnowrotatedby90 °intheclockwisedirection, movingfroma)tob)inFigure5, s;bwillequal s,andthe tow path will obey all intersection constraints. The transla- tion fromatobis given in Equation 6. b==90+ saf0 b==*90+ sa>0(6) It should be reiterated that the local thickness values are calculated with 0;aand1;anot0;band1;b. Furthermore, there are constraints on 0and1. For0, theconstraintisaccordingtoEquation7,whichwasrequired tolimittheamountofoverlapoutsideofthedesignatedstiﬀ- ener area;1is constrained by Equation 8, which was re- quiredtoavoidirregularitiescausedbyhavingtowsoverlaps ofmorethan3timesthetowthickness,astheoverlapheight will start to reduce the accuracy of Equation 1. In addition, the plate model also starts to become a poor approximation oftheshallowshellforsteeringanglesover60 °,asdiscussed by Groh and Weaver [12] . Additionally, 0is also constrained to be smaller than 1 inEquation9toconsideronlydesignsthatactuallyproduce a desired region with higher thickness. ð0ðf30° (7) ð1ðf60° (8)1>0 (9) Putting all of the previously mentioned constraints to- gether leads to the following design space for the assembly of the ply vectors: wsË.0;0:2/ [ m] xsË[0;a] [ m] sË[*90;90] [ °] if sË[0;90]: 1Ë[*60;0] [ °] if1>*30: 0Ë.1;0] [ °] else: 0Ë[*30;0] [ °] if sË[*90;0/: 1Ë[0;90] [ °] if1<30: 0Ë[0;1/ [ °] else: 0Ë[0;30] [ °] P=[0;1;ws; s;xs] 2.3. Minimum radius of curvature constraint The minimum radius of curvature that the tow paths can achieve is dictated by the manufacturing method. The main R Ummels, SGP Castro, preprint submitted to Elsevier Page 4 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints method considered in this paper is automated ﬁber place- ment(AFP),whichhasarelativelylargeminimumradiusof curvature requirement. This is because the ﬁbers are essen- tially bent into a curved shape, which causes buckling and wrinklingoftheﬁbresontheinsideofthecurvedtow,where compressive stresses develop. Asecondarymethodthatisconsiderediscontinuoustow shearing (CTS) [5]. This production method curves the ﬁbers through shear, avoiding residual stresses on the ﬁbres andthereforeavoidingtowwrinkling. Muchsmallerradiiof curvature can be achieved as a result. The stiﬀened ply as deﬁned in Figure 4 cannot be pro- duced with current AFP processes without compromising theﬁberssigniﬁcantly. Lookingattheidealizedstiﬀenedply ina)ofFigure6,onecanseethatattheedgesofthestiﬀener the tow path angle transfers instantaneously from 0to1 and vice versa. This turn eﬀectively has a radius of curva- tureof0mm. LaminatesproducedwithAFPtypicallyhave tohavepathswithaminimumradiusofcurvature, R0,ofat least 400 mm for state-of-the-art thermoplastic tows heated with a laser source [7] and 500 mm for thermoset tows [5]. FortheAFPdesign,theminimumradiusofcurvaturewill besetto450mmforthepurposesofthispaper. FortheCTS design, the minimum radius can be reduced to 50 mm [5]. xy θ0θ1Stiffener Ideal Tow Path xyx + R0Circular Arc Angle Transition Regionθ1 θ0a) b) Figure 6: a): Tow path of idealized stiﬀener thickness distribu- tion. b): Tow path of stiﬀener thickness distribution obeying the minimum radius of curvature manufacturing constraint. Makingsureallpossibleplygeometrieswhichcanbegen- eratedwithin thepreviouslymentioned designspacesatisfy theminimumradiusofcurvatureconstraintmightseemslike a very complex problem at ﬁrst. Fortunately, there is a sim- ple yet powerful solution which satisﬁes this constraint for all plies within the design space. The tow paths can be de- ﬁnedsuchthattheyfollowtheexactcirculararcswithradius R0required to go from 0to1when entering the stiﬀener, and to go from 1to0when exiting the stiﬀener on theother side. A tow path connected by these circular arcs is illustrated in b) of Figure 6. These circular arcs are straightforwardly implemented in theplygenerationalgorithmusedinthepresentstudy,which consists of three steps: Step 1: The idealized stiﬀener geometry is assigned on a 101 by 101 reference grid with width aand heightb. This isillustratedinFigure7a). Thegridusedintheimageis41 x 41 to make it easier to see individual grid spaces. A grid size of 101 by 101 ensures that the transitions are relatively smooth,andthattherearenogapsduetolargediscretization steps; Step 2: A circular arc grid stencil is generated based on the 0and1values of the current ply, and the size of the grid rectanglesinthereferencegrid. ThestencilusedinFigure7 can be found in Figure 8. The angle values the grid spaces in the stencil are assigned are the angles the ﬁbers need to follow the arc shape of the stencil at the location of those grid spaces.; Step 3: The algorithm cycles through the reference grid, identiﬁestheentryandexitedgesofthestiﬀener,andpastes the arc stencil along the edges of the stiﬀener in the cor- rectorientation,dependingonwhetheritsanentryoranexit edge. This results in the angle distribution grid seen in Fig- ure 7 b), which can be converted to a thickness distribution usingEquation1. Thesethreestepsareperformedforevery ply within a laminate. The idealized stiﬀener is now accompanied by an ’angle transitionregion’,wheretheanglevalueistransitioningfrom 0to1or vice versa. This in turn leads to a thickness tran- sitionregion,asthelocalﬁberangledictatesthelocalthick- ness. This can be seen in the three ply plots in Figure 9. 3. Fast buckling evaluation of variable stiﬀness plates The design concept herein proposed will be explored using a robust optimization scheme driven genetic algo- rithm, explained in the next section. Given the high number of individuals evaluated in the genetic algorithm scheme, a fast constraint evaluation is preferred. For lin- earbucklingofcompositeplates,fastanalytical[13,14,15]. Semi-analyticalmethodsareextremelyrecommendedwhen the structural matrices can be analytically integrated, be- cause these methods are able to approximate the contin- uum with less degrees-of-freedom when compared to other lower-order interpolation methods such as ﬁnite elements. However, when numerical integration is needed, the non- local support of the degrees-of-freedom involved in semi- analytical modelling requires integrands of the size of the entire structural matrices to be evaluated per integration point, making the integration process slower [16, 17]. Even tough the integration becomes slower, the reduced num- berofdegrees-of-freedomtodiscretizethecontinuummight compensate [10]. The complexity of the stiﬀness variation of the present designs motivated the authors to use an alternative method R Ummels, SGP Castro, preprint submitted to Elsevier Page 5 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints Exit edge Entry edge a) b) Figure 7: a) Initial idealized stiﬀener assigned on the reference grid. b) The stiﬀener after the algorithm has applied the cir- cular arc stencil to all edges. The reference grid is set to 41x41 to make the gridlines visible in this example. tosemi-analyticalapproaches,thatwouldallowahigh-order approximationofthecontinuumwhilekeepingthelocalsup- port of the degrees-of-freedom, as detailed next. 3.1. Bogner-Fox-Schmit ﬁnite element The Bogner-Fox-Schmit (BFS) ﬁnite element [18] is a classical C1 contiguous conﬁrming plate element known as oneofthemostaccuraterectangularﬁniteelementsforthin- walled shells, as stated by Zienkiewicz & Taylor [19, p. 153]. Recently, this element has been enriched by Castro andJansen[20]toallowforfastpost-bucklinganalysisusing the Koiter method. The ﬁeld approximation for the BFS is obtainedbytakingtensorproductsofcubicHermitesplines. Withonly4nodesperelement,thestandardBFSelementap- proximates the out-of-plane displacements using 3rd–order polynomials,whichisstillareasonablelow-orderinterpola- tionforplatesandverysimpletoimplement[21],incontrast with triangular elements which use higher order polynomi- /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000005b/uni00000003/uni00000056/uni00000053/uni00000044/uni00000046/uni00000048/uni00000056/uni00000013/uni00000015/uni00000017/uni00000019/uni0000001b/uni00000014/uni00000013/uni0000005c/uni00000003/uni00000056/uni00000053/uni00000044/uni00000046/uni00000048/uni00000056 /uni00000013/uni00000014/uni00000013/uni00000015/uni00000013/uni00000016/uni00000013/uni00000017/uni00000013/uni00000018/uni00000013/uni00000019/uni00000013 /uni0000002a/uni00000055/uni0000004c/uni00000047/uni00000003/uni00000036/uni00000053/uni00000052/uni00000057/uni00000003/uni00000029/uni0000004c/uni00000045/uni00000048/uni00000055/uni00000003/uni00000024/uni00000051/uni0000004a/uni0000004f/uni00000048/uni00000003/uni0000003e/uni00000047/uni00000048/uni0000004a/uni00000040Figure 8: Stencil used in Figure 7 to attach to the entry and exit edges to create the circular arc transition region. als[21],suchastheArgyriselement[22]. Thedisplacement ﬁeld is approximate as: u;v;w=³4 i=1SSSu;v;w iueueuei(10) whereueueueicontains the 6 degrees-of-freedom of the ith node. The shape functions SSSu;v;w iare deﬁned as: SSSu i=Li0 0 0 0 0 SSSv i=0Li0 0 0 0 SSSw i=0 0HiHx iHy iHxy i(11) withLi,Hi,Hx i,Hy i,Hxy icalculatedusingnaturalcoor- dinates [23, 24, 25]: Li=1 4.1+i/.1+i/ Hi=1 16.+i/2.i*2/.+i/2.i*2/ Hx i=*lx 32i.+i/2.i*1/.+i/2.i*2/ Hy i=*ly 32.+i/2.i*2/i.+i/2.i*1/ Hxy i=lxly 64i.+i/2.i*1/i.+i/2.i*1/(12) wherelx;lyare respectively the ﬁnite element dimen- sions along x;yThe values of i;igiven in Eq. 13 were R Ummels, SGP Castro, preprint submitted to Elsevier Page 6 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints a) b) c) Example Tow Path Figure 9: a): idealizedstiﬀenedplythicknessdistribution. Plot b): ply thickness distribution with transition regions due to radius of curvature constraint. c): ply angle distribution with transition region due to radius of curvature constraints. Tows follow the angle distribution, as illustrated with the example tow path. adopted for each of the four nodes. Node ii 1 *1 *1 2 +1 *1 3 +1 +1 4 *1 +1(13) Inthepresentstudyonlyrectangularelementswereused, such that the natural coordinates can be deﬁned simply as =2x_lx*1;=2y_ly*1. All required derivatives of SSSu;v;w ican then be calculated in terms of the natural coordi- nates using Eq. 14. All integrations over the ﬁnite element domains are performed numerically using standard Guass- quadrature and a mesh of 44integration points per ele- ment. ) )x=lx 2) ) ) )y=ly 2) )(14)3.2. Linear buckling analysis linear buckling behavior can be calculated applying the neutral equilibrium criterion [26]: 2=0 (15) whereis the total potential energy functional. Replac- ing the expressions for the elastic energy and external work into Eq. 15, the following generalized eigenvalue problem can be derived [26]: K0K0K0+KG0KG0KG0=0 (16) whereK0K0K0istheglobalconstitutivestiﬀnessmatrixofthe system,andKG0KG0KG0thegeometricorinitialstressstiﬀnessma- trix. Equation 17 gives the expression used to assemble K0K0K0 withthecontributionofeachplateﬁniteelement eofdimen- sions inxandydirection respectively given by lxandly. K0K0K0=kÉ e=1lxly 4ËBeBeBeTFFFBeBeBedd (17) InEq. 17,BeBeBeisdirectlyobtainedfromtheshapefunctions of Eq. 11 and the kinematic relations for plates, as given in Eq. 18. Since the BFS element has 6 degrees-of-freedom pernodeand4nodesperelement,theshapeof BeBeBeis246. BeBeBeT=4 )SuSuSu )x;)SvSvSv )y;)SuSuSu )y+)SvSvSv )x;*)2SwSwSw )x2;*)2SwSwSw )y2;*2)2SwSwSw )x)y5 (18) MatrixFFFcontainstheconstitutivestiﬀnessforaplateob- tained with classical laminated plate theory (CLPT): FFF=b f f f f f fdA11A12A16B11B12B16 A12A22A26B12B22B26 A16A26A66B16B26B66 B11B12B16D11D12D16 B12B22B26D12D22D26 B16B26B66D16D26D66c g g g g g ge(19) where the calculation of Aij;Bij;Dijcan be found, for instance, in Jones [27] or Kassapoglou [15]. KG0KG0KG0=kÉ e=1lxly 4Ëb f fd)SwSwSw )x )SwSwSw )yc g geT4NxxNxy NxyNyy5b f fd)SwSwSw )x )SwSwSw )yc g gedd (20) Note in Eq. 17 for K0K0K0thatFFFcontains the constitutive termsthatwillbeusedtorepresentthestiﬀnessvariationof theproposedplatedesigns. InEq. 20for KG0KG0KG0themembrane stress state given by Nxx,NyyandNxywill also depend on R Ummels, SGP Castro, preprint submitted to Elsevier Page 7 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints the local stiﬀness properties. Therefore, during the numeri- cal integration of K0K0K0andKG0KG0KG0one must know the local val- uesofFFF.;/andNxx.;/,Nyy.;/andNxy.;/,within eachﬁniteelement. Inthepresentstudy,thenumericalinte- grationisperformedusingameshof 44integrationpoints deﬁned with standard Guass-quadrature rules. 4. Optimization through a genetic algorithm Previous applications of genetic algorithms in the opti- mization of composites designs include laminate stacking sequence of several plates under buckling and strength con- straints[8]andstiﬀenedpanels[28]andwingboxstructures. Forthelatter,McIhaggaetal. [29]compareddiﬀerentsearch schemes against genetic algorithms. Wan et al. [30] per- formed aeroelastic tailoring to minimize the wingbox skin weight. Arizono et al. [31] used a GA to ﬁnd the mini- mum weight of a supersonic wingbox structure using static strength, local buckling, and minimum ﬂutter speed as con- straints. Castro et al. investigated: fast GA schemes with progressive mesh reﬁnement to allow for more exploration of the design space [32]; the eﬀect of GA parameters [33]; comparisonwithgradient-basedmethods[34];anddiﬀerent parameterization schemes [35]. Jin et al. [36] used parallel GAforlargewingboxoptimizations. Morerecently,Liguori et al. [37] applied GA in the optimization of a variable- angle tow wingbox construction considering post-buckling constraints. The optimization of the panels herein addressed is a sig- niﬁcant nonlinear problem made exponentially more com- plexbytheamountofvariablesthatbecomeinvolvedasthe ply count increases. However, the vector descriptions for plies and laminates, given in Equation 2 and Equation 3 re- spectively, led to a representation that could well be used as the genetic data in a genetic algorithm. The present ge- neticalgorithmwaswrittenusingtheDistributedEvolution- ary Algorithms in Python (DEAP) module [38]. The work- ings of the algorithm are illustrated in Figure 10. It should be noted that the previously mentioned con- straints for the variables describing a ply are maintained throughout all steps of the process. Randomization, muta- tion and crossover of individual ply variables thus become all constrained operations. The same goes for randomiza- tion of full plies, or full laminates. Some special operations are used during mutation and mating. Two types of crossover and three types of muta- tionareutilizedtotryandmaximizetheimprovementofthe genetic data from generation to generation. Formatingthemoreconventionalvariablecrossoverfrom one ply to another is described in Equation 21. A random variable in one ply in a laminate is replaced by that samevariable from a ply in another laminate. Pi;n=[0i;n;1i;n;wsi;n; si;n;xsi;n] + Pj;m=[0j;m;1j;m;wsj;m; sj;m;xsj;m] Úvariable crossover Pi;n=[0i;n;1i;n;wsj;m; si;n;xsi;n](21) Lessconventionally,afullplycrossoverofoneplyfroma laminate, fully replacing a ply in another laminate, can also be performed, as per Equation 22. Li=[Pi;1;Pi;2;Pi;3;Pi;4;Pi;5] + Lj=[Pj;1;Pj;2;Pj;3;Pj;4;Pj;5] Úply crossover Li=[Pi;1;Pi;2;Pj;5;Pi;4;Pi;5](22) Variableandplymutationworkthesameastheydoforply crossover, but the single variable/ply is instead replaced by a randomly generated variable/ply. Finally, full randomiza- tionrandomlygeneratesanentirelynewlaminatetoreplace an old laminate. This operation is introduced to ensure a steadystreamofgeneticdatafromthroughouttheentirede- sign space. The algorithm starts out with a randomly generated ini- tial population of nindividuals. These get tested for their bucklingstrengthandrankedinperformanceaccordingtoa ﬁtness equation. The best 0:5nlaminates immediately con- tinue to the next generation. The genetic data of the entire populationissubsequentlymixedthroughmatingandmuta- tion to create 0:5nnew laminates. These new laminates get evaluated for buckling performance, ranked, and the cycle continues. Aftermgenerations laminates remaining should in the- oryshowsigniﬁcantlybetterbucklingperformancethanthe laminates of the initial random population. The best panel in the ﬁnal generation is output as the optimized panel. 5. Benchmarking against Existing Literature Toprovethatthemethodologydescribedaboveiscapable ofgeneratingbetterperforminglaminatesthanconventional straight ﬁber laminates, a benchmark optimization problem is set up. 5.1. Benchmark Case Setup ThebenchmarkcaseistakenfromtheresultsofLeRiche and Haftka [8], where straight ﬁber laminates were opti- mizedforbucklingperformanceunderbi-axialcompression through a genetic algorithm. The stacking sequence of the optimal64plylaminateusedisasfollows: ( 9010,,452,902, ,453,902,,454)s. R Ummels, SGP Castro, preprint submitted to Elsevier Page 8 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints Random initial population of n laminatesFEM buckling evaluation0.5n laminates 0.5n best laminates 0.5n worst laminates Discarded0.25n Mutation 0.25n MatePly mutation V ariable mutation Full randomization Ply crossover0.5n laminates n laminates V ariable crossoverm generations Optimized panelAfter m generations Figure 10: Flow chart of the genetic algorithm. Thegeometryofthelaminateisasimpleplate,ofwhich a = 0.508 m and b= 0.254 m. The bi-axial load case is illus- trated in Figure 11. The load factor, (N/m), is applied to boththexandydirections[8]. Theplateissimplysupported on all sides. The material properties are those of graphite-epoxy car- bonreinforcedpolymer. With E1=127.59GPa, E2=13.03 GPa,12= 0.3,G12= 6.41 GPa. The tow thickness is tp= 0.127 mm. In the BFS ﬁnite element used in the present study, the achieved by the reference panel is 6:96105N/m. The load factor achieved in [8], converted from imperial units, was6:96105N/m, which shows correspondence to the method employed in [8] to within a 1% error, verifying the BFS ﬁnite element herein implemented for calculating the buckling constraint. The performance comparison against the benchmark is determined by dividing its buckling failure load, , by the total panel volume. The overlaps of the VAT stiﬀened panel add volume, and thus weight. Therefore, the volume- normalized buckling load becomes a fairer way to com- paretheoptimalstraight-ﬁberplateagainsttheVAToverlap- stiﬀenedpanelswithregardstotheirbucklingperformance. This comparison serves as a proof of concept. Once it is proven that the VAT overlap-stiﬀened panel can provide a better volume-normalized buckling performance than a straight ﬁber panel, there is a good probability that one is able to design a VAT overlap-stiﬀened panel to be lighter than a straight ﬁber panel, while constrained by a given de- sign load. 6. Overlap-stiﬀened panel optimizations The genetic algorithm is set to the same geometric and materialvaluesasthoseofthereferencepanel. Additionally, theminimumradiusofcurvature R0issetto450mmforthe AFP panel and to 50 mm for the CTS panel. The plies within the 64 ply laminates are generated and optimized with the genetic algorithm as described in the methodology. The population size nis set to 100, and the numberofgenerations missetto150. Withthesesettingsthe improvement of the laminates generally stagnates at around the 140th generation in test runs Fromgenerationtogenerationthebetterlaminatesarese- xy λλ λ λbaFigure 11: Bi-axial load case used to determine the buckling strength of the laminates. [8] lected according to the ﬁtness function in Equation 23. In this way, the genetic algorithm works towards optimizing the overlap-stiﬀened panels aiming at a maximum volume- normalized buckling load. max. panel volume/ (23) The way the design space is set up and constrained is ex- plainedinthemethodology. Intheplygenerationalgorithm all angle values 0,1, andsare set to only take on inte- gerdegreevalueswithinthegivenconstraints. Thevaluesof wsandxscanonlybeoneoutof20and100discretizedval- ues,respectively. Evensmallerlevelsofdiscretizationwould most likely not make a meaningful diﬀerence in the achiev- ablelevelofbucklingload,whilesigniﬁcantlyincreasingthe size of the design space. Taking into account all aforementioned ply generation constraints,thepresentoptimizationproblemleadstoanap- proximatedesignspacehaving 8 1010possible64plylam- inate individuals, with there being well over 1 109diﬀer- ent individual ply combinations. The unconstrained design spaceiscomprisedofapproximately 1:6 1011diﬀerentlam- inates. The relatively simple parameterization of the laminates andplieshereinproposedservestonarrowthedesignspace R Ummels, SGP Castro, preprint submitted to Elsevier Page 9 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints signiﬁcantly,asitcouldbemanyordersofmagnitudelarger with a more complex parameterization method. Withn= 150 andm= 100, and making the approx- imation that every mutation or crossover will result in a unique untested laminate, the algorithm will at most test 7600 unique laminates per run. As a result, the algorithm is almost guaranteed to end up in a highly local optimum. Amoresophisticatedalgorithmwillberequiredtoapproach global optima, or to be able to cycle through a larger part of the design space in a single run. However, for the pur- posesofthepresentstudy,thecurrentalgorithmissuﬃcient toshowaproofviabilityoftheVAToverlap-stiﬀeneddesign method. 7. Results The results of the reference panel, the optimized AFP panel,andtheoptimizedCSTpanelaredisplayedinTable1. The AFP panel resulted in a 154% improvement in the volume-normalized buckling load over the reference panel, followed by the CST panel with 111% improvement. These resultsarenotunexpected. ThereferencepanelandtheAFP and CTS panels all have the same amount of layers, with theAFPandCTSpanelsbothbeingabletoachieveahigher thickness due to the overlapping. The increase in thickness willgenerallyalwaysresultinapanelwithabettervolume- normalized buckling load. The ﬁnal thickness distributions of the AFP and the CTS panelareillustratedinFigure12andFigure13,respectively; where both show a relatively similar trend, with the thick- ness increasing towards the center of the panel. The AFP panelachievesahighermaximumandaverageoverallthick- nesscomparedtotheCTSpanel. Thediﬀerenceinthickness distributions is likely a direct result of the way in which the minimumradiusofcurvaturesettinginteractswiththethick- nessdistributionoftheplies. TheAFPpanel’s R0isalmost as big as the ply, meaning that the thickness transition re- gions accompanying the ply stiﬀeners will usually cause an increase in thickness over a majority of the ply area. The CTS panel, on the other hand, has a relatively small R0setting, thus creating small thickness transition regions. The thickness increase will, as a result, remain more local- ized to the actual idealized stiﬀener location, resulting in plies with a thinner average thickness. Consequently, the AFP panel result is likely to be thicker than the CTS panel result, as conﬁrmed by the present re- sults. Finally, it should be noted that the present results should not be taken as the maximum volume-normalized buckling values that can be achieved by the present design parame- terization, considering that the genetic algorithm does not guaranteeaglobaloptimum. Thereadershouldinterpretthe values in Table 1 as the best results of multiple runs of the genetic algorithm for each panel design.8. Conclusions Variable-angle tow overlap-stiﬀened panels were pre- sented as a new design possibility taking advantage of tow overlaps in automated ﬁber placement (AFP) and inher- ent steering-thickness variation in continuous tow shear- ing (CTS). In the design and optimization examples herein demonstrated, manufacturability constraints were included and a design parameterization using a reduced number of variableswasproposed,wherebyonly5variablesareneces- sary per ply containing one straight overlap-stiﬀener. A method for testing and optimizing such overlap- stiﬀened panels for buckling performance was developed, andsubsequentlyappliedtotheoptimizationofanAFPand a CTS overlap-stiﬀened panel, aiming at a maximized spe- ciﬁc buckling performance. The new overlap-stiﬀened de- signoutperformedthestraight-ﬁberdesignforthestudycase herein investigated. The minimum radius of curvature manufacturing con- straint was successfully satisﬁed in the ply generation algo- rithm by connecting two diﬀerent assigned thicknesses by circular arcs, of which the radius of curvature will always exceed the minimum allowable radius of curvature. A genetic algorithm was written for optimization of the volume-normalized buckling loads of both AFP and CTS panels simulating overlap-stiﬀened 64-ply designs. These panelswerebenchmarkedagainstastraightdesignalsowith 64plies,optimizedforbiaxiallycompressiveloadingcondi- tions. The AFP and CTS panels are capable of outperform- ing the reference panel on the volume-normalized buckling loadbyatleast154%and111%,respectively. TheAFPpanel was60%heavierthanthereferencepanel,andtheCTSpanel was 50% heavier. In conclusion, a solid foundation was laid out for the de- signoftheoverlap-stiﬀenedplies,withthemethodshowing a large potential in being able to improve the buckling per- formance of composite panels. 8.1. Future Studies Nowthatthedesignofoverlap-stiﬀenedpanelshavebeen shown their potential for maximizing volume-normalized buckling loads, a logical next step is to modify the opti- mization to design a lighter panel for a given design load. Thiscouldbeachievedbymakingthenumberofpliesinthe overlap-stiﬀened laminate variable. Additionally, the method could be modiﬁed with count- lessdiﬀerentdesignparametersthatwouldsuitdiﬀerentge- ometriesanddesireddesignassumptions. Multiplestiﬀeners perply,curvedstiﬀeners,amixofnormalstraight-ﬁberplies and overlap-stiﬀened plies, stiﬀeners which vary in thick- nessalongtheirlength,andincorporatingthesamestiﬀener in every single ply to create one big laminate stiﬀener; are all examples of possible designs to be explored. Acknowledgements The authors thank the Delft University of Technology AerospaceEngineeringHonoursProgramBachelorforpro- R Ummels, SGP Castro, preprint submitted to Elsevier Page 10 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000018 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 /uni00000013/uni00000011/uni00000013/uni00000014/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000013/uni00000018/uni00000013/uni00000011/uni00000013/uni00000014/uni00000014/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000014/uni00000018/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000018/uni00000013/uni00000011/uni00000013/uni00000014/uni00000016/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000016/uni00000018/uni00000013/uni00000011/uni00000013/uni00000014/uni00000017/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000017/uni00000018 /uni0000002f/uni00000052/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000033/uni0000004f/uni0000005c/uni00000003/uni00000037/uni0000004b/uni0000004c/uni00000046/uni0000004e/uni00000051/uni00000048/uni00000056/uni00000056/uni00000003/uni0000003e/uni00000050/uni00000040 Figure 12: Final thickness distribution of the optimized VAT overlap-stiﬀened panel using AFP with R0= 450 mm. /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000018 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 /uni00000013/uni00000011/uni00000013/uni00000013/uni0000001c/uni00000019/uni00000013/uni00000011/uni00000013/uni00000014/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000013/uni00000017/uni00000013/uni00000011/uni00000013/uni00000014/uni00000013/uni0000001b/uni00000013/uni00000011/uni00000013/uni00000014/uni00000014/uni00000015/uni00000013/uni00000011/uni00000013/uni00000014/uni00000014/uni00000019/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni00000017/uni00000013/uni00000011/uni00000013/uni00000014/uni00000015/uni0000001b/uni00000013/uni00000011/uni00000013/uni00000014/uni00000016/uni00000015 /uni0000002f/uni00000052/uni00000046/uni00000044/uni0000004f/uni00000003/uni00000033/uni0000004f/uni0000005c/uni00000003/uni00000037/uni0000004b/uni0000004c/uni00000046/uni0000004e/uni00000051/uni00000048/uni00000056/uni00000056/uni00000003/uni0000003e/uni00000050/uni00000040 Figure 13: Final thickness distribution of the optimized VAT overlap-stiﬀened panel using CTS with R0= 50 mm. viding the opportunity and the support for them to work to- gether on, and complete this piece of research.CRediT authorship contribution statement Raphael Ummels: Methodology, formal analysis, inves- tigation, validation, software, writing - original draft prepa- ration, writing - editing. Saullo G. P. Castro: Conceptual- Table 1 Final result comparison of buckling performance of the reference panel and the VAT stiﬀened panels. Panel [N_m]volume[m3]volume-normalized [N_m3]Improvement over reference panel [%] Reference Panel [8] 6.96e5 1.01e-3 6.63e8 0 AFP Panel 23.5e5 1.63e-3 16.9e8 154 CST Panel 21.4e5 1.53e-3 14.0e8 111 R Ummels, SGP Castro, preprint submitted to Elsevier Page 11 of 13 Overlap-Stiﬀened Panels for Optimized Buckling Performance Under Minimum Steering Radius Constraints ization,investigation,supervision,software,writing-origi- nal draft preparation, writing - editing.