A PSEUDO STRENGTH FUNCTION FOR THE GENERATION OF LOCAL LAMINATE REINFORCEMENT DOUBLERS

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1 18 H INERNAIONAL CONFERENCE ON COMPOSIE MAERIALS A PSEUDO SRENGH FUNCION FOR HE GENERAION OF LOCAL LAMINAE REINFORCEMEN DOUBLERS B. Schläpfer 1*, G. Kress 1 1 Centre of Structure echnologes, EH Zurch, Zurch, Swtzerland * Correspondng author (bschlaepfer@mavt.ethz.ch) Keywords: Local Renforcement, Pseudo Strength Functon, Senstvty Analyss, Structural Optmzaton 1 Introducton Due to ther superor mass-specfc mechancal propertes, lamnated composte structures are today well establshed for arcraft and spacecraft applcatons. he possblty to talor the propertes of lamnates to local requrements at dfferent regons of the structure s an addtonal advantage. he large number of desgn varables as well as the ansotropc behavor of the layers result n a complex and tme-consumng desgn process. he usage of an automated desgn procedure ncludng an optmzaton can reduce these costs and lead to better solutons. Stress concentratons caused by cut-outs or load ntroducton ponts may reduce the strength of a structure. hese areas can be renforced wth local lamnate doublers. Fndng effcent doubler geometres s not trval and hard to be done ntutvely. Automated desgn processes can help mprovng solutons qualty and reduce tme-costs. In contrast to egenfrequences, bucklng or complance, strength s a local phenomenon. he problem of fndng optmal, homogeneous lamnates for a gven loadcase has been consdered by many researchers [1-7] (and references theren). Consderng non homogeneous stress states, the strength of the structure s restrcted by the maxmum stresses. Allowng locally varyng lamnates, a desgn change may cause a sudden relocaton of the crtcally stressed regon what leads to a non-dfferentable objectve functon. Gradentbased optmzaton algorthms cannot be appled and stochastc algorthms are often used nstead [8-10]. Hansel and Becker [11] address the problem of fndng weght-mnmal solutons consderng strength constrants by takng advantage of local renforcement doublers. Materal s removed layerwse dependng on the prncpal stresses and the falure ndces. he heurstc approach s later enhanced wth a genetc algorthm [12]. he generaton of renforcement doublers for lamnated plates wth holes usng ths method s presented n [13]. he parameterzaton scheme ensures that the obtaned desgns are adapted for producton. In ths paper, a dfferentable pseudo objectve functon s ntroduced that untes the falure ndces of the structure. hs enables the determnaton of a gradent feld that expresses the nfluence of a layer thckness change on the strength. It s later used to generate renforcement doublers startng from an unrenforced ntal desgn. Materal s added stepwse based on gradent nformaton untl a requred strength s reached. Snce weght s lnearly dependng on the layer thcknesses, the placement of the renforcements s very effcent n terms of mass. 2 Pseudo Strength Functon he proposed pseudo strength functon s based on the well-known sa-hll crteron [14,15] for frst ply falure n lamnated composte structures. he Falure Index FI can be determned wth equaton (1), where σ 1, σ 2 and τ 12 stand for the plane-stress components n materal prncpal coordnates and X, Y and S are the correspondng strength values. Havng dfferent strength values for tenson and compresson, the choce of X, Y and S s dependng on the stress state FI 2 (1) X X Y S By rearrangng the stress components nto a stress vector, the equaton can be transferred to matrx notaton (Eq. (2)).

2 2 2 1 ( X) ( 2 X) FI 2 ( 2 X ) ( Y) 0 2 (2) ( S) 12 hs equaton agan can also be wrtten n a short form. FI S (3) 12 H 12 he strength matrx S H dffers for the sngle layers but not for the elements contanng a specfc layer. Equaton (3) can also be expressed wth element strans n global coordnates nstead of the prncpal stress components. hs enables an effcent evaluaton snce global element strans arse drectly from the dsplacement vector obtaned n the fnte element analyss. For that, strans are mapped to stresses wth the materal stffness matrx Q whch s also unque for a sngle layer. For smplcty, only plane stresses are consdered so that Q becomes the membrane stffness matrx. 12 Q ε 12 (4) Subsequently, the element strans n global coordnates are rotated to local coordnates by an angle φ wth a transformaton matrx and a so called Reuter-matrx R [16]. wth and ε RR ε (5) 1 12 xy 2 2 cos sn 2 sn cos 2 2 sn cos 2 sn cos (6) 2 2 sn cos sncos cos sn R Combnng equaton (4) and (5) leads to. (7) σ12 QRRε xy. (8) Equaton (8) can then be used to transform the stress-based strength matrx to a stran-based strength matrx applyng a both-sded multplcaton. 1 1 S H QRR S H QRR (9) hus the Falure Indces FI can be expressed by FI xys H xy. (10) Also the transformed strength matrx has to be determned only once for every sngle layer snce Q, R and are element ndependent and thus unque for a sngle layer. Based on the stran-based matrx formulaton of the falure ndces FI, a pseudo stran functon s defned. It s nspred by the optmalty crteron that a structure has an optmal materal dstrbuton when the stress dstrbuton s homogeneous. Kress [17,18] takes the stress standard devaton as objectve functon to fnd homogeneous desgns consderng dfferent sde constrants. Here, the objectve functon s formulated by summarzng the squares of the falure ndces of every ply of every fnte element, also called element-layers, subtracted by a gven value β. nel f 2 H S H (11) 1 he mnus sgn n front converts the mnmzaton problem to a maxmzaton problem, so that areas where the structure should be renforced show hgh gradent values. hs functon becomes maxmal (or equal to 0), f all falure ndces match the requred value β. he choce of β has to be done manually by the desgner. Choosng a value of 1 mples that the strength of the optmal desgn s crtcal n every part of the structure. o obtan non-crtcal desgns, β should be set to a low value (e.g. equal zero). It s clear that the optmal value of the functon may never be reached n a real optmzaton process. Moreover, t s not guaranteed that all falure ndces fall below the crtcal threshold of 1. However, ths formulaton unfes the strength whch s, as mentoned before, a local phenomenon, nto a global expresson that can be used as an objectve functon for optmzaton. Referrng to equaton (10), the pseudo stran functon can be expressed n terms of element strans. nel f 2 H S H (12) 1 o be able to perform an optmzaton of the lamnate, senstvtes wth respect to the thcknesses of the element-layers wll be determned. hcknesses are chosen as desgn parameters to have

3 A PSEUDO SRENGH FUNION FOR HE GENERAION OF LOCAL LAMINAE REINFORCEMEN DOUBLERS the possblty of fndng solutons wth a good strength-to-mass rato. Generally, also orentaton angles could be taken as desgn parameters. Dong that, only mass neutral solutons can be found. Addtonally, desgns may have orentaton angle dstrbutons that cannot be manufactured. Consderng equaton (12), only element strans are senstve to a thckness change. Applyng the chan rule, the dervatves form to nel f H 2 SH 2xy, SH (12) t 1 t he element strans ε xy are the product of the spacal dervatves of the shape functons B and the nodal dsplacements u. Bu (13) B contans the dervatves of the element shape functon Φ of the th element wth respect to the reference coordnates and s not dependng on the layer thcknesses t. he element stran dervatves can be determned wth equaton (14). t B u t (14) he dsplacement dervatves agan can be determned usng the senstvty equaton (15). u 1 r K K u (15) t t t here, K stands for the stffness matrx and r for the load vector. he evaluaton of the senstvtes can be done effcently snce they are analytcally derved. he numercal costs arse from the dsplacement senstvtes that are necessary for the calculaton of the stran dervatves. he embraced term of equaton (15) s smplfed by assumng that loads are not dependng on the desgn. he determnaton of the stffness matrx dervatves s straght forward consderng that the stffness of an element s only senstve to a change of t s own thckness. hus, t can be done on an element level snce the contrbutons of all elements except the currently consdered one are zero. he dsplacement dervatves are then the result of a lnear system of equatons wth multple rght hand sdes. It s a matrx wth the sze of number of degrees of freedom tmes the number of element-layers. However, computatonal capactes of today s workstatons are able to handle ths amount of data f the sze of the model s not too bg. Usng the senstvtes n an teratve optmzaton, the process can be accelerated by assumng that the changes of the dsplacement dervatves durng the optmzaton process are small enough to be neglected. hus, equaton (15) s only evaluated once at the begnnng of the process. he senstvtes of the pseudo stran functon gve an assumpton how and where the current desgn can be renforced most effcently n terms of addtonal mass. hey consder the effect of changng the thckness of a layer n one element to the falure ndces of all parts of the structure. hus, a possble strength ncrease n one element by changng the thckness of a layer n another element can be localzed. he strength problem wth local characterstcs becomes a global problem. An analogous pseudo strength functon can be formulated for the sa-wu [19] crteron by applyng slght modfcatons to the strength matrx. 3 Renforcement Patch Generaton Process he obtaned senstvtes can be used for an teratve generaton of local lamnate renforcement doublers. Senstvtes can also be calculated for layers wth predefned materal and orentaton but wth a thckness specfed to zero, also called Ghost Layers [20]. Addng an arbtrary number of Ghost Layers at the nterfaces of the lamnate stack does not affect the structural response of the desgn. However, an nfntesmal thckness change of these vrtual layers would change the behavour so that senstvtes dffer from zero. For the generaton of the lamnate renforcement doublers, a predefned set of Ghost Layers s added to the exstng lamnate. Subsequently, a real thckness s assgned to the layers havng the hghest senstvty values. he renforcement layers start to grow on the structure applyng the evaluaton of the senstvtes and the thckenng n an teratve procedure. he scheme of the patch generaton process s llustrated n Fgure 1. hs optmzaton process does not guarantee that the obtaned solutons are globally optmal. Moreover, the probablty that the process steps nto a local optma s hgh. However, the process s fast and t s shown n Secton 4 (and t has already been 3

4 shown n [20]) that the renforced desgns have superor propertes compared to the ntal desgns. Intal Layup FE Geometry,BC,Loads Read Generate Element Matrces Memory Update Modfed Element Matrces double lamnate layup of the basc plate ( (0,45,- 45) 2S ). For tme-savngs, only a quarter of the model wll be consdered. he boundary condtons have to be modfed n order to have an equvalent load case. Dsplacements n x-drecton on the left cuttng edge as well as n y-drecton on the lower cuttng edge are locked. Addtonally, out of plane rotatonal degrees of freedom on both cuttng edges are locked. he fnte element model s llustrated n Fgure 3. Assemble Global Matrces Adjust hcknesses FE Analyss ermnaton yes Optmzed Layup no Gradent Feld Senstvty Analyss Fg. 1: Patch generaton process scheme 4 Numercal Experment he potental of the patch generaton process takng advantage of the pseudo stran functon and the resultng gradents s llustrated on the common example of a unaxal tenson specmen made of carbon renforced plastcs wth a centered hole. he rectangular plate has an extenson of 200mm tmes 50mm and the hole has a dameter of 20mm. he plate s made of unaxal renforced carbon/epoxy ples (see able 1) wth a layup of (0,45,-45) S. Each ply has a thckness of 0.25mm what results n a total thckness of 1.5mm. Fg. 3: Fnte element model (quarter model) o predefne the materal and the orentaton angles of the renforcement doublers, three Ghost Layers wth a layup of 0, 45 and -45 degrees are stacked to each sde of the lamnate. Applyng the teratve process proposed n Secton 3, the renforcement doublers are generated. he process wll be stopped as soon as the maxmal falure ndces drop below the crtcal value of one. Consequently, the goal of the optmzaton s to reach the same strength, as a desgn wth double layup, wth a sgnfcant lower mass Fg. 2: Geometry of the notched plate he tenson load for the optmzaton s set to 19500N. hs s exactly the frst-ply-falure load (wth sa-hll crteron) of a plate havng the Fg. 4: Characterstcs of the patch generaton process In Fgure 4, the maxmal falure ndex s plotted over the mass for the patch generaton process. Intally, the maxmal falure ndex starts to ncrease sgnfcantly. here, only a few elements buld a renforcement doubler n the area wth hghest

5 A PSEUDO SRENGH FUNION FOR HE GENERAION OF LOCAL LAMINAE REINFORCEMEN DOUBLERS strans. he resultng dscontnuty of stffness causes an addtonal stress concentraton. Later on, the maxmal falure ndex drops untl the requred value of 1 has been reached. he unsteadness of the characterstcs s based on the choce of the fnte element mesh. akng a fner fnte element mesh wll lead to a smoother characterstcs snce the stress change wthn one step s smaller. he termnal soluton has been reached after 244 evaluatons. he geometres of the renforcement doublers for the allowed orentaton angles are shown n Fgure 5 and 6. Fg. 5: 0 degree renforcement doubler desgn wth fully coverng layers n terms of strength-to-mass rato. 4 Concluson he usage of the proposed pseudo strength functon does not guarantee that all falure ndces fall below a requred value. Nevertheless, t can be shown that the strength of a thn plate wth hole can be ncreased sgnfcantly. he renforcement doublers keep the weght low snce mass s respected n the senstvtes due to the lnear dependence on the thckness. he senstvtes consder the effect of a thckness change n one element to the whole structure. hus, regons can be localzed where a change n thckness affects the strength n other regons. hs s not possble when only local stresses are taken as decson parameters. Locally renforced desgns have superor strength-tomass ratos compared to desgns wth fully coverng layers. he proposed process has hgh potental to fnd extreme lght-weght structures. Locaton, geometry, as well as orentaton of the renforcement doublers are result of the teratve generaton process. he process s effectve and the created desgns can hardly be found ntutvely. Fg. 6: -45 degree renforcement doubler Obvously, the man strength ncrease arses from the 0 -layer. he -45 -layer has only developed n a small regon tangental to the hole and the 45 -layer has not developed at all. he connectvty of the doublers s hgh so that they can be manufactured wthout sgnfcant modfcatons. As requred, the maxmal falure ndex s below the crtcal value of 1. However, the ncrease of mass s only 5.17%. hs s remarkable keepng n mnd, that the strength of the soluton wth double layup (whch has a mass ncrease of 100%) s not hgher. Desgn Max FI Mass [kg] Mass [%] Intal Optmzed Double ab. 1: Desgn summary he proposed patch renforcement process provdes a locally renforced desgn that s superor to a Appendx: Materal Propertes E GPa E 22 9 GPa G GPa G GPa G GPa ν ρ 1556 kg/m 3 X t 1500 MPa X c 1250 MPa Y t 40 MPa Y c 200 MPa S 25 MPa ab. 2: Materal propertes CFRP/Epoxy 5

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