6.2 The Moment-Curvature Equations

Transcription

1 Secio The ome-crare Eqaios 6.. From Beam Theor o Plae Theor I he beam heor based o he assmpios of plae secios remaiig plae ad ha oe ca eglec he raserse srai he srai aries liearl hrogh he hickess. I he oaio of he beam ih posiie p / here is he radis of crare posiie he he beam beds p (see Book I Eq ). I erms of he crare / / here is he deflecio (see Book I Eq ) oe has (6..) The beam heor assmpios are esseiall he same for he plae leadig o srais hich are proporioal o disace from he eral (mid-plae) srface ad epressios similar o 6... This leads agai o liearl arig sresses ad ( is also ake o be ero as i he beam heor). 6.. Crare ad Tis The plae is iiiall deformed ad fla ih he mid-srface lig i he plae. Whe deformed he mid-srface occpies he srface ( ) ad is he eleaio aboe he plae Fig iiial posiio Fig. 6..: Deformed Plae The slopes of he plae alog he ad direcios are / ad /. 5 Kell

2 Secio 6. Crare ecall from Book I 7.4. ha he crare i he direcio is he rae of chage of he slope agle ih respec o arc legh s Fig. 6.. d ds. Oe fids ha / Also he radis of crare / / Fig. 6.. is he reciprocal of he crare 3/ (6..) /. s Fig. 6..: Agle ad arc-legh sed i he defiiio of crare As ih he beam he he slope is small oe ca ake a / ad d / ds / ad Eq. 6.. redces o (ad similarl for he crare i he direcio) (6..3) This impora assmpio of small slope / / meas ha he heor o be deeloped ill be alid he he deflecios are small compared o he oerall dimesios of he plae. The crares 6..3 ca be ierpreed as i Fig as he i icrease i slope alog he ad direcios. 6 Kell

3 Secio 6. A B C D C A A B Figre 6..3: Phsical meaig of he crares Tis No ol does a plae cre p or do i ca also is (see Fig. 6..3). For eample sho i Fig is a plae dergoig a pre isig (a applied isig momes ad o bedig momes). Figre 6..4: A isig plae If oe akes a ro of lie elemes lig i he direcio emaaig from he ais he frher oe moes alog he ais he more he is Fig Some of hese lie elemes are sho i Fig (boom righ) as eied lookig do he ais oards he origi (elemes alog he ais are sho boom lef). If a lie eleme a posiio has slope / he slope a is / ( / ) /. This moiaes he defiiio of he is defied aalogosl o he crare ad deoed b /T ; i is a measre of he isiess of he plae: T (6..4) 7 Kell

4 Secio 6. C A D B D D C B A C A B Figre 6..5: Phsical meaig of he is The sigs of he momes radii of crare ad crares are illsraed i Fig Noe ha he deflecio ma or ma o be of he same sig as he crare. Noe also ha he / he /. / / Figre 6..6: sig coeio for crares ad momes O he oher had for he is ih he sig coeio beig sed he / as depiced i Fig Pricipal Crares Cosider he o Caresia coordiae ssems sho i Fig he secod ( ) obaied from he firs ( ) b a posiie roaio. The parial deriaies arisig i 8 Kell

5 Secio 6. he crare epressios ca be epressed i erms of deriaies ih respec o ad a icreme i is as follos: ih Also referrig o Fig ih Ths (6..5) si (6..6) si (6..7) Similarl for a icreme oe fids ha si (6..8) Eqaios ca be iered o ge he ierse relaios si si (6..9) o Figre 6..7: To differe Caresia coordiae ssems The relaioship beee secod deriaies ca be fod i he same a. For eample si si si si (6..) 9 Kell

6 Secio 6. Kell 3 I smmar oe has si si si si si (6..) ad he ierse relaios si si si si si (6..) or T T T T si si si si si (6..3) These eqaios hich rasform beee crares i differe coordiae ssems hae he same srcre as he sress rasformaio eqaios (ad he srai rasformaio eqaios) Book I Eqs As ih pricipal sresses/srais here ill be some agle for hich he is is ero; a his agle oe of he crares ill be he miimm ad oe ill be he maimm a ha poi i he plae. These are called he pricipal crares. Similarl js as he sm of he ormal sresses is a iaria (see Book I Eq. 3.5.) he sm of he crares is a iaria : (6..4) If he pricipal crares are eqal he crares are he same a all agles he is is alas ero ad so he plae deforms locall io he srface of a sphere. hese eqaios are alid for a coios srface; Eqs. 6.. are resriced o earl-fla srfaces. his is ko as Eler s heorem for crares

7 Secio Srais i a Plae The srais arisig i a plae are e eamied. A comprehesie srai-sae ill be firs eamied ad his ill he be simpilfied do laer o arios approimae solios. Cosider a lie eleme parallel o he ais of legh. Le he eleme displace as sho i Fig Whereas as sed i he preios secio o crares o deoe displaceme of he mid-srface here for he mome le ( ) be he geeral erical displaceme of a paricle i he plae. Le ad be he correspodig displacemes i he ad direcios. Deoe he origial ad deformed legh of he eleme b ds ad ds respeciel. The i chage i legh of he eleme (ha is he eac ormal srai) is sig Phagoras heorem ds ds pq pq ds pq (6..5) ( ) p q mid-srface p q ( ) ( ) Figre 6..8: deformaio of a maerial fibre i he direcio I he plae heor i ill be assmed ha he displaceme gradies are small: of order ε sa so ha sqares ad prodcs of hese erms ma be egleced. Hoeer for he mome he sqares ad prodcs of he slopes ill be reaied as he ma be sigifica i.e. of he same order as he srais der cerai circmsaces: 3 Kell

8 Secio 6. Eq o redces o (6..6) Wih / for oe has (ad similarl for he oher ormal srais) (6..7) Cosider e he agle chage for lie elemes iiiall lig parallel o he aes Fig Le be he agle rpq so ha / is he chage i he iiial righ agle rpq. r p s q r p r r q q q Figre 6..9: he deformaio of Fig shoig shear srais Takig he do prodc of he of he ecor elemes p q ad p r : 3 Kell

9 Secio 6. pq rr qq r r qq pr pq pr (6..8) Agai ih he displaceme gradies / / / / / of order ε (ad he sqares / a mos of order ε ) (6..9) For small si so (ad similarl for he oher shear srais) (6..) The ormal srais 6..7 ad he shear srais 6.. are o-liear. The are he sarig poi for he arios differe plae heories. Vo Kármá Srais Irodce o he assmpios of he classical plae heor. The assmpio ha lie elemes ormal o he mid-plae remai ieesible implies ha (6..) This implies ha so ha all paricles a a gie hrogh he hickess of he plae eperiece he same erical displaceme. The assmpio ha lie elemes perpediclar o he mid-plae remai ormal o he mid-plae afer deformaio he implies ha. The srais o read 33 Kell

10 Secio 6. Kell 34 (6..) These are ko as he Vo Kármá srais. embrae Srais ad Bedig Srais Sice ad ) ( oe has from Eq. 6..b ) ( ) ( (6..3) I ca be see ha he fcio ) ( is he displaceme i he mid-plae. I erms of he mid-srface displacemes he (6..4) ad he srais 6.. ma be epressed as (6..5) The firs erms are he sal small-srais for he mid-srface. The secod erms iolig sqares of displaceme gradies are o-liear ad eed o be idered he he plae bedig is fairl large (he he roaios are abo 5 degrees). These firs o erms ogeher are called he membrae srais. The las erms iolig secod deriaies are he fleral (bedig) srais. The iole he crares. Whe he bedig is o oo large (he he roaios are belo abo degrees) oe has (droppig he sbscrip from )

11 Secio 6. Kell 35 (6..6) Some of hese srais are illsraed i Figs. 6.. ad 6..; he phsical meaig of is sho i Fig. 6.. ad some erms from are sho i Fig Figre 6..: deformaio of maerial fibres i he direcio Figre 6..: he deformaio of 6.. ieed from aboe ; a b are he deformed posiios of he mid-srface pois a b p p a q b q b a c p a a p b q q b q mid-srface

12 Secio 6. Fiall he he mid-srface srais are egleced accordig o he fial assmpio of he classical plae heor oe has (6..7) I smmar he he plae beds p he crare is posiie ad pois aboe he mid-srface eperiece egaie ormal srais ad pois belo eperiece posiie ormal srais; here is ero shear srai. O he oher had he he plae dergoes a posiie pre is so he isig mome is egaie pois aboe he mid-srface eperiece egaie shear srai ad pois belo eperiece posiie shear srai; here is ero ormal srai. A pre shearig of he plae i he plae is illsraed i Fig op mid-srface boom Figre 6..: Shearig of he plae de o a posiie is (eaie isig mome) Compaibili Noe ha he srai fields arisig i he plae saisf he D compaibili relaio Eq..3.: (6..8) This ca be see b sbsiig Eq. 6.5 (or Eqs 6.6-7) io Eq The ome-crare eqaios No ha he srais hae bee relaed o he crares he mome-crare relaios hich pla a ceral role i plae heor ca be deried. 36 Kell

13 Secio 6. Sresses ad he Crares/Tis i a Liear Elasic Plae From Hooke s la akig E E E E E (6..9) so from 6..7 ad solig 6..9a-b for he ormal sresses E E E (6..3) The ome-crare Eqaios Sbsiig Eqs io he defiiios of he momes Eqs ad iegraig oe has D D D (6..3) here 3 Eh D (6..3) Eqaios 6..3 are he mome-crare eqaios for a plae. The momecrare eqaios are aalogos o he beam mome-deflecio eqaio / / EI. The facor D is called he plae siffess or fleral rigidi ad plas he same role i he plae heor as does he fleral rigidi erm EI i he beam heor. Sresses ad omes From he sresses ad momes are relaed hrogh 3 3 (6..33) 3 h / h / h / 37 Kell

14 Secio 6. 3 Noe he similari of hese relaios o he beam formla / I ih I h / imes he idh of he beam Pricipal omes I as see ho he crares i differe direcios are relaed hrogh Eqs I comes as o srprise eamiig 6..3 ha he momes are relaed i he same a. Cosider a small differeial eleme of a plae Fig. 6..3a sbjeced o sresses ad correspodig momes gie b O a perpediclar plaes roaed from he orgial aes b a agle oe ca fid he e sresses Fig. 6..3b (see Fig. 6..5) hrogh he sress rasformario eqaios (Book I Eqs ). The d si d si d d (6..34) si si ad similarl for he oher momes leadig o si si si si si (6..35) Also here eis pricipal plaes po hich he shear sress is ero (righ hrogh he hickess). The momes acig o hese plaes ad are called he pricipal momes ad are he greaes ad leas bedig momes hich occr a he eleme. O hese plaes he isig mome is ero. Figre 6..3: Plae Eleme; (a) sresses acig o eleme (b) roaed eleme 38 Kell

15 Secio 6. omes i Differe Coordiae Ssems From he mome-crare eqaios 6..3 { Problem } D D D (6..36) shoig ha he mome-crare relaios 6..3 hold i all Caresia coordiae ssems Problems. Use he crare rasformaio relaios 6.. ad he mome rasformaio relaios o derie he mome-crare relaios Kell