Element displacement vector

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1 Buklng Analyss of Rng-Strnger Stffened Composte Cylnder Nawras Hadar Mostafa Babylon Unversty, College of Engneerng, Appled Mehans Abstrat Buklng of the stffened omposte ylnder s a very omplex phenomenon that nvolves omplex nteratons between the skn and the stffeners. Dependng on dfferent parameters, dfferent buklng falure modes and falure loads are observed n stffened omposte ylnders. A 3-D fnte elements model was bult usng ANSYS, whh takes nto onsderaton the exat geometr onfguraton and the orthotrop propertes of the stffeners and the shell. The effets of several parameters suh as fber orentaton, shell thkness, and elast materal propertes were arred out and general onlusons were drawn regardng the optmum onfguratons of the dfferent parameters of the rng-strnger stffened omposte ylnder. The study orroborated that global buklng falure mode should be the desgn rtera for the stffened omposte ylnder. الخالصة: يعتبررا بعبعررألس بانرر بعة بلمقرر با بلموررع نة مررا بلمرر بة بلماةبررة ةررألتاا معقررةا عألتاررة نررا بلتررةب و بلمعقررة برريا بلق رراا بلمق يررأل بألالنتمألة نلى بعض بلمتغياب ل حة أعمأل أحمألو م تلفة للف و تم بعألء بلعم ذس ثالثي بابعألة بألنت ةبم ايقرة بلععألورا بلمحرةةا بلمتمثلررة بأللباعررألمت بلتحليلرري )ANSYS, بلررذأ أ ررذ بعةررا بالنتبررألا بل ررةو بليعةنرري بلحقيقرري رر ب بلمررألةا بلماةبررة للق رراا بلمق يررأل أاايرر ةابنررة نلررى بعررض بلمتغيرراب مثررو لب يررة تررة يا باليررألو نررمل بلق رراا, م بوررفأل بلمررألةا بلماةبررة نلررى بالن بعة بلمق با بلتي أة إلى ةي ا عتألئت تتعلق بأفضو تاتيب ان بعة موع نة مرا مر بة ماةبرة تحتر أ نلرى مق يرأل ليرة حلقية أةة بلةابنة نلى ا ب بلتوميم ضما ع ألق عم بلف و بألالعبعألس بلعألم Nomenlatures x-y-z X-Y-Z C j f T m Q j u u [K] e {a} e [K] s-t-r {ε} {σ} υ j [E] E Materal oordnate systems Nodal oordnate systems Global oordnate systems Element of elastty matrx to materal axs N/m 2 Shear orreton fator -- Transformaton matrx Element of elastty matrx. N/m 2 Dsplaement m Nodal dsplaement n x-dreton m Element stffness matrx N/m, N.m/rad Element dsplaement vetor m Overall stffness matrx N/m, N.m/rad Loal oordnate system Stran vetor -- Stress vetor N/m 2 Posson s rato through j plane -- Elastty matrx N/m 2 Elastty modulus along th dreton N/m 2 1. Introduton A omposte materal an be defned as a materal that s omposed of two or more dstnt phases, usually a renforng materal (flament) supported n ompatble matrx, assembled n presrbed amounts to aheve spef physal and hemal propertes (Stegman, J. and Lund, L, 21). A bas ply or lamna of a fberrenfored omposte materal an be onsdered from maro-mehanal pont of vew as orthotrop materal wth two prnpal materal dretons or natural axs parallel and perpendular to the dreton of the flaments. By bondng these lamnas together, a mult-lamnas omposte alled lamnate s formed. Composte materals are desrable n lghtweght strutures due to ther hgh spef stffness and strength. Lamnated omposte materals provde the desgner 1

2 wth freedom to adapt the propertes of the struture for gven loads to obtan the maxmum weght effeny. However, hgh modulus and strength haratersts of ompostes result n strutures wth very thn setons that are often prone to buklng. Stffeners are requred to nrease the bendng stffness of suh thn walled members (plates, shells). Consequently, stffened panels are often used n arraft and launh vehles to obtan lghtweght strutures wth hgh bendng stffness. Stffened shells are also more tolerant to mperfetons and resst atastroph growth of raks. Cylndral strutures made of omposte materal are wdely used n the above mentoned ndustres. Arraft fuselage and launh vehle fuel tanks are some of the many applatons of these strutures n aerospae and arraft ndustres (Hatem H. O., 22) suh as n Fgure (1). The advent of new manufaturng tehnques n flament wndng and automated fber plaement tehnques as well as new nnovatve toolng onepts have dereased the manufaturng dffultes and hene have boosted the applaton of these omposte ylnders (Langly, P. T.,1989). Payload shroud Skn panel Oxdzer tank Fuel tank Intermedate rng frame Engnes Fgure (1): Stffened panel onfguraton (Venkataraman, S., 1999 and Horak,J., et al, 25). Cylndral shells are subjeted to any ombnaton of n plane, out of plane and shear loads durng applaton. Mnmzng the mass of shell strutures has beome a more mportant desgn onsderaton as the sze of flght strutures has nreased. Large strutures are often lghtly loaded, and falure s frequently the result of buklng rather than of yeldng (John, L et al 1972). Buklng falure mode of a stffened ylndral shell an further be subdvded nto global buklng, loal skn buklng and stffener rpplng. Global buklng s ollapse of the whole struture,.e. ollapse of the stffeners and the shell as one unt. Loal skn buklng and stffeners rpplng on the other hand are loalzed falure modes nvolvng loal falure of only the skn n the frst ase and the stffener n the seond ase. A stffened ylnder wll fal n any of these falure modes dependng on the stffener onfguraton, skn thkness, shell wndng angle and type of appled load. Over the past four deades, a lot of researh has been foused on the buklng, ollapse, and post buklng behavor of ylndral shells (Knght, N., and Stranes, J. 1997). A good porton of ths work was devoted to the study of stffened ylnders. The smplest stffened ylnder onssts of only axal stffeners or strngers. A rng struture an be added to the strngers to aheve a better stffened orthogrd onfguraton. A work by Graham,J. (1993) presents analyss method for determnng the buklng loads of rng and strnger stffened ylnders. Another type of stffener arrangement s the ross stffeners arrangement (damond shaped pattern). Phllps J.L. and Gurdal Z. (199), dsussed a smearng method for determnng the global buklng load of ths type of stffened panels. Mostafa, N. H. and Waheed, S. O. (28) studed the buklng 2

3 phenomenon of a omposte ylnder wth strnger stffeners. The effets of shell wndng angle, skn thkness, and materal propertes were nvestgated. Sne the omposte stffened ylnder has mportant several applatons espeally n the aeronautal ndustres, thus t s an mportant to explore the optmum desgn (lght weght wth hgh strength) of suh strutures. The man objet of ths researh s to extend the study that aheved by Mostafa, N. H. and Waheed, S. O. (28) to nvestgate the effets of several parameters on the buklng load of a more omplex stffener onstruton nludng rng-strnger stffened omposte ylnder. 2 Fnte Element Dsrtzaton At present, the fnte element method s the most powerful numeral tehnque, whh offers an approxmate soluton to realst types of strutures. In the present study, the 8 nodded soparametr quadrlateral and 2-noded strutural sold elements are used for dsrtzaton of layered shell and layered stffener respetvely. 2.1 Element Parameters A quadrat element named SHELL99 of quadrlateral shape as shown n Fgure (2) onssts of eght nodes, all of whh are loated on the element boundary have been used to defne the shell fnte element model, whle that of retangle shape onssts of sx nodes. These types of elements are used for plates and shells struture applatons for membrane and flexural load ondtons (Yunus, et al 1989). SOLID191 element shown n Fgure (2) s a layered verson of the 2-node strutural sold desgned to model layered thk shells or solds. The element s defned by 2 nodes havng three degrees of freedom per node: translatons n the nodal x, y, and z dretons. SOLID191 has stress stffenng apabltes. Ths type of element s used to model the stffeners that onneted n the panel. r t s LN = Layer Number SHELL99 SOLID191 NL = Total Number of Layers Fgure(2):Types of Elements geometry (ANSYS Element Referene,24). The dsplaement at any pont wthn the element s wrtten n terms of nodal translatons and rotatons as: u u u u n a n b x, rt v v v N v N a b (1) w w w w a b y, Where, n s the number of element nodes. 3

4 N = shape funtons. u, v, w =global nodal dsplaements of node. {a}= unt vetor n s dreton. {b}= unt vetor n plane of element and normal to {a}. = rotaton of node about vetor {a}. x, y, = rotaton of node about vetor {b}. t = nodal thkness. r = natural oordnate along thkness dreton but normal to the shell surfae. 2.2 Mesh Generaton Fgure (3) shows the proposed fnte element model (stffened ylnder), reated usng soparametr shell elements of quadrlateral shapes. ANSYS * fnte element program s used as a mathematal tool n the analyss of ths model. Fgure (3): Meshed geometry 2.3 Stress Stran Relatonshp Beause of the lnearly elast behavor (assumpton), the generalzed Hooke s law s used for relatng the stresses to strans. A materal oordnate system s x-y-z, as shown n Fgure (4) whh ntrodued for the undretonal renfored lamna. Hook s law gves the general ansotrop onsttutve relaton wth respet to a materal oordnate system as follows (Barbero E.J, 1999) Fgure (4): In-plane and Interlamnar Shear Stresses (Barbero E.J., 1999) * ANSYS s hghest qualty engneerng tools used to analyze the desgn wth multple applatons 4

5 (2) The stress stran relatons n oordnates algned wth prnpal materal dretons are gven by, (Yunus, S.M, Khonke, P.C.,1989): or E where (3) (4) 11 = A E 1,j 33 =E 3,j B 12 = E 1,j 66 =G 12,j 22 = A E 2,j 44 =fg =f G 13 A=E 2,j / (E 2,j - j E 1,j ), B=1-6 1/ f To obtan the stress stran relatons for the lamna of arbtrary orentaton, the transformaton equatons are used for expressng stresses n x-y-z oordnate system n terms of stresses n oordnate system (Calote L.R., 1969): x x y y z -1 z [Tm ] [E][Tm ] (5) yz yz xz xz xy xy where 5

6 2 2 s s 2 2 s - s 1 T m (6) 2 2-2s 2s - s - s s In whh, = os, s = sn and s the fber orentaton angle n degrees. Usng the above transformatons, the stress stran relatons for arbtrary lamna orentaton an be wrtten as: x Q11 Q12 Q14 x Q Q Q y y Q33 z z yz Q55 Q56 yz xz Q65 Q 66 xz xy Q41 Q42 Q44 xy where: Q 1j = 2 F 1j + s 2 F 2j - 2s F 4j ( j =1, 2, 4) Q 21 = 2 F 2j + s 2 F 1j + 2s F 4j ( j =1, 2, 4) Q 33 = F 33 Q 4j = s F 1j - s F 2j + ( 2 -s 2 ) F 4j ( j =1, 2, 4) Q 5j = F 5j + s F 6j ( j =5, 6) Q 6j = -s F 5j + F 6j ( j =2, 6) F 11 = C C 12 s 2 F 12 = C 11 s 2 +C 12 2 F 14 = s (C 11 - C 12 ) F 21 = C C 22 s 2 F 22 = C 12 s 2 +C 22 2 F 24 = s (C 12 C 22 ) F 33 = C 33 F 41 = -2s C 44 F 42 = 2s C 44 F 44 = C 44 ( 2 -s 2 ) F 55 = f C 55 F 56 = -sf C 55 (7) F 65 = sf C 66 F 66 = f C 66 A generally orthotrop omposte lamna s an orthotrop lamna n whh the prnpal materal axes are not algned wth the strutural axes and the onsttutve matrx (Q) s as defned above n Equaton (7). 2.4 Stran Dsplaement Relatonshp The stran-dsplaement relatonshp an be wrtten wth ts dependene expltly expressed as: B u (8) All elements of the stran dsplaement matrx, B, are derved n terms of the shape funton dervatves and the Jaoban matrx. 6

7 2.5 Element Stffness Matrx In general, the bas onept of the fnte elements method s to dsrete the ontnuum nto a defnte numbers of small elements onneted together at ther ommon nodes. The stran-dsplaement matrx,b, as shown prevously s gven by: Bu, and therefore; the element stffness matrx an be wrtten as: e T [B] [E][B] dv V T [ k] [B] [E][B]det dr ds dt (9) n whh, [K] e : s the element stffness matrx, J : s the determnant of the Jaoban matrx. J For layered element, equaton (9) an be wrtten as (Yunus, et al 1989): tp 1 1 N r j K j ds dt (1) 1 1 j1 r bt j e 1 T [B] [E] [B] dr det J where, N 1 = number of layers. r j tp and r j bt =oordnates of the top and bottom of the layer j, respetvely. 2.6 Buklng Analyss Lnear buklng analyss n ANSYS fnte-elements software s performed n two steps. The frst step a stat soluton to the struture s obtaned. In ths analyss the prebuklng stress of the struture s alulated. The seond step nvolves solvng the egenvalue problem gven n form of Equaton (11). Ths equaton takes nto onsderaton the prebuklng stress effet matrx [S] alulated n the frst step (ANSYS, 24). K ) (.(11) S where k stffness matrx S Stress stffness matrx = th egenvalue (used to multply the loads whh generated [S]) = th egenvetor of dsplaements. The Blok Lanzos method was used to extrat the egenvalue resultng from Equaton (11). The buklng analyss was done to obtan the ylnder falure behavor under whh gve an dea on the ablty of use of the omposte ylnder n dfferent applaton n dfferent load manner. In addton to defne the regon of falure to deal 7

8 t arefully. The rtal buklng load of the model presents the more mportant ndex n the buklng analyss. 3. The flow-hart of the Computer Program. The flow-hart of the omputer program ANSYS V9. that s used n ths researh wll be summarzed n Fgure (5) below. Start Buldng the model whh nludes the defntons of geometry, element types, materal propertes, boundary ondtons nludng the load and mesh generaton. Obtan the stat soluton wth atvate the prestress effet Obtan the egenvalue buklng soluton usng 'Blok Lanzos' method Expand the soluton Revew the results Fgure (5): The flow-hart of the omputer program. 4. Results and Dsusson Effet of several parameters are nvestgated for a sx layers, eght strnger stffeners and sx rng stffeners wth ross seton (6*3 mm 2 ) stffened omposte ylnder of 18 mm n length and 146 mm n dameter for a style of stakng sequene ( 9/- /+ ). Table 1: Maromehanal Propertes of Stffened Layers n a Lamnate glass/ epoxy (Jones, R.M, 1999) E 1 GPa E 2 =E 3 GPa v 12 G 12 GPa End G 23 =G 13 GPa v 13 = v 23 Densty, kg/m Effet of Shell Thkness The effet of shell thkness on buklng load was nvestgated usng the fnte elements model. Ten analyses were performed to smoothly nrease the skn thkness from.5 mm to 7 mm. Fgure (6) shows plot of the results obtaned from these analyses for wndng angle sequene of 9/-15/+15. It s observed that the buklng resstane of the stffened ylnder steadly nreases wth nrease n shell thkness. Even though a steady nrease n buklng load s observed wth skn thkness nrease, the gan per mass added reahes a 8

9 maxmum and then delnes after a ertan pont (Fgure (7)) beause the nrease n mass s larger than the nrease n the strength of the ylnder as depted. The gan per mass measures the effeny of the mass added,.e., the addtonal load arred by the added mass. For the analyss performed on the stffened ylnder, the optmum skn thkness at whh the spef load (load per mass) reahes maxmum s found to be 2.48 mm (Fgure 7). It an be observed from the same fgure that the optmum skn thkness les n the global buklng falure mode regon. Ths result s very sgnfant as t onfrms the observaton of other researhers, (Bran, F. T. (1998), and Mostafa, N. H., Waheed, S. O. (28)) that only global buklng falure mode results n the maxmum spef buklng load, and onsequently leads to the onluson that the global buklng falure mode should be the desgn rtera for a stffened ylnder and t must be taken n the onsderatons. 7 8 Buklng Load (kn) Spef Buklng Load (kn/kg) Loal Skn Buklng Mxed Global Buklng Mxed Stffener Crpplng Skn Thkness (mm) Fgure (6): Buklng load due to nrease the skn thkness Skn Thkness (mm) Fgure (7): Spef buklng load under dfferent bukled regons The ylnder wth the thnnest shell thkness of.5 mm up to 1.53 mm was observed to fal purely due to loal skn buklng Fgure (8a) where the stffness s largely depends on the thkness value. When the skn thkness was nreased, the falure mode gradually hanged to global buklng at about 1.85 mm skn thkness. At ths pont n addton to loal buklng of the skn, the adjaent stffeners started to bukle as well. Wth further skn thkenng of the shell, the loalzed skn and stffener falure spread to adjaent ells and gradually transformed to a more global bukled falure mode Fgure (8b). At about a skn thkness of 3.23 mm, the shell was observed to be relatvely stronger than the stffeners and hene loalzed stffener rpplng started to our. For any skn thkness more than 3.23 mm the loal stffener rpplng falure mode prevaled Fgure (8). 9

10 ( a ) ( b ) ( ) Fgure(8): Buklng falure modes :(a) loal skn buklng, (b) global buklng,() loal stffener rpplng. Fgures (9) and (1) that found by Mostafa, N. H. and Waheed, S. O. (28) used a omposte ylnder wth only strnger stffeners. Kdane, S., (22) used fnte elements method to fnd the rtal buklng load and the spef load wth thkness for a omposte ylnder wth grd stffeners as shown n Fgure (11). It should be noted that the problem dmensons and materal propertes that used by, N. H. and Waheed, S. O. (28) and Kdane, S., (22) were dfferent from the present researh, but the behavor of the results are the same when they ompared wth present study Buklng Load (kn) Buklng Load per Volume (N/m^3) Loal Skn Buklng Mxed Global Buklng Mxed Stffener Crpplng Skn Thkness (mm) Fgure (9): Buklng load due to nrease the skn thkness Mostafa, N. H. and Waheed, S. O. (28) Skn Thkness (mm) Fg.(1): Buklng load under dfferent bukled regons Mostafa, N. H. and Waheed, S. O. (28) 11

11 Fgure (11): Buklng load and spef buklng load wth thkness, Kdane, S., (22). 4.2 Effet of Shell Wndng Angle Shell wndng angle s one of more mportant desgn varables that an be easly vared usng the fnte-elements model. The shell wndng angle an be vared by just hangng the nputs of the real onstants table, wthout hangng the model. The effet of shell wndng angle was nvestgated for the three types of buklng falure modes. The analyss was performed on models havng skn thkness of.5 mm, 2.4 mm and 4.5 mm. These three skn thknesses orrespond to loal skn buklng, global buklng and stffener rpplng falure modes respetvely. The analyses results are presented n Fgure (12) below. 6 Stffener rpplng Skn thkness.5 mm 2.4 mm 4.5 mm Buklng Load (kn) 4 2 Global buklng Loal skn buklng Shell Wndng Angle (Degrees) Fgure (12): Effet of shell wndng angle on buklng load It an be observed that the varaton n shell wndng angle has dfferent effets on eah type of falure modes. For loal skn buklng falure mode, whh orresponds to.5 mm skn thkness urve, t s lear that buklng load seems nsenstve to hanges n shell wndng angle and show almost flat urves for buklng load versus 11

12 mat orentaton. Whle, the optmum shell wndng angle for a stffened ylnder falng n stffener rpplng falure mode s about 3 o. The buklng load resstane nreases steadly wth wndng angle nrement. On the other hand, for global buklng falure mode (2.4 mm skn thkness), wth nrease n shell wndng angle the load resstane of the struture frst nreases and then goes down after reahng a maxmum. Hene, t was onluded that there exsts an optmum shell wndng angle for a stffened ylnder falng n global buklng falure mode. The optmum shell wndng angle for a stffened ylnder havng a skn thkness of 2.4 mm s found to be about 45 o. 4.3 Effet of Modulus The effet of modulus on the buklng load of a stffened omposte ylnder was nvestgated usng ANSYS pakage. The analyss was performed for a wde range of skn thkness. It has been shown n Seton 4.1 that buklng falure mode hghly depends on the skn thkness. The longtudnal modulus of the omposte system was vared from 54 GPa to 1 GPa dependng on the temperature and volume fraton (Valery, V., Evgeny, V., 21). The effet of modulus was studed on stffened ylnders havng shell thkness varyng from.5 mm to 3.5 mm wth shell wndng angle equal to 15. Fgure (13) below summarzes the results obtaned. The buklng load was observed to nreases lnearly wth nrease n longtudnal modulus for all skn thkness. It appears from the plot that the gan n buklng resstane nreases as the skn thkness nreases. But a lose look at Table (2), whh tabulates the perentage gan n the buklng load wth nrease n modulus, shows a hgher gan n buklng load s obtaned for a skn thkness 2 mm (global buklng mode).hene t an be onluded that a better gan n buklng load resstane s aheved f the longtudnal modulus s nreased for a stffened ylnder falng n global buklng falure mode than for a stffened ylnder falng n a stffener rpplng falure mode or n loal skn buklng falure mode. 6 5 Skn thkness (mm) Buklng Load (kn) Longtudnal Modulus E1(GPa) Fgure (13): Effet of longtudnal modulus on buklng loads. 12

13 where % Load Buklng gan Load Buklng E11 Buklng Load E154 Load E154 *1 (12) Table (2): Gan n buklng load wth thkness nreases Skn thkness.5 mm 1 mm 1.5 mm 2 mm 2.5 mm 3mm 3.5 mm % Load gan Conlusons In the buklng analyss of stffened omposte ylnders, a number of mportant problems reman to be resolved. In ths paper, attenton has been pad to nlude several parameter that affet the buklng behavor of the problem wthn hand suh as fber orentaton, shell thkness, and elast materal propertes. Three dfferent falure modes named global buklng, loal skn buklng, and stffener rpplng were studed wth dfferent parameters. Eah falure mode gave dfferent results of the other mode, and the results were ompared to fnd optmum desgn of the stffened omposte ylnder. The numeral results presented heren show that global buklng falure mode was the optmum desgn rtera of the stffened omposte ylnder. Referenes Barbero, E.J., 1999, Introduton to Composte Materal Desgn, Taylor & Frans, Phladelpha, PA. Bran, F. T., 1998, "Analyss and Desgn of Varable Stffness Composte Cylnders" Ph.D. Thess, Vrgna Polytehn Insttute and State Unversty, Blaksburg, Vrgna, USA. Calote, L. R., 1969, The Analyss of Lamnated Composte Strutures, Van-Nostrand Renhold, New York. Graham, j., 1993," Prelmnary Analyss Tehnques for Rng and Strnger Stffened Cylndral Shells", NASA report TM Horak,J., Lord, G. J.,and Peleter,M.A., 25," Cylnder Buklng: the Mountan Pass as an Organzng Center" arxv:math.ap/57263 V1. Hatem, H. O., 22, Investgaton of Performane and Optmum Desgn of Fber- Renfored Composte Shell of Arraft Struture,, Ph.D. Thess, The Mltary College of Engneerng, Baghdad, Iraq. Stegman, J., and Lund, L, 22, Notes on Strutural Analyss of Composte Shell Struture, Frst Edton. Jones, R. M., 1999, Mehans of Composte Materals, Seond Edton, Taylor & Frans, In.. John, L., Melvn, S., and Robert, L., July 1972, "Optmum Mass-Strength Analyss for Orthotrop Rng-Stffened Cylnders Under Axal Compresson" NASA TN D Kdane, S., 22 " Buklng Analyss of Grd Stffened Composte Strutures" M.S. Thess, Lousana State Unversty,USA. Knght N.F., Stranes J.H., 1997, Development n Cylndral Shell Stablty Analyss, NASA report. Langley, P. T., 1998, Fnte Element Modelng of Tow-Plaed Varable-Stffness Composte Lamnate, M.S. Thess, Vrgna Polytehn Insttute and State Unversty, Vrgna, U.S.A. Mostafa, N. H and Waheed, S. O., 28,"Buklng Analyss of Strnger Stffened Composte Cylnder" The Iraq Journal for Mehanal & Materals Engneerng. 13

14 Phllps, J.L., Gurdal, Z., 199, Strutural Analyss and Optmum Desgn of Geodesally Stffened Composte Panels, NASA Report CCMS-9-5 July. Valery, V., Evgeny, V., 21 Mehans and Analyss of Composte Materals Frst edton, Elsever Sene Ltd. Venkataraman, S., 1999, " Modelng, Analyss and Optmzaton of Cylndral Stffened Panels for Reusable Launh Vehle Strutures" Ph.D. Thess, Unversty of Florda, USA. Yunus, S. M., and Khonke, P. C., 1989, An Effent Through-Thkness Integraton Sheme n an Unlmted Layer Doubly Curved Isoparametr Composte Shell Element, Internatonal Journal for Numeral Methods n Engneerng, Vol. 28, p.p ANSYS, Engneerng Analyss System Theoretal Manual, www. Ansys. Com, Ansys Verson 9.,