4. Classical lamination theory

Transcription

1 O POSI TES GROUP U I V ERSIT Y OF T W ET E omposites orse 8-9 Uniersit o Tente Eng. & Tech lssicl lmintion theor - Lrent Wrnet & Remo ermn. 4. lssicl lmintion theor 4.. Introction hpter n ocse on the ehior o single ler. The thermomechnicl properties o sch ler cn e preicte n the stress-strin reltionship is non. This ill no e etene to the more generl cse o lminte plte. The most common tpe o nlsis ill e eelope net n is non s the clssicl lmintion theor. This plne stress theor mes it possile to relte eternl los in-plne orces n moments to the composite plte eormtions. The nlsis o lmintes ill irst e introce consiering the ehior o simple lminte em ner pre ening. 4.. Lminte ems in pre ening onsiering lminte ems ner pre lerl loing gies n introction to the more generl lminte plte theor. Elementr principles o mechnics o mteril pplie or em clcltions ill e se here. We ill ocs on the min ierences occrring ith isotropic ems. rectnglr lminte em o thicness h n ith is sjecte to ening moment hich is constnt oer the hole length o the em pre ening. The em is me o lers. s shon in ig. 4. the position o ler in the em is eine the istnces - n. It is orth emphsising tht these istnces re ten rom the mile is o the em n not rom the netrl is. The olloing ssmptions re ten:. The plies re perectl one together.. Plne cross-sections hich re initill perpeniclr to the longitinl is o the em remin plne n norml ring lere. c. Ech pl ehes liner elsticll ith no in-plne sher copling 6 ] We ill or simplicit tht the em hs geometricl n propert smmetr ot the mile is. ig. 4.: omposite em ening ith ler nmering sstem ig. 4. shos em element o length sjecte to moment n thereore hing ris o crtre n n ngle eteen the normls to the em is θ. From ssmption n epression or the longitinl strin t istnce rom the mile is is: θ - h/ n h mi -plne

2 O POSI TES GROUP U I V ERSIT Y OF T W ET E omposites orse 8-9 Uniersit o Tente Eng. & Tech lssicl lmintion theor - Lrent Wrnet & Remo ermn. θ θ 4. θ ccoring to ssmption c the longitinl stress t istnce rom the mile is ecomes: σ E 4. Sttic eqilirim gies n epression or the ening moment: h σ 4. h ting into ccont the stress e otin: E σ n elsticit mols E E in ech ler 4.4 The ening moment cn lso e epresse s nction o the elsticit mols o the lminte em E : l l E I ith I 4.5 With I the moment o inerti ith respect to the -is. From 4.4 n 4.5 n epression or the elsticit mols o the em cn e otine: l E E 4.6 I Using this epression it is possile to otin epressions or the election o lminte ems rom elementr mechnics o mteril. Epression or the stress in the th ler σ cn e ritten eliminting in reltions 4. n 4.5: E σ l 4.7 I E This reltion or the stress is similr to the one se or isotropic em correcte the imensionless term in rcet. The stress is thereore iscontinos nction o the em epth in contrst to the stress in n isotropic em. n emple o stress proile o smmetric 6 lers em sjecte to ening moment here E > E E is gien in ig. 4.. σ ig. 4.: Emple o stress proile o em em sjecte to ening moment.

3 O POSI TES GROUP U I V ERSIT Y OF T W ET E omposites orse 8-9 Uniersit o Tente Eng. & Tech lssicl lmintion theor - Lrent Wrnet & Remo ermn. 4.. Theor o lminte pltes The more generl cse o lminte plte ner plne stress conition ill no e nlse. In-plne loing il n sher ill e consiere s loing s ell s moments ening n torsion. s or the lminte em the lers re ssme one together. o restriction is set on the l-p se hich mens tht rios copling eects ill e present. opling eects men tht n in-plne stress pplie to lminte plte m reslt in comple comintion o etensionl lerl n torsionl eormtions. The ierent nottions re eine in ig. 4.. This igre lso eines the ierent coorinte sstems. The position o the lers in the norml irection is eine ith the mi-plne s reerence n not the netrl plne. The ssmptions relent or the nlsis re similr to the one se or the em nlsis n cn e trnslte to:. The isplcements corresponing to the irections n re smll compre to the plte thicness h. The isplcement n re liner nction o the epth plne cross-sections hich re initill perpeniclr to the longitinl is o the plte remin plne n norml ring eormtion. Trnserse sher strins n norml strin re negligile Strin in ler ccoring to ssmption in-plne isplcements re liner nction o the epth. oth cn e epresse ith the isplcement in the mile srce n s reerence: orml strin is negligile ssmption n the norml or trnserse isplcement is constnt or n coorinte. ccoring to the sme ssmption trnserse sher strins re negligile n cn e se in the strinig. 4.: The ierent coorinte sstems se in the lminte plte theor 4.8 n -/ h - n / h θ : Lminte S t mi-plne : Ler S : teril S t θ

4 omposites orse 8-9 Uniersit o Tente Eng. & Tech lssicl lmintion theor - Lrent Wrnet & Remo ermn. O POSI TES GROUP U I V ERSIT Y OF T W ET E isplcement reltions in orer to in n epression or n. The strin in ler n in its coorinte sstem cn e ritten s: 4.9 Using the epressions or the in-plne isplcements 4.8 the in-plne strins re: 6 γ 4. With γ the in-plne strins in the mi-plne. It cn e shon rom geometricl consiertions tht: ; ; 4. ith the plte crtre /. The eqtion 4. reltes the strin in ler ith qntities relte to the lminte: the mi-plne strin n the crtre Stress-strin reltionship o ler n epression or the stress in the th ler s nction o the mi-plne strin n the plte crtre is otine comining 4. ith the reltion.7 otine in chpter. In mtri orm this cn e ritten s: σ Lminte loing-eormtion reltions Eternl orces n moments cting on lminte plte cn e relte to the stress in the ler n then to the lminte eormtion. For emple the il orces per nit ith cn e otine smming the il stresses σ cting on ech ler: σ 4. here σ is the stress in the th ler in the irection ler coorinte sstem. similr epression cn e ritten or the norml orce in the -irection s ell s or the in-plne sher orce. Sstitting 4. in the orce resltnts gies in mtri orm: 5 4

5 omposites orse 8-9 Uniersit o Tente Eng. & Tech lssicl lmintion theor - Lrent Wrnet & Remo ermn. O POSI TES GROUP U I V ERSIT Y OF T W ET E 4.4 This is mostl reritten in the olloing : sm sm γ The -mtri is lso clle the lminte etensionl stiness mtri is smmetric n its components re eine s: 4.6 The -mtri is clle the lminte copling stiness mtri is smmetric n its components re eine s: 4.7 similr eelopment cn e perorme or the moment resltnts. This gies s en reslt the olloing reltions: sm sm γ The -mtri is lso present here. The -mtri is clle the lminte ening stiness mtri is smmetric n its components re eine s: 4.9 The reltions 4.5 n 4.8 re oten ritten in prtitione orm: 4. lthogh the three components o the '' mtri he similr ppelltions stiness the he istinct nits. s loing is mostl epresse "per nit ith" orce resltnts in /m n moments in the -components he or nit /m the -components in n the -components in m.

6 O POSI TES GROUP U I V ERSIT Y OF T W ET E omposites orse 8-9 Uniersit o Tente Eng. & Tech lssicl lmintion theor - Lrent Wrnet & Remo ermn Inersion o the lminte loing-eormtion reltions Inersion o reltion 4. is oten necessr s the loing is generll non n not the other ron. This cn e one simpl inerting the ] mtri s hole. Hoeer perorming the inersion inerting the s mtrices n cn e interesting or prticlr lminte l-p. The reslt is then o the orm: T 4. For emple it cn e shon tht the etensionl complince mtri ] cn e ritten: ] ] ] ] ] ] ] ] ] ] 4. This reltion reces or to ] - hen the components o the ]-mtri re ero lminte is smmetric. omponents o the ]- n ]-mtri cn e elte s ollo: ] ] ] ] ] ] ] ] ] ] ] ] 4. Similrl to the ]-mtri the inerte ening stiness ] components cn e clclte irectl inerting the ]-mtri onl i the components o the ]- mtri re ero. It is orth ing tht the copling complince mtri ] is not smmetric. The reltion eteen the crtre ector n the orce ector is eine the trnspose o the ]-mtri Prolems 4. erie or the ]-mtri or lmintes -45/45] s smmetric n -45/45/-45/45] ntismmetric se on mm thic lers o ron-pei hing the olloing ler properties: E GP E 8GP G 5GP ν. In hich ] terms o these to l-p ier n ht oes this men? 4. Gie on the net igre the phsicl signiiction o the olloing stinessmtri terms: 6 n 6 ; n 6 n 6 6 n 6

7 O POSI TES GROUP U I V ERSIT Y OF T W ET E omposites orse 8-9 Uniersit o Tente Eng. & Tech lssicl lmintion theor - Lrent Wrnet & Remo ermn