Linear Transient Analysis of Laminated Composite Plates using GLPT

Transcription

1 Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56, o. (13) Joural homepage: hp://cosrucii.ucluj.ro/acacivileg Special ssue: Firs eraioal Coferece for PhD Sudes i Civil Egieerig, CE-PhD 1. Liear Trasie Aalysis of Lamiaed Composie Plaes usig GLPT Miroslav S. Marjaović* 1, Đorđe M. Vuksaović 1, Uiversiy of Belgrade, Faculy of Civil Egieerig, Bulevar kralja Aleksadra 73, 11, Belgrade, Serbia (Acceped 8 Augus 13; Published olie 3 Sepember 13) Absrac The objecive of his work is o sudy he rasie respose of lamiaed composie plaes uder differe ypes of dyamic loadig. For his purpose, lamiaed composie plae is modeled usig Reddy s geeralized layerwise plae heory (GLPT). This heory assumes layerwise liear variaio of displacemes compoes. Trasverse displaceme is cosa hrough he hickess of he plae. Usig he assumed displaceme field, liear kiemaic relaios, as well as Hooke s cosiuive law, equaios of moio are derived usig Hamilo s priciple. Aalyical soluio for cross-ply lamiaes is derived usig he avier mehod. umerical soluio is obaied usig FEM. Goverig parial differeial equaios i boh soluios are reduced o a se of ordiary differeial equaios i ime usig ewmark iegraio scheme. The equaios of moio are solved usig cosa-average acceleraio mehod. Effecs of ime sep, mesh refieme ad lamiaio scheme o accuracy of rasie respose are cosidered. llusraive commes are give abou he ifluece of shear deformaio o rasie respose. Fially, differe schemes of dyamic loadig are ivesigaed. Good agreeme is obaied wih resuls from he lieraure. Rezuma Obiecivul acesei lucrări ese de a sudia răspusul razioriu al plăcilor compozie lamiae sub diferie ipuri de îcărcare diamică. Î aces scop, placa compoziă lamiaă ese modelaă folosid eoria geeralizaă a plăcilor propusă de Reddy (GLPT). Aceasă eorie presupue variația liiară a compoeelor deplasării î rapor cu sraurile plăcii. Deplasarea rasversală ese cosaă î grosimea plăcii. Ecuațiile de mișcare su derivae folosid pricipiul lui Hamilo, uilizâd câmpul de deplasare asuma, relațiile liiare ciemaice, precum și legea cosiuivă a lui Hooke. Soluția aaliică peru plăci lamiae di fibre di lem îcrucișae ese derivaă folosid formula lui avier. Soluția umerică ese obțiuă cu ajuorul meodei elemeului fii. Ecuațiile cu derivae parțiale î ambele soluții su reduse la u se de ecuații diferețiale ordiare î imp, uilizâd Meoda ewmark. Ecuațiile de mișcare su rezolvae folosid meoda de iegrare impliciă ewmark β. Su luae î cosiderare efecele iegrării umerice î imp, pas cu pas, ale rafiării discreizării și ale sisemului de lamiare asupra acuraeței răspusului razioriu. Su prezeae comearii ilusraive despre iflueța deformării cauzae de forfecare asupra răspusului razioriu. Î cele di urmă, su ivesigae diferie scheme de îcărcare diamică. Rezulaele obțiue su î cocordață cu cele di lieraura de specialiae. Keywords: lamiaed plae, rasie aalysis, avier soluio, FEM, ewmark iegraio

2 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) roducio Lamiar composies play a impora role i he desig ad cosrucio of aircrafs, ships, ad may oher pars i machie idusry. They arac grea aeio i a field of Civil Egieerig, oo, ad heir massive use i srucural desig is expeced. They ca be used as mai load carryig members i he form of hick lamiaed ad sadwich plaes [1]. Suiabiliy for differe desig purposes due o heir grea siffess o weigh raios is highly valued. Lamiaes are ofe composed of several orhoropic layers (lamias, plies). Layer orhoropic behavior comes from he highsregh fibers, which are orieed i predefied direcio for each layer idividually. he case of cross-ply lamiaed composie plaes, ply fibers are orieed aleraely, wih agles of о or 9 о. Differe orieaio of plies forms symmeric or ai-symmeric lamiaio schemes. Composie lamiaes are ofe exposed o differe ypes of saic ad rasie dyamic loadig. They are characerized by sigifica rasverse shear deformaios. This all leads o he eed for he accurae modelig of hick composie plaes: displaceme coiuiy codiios mus be fulfilled a layer ierfaces. This crucial codiio is saisfied by he use of layerwise plae heory, which will be explaied i his work. Differe plae heories are derived for he aalysis of composie lamiaes. Global behavior ca be accuraely deermied by he use of relaively simple equivale-sigle-layer lamiae heories (ESL) [], especially for hi lamiaes. he case of hicker srucural compoes, ESL heories are o adequae, so refied heory is eeded o accou for he hickess (shear) effecs. Vuksaović ivesigaed sigle layer models of higher order, which represe plae kiemaics wih improved accuracy [3]. ESL heories cao accou for discoiuiies i rasverse shear srais a he ierfaces bewee layers of differe sifesses. Aoher problem is he aalysis of local effecs, such as marix crackig, delamiaio or free edge effec. For hese reasos here is a eed for applyig layerwise plae heories. he layerwise approach, i is assumed ha C -coiuiy hrough hickess of he lamiae is saisfied. The plae is aalyzed as a mulilayered i he rue sese of word (each layer is cosidered separaely). Cross-secioal warpig is ake io accou, which is much more kiemaically correc represeaio of displacemes. Shear deformaio (cosiderable as a resul of plae s aisoropic srucure) is icluded. his paper, Geeralized Layerwise Plae Theory of Reddy [4] is used o aalyze rasie respose of lamiaed composie plaes. allows idepede ierpolaio of i-plae ad ou-ofplae displaceme compoes. Piece-wise liear variaio of i-plae displaceme compoes, ad cosa rasverse displaceme hrough he hickess are imposed. Cosise mass marix is employed i he dyamic aalysis. he displaceme-based FE formulaio, oly C -coiuiy is eeded, so he odal variables are raslaio compoes. The goal of his work is o prese he boh aalyical ad umerical soluio for he rasie respose of lamiaed plae. Soluios are obaied usig MATLAB ad compared wih exisig daa from he lieraure.. Formulaio of he geeralized layerwise heory.1 Assumpios We will cosider a lamiaed plae composed of orhoropic layers. Physical layers i he lamiae are umbered sarig from he boom layer. Mid-plae coordiaes are (x, y, z). Typical lamiaed composie plae i global coordiae sysem is show i Figure 1. 59

3 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) Figure 1. Lamiaed composie plae (4 layers) Derivaio of Geeralized Lamiaed Plae Theory is based o followig assumpios: (1) all lamias are perfecly boded ogeher (o relaive displacemes exis a layer ier-faces), () all layers are of uiform hickess, (3) maerial is liearly elasic, (4) all layers are orhoropic, (5) srais are small ad (6) iexesibiliy of ormal is imposed..1 Displaceme field Displaceme field (u 1, u, u 3 ) i he poi (x, y, z, ) of lamiaed plae ca be wrie as: u ( x, y, z, ) u( x, y, ) u ( x, y, ) ( z) u ( x, y, z, ) v( x, y, ) v ( x, y, ) ( z) u ( x, y, z, ) w( x, y, ) 1 (1) Eq. (1), (u, v, w) are displaceme compoes i hree orhogoal direcios i he mid-plae of he plae, (u, v ) are coefficies which will be calculaed laer, ad (z) are layerwise coiuous fucios of he hickess coordiae (liear, quadraic or cubic Lagragia ierpolaios of hickess coordiae), which ca be foud i []. Eq. (1), deoes arbirary ime poi. For he purpose of his work, liear ierpolaio is chose hrough he hickess coordiae. he FEM aalysis, (u, v ) are he odal values of (u 1, u ) i he h umerical layer hrough he plae hickess. is he umber of layers hrough hickess of he lamiae (or he umber of odes i z-direcio i FE discreizaio). Geeralized displacemes hrough he plae hickess are show i Figure.. Kiemaic relaios of lamia Figure. Displaceme compoes i GLPT, for 4-layer plae Liear srai displaceme relaios () are assumed as follows: i-plae deformaio compoes are coiuous hrough he plae hickess, while he rasverse srais eed o o be. Cosiuive equaios of sigle ply are used i derivig cosiuive equaios of lamiae. u u u 1 x x x 1 x xz u u d w dz dx dz dx 1 3 u 1 6

4 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) u v v y y y 1 y u u u v u v 1 xy y x y x 1 y x.3 Cosiuive equaios of lamia yz u u d w dz dy dz dy 3 v 1 () The sress-srai relaios for k h lamia ca be wrie as: ( k) ( k ) ( k) x Q11 Q1 Q16 x y Q1 Q Q6 y xy Q16 Q6 Q66 xy Q Q xz xz yz Q45 Q 44 yz (3) Eq. (4), Q ij (k) are rasformed elasic coefficies of k h lamia i global coordiae sysem..4 Equaios of moio Whe derivig he dyamic equilibrium of virual srai eergy (U), virual work of exeral forces (V) ad virual kieic eergy (K), i is assumed ha loadig q is acig i he middle plae of he plae. This loadig works o virual displaceme w i he mid-plae of he plae. Homogeuous boudary codiios o he surface are imposed. Dyamic expressios for U, V ad K are give i Eq. (4): x x y y xy xy xz xz yz yz dvd (,, ) V V U V K u u u u u u dvd V q x y w dvd (4) Dyamic versio of virual work saeme ca be derived usig Hamilo priciple: u v u v w w x y xy Qx Qy x y y x x y U d d u v u v x y xy Qxu Qyv 1 x y y x V qw dd u u v v w w u u v v u u v v 1 K dd J J J u u v v 1J1 (5) U V K (6) 61

5 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) Sress resulas As saed before, q is he uiform rasverse loadig ime fucio - acig i he mid-plae. Sress resulas {} ad ieria erms {} are derived i [1,, 5]: k1 zk1 zk dz x zk1 x y y k 1 zk xy xy k ( k ) ( k ) zk1 Qx xz Q y k1 z yz k dz dz k1 zk1 zk dz k y y k1 zk xy xy zk1 J J k k1 zk ( k) x z x k1 dz Q k1 d z ( k) x xz Qy k1 z yz k dz dz (7) dz 3. Aalyical (avier) soluio for simply suppored cross-ply lamiaes avier soluio of GLPT is derived for simply suppored recagular cross-ply lamiaes, wih dimesios a b [8]. Boudary codiios i his case are: v w v x, a x u w u y, b y x y (8) We have o choose appropriae displaceme field o saisfy boudary codiios (8) o he edges of he simply suppored lamiaed composie plae ad Euler-Lagrage equaios of moio (6): mx y u( x, y, ) Xm( ) cos si a b m1 1 mx y v( x, y, ) Ym( ) si cos a b m1 1 mx y w( x, y, ) Wm( ) si si m1 1 a b mx y u ( x, y, ) Rm( ) cos si a b m1 1 mx y v ( x, y, ) Sm( ) si cos a b m1 1 (9) Loadig should be expaded i double rigoomeric series i a same maer. q( x, y, ) m1 1 q m ( ) si x si y (1) (9, 1), m ad deoe umber of members i Fourier series. X m, Y m, W m, R m, S m are Fourier coefficies - ime fucios - which are chose oly i a way such ha u, v, w, u ad v saisfy Eq. (6). Secod par of he expasio deermies he spaial variaio of he rasie 6

6 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) soluio. Afer icorporaio of Eq. (9) ad Eq. (1) i Eq. (6), we derive he marix form of virual work saeme. f cross-ply lamiaes are aalyzed, some elemes i marix of elasic coefficies are ideically zero, as show i [5], so compaced marix form of Eq. (6) becomes: Xm() Xm() Ym() Ym() k k m m Wm() Wm() qm() J J k k m m Rm() Rm() S () () m Sm (11) Submarices [k], [k ] ad [k J ] are i deail derived i [5]. Here we will derive he submarices of cosise mass marix: m m m J J J Eq. (11), {} is a vecor of Fourier coefficies ad is a vecor of secod derivaios of Fourier coefficies. f we observe discree ime poi, followig marix equaio which saisfies equilibrium codiios is: (1) K M F (13) Eq. (13), subscriped deoes appropriae value i discree ime poi. Superposed dos deoe differeiaio wih respec o ime. Global siffess marix [K], as well as cosise mass marix [M], remais cosa i all ime pois. f homogeuous iiial codiios (displacemes ad velociies) are assumed, X m, Y m, W m, R m ad S m ad heir firs derivaives i ime are zero. Disribued loadig {F} acs perpedicular o he mid-plae of plae. 4. Fiie eleme model Layerwise fiie eleme cosis of mid-plae ad umerical layers (excepig he middle plae) hrough he hickess of he plae, as show o Figure 3. Adoped odal degrees of freedom are raslaios i hree orhogoal direcios i he mid-plae (u i, v i, w i ) ad relaive raslaios (u i, v i ) i h umerical layer hrough he plae hickess. Figure 3. Layerwise FE wih layers ad m odes Geeralized displacemes saisfy C coiuiy codiio o eleme boudaries. Displaceme field is ierpolaed usig he Lagragia ierpolaio fucios: 63

7 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) 58-71,,,, m u v w u v,,,, u v w u v i i i i i i i1 (14) Eq. (14), idex m deoes he umber of odes per eleme. For his purpose, 4-ode Lagrage quadrilaeral is chose. erpolaio is obaied usig sadard D Lagragia polyomials i. f we icorporae Eq. (14) io he virual work priciple give i Eq. (6), we will derive equilibrium equaios of sigle FE i marix form. is possible o derive he siffess marix ad he cosise mass marix of he sigle layerwise FE: K M T T B A B B B B d 1 e T T J B B B B D B 1, J 1 d T T 1 e T J T 1, J 1 e e (15) (16) Eq. (15), [A], [B ] ad [D J ] are cosiuive marices of he lamiae, which are derived i [5]. Kiemaic marices are: B 1, x 1, y 1, y 1, x 1, x 1, y 1, x 1, y B 1, y 1, x 1 1 5m (17) Eq. (16),, ad J are ieria erms previously derived i Eq. (7), ad marices of ierpolaio fucios are: m 1 1 m 1 1 3m (18) Dyamic equilibrium equaio of he sigle FE is give i followig marix equaio: M K f (19) Eq. (19), {} ad {f} are vecors of odal displacemes ad forces of he sigle FE, respecively. f we observe discree ime poi, marix equaio which saisfies Eq. (19) is: M K f () Layerwise elemes suffer from he pheomea such as spurious shear siffess. Because of his, Selecive iegraio scheme S1 is used o avoid he shear lockig i calculaio. his scheme, all erms i he eleme siffess marix which coai he rasverse shear sifesses Q 44, Q 45 ad Q 55 64

8 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) are compued usig reduced (11) iegraio. All remaiig erms are calculaed usig full () iegraio. egraio over each FE is performed usig Gauss-Legedre quadraure. 5. Trasie aalysis he precedig secios, we have derived he goverig parial differeial equaios of he problem. he aalyical soluio, spaial variaio of displacemes is assumed usig avier mehod, ad i umerical (FE) soluio displaceme field is ierpolaed usig he Lagragia polyomials. his way we have derived he se of ordiary differeial equaios i ime (13 ad ). These equaios ca be solved exacly usig eiher he Laplace rasform mehod or he modal aalysis mehod []. These mehods require he deermiaio of eigevalues ad eigefucios. Aalyical soluios ca be foud i previous works of Vuksaović, Hio [6-7]. his work, umerical soluio will be adoped, usig ewmark s iegraio scheme for secod-order differeial equaios. 5.1 Time discreizaio he ewmark mehod, acceleraios ad velociies are approximaed usig rucaed Taylor s expasios ad oly erms up o he secod derivaive are icluded []. Therefore soluio is obaied oly for discree imes ad o as a coiuous fucio of ime. Amog several wellkow ewmark iegraio schemes, cosa-average acceleraio mehod is chose for his purpose. This sable scheme provided ha iroduced approximaio error does o grow uboudedly. Deail revisio of umerical ime iegraio is give i []. Here he cosa average acceleraio mehod is explaied. Approximaed ime fucios ad heir derivaives are: (1) Eq. (1), is he ime icreme, is he curre ime poi ad +1 is he ex ime poi i which we seek he soluio. Afer subsiuio, we obai: () Premuliplyig he secod Equaio from () wih [M] +1 ad usig Eq. (13) or Eq. () i +1, we obai (if siffess marix ad cosise mass marix is cosa i all ime pois): Kˆ F ˆ 1 (3) (4) ˆK K M F ˆ F M 1 Eq. (3) represes a sysem of algebraic equaios amog he discree values (odal displacemes vecor i FEA or vecor of Fourier coefficies i avier soluio), a ime +1 i erms of kow values a ime. is obvious ha iiial values of displacemes, velociies ad acceleraios are eeded for obaiig rasie respose of he srucure. Firs wo are kow from homogeuous iiial codiios. However, acceleraio vecor should be calculaed from followig expressio: 65

9 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) M F K M F (5) 6. umerical examples ad discussio Proposed mehodology of obaiig he rasie respose hrough he aalyical ad umerical mehod was ivesigaed o several examples preseed i his chaper. Homogeeous iiial codiios (zero displacemes ad velociies) were assumed i all cases. Prelimiary calculaios showed ha umber of members i double rigoomeric series does o affec he resuls severely [8]. he FE calculaios, 1x1 mesh of 4-ode quadrilaerals is chose. All plaes are simply suppored o all sides. Wheever is possible, obaied resuls from he prese model are compared wih exisig soluios from he lieraure. Some ew resuls are preseed. odimezioalized ceer rasverse deflecio is preseed i all examples: 3 1Eh w w qa 4 (6) all calculaios i was assumed ha lamiaed srucure is composed from arbirary umber of layers, which have he same mechaical properies, as i examples from he works of Reddy []: E 1 = /cm G 1 = G 13 = /cm 1 =.5 E = /cm G 3 = /cm = s /cm 4 (7) 6.1 fluece of ime icreme fluece of ime icreme was ivesigaed wih hi cross-ply composie plaes wih characerisics give i (7): -layer plae (/9) ad 4-layer plae (/9). Differe ime seps were used. crease of ime icreme reduced he ampliude of oscillaios, bu icreased he period, as showed o Figures 4-7. Plaes were exposed o uiformly disribued sep loadig, ad calculaio was made usig boh aalyical mehod ad FEM, as explaied i precedig secios. Maximum rasie ceer rasverse deflecios i boh cases are abou imes ha of he saic ceer rasverse deflecio. Boh plaes are squared. Plae dimesios are: a = b = 5 cm. Overall plae heigh is h = 1 cm (a/h = 5 hi plae). Figure 4. Aalyical soluio for (/9) lamiae for differe ime seps Figure 5. FEM soluio for (/9) lamiae for differe ime seps 66

10 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) Figure 6. Aalyical soluio for (/9) 4 lamiae for differe ime seps Figure 7. FEM soluio for (/9) 4 lamiae for differe ime seps -layer plae (Aalyical): -layer plae (FEM): 4-layer plae (Aalyical): 4-layer plae (FEM): wmax, dyamic wmax, saic wmax, dyamic wmax, saic wmax, dyamic w 8936 max, saic. wmax, dyamic w 8554 max, saic. w d /w s = w d /w s =.363 w d /w s = 1.91 w d /w s = fluece of FE mesh refieme fluece of FE mesh refieme was ivesigaed wih cross-ply plaes made of 4 layers (/9), wih characerisics give i (7). Two side-o-hickess raios were used: a/h=5 ad a/h=5. Figure 8. ormalized deflecio versus ime for (/9) lamiae wih a/h = 5 Figure 9. ormalized deflecio versus ime for (/9) lamiae wih a/h = 5 67

11 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) Plaes were exposed o uiformly disribued sep loadig. Time sep of = 5 s was chose i all calculaios. Covergece of soluio is reached wih 11 FE mesh, so his refieme is adoped as fie mesh ad used i all followig calculaios. Coarse mesh of 44 FE showed a lile uderesimaio of ampliude ad he period i hi plae siuaio (Figure 8). he hick plae siuaio (Figure 9), coarse mesh overprediced he ampliude of oscillaios, ad uderprediced he period slighly. 6.3 fluece of shear deformaio Calculaio was performed wih cross-ply (/9) composie plaes wih characerisics give i (7). Plae dimesios are: a = b = 5 cm. Two side-o-hickess raios were used: a/h=1 ad a/h=5. Plaes were exposed o uiformly disribued sep loadig, ad calculaio was made usig boh aalyical mehod ad FEM. Time sep of = 5 s was chose. The accuracy of he prese model is verified wih exisig resuls from he lieraure []. As we ca see from Figures 1 ad 11, shear deformaio affecs he resuls i a followig way: period of oscillaios is icreased, ad he ampliude of oscillaios is cosa or icreased. Slighly sroger ifluece is obaied by usig he FEM soluio. is obvious ha ifluece of shear deformaio is much more proouced i he case of hick plaes, while i he hi plae siuaio i is almos eglible. Figure 1. ormalized deflecio versus ime for (/9) lamiae wih a/h = 5 Figure 11. ormalized deflecio versus ime for (/9) lamiae wih a/h = fluece of lamiaio scheme The effec of he lamiaio scheme o he rasie respose of lamiaed srucure is ivesigaed usig a cross ply lamiaes wih differe umbers of layers. all cases, uiformly disribued sep loadig was used. Calculaio was performed usig hi lamiaed composie plae, wih a = b = 5 cm, ad a/h = 5. Figure 1 shows ha reducio i he umber of layers hrough he plae hickess leaded o he more flexible plae respose i is icreasig he ampliude as well as he period. By usig more cross-ply layers hrough he same overall plae hickess, siffer respose is obaied. Very good agreeme was obaied bewee aalyical ad FE soluio. 68

12 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) Figure 1. ormalized deflecio versus ime for (/9) lamiae wih a/h = 5 Figure 13. Deformed FE mesh (11) of (/9) lamiae wih a/h = Resuls for differe forcig fucios Forcig fucios, which describe he load chage hrough ime, are aalyzed i his secio. Four differe paers of load chage hrough ime are adoped as show i Figure 14 ad i Eq. (8): Sepped Pulse F( ) F Sie Pulse Triagular Pulse Blas Pulse F( ) F F( ) F 1 F( ) si T F e T (8) Figure 14. Differe forcig fucios Eq. (8), F represes he ampliude of he dyamic loadig, is he curre ime variable, T is he overall ime i which loadig acs, ad is he dampig parameer. For he purpose of his work, =.5 ad T=1s. The effec of he applied forcig fucios o he rasie respose of lamiaed plae is ivesigaed usig a cross ply (/9) lamiaes wih a/h = 1 ad a = b = 5 cm. all cases, uiformly disribued sep loadig was used. Resuls are obaied usig FEM wih 11 mesh, for simply suppored (SS) ad clamped (CC) lamiaed plae. Figures 15 ad 16 illusrae he resuls of he calculaio. Aalyical soluios usig Fourier series ad Covoluio egral for simply suppored Midli plaes are i deail explaied i previous work of Hio, Vuksaović [6]. Also, FE resuls are give i Hio, Vuksaović [7]. As saed i [6], riagular pulse is used o simulae a uclear blas loadig. The expoeial pulse may be used o simulae a high explosive blas loadig. Dampig parameer is adjused o approximae he pressure curve from he blas es. 69

13 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) Figure 15. ormalized deflecio versus ime for (/9) SS lamiae wih a/h = 1 Figure 16. ormalized deflecio versus ime for (/9) CC lamiae wih a/h = 1 7. Coclusios his paper aalyical ad FE soluios are preseed for liear rasie aalysis of lamiaed composie plaes. Geeralized Lamiaed Plae Theory is iroduced usig dyamic versio of virual work priciple. Origial MATLAB code is applied for obaiig he rasie respose. fluece of ime sep legh o he soluio accuracy was ivesigaed ad resuls for differe sackig sequeces are preseed. is obvious ha larger ime sep icreases he period of oscillaio, ad reduces he ampliude. Before he furher umerical simulaios, ifluece of FE mesh refieme was ivesigaed. Oce he covergece was reached, fie mesh was chose for all followig calculaios, as show i examples. Usig he exisig examples from he lieraure, i is show ha usig he ESL plae heories, especially for he hick plae problem, uderpredics he ampliudes of oscillaio. Sackig sequece affecs he plae rasie respose i a followig maer - reducio i a umber of lamias hrough he same overall plae hickess leads o a more flexible plae behavior (icreasig he ampliude as well as he period). Fially, differe forcig ime fucios for uiformly disribued pressure load were applied, ad plae respose was obaied. These ew resuls are preseed for simply suppored ad clamped cross-ply lamiaes, wih a/h = 1. Auhors furher research will be aimed o he umerical aalysis of lamiar composies wih he presece of delamiaios (aural frequecies, rasie respose, liear ad oliear bedig ad bucklig loads). Some ivesigaios are already made i he field of crack propagaio, oo. Ackowledgemes The fiacial suppor of he Goverme of he Republic of Serbia - Miisry of Educaio ad Sciece, uder he Projec TR-3648, is graefully ackowledged. 7

14 Miroslav S. Marjaović, Đorđe M. Vuksaović / Aca Techica apocesis: Civil Egieerig & Archiecure Vol. 56 o (13) Refereces [1] Ćeković M, Vuksaović Đ. Bedig, free vibraios ad bucklig of lamiaed composie ad sadwich plaes usig a layerwise displaceme model. Composie Srucures, Vol. 88, pp. 19 7, 9. [] Reddy J. Mechaics of Lamiaed Composie Plaes: Theory ad Aalysis. CRC Press, [3] Vuksaović Đ. Liear aalysis of lamiaed composie plaes usig sigle layer higher-order discree models. Composie Srucures, Vol. 48, pp. 5 11,. [4] Reddy J, Barbero EJ, Teply JL: A plae bedig eleme based o a geeralized lamiae plae heory, eraioal Joural for umerical Mehods i Egieerig, Vol. 8, pp. 75-9, [5] Vuksaović Đ, Ćeković M. Aalyical soluio for mulilayer plaes usig geeral layerwise plae heory. Faca Uiversiais, Vol. 3, pp , 5. [6] Hio E, Vuksaović Đ. Closed form soluios for dyamic aalysis of simply suppored Midli plaes. : Hio E, edior: umerical Mehods ad Sofware for Dyamic Aalysis of Plaes ad Shells. Swasea, UK: Pieridge Press, 1988, pp [7] Hio E, Vuksaović Đ. Explici rasie dyamic fiie eleme aalysis of iiially sressed Midli plaes. : Hio E, edior: umerical Mehods ad Sofware for Dyamic Aalysis of Plaes ad Shells. Swasea, UK: Pieridge Press, 1988, pp [8] Marjaović M, Vuksaović Đ. Liear Trasie Aalysis of Lamiaed Composie Plaes. Proceedigs of he Firs ieraioal coferece for PhD sudes i Civil Egieerig CE-PhD Cluj-apoca, Romaia, 1, pp