Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 www.seipu.org/rme Design of Anis Grid Composite Lttie Conil Sell Strutures Jfr Eskndri Jm 1, Mild Noordi 1, Hossein Tgvin, Msoud Mommdi, Nder Nmdrn 1 1 Composite Mterils nd Tenology Center, Tern, Irn Deprtment of Menil Engineering, Buinzr rn, Islmi Azd University, Buinzr, Irn Corresponding Autor: Msoud Mommdi, Emil: Mssoudmommdi8@yoo.om Astrt Lttie strutures due to teir low weigt nd ig performne s struturl elements ve een widely used in different erospe pplitions. In tis pper, effetive prmeters on te design of nisogrid lttie onil sells re investigted. First, filment winding ptterns regrding te desired xil strengt is deided. Ten, regrding geometril reltions, effetive prmeters in order to form te nisogrid ell re identified. Distne of irulr ris from e oter s n importnt role in determining te onil lttie struture nd eventully in deriving te stiffness mtrix. Finlly, onsidering te reltions, finite element nlysis model of te lttie onil struture s een performed y ABAQUS softwre nd ukling nlysis under xil loding is done. In deriving te strengt results, te verifitions of referenes of tis pper nd te lssi teory ve een used. Keywords Ri; Stiffness Mtrix; Lttie Struture; Finite Element Introdution Composite lttie strutures re one of te most omplex nd newest strutures, wi re designed nd produed in ot isogrid nd nisogrid forms. In te pst dede, reseres done on polymeri mtrix fier omposite lttie strutures ve eome one of te sientifi nd widely used enters of ttention, nd tese strutures ve widespred pplition in different erospe strutures. Tese strutures s lttie lmin re used lonely or wit internl or externl sells. Lttie lmin in tese strutures inludes different systems of ris, wi re produed from ontinuous fiers, using utomti filment winding metod. In tese strutures under ritrry loding fiers of te lttie re under tension or ompression, wi is quite desirle stte for omposites, nd is one of te mjor dvntges of tese strutures ompred to te onventionl strutures for omposites. Mjor rteristis of tese omposites in te fe of xil ompression nd ending moment lodings, re different filures ourring due to totl ukling of te struture, filure due to mximum stress in elil ris, nd filure due to lol ukling of te elil ris in te lttie. Bukling lod relted to e of te filure sttes depends upon geometri prmeters nd totl dimensions of te struture. In te nlysis of tese strutures it is supposed tt elements of te lttie re two fore memers, eving s n ortotropi ody, nd te reltions of ortotropi strutures re used ere for te nlysis of te totl ukling of tese strutures. Mjor pplition of tese strutures is in erospe industry, wi widely uses different forms of tem, inluding: urved sped nd nnulr sells (espeilly onil sells). For te purpose of improving te menil properties of te struture, nd its weigt optimiztion, omposite mterils ve een used. In te midst of 196's te finite element metod ws developed for te numeril nlysis of tese strutures. Generlly, tese strutures re mnuftured y dvned forming of fiers. Also, te ris use in tese strutures n move in up to 4 diretions. Design, nlysis nd mnufture of tese strutures re investigted in referene [1]. More detils of teir nlysis re inluded in referenes [, 3]. In referenes [4, 5] ukling lod nlysis of grid stiffened ylindril sells is performed. Also, te results of te experimentl test re ompred nd verified in omprison wit te nlytil results. Kim [6, 7] s studied te frition nd testing of tin omposite isogrid stiffened pnels nd omposite isogrid stiffened ylinders, so s to investigte teir ukling evior. In referene [8] design nd optimiztion of lminted onil sells for ukling nd mximum ukling lods s een performed. In tis study, optimiztion s een performed in two 5
www.seipu.org/rme Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 sttes of using te mximum ukling lod t ertin weigt, nd using te minimum weigt under onstnt ritil lod. In referene [9] ritil ukling lod is derived from te solution of te governing nonliner prtil differentil eqution wit different oeffiients. Referene [1] s investigted te optimized stking sequene design, so s to ieve te mximum fundmentl eigenfrequeny in onil sells. In referene [11] ukling of te ylindril lttie struture s een done, using finite element metod. Also, in referene [1] finite element nlysis for ukling of te onil lttie sell is studied. In tis pper, numeril ode is developed for te onil lttie struture, wi determines te si prmeters, ukling under ritil xil loding, torsion moment, nd ending moment. In tis pper, design of te onil lttie struture wit te nisogrid ell is investigted. First, te governing differentil equtions of te nisogrid ell of te onil lttie struture re derived, regrding te filment winding tenique used. Ten vritions of te prinipl prmeters of te design re investigted in reltion to te inrese of te numer of elil ris. Lter, y reting finite element model of te onil lttie struture wit te nisogrid ell, ukling nlysis of te struture under xil loding is performed. Also, te vritions of te ritil ukling lod, reltive to inrese of tikness, widt, nd te distne etween ris is onsidered. Governing equtions of design Lttie strutures re widely used in different industries. Tese strutures re igly strong ginst te destrution used y impt, delmintion, nd rze propgtion in te struture. One of teir dvntges over oter strutures is redution of weigt nd mnufture time. Lttie struture is mde of numer of elil nd irulr ris. Tese strutures n ve up different forms, inluding isogrid nd nisogrid. Generlly, te min purpose of using lttie strutures is optiml usge of longitudinl properties of te omposite mterils used. Also, semti of te onil lttie sell used s stellite rrier dptor, t te lower setion of te stellite is sown in figure 1. All te prmeters re determined on te sis of te lotion of elil nd irulr ris. Some of tese prmeters depend on te ngel of te elil ris. Also, some of tese prmeters s te eigt of te struture inreses ve vritions wit ertin rte. Beuse of te omplexity of te geometry of onil lttie sells, in order to design, one first must divide te effetive design prmeters of te struture in two dependent nd independent tegories, nd ten perform te omplete investigtion proedure of design. FIGURE 1 SCHEMATIC OF THE SATELLITE CARRIER ADAPTOR [13] Independent prmeters Te numer of irulr nd elil ris (n, n ) re independent prmeters in te design of onil lttie struture. Dependent prmeters ϕ λ Dependent prmeters inlude: ψ. A semti of te prmeters stted ere in te design of nnulr lttie struture, is sown in figure. FIGURE GEOMETRIC PARAMETERS OF THE LATTICE STRUCTURE Sine te design of tese strutures is intended to stnd xil ompression lodings, terefore te most importnt prolem regrding tese strutures is te nlysis of ukling nd elsti stility. Considering 6
Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 www.seipu.org/rme tese fts, design of te lttie struture nd eventully mnufture of it, must e sed on stte tt sows te mximum strengt ginst ukling. Amongst different filment winding ptterns, geodesi pttern is te est pttern of filment winding for lttie strutures, rrying xil ompression lodings. In order to derive te governing differentil equtions of te struture, first y developing setion of te one, te geometri equtions, wi ve diret effet on te stiffness mtrix, re ieved (figure 3). ssumption tt te irulr ris re loted etween te intersetion of te elil ris. Considering te geometry of figure 3, te following eqution is ieved: ψ γ ( n 1) (4) Geodesi ngle t te smller ross-setion ( ϕ ) is omputed s follows: ϕ tn 1 ρ F sinγ ρf osγ ρ (5) And lso, te geodesi ngle t te igger ross-setion ϕ ) is equl to: ( ϕ F ϕ 1 (6) Vritions of ρ versus ϕ re ieved y tis eqution: FIGURE 3 DEVELOPED SECTION OF THE CONICAL LATTICE SHELL Using geodesi reltions, te following eqution for e point of te lttie struture exists [6]: ρ sin ϕ ρ sinϕ C onstnt Werein ρ r sinα (1), nd r is te rdius of te ross setion of te one wi is different t e point, nd α is te ngle etween te line pssing troug te pex of te one nd te xis of symmetry of te one. Tking derivtion of eqution (1) gives: dρ dϕ ρ tnϕ () ψ is te ngle etween te two elil Were ris of figure 3, nd is found ere: π sinα ψ n Geometril design nd finite element nlysis of te onil lttie struture is performed wit te (3) ρ C d osϕ dϕ sin ϕ Vertil spe etween irulr nd elil ris for ell t e row is derived from te following equtions: sinϕ ρ i ρ i + 1 ( ) i, i+ 1 ϕi+ 1 ϕ i ψdρ dϕ In eqution (9), i indites te numer of ris. Te minus sign in eqution (9) sows tt te xis of te system of oordinte system is loted on te smller ross-setion. Stiffness properties of one ell, wi s repeting pttern, is te representtive of te totl repeting setion. Te ortotropi properties ieved for nnulr odies, is long te xil diretion. For exmple, for te ylindril struture, A stiffness long te xil diretion is equl to E x, L were A πrh is te ross-setion of te struture. (7) (8) (9) 7
www.seipu.org/rme Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 4 E E [ Q] s E E s s sin ϕ, osϕ 4 s E + E s (1) E nd E re te moduli of elstiity of elil nd irulr ris, respetively. Eqution (1) is diretly derived from teory, formultion nd ssumptions, wi re relted to te stiffness of te lyer nd te properties of fiers. Also, properties of te equl stiffness long te xil diretion re derived from eqution (11), [6]: E x 1 q H 11 q q q 1 (11) Finite element model Regrding te ft tt tese strutures nnot e trnsformed into smller smples, in order to determine te menil speifition of te struture, one n only use te possile solution of experimentl test. Numeril nlysis of te lttie struture is performed, using Timosenko's em element, nd te sell's element, s it is seen in figure 4. Generlly, onsidering te sitution tt mjor lods pplied to te lttie struture re xil lods, terefore ukling is te most importnt prmeter in strengt evlution of tese strutures. Coosing of te em element is desried in referene [11, 1]. Cirulr nd irumferentil ris of te lttie struture re ompletely onded to e oter t te intersetion points. Te distne etween irulr nd elil ris from e oter is ieved, onsidering te governing reltions. qij used in eqution (11) re te omponents of te stiffness mtrix Q, nd s it is seen, tey depend upon te ngle of elil ris, widt of te ris, nd te distne etween elil nd irulr ris. Te ritil ukling xil lod for onil sell is given y [14]: P yl E t 3(1 v π x os ) α (1) Aording to eqution (1), te ritil ukling xil lod for onil sells is te sme s tt of ylindril sells, wit tis differene tt te ngle for lf-pex of te one is lso effetive in te ritil lod. By omprison of te experimentl, nlytil nd finite element nlysis, it is found out tt in order for te nlytil result to ve n eptle nswer, ukling orretion ftor of C must e multiplied wit it. Eventully, y pplying te ukling orretion ftor, te ritil ukling lod of te onil sell is equl to: P yl C 3(1 v πe xt os ) α (13) Aording to referene [14], te mgnitude of prmeter C depends on te ngle of te one's inlintion. Te orretion ftor of C for one, ving te pex ngle of 1 up to 75 degrees, is equl to.33. At rnge outside tese ngles, C is derived ording to experimentl test. FIGURE 4 A SCHEMATIC OF THE BEAM ELEMENT USED IN FINITE ELEMENT ANALYSIS In order to design te winding pttern of te fiers, it is neessry to derive te geodesi ngles of te elil ris in ig nd smll ross-setions, nd te eigt lotion of te onil struture sould e known. For te design of onil lttie struture, nd regrding equtions (6), (7), nd (1) te totl dimensions inluding eigt, igger nd smller ross-setions, nd te pex ngle must e determined. By tis explntion, ig nd smll dimeters of te one re 5 mm nd 15 mm, respetively, nd te pex ngle is 34.7. Now, regrding te geometri speifition stted, geodesi ngle vritions t different eigt lotions snd different numer of irulr ris is derived. Considering te model t te previous setion, geodesi ngle vritions t te igger nd smller ross-setions wit te inrese of te eigt of te one, nd vritions of te numer of irulr ris, re sown in figure 5 nd 6, respetively. As it is seen, wit te inrese of te 8
Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 www.seipu.org/rme eigt of te one, geodesi ngles re non-linerly deresing. Also, wit te inrese of te numer of irumferentil ris t te sme geometri onditions, geodesi ngles t e setion re redued. Results sow tt y inresing te numer of irumferentil ris, vritions of te geodesi ngle t te igger setion re reduing wit more inlintion, ompred to te smller setion. Regrding te totl geometri speifitions of te onil struture, geodesi ngle n e ieved t te igger nd smller ross-setions for ertin numer of elil nd irulr ris from te digrms of figure 5 nd 6. For exmple, for te onil lttie struture wit ρ F ρ 1 irulr ris, nd te rtio of, geodesi ngle t te igger nd smller ross-setions re 15.98 nd 33.44, respetively. Also, vritions of λ wit te inrese of te numer of elil ris, is sown in figure 7. Te finite element model wit totl speifitions nd geodesi ngle is sown in figure 8. FIGURE 7 VARIATION OF λ WITH THE INCREASE OF THE NUMBER OF HELICAL RIBS FIGURE 8 FINITE ELEMENT MODEL OF THE LATTICE CONICAL SHELL, MADE BY ABAQUS SOFTWARE Verifition FIGURE 5 VARIATION OFϕ WITH THE INCREASE OF FIGURE 6.VARIATION OF ( α 34.7 ) ϕ F WITH INCREASE OF ( α 34.7 ) ρ F ρ ρ F ρ Finite element model of te lttie onil sell wi s undergone ukling nlysis under xil nd ending lodings in referene [1] is used ere nd ukling nlysis is performed on it, using ABAQUS softwre nd Timosenko's em element. Geometri speifitions nd te mterils used in te finite element nlysis re osen ording to referene [1]. A semti of te finite element nlysis of te lttie onil sell wit te defined geometri speifitions is sown in figure 9. Critil lods ere, re in Newton, nd for etter disply of te deformtion of te struture, te spet rtio of 4 is pplied. Results presented in tle 1 sow tt te results ieved from te numeril finite element solution nd te nlytil results re of ig ury. Regrding te verifition performed in te following on te lttie onil nd ylindril strutures wit ertin geometril speifitions, nd te equtions stted ove, weigt optimiztion of te struture is performed. 9
www.seipu.org/rme Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 TABLE 3 PROPERTIES OF THE MATERIALS USED IN THE LATTICE CONICAL SHELL E (GP) σ (MP) m ( Kg 3 m ) E (GP) m ( Kg 3 m ) 8 35 145 64 141 Aording to te digrm of figure 1, nlytil results sow tt y inresing te ukling lod inreses. rtio, ritil FIGURE 9 DERIVATION OF THE CRITICAL BUCKLING LOADING, USING FINITE ELEMENT METHOD (HB4MM, α 5 ) TABLE1 COMPARISON OF THE RESULTS OF REFERENCE [1] WITH THE RESULTS OF ANALYTICAL ANALYSIS, AND THE NUMERICAL ANALYSIS PRESENTED HERE 8mm mm 44.334 49.11 47.997 4mm 4mm 34.416 33.831 36.16 Results nd disussion mm 8mm 13.863 15.3 15.6 Referene [1] ABAQUS Anlytil Critil ukling lod in strutures depends upon stking sequene of te ris, ngle of elil ris, ross-setion of te dris, nd et. Lter in tis pper, results of vritions of xil loding under lmped support onditions re investigted. Geometri speifitions of te lttie onil sell re summrized in tle wit regrds to figures 5 nd 6. Also, te properties of mterils used in te lttie onil sell re inluded in tle. FIGURE 1 VARIATION OF BUCKLING LOAD, VERSUS INCREASE OF. Also, wit te inrese of rtio, ritil ukling lod dereses (figure 11). Also, vritions of te ritil ukling lod, y inresing te rtio, nonlinerly dereses (figure 1). In figure 13, vrition of te ritil ukling lod versus inrese of tikness, s een sown for two sttes of nlytil nd finite element metods. Results indite tt, y inresing te tikness of te ris of lttie sell, te ritil ukling lod of te struture, non-linerly inreses. TABLE GEOMETRIC PARAMETERS OF THE LATTICE CONICAL SHELL STUDIED n 1 n 53 α f 15.98 α 33.44 H (mm) 18 (mm) 4 (mm) 5.75 FIGURE 11 VARIATION OF CRITICAL BUCKLING LOAD, VERSUS INCREASE OF RATIO 1
Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 www.seipu.org/rme Also, wit te inrese of te widt of elil nd irulr ris, ritil ukling lod inreses nonlinerly (figure 15). FIGURE 1 VARIATION OF CRITICAL BUCKLING LOAD, VERSUS INCREASE OF RATIO FIGURE 15 VARIATION OF CRITICAL BUCKLING LOAD, VERSUS INCREASE OF WIDTH OF CIRCULAR AND HELICAL RIBS Regrding te results presented, one n derive te ritil xil lod wit good ury, using numeril finite element metod nd nlytil metod. A semti of te first up to sixt modes of ukling of te lttie onil struture, under xil loding is presented in figure 16. FIGURE 13 VARIATION OF CRITICAL BUCKLING LOAD, VERSUS INCREASE OF THICKNESS OF CIRCULAR AND HELICAL RIBS Regrding te digrm of figure 13, it is relized tt wit te inrese of te tikness of te ri (pproximtely mm), te em element does not possess ig ury in te onvergene of te results of nlytil nd numeril nlysis. 1mm 5mm 1mm 5mm mm 15mm FIGURE 16 FIRST MODE OF BUCKLING FOR THE LATTICE CONICAL SHELL UNDER AXIAL LOADING, VERSUS WIDTH VARIATION OF THE RIB (THICKNESS18MM) Finl onlusion mm 15mm FIGURE 14.A SCHEMATIC OF THE FIRST MODE OF BUCKLING FOR THE BUCKLING OF LATTICE CONICAL SHELL UNDER AXIAL LOADING, VERSUS THICKNESS VARIATION OF THE RIB (WIDTH5.75MM) Design of lttie onil sell, requires onsidering different prmeters. Tese prmeters re independently osen y te designer, or re dependent. For ukling nlysis of te lttie onil struture under xil loding, te equl stiffness of te sell in te xil, irumferentil, nd troug te 11
www.seipu.org/rme Reseres nd Applitions in Menil Engineering (RAME) Volume 1 Issue 1, Deemer 1 tikness diretions sould e determined. Stiffness mtrix of te lttie struture, esides te menil properties of elil nd irulr diretions, depends upon te geometry of te ells of te lttie struture, nd lso te ngle of te elil ris t different points of te eigt. In ukling nlysis of te lttie onil sell under xil loding, te following results re ieved: By inresing te lod inreses. By inresing te lod dereses. By inresing te lod dereses. rtio, te ritil ukling rtio, te ritil ukling rtio, te ritil ukling By inresing te tikness of te irulr nd elil ris, te ritil ukling lod inreses non-linerly. By inresing te widt of te irulr nd elil ris, te ritil xil loding inreses. REFERENCES Adrin W. Blom, Srir Setoode, Jn M.A.M. Hol, Zfer Gurdl, " Design of vrile-stiffness onil sells for mximum fundmentl eigenfrequeny", Computers nd Strutures (7). E.V. Morozov, A.V. Loptin, V.A. Nesterov. "Bukling nlysis of nisogrid omposite lttie onil sells", Composite Strutures 93 (11) 315 316. Goldfeld Y, Aroz J. Bukling of lminted onil sells tking into ount te vritions of te stiffness oeffiients. AIAA J 4;4(3):64 649. HuyretsS, Meink TE. "Grid stiffened strutures: survey of frition, nlysis nd design metod". 1999 Interntionl onferene on omposite mterils(iccm) proeedings.1999. HuyretsS, Meink TE. "Advned Grid stiffened strutures for te next genertion of lun veiles". 1999 IEEE Aerospe onferene Proeding.1997. HuyretsS, Meink TE. "Hyrid tooling for dvned grid stiffened strutures (AGS). 1996 SAMP Tenil Conferene Proeedings. 1996. Kidne, S., Li, G., Helms, J., Png, S. nd Woldesenet, E., Bukling Lod Anlysis of Grid Stiffened Composite Cylinders. Composites Prt B: Engineering, 34, 1-9 (3). Kim, T.D., Frition nd Testing of Tin Composite Isogrid Stiffened Pnel.Composite Strutures, 49, 1-5 (). Kim, T.D., Frition nd Testing of Composite Isogrid Stiffened Cylinder. Composite Strutures, 45, 1-6 (1999) Morozov EV, Loptin AV, Nesterov VA. Finite-element modelling nd ukling nlysis of nisogrid omposite lttie ylindril sells. Compos Strut 11;93:38 3. V. I. Weingrten, E.J. Morgn nd seide, p.: Elsti Stility of tin-wlled Cylindril nd Conil Sells under Axil Compression. AIAA J., vol. 3, no. 3, Mr. 1965, pp. 5-55. V. V. Blepin,M. Mit, S. N. B. Murty, Tird Wy of Development of Single-Stge-to-Orit ropulsion, JOURNAL OF PROPULSION ANO POWER, Vol. 16, No. 1, Jnury-Ferury. Wodesenet, E., Kidne, S. nd Png, S., Optimiztion for Bukling Lods of Grid Stiffened Composite Pnels. Composite Strutures, 6, 159-169 (3). Yisk Goldfeld, Jonn Aroz, Aln Rotwell, "Design nd optimiztion of lminted onil sells for ukling", Tin-Wlled Strutures 43 (5) 17 133. 1