ACCURATE COMPUTATION OF CRITICAL RESPONSE QUANTITES FOR LAMINATED COMPOSITE STRUCTURES

Transcription

1 ACCURATE COMPUTATION OF CRITICAL RESPONSE QUANTITES FOR LAMINATED COMPOSITE STRUCTURES C.S. UPADHYAY, P.M. MOHITE and A. K. ONKAR Department of Aerospace Engneerng Indan Insttute of Technoogy Kanpur Kanpur 86 ABSTRACT Computatons are often empoyed n the desgn and certfcaton phase for arcraft components. Wth the advent of composte materas, conventona anayss toos need to be refned to accuratey account for the compex response characterstcs of components made of composte materas. For amnated composte pates, the frst py faure oad and buckng oad are taken as some of the crtca quanttes of nterest. For these quanttes, t w be shown that uness the mode and the fnte eement dscretsaton are propery chosen, the predcted crtca response quanttes can have sgnfcant error. Further, the effect of randomness n the matera and oadng data on the crtca quanttes w aso be shown. It w be demonstrated that a random anayss s essenta to obtan the dsperson n the crtca response quanttes, due to nherent dsperson n the matera and oadng data. INTRODUCTION Composte materas are beng used ncreasngy n the manufacture of structura components. Due to ther hgh strength to weght rato, composte materas are repacng conventona metas n the fabrcaton of arcraft components. Components wth taored response features are now possbe. For exampe, the whoe wng-box structure of an arcraft can now be co-cured, emnatng the need to produce the sub-components separatey foowed by assemby of the component. In the Indan aerospace ndustr a-composte vehces have become very popuar. The ght combat arcraft and the advanced ght hecopter are exampes of such deveopments. Durng detaed tras, these vehces have proved to have better handng quates. Composte materas are heterogeneous meda. Further, the composte structure s ayered n nature, wth severa ndvdua amnae stacked together. These features of composte structures make the anayss of the response more compcated. The response characterstcs depend on the fbre and matrx matera, the curng process, py orentatontackng sequence, nherent amna and amnate eve faws ntroduced durng the manufacturng process. Hence, the anayss of these structures becomes very chaengng. Specazed anayss toos have to be deveoped for the anayss of composte structures. Ths study focuses on the ssues of reabty of the computatona toos. As exampes, the statc probem s dscussed, wth the goa beng to accuratey represent the macro-eve response of thn, pate-ke amnated structures. Lnearzed eastcty modes w be empoyed throughout ths study. The effect of the mode and the fnte eement dscretsaton w be dscussed. It w be shown that wth proper mode seecton and mesh desgn, accurate computaton of the desred response quanttes s possbe. Here, the frst py faure oad and the buckng oad have been taken as exampes of desred macro-eve quanttes. Sgnfcant dfferences wth the resuts reported n the terature w be demonstrated for these quanttes. Most often a determnstc anayss s reported. However, the matera propertes and oadng data are random n reaty. Dsperson n the data can severey ater the crtca response characterstcs of a structure. In order to get a reastc pcture, dsperson n the crtca quanttes shoud aso be reported aong wth the mean data. A perturbaton based anayss mode w be dscussed n ths study. Ths mode w be empoyed to report the dsperson n the frst py faure oad and the buckng oad for a composte structure. PLATE MODELS The system of parta dfferenta equatons of three dmensona eastcty s generay ntractabe anaytca especay for a ayered medum. The deveopment of cassca theores was motvated to aevate these probems by reducng the dmenson for anayss. For exampe, n case of pates and shes

2 reducton from three to two dmenson reduces the computatona cost and enabes the handng of a arge cass of probems. Tradtona for the pate and she ke thn structuresevera pate theores have been proposed. These can be broady cassfed as: () Shear deformabe theores (HSDT); () Herarchc pate theores and (3) Layerwse theores Shear Deformabe Theory Here, one such theory due to Reddy [] s taken as representatve theory from ths group. It s a thrd order shear deformabe theory. And mposes the condton of paraboc dstrbuton of transverse shear strans through thckness of the pate to satsfy the zero transverse shear stress on the top and bottom face of the pate. Herarchc Pate Theory In these, the dspacement components have a zg-zag or herarchc representaton through the thckness. The herarchc pate modes are a sequence of mathematca modes, the exact soutons of whch consttute a convergng sequence of functons n the norm or norms approprate for the formuaton and obectves of anayss. The constructon of herarchc modes for amnated pates by Babuška et a. [] and Acts et a. [3]. The soutons of the ower order modes are embedded n the hghest order mode and these modes can be adapted accordng to the requrement. In these modes the dspacement fed s gven as product of functons that depend upon the varabes assocated wth the patehe mdde surface, and functons of the transverse varabe. The transverse functons are derved on the bass of the degree to whch the equbrum equatons of threedmensona eastcty are satsfed. Layerwse Theory In these theores, the ndvdua amna has contnuous through thckness representaton of dspacements. In the present stud the ayer-by ayer mode proposed by Ahmed and Basu [4] s adapted. In ths mode, a the dspacement components are represented as product of n-pane functons of same order and out-ofpane approxmatng functons of dfferent order for (,v)) dfferent transverse approxmaton order can be taken for ( u,v)) and w. MATHEMATICAL FORMULATION OF PLATE MODELS u and w. In the ayerwse mode proposed n [9] The generc representaton of the dspacement fed for the pate modes s gven as: u( x, u( x, v( x, φ, w( x, [ () z ] U( x y) where φ ( φ3( φ6( L [ φ( ] φ( φ4( φ7 ( L ( ) ( ) φ5 z φ8 z L () () and { U ( x y) } { U ( x, y) U ( x, y) U ( x, y) U ( x, y) U ( x, y) } T, 3 4 L 8 (3) Note that ( x y), U ( x, y), U ( x y)l U 3 6, U x y, U 4 x, y, U 7 x,,, are the n-pane components of dspacement terms u ( x,. Smar (, ) ( ) ( y)l are the n-pane components of dspacement terms v ( x,. The n-pane components of transverse dspacement w( x, are gven by

3 ( x y), U ( x y)l U 5, 8,. The transverse functons are gven n terms of the normazed transverse coordnate z ˆ ( / t) z (where t s the thckness of the amnate). For the hgher order shear deformabe mode the functons φ ( ẑ) are gven as: φ () φ () z φ () z, φ () φ () z z, z 5 3 z 4 3 φ 6 () z φ7 () z φ8() z φ() z, φ 9 () z φ ( z For the herarchc famy of the pate modes the transverse functons ( ẑ) φ ( zˆ) φ ( zˆ) φ ( zˆ) ; t φ6( zˆ) where ϕ t φ3( zˆ) φ4( zˆ) zˆ ; t 7 { ϕ ( zˆ) -ϕ ()}; φ ( zˆ) { ψ ( zˆ) -ψ ()}; φ ( { ρ ( zˆ) - ρ ()}; Q - 5 Q zˆ zˆ zˆ ( zˆ) dzˆ; ψ ( zˆ) dzˆ; ρ( zˆ) dzˆ Q Q Q Q Q Q Q3 Q - Q 8 t φ are gven as: Here Q are the coeffcents of the goba consttutve reaton, n the goba xyz -coordnate system. For other transverse functons see Acts et a. The present ayerwse pate mode s an mprovement over the mode gven In [4] as the orgna ayerwse mode had same order transverse representaton for a three dspacement components, whereas the present ayerwse mode can have dfferent approxmaton n transverse drecton for ndvdua dspacement components. The dfferent approxmaton for dspacement components s used as suggested by Schwab [5] for a snge amna, to separatey account for bendng and membrane actons. The th dspacement component u, for a prsmatc eement (.e. tranguar n-pane proecton) n the ayer gven as u ( x, where ( pxy + )( pxy + ) u p + p xy and and ( x y) z k u k N ( x, y) M k ( u p z are the n-pane and transverse approxmaton order (for component u ) and N ( x, y) M k, are n-pane and transverse approxmaton functonsespectvey. Smary the other components v and w can be expressed. The transverse approxmaton orders for u and v dspacement components w be the same, whe that for the component w can be dfferent. In ths stud p xy or 3; u z p,, 3; p w z,, 3 w be used. p v z FINITE ELEMENT FORMULATION For a gven th amna, the consttutve reatonshp n prncpa matera drectons s gven as: { ( ) } [ C ( ) ]{ ε ( ) } (4) ( ) ( ) ( ) ( ) ( ) ( ) where { } T ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } { ε ε ε γ γ } T are the stress components for the ayer, and ε are the components of stran. The subscrpts, and 3 denotes ( ) γ the three prncpa matera drectons. The consttutve reatonshp n goba xyz coordnates can be obtaned by usua transformatons. The potenta energ Π, for the amnate s gven by Π {}{} ε dv - q w ds (5) + V R U R Where V s the voume encosed by the pate doman, R + and R - are the top and bottom faces of pate and q(x,y) s the transverse apped oad. The souton to ths probem u ex s the mnmzer of the potenta energy Π.

4 ERROR ESTIMATOR FOR LOCAL QUANTITY OF INTEREST In the anayss of amnates for frst-py faure, accurate computaton of state of stress at a pont s essenta. When the fnte eement anayss s empoyed, the ssue of contro of modeng error (error due to mode empoyed n the anayss of amnate, as compared to three dmensona eastcty) and dscretzaton error becomes mportant. Varous smoothenng based a-posteror error estmaton technques for amnated compostes have been proposed by the authors for the oca quantty of nterest [6]. Further, estmaton and contro of the error n the quantty of nterest and one shot adaptve approach for the contro of dscretzaton error was proposed n [7],[8]. Further, ths approach s used for accurate computaton of crtca oca quanttes n [9]. The quantty of nterest s the stress component, whch contrbutes maxmum to the Tsa-Wu [] frst py faure ndex. Smar procedures can be used for the Hoffman and other faure theores. TSAI-WU FAILURE CRITERION The Tsa-Wu crteron s gven by FITW F + F (6) where F, F are the strength tensor terms and are the stress components. NUMERICAL RESULTS One of the maor goas of ths paper s to do a crtca anayss of varous fames of pate modes, wth respect to the quaty of the pont-wse stresses obtaned usng the modes. The effect of n-pane approxmaton order, mode order and type w aso be nvestgated here. A the modes are subected to rgorous numerca studes to compare the transverse defecton and stress profes for numerous py orentatonstackng sequences and boundary condtons under transverse oadngs. In the present study unform pressurenusoda transverse and constant n-pane oadngs are consdered. Effect of Mode on Accuracy of Pont-wse Data Comparson of Transverse Defectons The goa of ths numerca experment s to compare the vaue of transverse dspacement components obtaned usng varous modes, and n-pane dscretzaton, wth the exact three-dmensona eastcty resuts reported n [], for cross-py amnate sequence wth matera propertes gven n tabe. The pate has dmenson a aong x -axs and b aong y -axs, and s subected to snusoda oadng of the form ( x, y) q ( x, y) sn( πx / a) sn( πy b) q / (7) A edges of the pate are smpy supported (see tabe for a BC s used). The transverse defecton at a / b / s reported n tabes 3 and 4. Note that n a the computatons the ayerwse mode uses ( ) (3,3,) mode (uness specfed), that s, transverse approxmaton for u and v s cubc and quadratc for w. For the herarchc famy fed mode s used, whe for the HSDT mode (3,3,) approxmaton s used. Tabe Matera Propertes for [, ]. Property E E G G 3 υ υ 3 Vaue 5 6 ps 6 ps.5 6 ps. 6 ps.5 Tabe Boundary condtons Boundary Condton At y and yb At x and xa Soft Smpe Support vw uw Camped uvw uvw Free u, v, w u, v, w

5 In ths study foowng case has been studed: o o Square pate wth cross py amnaeuch that outer amnae wth orentaton and tota thckness of o amnae s equa to tota thckness of 9 amnae. Aso amnae wth same orentaton have equa thckness. In ths stud 7 ayered amnate s used. Transverse defecton s nondmensonased as 4 * π Qw w, Q 4G 4 + [ E + E ( + ν 3 )]/( νν ). Here, p xy 3 s used for a modes. q S t Numbers n parenthess show the % error wth respect to exact souton gven n []. * Tabe 3: Non-dmensona transverse defecton ( w ) for 7 ayered cross-py amnate. S Exact [] Layer-wse HSDT Herarchc (.).47 (7.33).444 (5.56) 5.. (.).5 (.56).7 (.39).5.5 (.).993 (.9).4 (.9) From these tabes we observe that:. The ayerwse mode predcts the transverse defecton accuratey for a the aspect ratos.. For the HSDT and herarchc mode wth aspect ratos S > the dspacement s cose to exact. The error s.-3 %. Comparson of Stresses In ths case [/9/]quare amnate wth a edges smpe supported s consdered. A the amnae are of equa thckness. The snusoda oadng s of the form as n above subsecton. The n-pane stresses are, τ / q S a / b / z, τ, z and the transverse stresses as ( ) nondmensonased as ( xx xy ) ( ) xx ( ) xy (, ) ( τ xz ) /( q S) ( τ (, b / ) xz. The n-pane stress components are shown n fg. and transverse stress component s shown n fg.. These are compared wth exact ones gven n []. Fg. [/9/] amnate; SSSS b.c. Drect stresses Equbrum stresses Fg. [/9/] amnate; SSSS, transverse stress

6 Effect of Modes on Accuracy of Predcted Faure Load The amnates consdered are [/9] S and [-45/45/-45/45]. The pate s camped on a edges. The top face q x, y q. The pate dmensons are of the pate s subected to unform transverse oad ( ) a 8.9 mm (9n) and b 7 mm (5 n). The matera propertes are gven n tabe 4. The frst-py 4 faure oad s nondmensonased as FLD q S / E. The resuts obtaned from the present anayss are compared wth those reported n [3]. Tabe 4: Matera propertes for T3/58 Graphte/Epoxy (Pre-preg) [3]. Property Vaue Property Vaue E 3.5 GPa X T 55 MPa E E 33.8 GPa X C 697 MPa G G GPa Y T YC ZT ZC 43.8 MPa G GPa R 67.6 MPa ν.4 S T 86.9 MPa ν 3 ν.49 Py thckness, t 3.7 mm The computed faure oad depends on the accuracy of the amna eve stress. Hence, an adaptve approach wth the capabty to estmate error n the oca stresses and refne mesh accordngy to brng the error down to acceptabe toerance requred. For the fxed mode, the focussed adaptve approach s empoyed to recompute the faure oad. Here, the stress component contrbutng maxmum to the Tsa-Wu frst-py faure crteron s used as the quantty of nterest. In tabe 5 the frst-py faure oads are gven. In these tabes, the superscrpt a shows a the vaues of faure oads and correspondng faure ndex obtaned usng unform mesh, the superscrpt b shows the vaue of the faure ndex obtaned wth the same oad as n a (see fg. 3) and the adapted mesh and the superscrpt c shows the frst-py faure oad for the adapted mesh. Fg. 3 Adapted mesh wth -fed herarchc for [-45/45/-45/45] camped amnate. Tabe 5: Frst-py faure oads; a edges camped, [-45/45/-45/45] amnate under unform transverse oadng, (equbrum stresses) p xy. Mode FLD Xco Yco Layer Locaton FI TW Max. Ref bottom - HSDT a top. yy HSDT b top.8 HSDT c top. -fed a top. yy -fed b top.75 -fed c top. Layer bottom. yy

7 Wth equbrum stresses we observe that:. For the nta mesh, the faure oads predcted by a the modes are ower than those obtaned by [3] (shown wth superscrpt a ) and those obtaned by usng drect stresses.. The ocatons predcted by a the modes are ether cose to one obtaned by [3] or are correspondng symmetrc ponts. The ocatons for both drect stresses and equbrum stresses are same (or correspondng symmetry ponts). 3. Faure oads predcted by the HSDT and herarchc modes are cose whe those predcted by ayerwse are sghty hgher than these. 4. When the dscretzaton error contro s used the faure ndex, for the faure oad obtaned usng adapted mesh, ncreases upto 8%. Ths s due to the ncreased fexbty of the numerca souton for the adapted mesh. 5. Wth the adapted mesh the error n the faure oad computatons can be cose to 5%. 6. The faure ocatons for the HSDT and herarchc modes are n the same regon before and after the use of dscretzaton error contro. It s obvous that a sutaby refned mesh, aong wth proper post-processed vaues of the transverse stresses necessary to obtan reabe vaues of the frst-py faure oad. Here, t can be seen that a sutaby refned anayss s necessary for the standard anayss. In rea fe probems, however, even the matera and the oad data s not determnstc. A proper random anayss s requred to get the true pcture of the nfuence of the dsperson of the nput data. In the foowng, propery controed fnte eement computatons are empoyed to carry out a random anayss. The effect of randomness on crtca desgn data, ke the frst-py faure oad and the buckng oad, are studed. STOCHASTIC FINITE ELEMENT APPROACH Consder a pate wth stochastcay varyng matera propertesubected to random transverse oadng. The pate s assumed to be neary eastc wth a stochastc eastcty tensor fed C k. The goa s to fnd the statstcs of faure ndex under transverse oadng and the statstcs of buckng strength n the case of n-pane oadng. The foowng Green s stran-dspacement reatonshp s used: L NL { ε } { ε } + { ε } (8) Here { ε L } s the near stran and { ε NL } s the geometrc nonnear stran wth components: L u u NL u k u k ε { + } ; ε { } (,, k, 3 ) (9) x x x x The tota potenta energy correspondng to the near state of the system for uncertan stffness can be wrtten as [5]: L L Π u ) Ckε ε kd Γ ( q u dγ (,, k,, 3 ) () The exact souton mnmzes Π on the set of a knematcay admssbe functons denoted by V,.e. u V such that V u H ( ) : u on Γ }. Ths yeds: { L L u k k Π( ) C ε δε d q δu dγ (,, k,, 3 ) () δ Γ where Γ Γ Γ represents the surface of the body. Γ denotes the Drchet part and Γ denotes the Neumann part of the boundary of the body. Accordng to the standard stochastc varatona formuaton n conuncton wth Tayor seres expanson, the zeroth- and frst-order varatona statements are wrtten as:

8 k ( ( k Zeroth-order: C ε δε d q δu dγ ( ( k Frst-order: Ckε δε d + C ε δε d q δu dγ Γ k ( ( k Γ (,, k,, 3 ) () (,, k,, 3 ; r,..., R ) (3) In the case of buckng anayss a near eastostatc probem s frst soved for the reference ref oadng q wth the gven constrants for the pate. The near souton s used for computng the nta ref stress tensor and then these stresses are used to compute the geometrc matrx. The zeroth- and frstorder varatona statements can be wrtten as: ( ( ( ref ) ( N k k cr Zeroth-order: C ε δε d + λ δε d (,, k,, 3 ) (4) ( ( ( ( ref ) ( N ( ref ) ( N k k k cr cr Frst-order: ( C ε + C ε ) δε d + λ δε d + λ δε d (,, k,, 3 ; r,..., R ) (5) where R s the number of basc random varabes chosen for the anayss. Statstcs of frst-py faure oad under transverse oadng Tsa-Wu and Hoffman faure crtera are adopted for the present anayss to predct the faure of a amnate based on frst-py faure anayss. f () F + F (,,..., 6 ) (6) The mean and varance of the faure ndex are expressed as: E[ f ()] F + F (7) + F F + F F m m Var[ f ( )] [ F F Statstcs of buckng strength + F F m m m + F F + F F m m m + F F m m m m ] E[( b r + F F r + F F b )( b The tota mnmum egenvaue or oad parameter takes the foowng form: cr ' λ cr λ ( b ) + λ cr ( b b ) (9) The mean and varance of the mnmum egenvaue can be expressed as: ' r ' s E[ λ ] λ ; Var[ λ ] λ λ E[( b b )( b b )] () cr cr Resuts and dscussons Statstcs of faure oad under transverse oadng cr cr cr The amnates consdered for generatng the resuts are made of T3/58 graphte/epoxy matera wth propertes as sted beow: E 3.5 GPa ; E E33.8 GPa ; G GPa ; G G3 5.7 GPa ; ν ν 3. 4 ; ν 3.49 r r s s s b s )] (8)

9 The utmate strengths for the above matera whch are used to cacuate the strength parameters are defned as: X T 55 MPa ; X C 697 MPa ; YT ZT 43.8 MPa ; YC Z C 43.8 MPa ; R 67.6 MPa ; S T 86.9 MPa In the present anayss the eastc modu ( E, E, G, G 3, ν, ν 3 ) of the matera are treated as ndependent random varabes. The amnated pate s subected to a unform dstrbuted random oad. Vadaton - Smpy supported (SSSS) symmetrc cross-py amnated pate A thn square amnated pate consstng of four ayers [ / 9 / 9 / ] of equa thckness wth b / h havng a edges smpy supported s consdered for the present vadaton. A unformy dstrbuted oad s apped on the top surface. Both Tsa-Wu and Hoffman faure crtera are used to compare the mean faure oad and the statstcs of the faure ndex. In order to vadate the ayer-wse mode mpementaton, the faure oads obtaned usng the ayerwse mode are compared wth that obtaned usng a cosed form souton and Krchoff-Love (K mode. From the resuts gven n Tabe 6, t can be observed that the mean faure oad s cose to that obtaned usng a cosed-form souton. The ayerwse mode gves ower faure oads because ths mode s ess stffer than the KL mode. Further for thn pates the behavor s accuratey predcted by the KL mode,.e. shear effect are neggbe. Ths s the reason why the ayer-wse mode and KL mode gve very cose vaues of the faure oad. Tabe 6: Comparson of the mean faure oad for [ / 9 / 9 / ] square amnate wth b / h havng SSSS boundary condton. Mean faure oad (MPa) Faure crtera SFEM Cosed form Tsa-Wu Hoffman In order to vadate the SFEM mpementaton the effect of randomness of the matera data and oadng, on the cacuated faure oad cacuated. In Tabe 7 the coeffcent of varaton (COV) of the faure oad s reported, wth respect to change n the random nput varabes, for the thn symmetrc cross py amnate consdered above. From the resut t can be noted that: () The COV of faure oad obtaned usng SFEM s cose to that obtaned usng the cosed form souton. () The faure oad s very senstve to change n the nput data. For exampe, for a COV of 4% n nput matera data, the COV of faure oad s 3%. (3) The Tsa-Wu and Hoffman measures gve smar COV for the faure oad Ths exampe vadates the SFEM mpementaton of the present study. It shoud be noted that the two faure modes,.e. Tsa-Wu and Hoffman, gve dfferent vaues of the mean faure oad (Tabe 6). It s aso note worthy that effect of matera defects on the faure oad s sgnfcant. Tabe7: Comparson of COV of faure ndex for [ / 9 / 9 / ] square amnate wth b / h havng SSSS boundary condton. sd/mean of a random varabes Tsa-Wu sd/mean of faure ndex f () Hoffman SFEM Cosed form SFEM Cosed form

10 Second order faure statstcs of ant-symmetrc amnated pates The probabstc faure of the ange-py amnated pates wth three dfferent boundary condtons SSSS, SCSC and SFSF are used n ths study. The pate thckness rato b / h has been used n ths study. Two dfferent py schemes for the pate used n the study are: Py scheme : [/9]; Py scheme : [-45/45] The mean faure oad predcted by Tsa-Wu and Hoffman crtera by keepng faure ndex equa to.8 are sted n Tabe 8 for square pate wth dfferent ay-ups and boundary condtons. These faure oads ensure that f a factor of safety of. were chosen for the desgn, the pates woud st be assumed to be safe n the determnstc envronment. Tabe 8: Mean faure oad for square pate wth dfferent ay-ups and boundary condtons. Py schemes [/9] [-45/45] Faure crtera Tsa-Wu Hoffman Tsa-Wu Hoffman Mean faure oad for boundary condtons SSSS SCSC SFSF Smutaneous Varaton of A BRVs The effects of matera propertes on faure ndex of composte amnated pates under transverse random oadng are now presented. The varatons of non-dmensonased faure ndex (FI) wth dsperson n a the basc random varabes (BRV) changng smutaneousy for py schemes and havng SSSS and SCSC boundary condtons are presented n Fgures 4 and 5 respectvey. It s found that: () Ange py s more affected by dsperson n the nput varabes compared to cross-py amnate. () Boundary condtons aso pay an mportant roe n the stochastc anayss. (3) The varaton n FI s most senstve for SCSC boundary condton. Fgure 4 - Infuence of SD/mean of a basc random nputs changng smutaneousy on COV of faure ndex for dfferent amnates wth SSSS boundary condton and b/h. Fgure 5 - Infuence of SD/mean of a basc random nputs changng smutaneousy on COV of faure ndex for dfferent amnates wth SCSC boundary condton and b/h.

11 Statstcs of buckng strength under n-pane compressve oadng Vadaton A graphte/epoxy antsymmetrc cross-py amnated pates wth E / E 4, G / E.6, G 3 / E.5, ν ν 3.5 and b/h5 are consdered under unaxa compressve oadng. Tabe 4 shows mean normazed buckng oad of two-ayer antsymmetrc cross-py square pate wth b/h5, havng dfferent boundary condtons. The effect of both pre-bucked stress and unform stress assumpton on the buckng oad s compared wth those obtaned usng Reddy s pate theores [4]. Further the effect of equvaent ayer, by assumng both matera ayers as an equvaent souton ayer, on the buckng oad usng unform stress assumpton s aso presented n the tabe. It s observed that conventona D pate modes overpredct buckng oads compared to ayer-wse pate mode for the pate consdered. Aso the reference uses unform stress assumpton whch s ony true for specfc amnates wth specfc boundary condton and s not true n genera. Buckng oad of a pate usng pre-bucked stress s found to be smaer compared to anaytca souton usng unform stress assumpton wth equvaent snge ayer consderaton. Tabe 4: Comparson of mean buckng oad for [/9] amnates wth dfferent supports and b/h5 Varous Boundary Condtons ) cr 3 Mean Buckng oad N λ b /( E h ) Present resut (ayer-wse pate mode) wth pre-bucked stress (ayer-by-ayer) wth unform stress (equvaent ayer) SSSS SCSC Varance of buckng strength Reddy s resut [4] (varous D pate mode) (HSDT) 8.77 (FSDT).957 (CPT).49 (HSDT) (FSDT) 3.8 (CPT) Second order statstcs of buckng oad s evauated usng stochastc fnte eement method. In the present anayss the eastc modu ( E, E, G, G 3, ν, ν 3 ) of the matera are treated as ndependent random varabes. The varatons of buckng oad wth % dsperson n a the basc nputs for antsymmetrc cross-py and ange py square pate wth SSSS boundary condtons s presented n Fgures 6 (a) and (b). mt s used to study the effect of matera propertes on crtca buckng oad. It s observed that the effect of dsperson n matera propertes on crtca buckng oad s sgnfcant for Fgure 6: Infuence of a basc random nputs changng smutaneousy on crtca buckng oad parameter for dfferent amnates wth SSSS boundary condton (a) [/9] (b) [-5/5].

12 pates wth ower aspect ratos and t decreases wth ncrease n aspect rato of the pate. CONCLUSIONS In ths stud a systematc procedure s outned for the reabe anayss of composte amnated structures. It has been shown that the dscretsaton error has to be controed n order to get accurate vaues of the response quanttes of nterest. A-pror the rght choce of the mesh, mode and approxmaton are not obvous. Heurstc reasonng can be dsastrous. As has been demonstrated a feedback-based anayss, where the error n the desred quantty of nterest s controed mandatory. A more fundamenta ssue s the choce of the rght mode, to represent the physcs (at the desred scae) accuratey. Ths s the subect of ongong research ntatves. Most often a determnstc engneerng anayss s done. Ths can be very far from the reat especay for composte materas, where dsperson n the amna and amnate eve matera propertes can be sgnfcant. Further, the oadng can never be determnstc. Hence, a meanngfu anayss shoud gve an ndcaton of the dsperson n the computed response quanttes, aong wth the mean vaues. Here, we have shown that the effect of the dsperson n the matera and oad data can be sgnfcant for the faure oad, whch s a oca quantty. Whe for the buckng oad, whch s a goba quantt the effect of the dsperson s sma. REFERENCES [] Redd J. N. (984), A smpe Hgher order Theory for amnated composte pates, J. App. Mech,. 5, [] Babuška, I.,. Szabó, B. A. and Acts, R. L, (99), Herarchc modes for amnated compostes, Int. J. Numer Methods Engrg.,33, [3] Acts, R. L, Szabó, B. A. and Schwab, C. (999), Herarchc Modes for amnated pates and Shes, Comput. Methods App. Mech. Engrg, [4] Ahmed, N. U., and Basu, P. K. (994), Hgher-order fnte eement modeng of amnated composte pates, Int. J. Numer Methods Engrg., 37, 3-9. [5] Schwab, C. (996), A-Posteror Modeng Error Estmaton for Herarchc Pate Mode, Numer. Math., 74, -59. [7] Mohte, P. M. and Upadhya C. S. (), Loca quaty of smoothenng based a-posteror error estmators for amnated pates under transverse oadng. Computers and Structures, 8(8-9), [8] Mohte, P. M. and Upadhya C. S. (3), Focussed adaptvty for amnated pates. Computers and Structures, 8, [9] Mohte, P. M. and Upadhya C. S. (4), Accurate computaton of crtca oca quanttes n composte amnated pates under transverse oadng. Communcated to Computers and Structures. [] Tsa, S. W. and Wu, E. M.(97), A genera theory of strength for ansotropc materas. Journa of Composte Materas,5,55-8. [] Pagano, N. J. and Hatfed, S. J. (97), Eastc behavor of mutayered bdrectona compostes. AIAA Journa, (7), [] Pagano, N. J. (97), Exact soutons for rectanguar b-drectona compostes and sandwch pates, J. Composte Materas, 4, -35. [3] Redd Y. S. N., and Redd J. N. (99), Lnear and non-near faure anayss of composte amnates wth transverse shear. Composte Scence and Technoog 44, [4] Reddy J. N. and Khedr A. A. Buckng and vbraton of amnated composte pates usng varous pate theores. AIAA, 989, 7(), [5] Keber M. and Hen T.D. The Stochastc Fnte Eement Method. John Wey & Sons, New York, 99.