NON-HOMOGENEOUS COMPOSITE BEAMS: ANALYTIC FORMULATION AND SOLUTION

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1 TH NTERNATONAL CONGRESS O THE AERONAUTCAL SCENCES NON-HOMOGENEOUS COMPOSTE BEAMS: ANALYTC ORMULATON AND SOLUTON M Greshten V Rovens M r (eershvl nd O Rnd Technon srel nsttute of Technolog cult of Aerospce Engneerng Hf SRAEL ewords: Non-homogeneous monoclnc em Neumnn prolem hrmonc prolem ulr prolems of plne deformton Astrct The pper presents dervton for the nlss of monoclnc ems of nonhomogeneous cross-secton tht undergo ll dstruted surfce lods nd od forces The dervton hs een proved to e ect smolc computtonl tools nd ncludes llustrtve numercl emples ntroducton Ths pper dels wth nsotropc ems of non-homogeneous cross-secton nmel tht consst of vrous Z-monoclnc mterls where the (- plnes of the cross-sectons concde wth the plnes of elstc smmetr The orgn of the method developed seems to e [] where the prolem soluton for homogeneous sotropc em under surfce lods tht do not vr long the genertors s epressed three hrmonc nd one hrmonc functons n n ndependent w [] presented n nltcl (level sed soluton for the sme prolem when the surfce lods re polnomls of the em s vrle Hence mn of the relted wors re generll referred to s 'Mchell-Almns (recursve method' An nlogous methodolog founded on prescred stress (nd not deformton dstrutons hs een presented much lter [] Smlr prolems were lso dscussed []-[6] nd others The ove solutons were further evolved [7] for homogeneous sotropc ems undergong oth surfce nd od lods The formulton ws frst vld for lods tht m e longtudnll epressed thrd order polnomls onl Lter on [8] [9] etended the method for the cse of Z- monoclnc ems nd generc polnoml lodng Ths pper presents n mproved dervton of the ove pproch We use nottons from [] for non-homogeneous domn nterlmnr nd oundr condtons generc Neumnn-tpe prolems nd the ulr prolems of plne deformton To prove the smolc ectness of the epressons the entre methodolog s documented nd verfed smolcll the Mple progrms The llustrtve emples re lso produced the Mple progrms The soluton hpothess nd procedure The consttutve reltons for Z-monoclnc mterl re gven the lner Hoo's low [ ε ε ε γ γ γ ] [ σ σ σ γ γ γ ] wth postve defnte mtr (wth ndependent coeffcents L N M 6 6 Sm 6 66 O Q P ( Such tpcl mterls m e otned rottng orthotropc mterl out the em ( s The reduced elstc constnts re defned

2 M Greshten V Rovens M r (eershvl nd O Rnd s j j j We ssume the most generl surfce lodng form (per unt re s {X s Y s Z s } nd dstruted od forces (per unt volume {X Y Z } see gure whch re epressed s vector polnomls of degree s { X Y Z }( ( ( ( s s s { X Y Z }( ( ( ( ( Ech soluton level s drven ts level lodng nd the qunttes otned n prevous (hgher levels Generl Vew A Cross-Secton g Bem Hence the cse of stnds for unform dstruted lods n the -drecton stnds for lner dstruton etc n generl we let ech of the two stress components to e polnomls of degree n whle σ σ nd σ re polnomls of degree n s ( ( { } { } ( ( ( ( { σ σ σ } { σ σ σ } ( ( ( whereσ re functons of n j homogeneous cse we let ech of the three stress components σ σ to e epressed s polnomls of degree n A scheme of the level-sed soluton methodolog s presented n Tle The process s ntted for nd contnues for lower levels down to or ech level set of the hrmonc nd the Neumnn prolems n nonhomogeneous domn should e solved Tle Soluton Procedure for All Non-Unform Lodng of Z-Monoclnc Bem As shown Tle the soluton ngredents re grdull ntroduced ccordng to ther level of ppernce Component Level σ σ γ γ σ σ ε ε ε ε γ γ u v w w H S S p q d d Φ L M X Y Z Xs Ys Zs U U Q P Q P Q P Tle Mml Level of Appernce nd Smmetr of Vrous Soluton Components

3 NON-HOMOGENEOUS COMPOSTE BEAMS: ANALYTC ORMULATON AND SOLUTON t should e noted tht the soluton s 'smmetrc' under the followng prmeter nterchnge: 5 X Y X s Y s etc One m otn smmetrc epressons usng operton 'Sm' or emple u v mens u Sm( v nd v Sm( u Also χ χ ϕ ϕ Q P Q χ P χ Q P D Sm( D nd Sm( σ Sm( σ Ths soluton does not ensure tht the three forces nd three moments t the em tp vnsh see stress ntegrl { σ σ σ } ( { P M M P P M } Hence one needs to supermpose seres of solutons for tp lods (see [] n order to cncel out these resultnts Stress components The detled stress epressons re σ U ( ( ( { Φ [ H d ( S ] ( d σ p σ ( ( ( q σ } ( ( ( ( ( ( ( σ { ( Φ U ( Φ U p q ( 6Φ ( [( d ( ( 6 H 6 ( ( d S S σ ( ] ( p σ ( ( [ ( ( ( q( σ ]} H ( H ( d ( S ( H d [ Φ ( ( ( ( p q ( ( ( ( ( ] σ Sm( σ Sm( (5 ( The od force potentls U ( tht pper n the ove terms re ( U X d U ( U Sm( U (6 j ( ( The epressons for S S ( re ( ( ( S [( ( L M ~ ( χ χ ϕ δ p P q P P (7 ( ( d ( u v ] d S Sm( S ( ( where ~ / δ χ P Q χ ~ for nd δ the polnomls P ϕ ϕ Q re defned n [] n (5 nd n ll epressons elow we ( replce H ( the RHS of ( ( ~ H δ ( p χ q χ ϕ (8 where ( ( s n ddtonl seres of longtudnl stress functons s requred the sngle-vlue condtons for the present prolem see tem (e n Secton Remr The prolem wthout sngle-vlued requrements for the hrmonc stress functon m e consdered n ths cse one m ssume p q nd use the hrmonc stress functons H ( onl e wthout defnton (8 Dsplcements The strn components re derved from (5 usng Hoo s low see ( Dsplcements re determned v ntegrton of strns The rgd od dsplcements re not ncluded n these epressons nd re ntroduced the tp lods correcton see Secton :

4 M Greshten V Rovens M r (eershvl nd O Rnd d 6 u { L ( u ( ( p ( ( ( ( p u ( ( [ ( ( q 6 ( ( u ]} ( d w { H [( v Sm( u (9 d ( 6 ( ] ( ( ( p q ( q p ( ( ( } Bhrmonc stress functons Q Y U [ H ( ( ( ( d ( S ( ] ( ( ( ( Q Sm( P P Sm( Q ( ( ( ( ( U U ( ( ( ( ( U U ( U U p q ( 6 ( 6 6 ( {( H 6 ( H ( H ( ( ( d 6 [ ( 6 ( ( ( ( ] [( S S ( ( ( S S ( S S ( ( ]} To enle further hndlng of the oundr condtons we shll lso use the followng generl denttes for C -dfferentle functons: d ( ( r ( r Φ Φ cos( n Φ cos( n ds d ( ( r ( r Φ Φ cos( n Φ cos( n ds The hrmonc stress functons Φ ( ( re governed The hrmonc opertor of ( s defned s ( 6 ( 66 6 ( ( The ellptct of follows from postve defnte Hoo's low ( see [] The / s tpe oundr condtons (c leve the vlues of the functons Φ ( nd the ( dervtves Φ ( Φ undetermned ( ( ( constnt for ech homogeneous domn Φ over ( d component As generl rule we select pont ( ( ( ( { Φ j j Φ } { } n ( ds over ech dvdng curve s ( j d ( ( j ( ( j where we force the functons to e equl for { Φ Φ } { } n j (c ds oth neghor domns nmel { ( ( j L M } { } n j (d ( ( ( j j [ j] { Φ Φ Φ }( [] { } r r where P cos( n Q cos( n nd or the se of smplct we lso ssume P Y ( ( ( H ( ( ( ( { Φ Φ Φ }( { }

5 NON-HOMOGENEOUS COMPOSTE BEAMS: ANALYTC ORMULATON AND SOLUTON The sngle-vlued condtons for hrmonc functon Φ ( on smpl connected (homogeneous domn re see [] ( ( ( ( { } { } ( Susequentl the sngle-vlued tpe condtons necessr for soluton of ( estence on non-homogeneous domn re ( ( ( ( { } j ( ( ( ( [ j] { } { } j [] ( These equltes ecome clerer f ( re wrtten frst for ech domn component [ j ] nd then summed up Longtudnl stress functons The Lplce-tpe opertor nd Neumnn-tpe oundr opertor re see [] ( (5 n D n r ( cos( (5 n r ( cos( ( The longtudnl stress functons ( re governed the Neumnn prolem ( ( ( over (6 n D ( ( on (6 n ( ( j [ ( D ] on j (6c ( j ( j [ ] on j (6d ( ( ( r r where P cos( n Q cos( n nd P ( ( Z d u ( v ( ( ( ( ( ( L M Q ( ( Sm( P (7 ( ( Z ( { ( Φ U ( ( ( ( ( ( Φ U Φ ( p q ( ( ( ( ( L M L M ( [ d H ( S ( S ( ( ]} d ( ε ε γ σ ( ( ( ( ( j ( d { [( ( 6 [ j] ( ]} [] or convenence we ssume ( ( The necessr condton for oundr vlue prolem (6 soluton estence see [] s ( ( [ j] [ ] [] j j ( (8 n contrst to torson nd endng stress ( functons m e dscontnuous long dvdng contours j see (6d 5 Aulr functons The functons L ( ( ( re ( ( ( 6 ( ( ( L Φ Φ { ( Φ U U 6 ( ( [( H S S ( d p ]} {[ (( 6 ( q ( ] } { ( [ ( ( ( 6 ] } (

6 M Greshten V Rovens M r (eershvl nd O Rnd d ( ( {[ ] ( 6 ( [ ] ( ( 6 6 [ ] } ( 6 l ( d { ( ( ( ( l ( [ ( Φ U U ] d 6 ( ( [( H d ( H ( ( ( ( ( [ Φ Φ ( ] [ Φ Φ ( U ] ( 6 ( ( ( ( ( 6[ Φ Φ ( U ] 66[ Φ Φ ( ] ( ( ( ( [ ( L M d ( ( ( ( 6 ( ( ( M L d] M L d d 6 ( u v d ( v u d ( ( v u d d} ( ( ( ( nd M Sm( L m S m( l n the ove for effcent wrtng we hve ntroduced the notton u p u q u ( d u ( ( ( v p v q v ( d v ( ( ( 6 Lodng constnts The lodng constnts p q d for p q X { X ( ( s ( [ ( ( L re defned ( M d u ( v ( ( ]} ( ( (9 q Sm( p p D s s D q D ( ( Y X D D ( Y X ( ( D { [( ( ( ( ( M ( ( ( L ( ( ( ( ( ( d (( u v ( u v ]} d Zs Z ( ( { [ ( ( ( ( ( Φ U ( Φ U ( p q ( ( S ( d ( ( 6 6 H where j re defned n [] ( ( σ S ( ]} 6 Φ ( { } ( σ { } ( { } ( σ { } nd DD D re ( ( D Q ϕ P ϕ [ ( ϕ ( ϕ ] ( D χ χ ϕ ϕ ( P Q P χ Q χ As shown n [] the estence condtons for χ χ requre selecton of the coordnte sstem orgn so tht ( ( { σ σ } { } Verfcton of soluton hpothess Equtons (5 provde n ect soluton tht stsfes ll requrements of the theor of elstct To crr out the ove ts we 6

7 NON-HOMOGENEOUS COMPOSTE BEAMS: ANALYTC ORMULATON AND SOLUTON emplo the equlrum equtons the comptlt equtons nd the outer surfce oundr condtons The process s roen nto susequent steps where n generl t ech one we use ll relevnt reltons tht were found n prevous steps ( Equlrum equtons Consderng the stress terms of (5 nd usng the fct tht the solutons of the ulr prolems stsf the equlrum equtons one m verf tht the equlrum equtons wth X nd Y re stsfed dentcll rom the thrd equlrum ( equton (wth Z the terms of n (7 re etrcted s the coeffcents of the 'free term' ( Boundr condtons rom the outer contour condton of X s Y s one m deduce the The out-of-plne dsplcement component: Contnut of w m e verfed emnng (9c Ths equton shows tht the coeffcents of nd do not contrute n dscontnut due to the fct tht the lodng constnts re not domn dependent or lower levels ths contnut condton s mposed s prt (6d of the Neumnn prolem for ( (e Sngled-vlue condtons The snglevlued-tpe condtons ( for Φ ( nd ts frst dervtves eld the epressons (9-c for the constnts q p respectvel (f Estence of the longtudnl stress functon The estence condton (8 of longtudnl stress functon ( elds the defnton (9d of the constnts d terms for the dervtves d ( Φ d ( Φ over the ds ds Applctons contour see ( rom the thrd oundr condton of Zs the ddtonl terms for the norml dervtves D n ( Homogeneous em under constnt presented n (7 re otned s the coeffcents of dstruted od force The lodng constnts of homogeneous em (c Comptlt equtons The terms of the re p X s X ( ( ( strn components stsf the comptlt { [ equtons due to the fct tht the solutons of the ( ( ( ulr prolems re consstent nd nherentl ( ( L M ]} stsf the comptlt equtons q Sm( p d Zs Z ( ( ( (d Dsplcement nterfce contnut S S { [ 6Φ The n-plne dsplcement components: ( ( ( ( ( Φ U ( Φ U Contnut for levels > see (9: follows ( from the fct tht lodng constnts of Secton 6 ( ( ( H 6 re not domn dependent (e constnts over the entre non-homogeneous domn p q S ( S ( Dsplcement contnut for levels ( ]} s cheved the terms of (9: whch cncel out the dscontnut of u v or We derve here the longtudnl nd levels dsplcement contnut s prt of hrmonc stress functons Φ for the hrmonc prolem snce eplct use of homogeneous Z-monoclnc em tht (9: the nterfce condtons of ( m undergoes constnt od lods e lso e wrtten s Hence we set X X Y Y Z Z where X Y Z re constnts whle no other surfce or ( ( [ j] { u v } { } on [] j 7

8 M Greshten V Rovens M r (eershvl nd O Rnd tp lods re ppled The lodng constnts p q nd d n ( re p S S X Y d Z q We emplo (7 to fnd Z P Z Q Z The soluton s ndependent of the domn shpe Z ( ( or the hrmonc stress functon Φ ( show the followng epnsons: X Y S [ ( 6 ( 6] r [ Y ( pχ qχ]cos( n Sm( Here we used epresson (8 for H ( nd (6 for U U Homogeneous em under lner od force dstruton We consder here Z-monoclnc em of generc cross secton when the od force dstruton s gven ( ( ( X γ Y Z γ whle ll other surfce nd tp lods vnsh A phscl emple for such lodng s the rottng em shown n gure g Notton for Rottng Bem n such cse γ γ ρ / where ρ s the specfc weght (denst of the mterl nd s the ngulr veloct We shll now dscuss the two levels of ths prolem soluton one one Level : Equtons (9 show tht ( p q d Z S γ or the hrmonc functon ( we fnd ( γ P ( γ Q ( γ Anlogousl to the soluton presented n ( ( s ndependent of the domn shpe ( γ ( ( Level : Equtons (9 nd Green Theorem show tht ( ( ( p [ X ( ] q Sm( p nd tht d Equtons (7 show tht ( At ths stge one should solve the hrmonc prolem ( for Non-homogeneous em under constnt l od force Consder non-homogeneous Z-monoclnc em tht undergoes constnt od force n the -drecton nmel Z Z const X Y whle no other surfce or tp lods re ppled The lodng constnts of (9 re p q d ZS / nd ZS ( ( D ( ] ( ( v u [( u v where D re gven (: (: Snce onl the stress functons Φ nd the ulr functons L M of level should e consdered nd hence for the se of convenence n wht follows we shll omt the 8

9 NON-HOMOGENEOUS COMPOSTE BEAMS: ANALYTC ORMULATON AND SOLUTON nde superscrpt The Neumnn prolem (6 for ( should e wrtten wth P d u ( v ( ( j d { [( [ j] 6 [] where B depends on ϕ nd A s lner dfferentl opertor of Φ We ssume ( ( ( { Φ Φ Φ }( { } The stresses ecome σ Φ d σ σ Sm( σ Φ ( Sm( Q Sm( P d ( ε ( ε ( γ ( σ ( Z ( ( ]} whle for convenence we ssume ( The hrmonc prolem for Φ( should hve the tpe of ( wth nd L A B M Sm( L d ( Once Φ re determned one needs supermpose sutle St Vennt s solutons of [] n order to cncel out the tp resultnts whch re nduced the ove stresses Let em cross-secton s geometrcll smmetrc out the -s wth nt-smmetrc lmnton sed on orthotropc mterl turned out ngles ±θ for emple nonhomogeneous rectngle see gure n ths cse the elstc modul re dentcl n two domns [] [ ] ecept for nd tht re of dentcl mgntude ut opposte sgns The sme s true for reduced elstc constnts j B (ntll plcng the coordnte sstem t the cross-secton mdpont show tht the soluton for the thrd ulr prolem see [] s ero Φ ( nd ( ( ( ε ε σ Hence S / nd the dsplcements re rgd ( 6 u v ( 6 The lodng constnts re p q d Z Z 6 D ( Snce n ths cse the hrmonc prolem ( s homogeneous Φ We otn σ 6 ( Z ( Φ Φ Φ ( σ d Z [] ( 6ϕ d v ( u ( ( ( where ϕ s nown hrmonc functon wth the smmetr ϕ( ϕ( see gure ϕ d - ( ϕ ϕ Q g The uncton ϕ g A -Lmnted Rectngle N The stress soluton tht does not produce n tp lods ecome σ σ σ Z ( l [] Z 6 6 [ ϕ ϕ ( ] [] Sm( 6 9

10 M Greshten V Rovens M r (eershvl nd O Rnd Referrng to (9 the s etenson ecomes u( v( w( Z( l / The sher stresses nd for Z re presented n gures 5 Acdem of Scences of the Georgn SSR 98 v n (n Russn [9] Zvvde RT nd Bereshvl RAGenerlton of Almns prolem for compund nsotropc clndrcl ems Trnsctons of Georgn Poltechncl nsttute 98 v 9 n79-5 (n Russn [] r (eershvl M Rovens V Greshten M nd Rnd O Uncoupled homogeneous em under tp lods Scence nd Engneerng of Composte Mterls To e pulshed [] Novohlov V V Theor of elstct Pergmon Press Ltd London 96 [] Lehnts S G Theor of elctct of n nsotropc od Mr Pulshers Moscow 98 ( ( ( ( g 5 The Sher Stresses 5 References [] Mchell J H The theor of unforml loded ems Journl of Mthemtcs9 v 8- [] Almns E Sopr l deformone de clndr sollectt lterlmente Att dell Acdem Nonle de Lnce 9 v n6 Not : - 8 Not : -8 [] Dhnelde G Y The Almns prolem Proceedngs of Lenngrd Poltechnc nsttute 96 v 5-8 (n Russn [] Love A E H A tretse on the mthemtcl theor of elstct ourth Edton New Yor NY USA: Dover Pulctons nc 9 [5] Mushelshvl N Some sc prolems of the mthemtcl theor of elctct P Noordhoof Ltd Gronngen The Netherlnds 95 [6] osmodmns A S Bendng of nsotropc em under generc dstruted lod Engneerng Bulletn nsttute of Mechncs Acdem of Scences SSSR 956 v -6 (n Russn [7] Ruchde A On one prolem of elstc equlrum of homogeneous sotropc prsmtc r Trnsctons of Georgn Poltechncl nsttute 975 v n (n Russn [8] Ruchde A nd Bereshvl R A On one generled prolem of Almns Bulletn of the