Module 3 : Analysis of Strain

Transcription

1 Mod/Lsso Mod : Aasis of trai.. INTROUCTION T o dfi ora strai rfr to th fooi Fir. hr i AB of a aia oadd br has sffrd dforatio to bco A B. Fir. Aia oadd bar Th th of AB is. As sho i Fir.(b) poits A ad B ha ach b dispacd i.. at poit A a aot ad at poit B a aot. Poit B has b dispacd b a aot i additio to dispact of poit A ad th th has b icrasd b. No ora strai a b dfid as i d d (.) I i of th iiti procss th abo rprsts th strai at a poit. Thrfor "trai is a asr of rati cha i th or cha i shap". Appid Easticit for Eirs T.G.ithara & L.GoidaRaj

2 Mod/Lsso.. TYPE OF TRAIN trai a b cassifid ito dirct ad shar strai. (a) (b) (c) (d) Fir. Tps of strais Appid Easticit for Eirs T.G.ithara & L.GoidaRaj

3 Mod/Lsso Fir.(a).(b).(c).(d) rprst odisioa todisioa thrdisioa ad shar strais rspcti. I cas of todisioa strai to ora or oitdia strais ar i b si appis to oatio; si to cotractio. (.) No cosidr th cha pricd b riht a AB i th Fir. (d). Th tota aar cha of a AB bt is i th ad dirctios is dfid as th shari strai ad dotd b. \ a a (.) Th shar strai is positi h th riht a bt to positi as dcrass othris th shar strai is ati. I cas of a thrdisioa t a pris ith sids d d d as sho i Fir.(c) th fooi ar th ora ad shari strais: (.) Th raii copots of shari strai ar siiar ratd: (.4) Appid Easticit for Eirs T.G.ithara & L.GoidaRaj

4 Mod/Lsso.. EFORMATION OF AN INFINITEIMAL LINE ELEMENT Fir. Li t i dford ad dford bod Fir. Li t i dford ad dford bod Cosidr a ifiitsia i t i th dford otr of a di as sho i th Fir.. Wh th bod dros dforatio th i t passs ito th i t P Q. I ra both th th ad th dirctio of ar chad. Lt th coordiats of P ad Q bfor dforatio b ( )( ) rspcti ad th dispact ctor at poit P ha copots ( ). Th coordiats of P P ad Q ar ( ) P : ( ) P : Appid Easticit for Eirs 4 T.G.ithara & L.GoidaRaj

5 Mod/Lsso 5 Appid Easticit for Eirs T.G.ithara & L.GoidaRaj Q : Th dispact copots at Q diffr siht fro thos at poit P sic Q is aa fro P b ad. \ Th dispacts at Q ar ad No if Q is r cos to P th to th first ordr approiatio (a) iiar (b) Ad (c) Th coordiats of Q ar thrfor Q Bfor dforatio th st had copots ad ao th thr as. Aftr dforatio th st Q P has copots ad ao th thr as. Hr th trs ik ad tc. ar iportat i th aasis of strai. Ths ar th radits of th dispact copots i ad dirctios. Ths ca b rprstd i th for of a atri cad th dispactradit atri sch as j i

6 Mod/Lsso 6 Appid Easticit for Eirs T.G.ithara & L.GoidaRaj..4 CHANGE IN LENGTH OF A LINEAR ELEMENT Wh th bod dros dforatio it cass a poit P( ) i th bod dr cosidratio to b dispacd to a positio P ith coordiats hr ad ar th dispact copots. Aso a ihbori poit Q ith coordiats ts dispacd to Q ith coordiats. No t b th th of th i t ith its copots. \ iiar b th th Q P ith its copots P Q \ Fro qatios (a) (b) ad (c) Taki th diffrc bt ad t Q P { } (.5) hr (.5a)

7 Mod/Lsso 7 Appid Easticit for Eirs T.G.ithara & L.GoidaRaj (.5b) (.5c) (.5d) (.5) (.5f) No itrodci th otatio hich is cad th rati tsio of poit P i th dirctio of poit Q o Fro Eqatio (.5) sbstitti for t If ad ar th dirctio cosis of th bstitti ths qatitis i th abo prssio Th abo qatio is th a of th rati dispact at poit P i th dirctio ith dirctio cosis ad.

8 Mod/Lsso..5 CHANGE IN LENGTH OF A LINEAR ELEMENTLINEAR COMPONENT It ca b obsrd fro th Eqatio (.5a) (.5b) ad (.5c) that th cotai iar trs ik tc. as as oiar trs ik. tc. If th dforatio iposd o th bod is sa th trs ik tc ar tr sa so that thir sqars ad prodcts ca b ctd. Hc rtaii o iar trs th iar strai at poit P i th dirctio ca b obtaid as bo. (.6) (.6a) (.6b) If hor th i t is para to ais th ad th iar strai is iiar for t para to ais th ad th iar strai is ad for t para to ais th ad th iar strai is Th ratios prssd b qatios (.6) ad (.6a) ar ko as th strai dispact ratios of Cach...6 TRAIN TENOR Jst as th stat of strss at a poit is dscribd b a itr arra th strai ca b rprstd tsoria as bo: Appid Easticit for Eirs 8 T.G.ithara & L.GoidaRaj

9 ij i j j i Mod/Lsso (i j ) (.7) Th factor / i th abo Eqatio (.7) faciitats th rprstatio of th strai trasforatio qatios i idicia otatio. Th oitdia strais ar obtaid h i j; th shari strais ar obtaid h i ¹ j ad. It is car fro th Eqatios (.) ad (.) that (.8) ij ji Thrfor th strai tsor ( ij ji ) is i b ij (.9)..7 TRAIN TRANFORMATION If th dispact copots ad at a poit ar rprstd i trs of ko fctios of ad rspcti i cartsia coordiats th th si strai copots ca b dtrid b si th straidispact ratios i bo. ad If at th sa poit th strai copots ith rfrc to aothr st of coordiats as ad ar dsird th th ca b cacatd si th cocpts of ais trasforatio ad th corrspodi dirctio cosis. It is to b otd that th abo qatios ar aid for a sst of orthooa coordiat as irrspcti of thir oritatios. Hc Appid Easticit for Eirs 9 T.G.ithara & L.GoidaRaj

10 Mod/Lsso Appid Easticit for Eirs T.G.ithara & L.GoidaRaj Ths th trasforatio of strais fro o coordiat sst to aothr ca b ritt i atri for as bo: I ra [ ] [ ][ ][ ] T a a..8 PHERICAL AN EVIATORIAL TRAIN TENOR Lik th strss tsor th strai tsor is aso diidd ito to parts th sphrica ad th diatoria as E E E hr E sphrica strai (.) E ) ( ) ( ) ( diatoria strai (.) ad It is otd that th sphrica copot E prodcs o o chas ithot a cha of shap hi th diatoria copot E prodcs distortio or cha of shap. Ths copots ar tsi sd i thoris of fair ad ar sotis ko as "diatatio" ad "distortio" copots.

11 ..9 PRINCIPAL TRAIN TRAIN INVARIANT Mod/Lsso ri th discssio of th stat of strss at a poit it as statd that at a poit i a coti thr ists thr ta orthooa pas ko as Pricipa pas o hich thr ar o shar strsss. iiar to that pas ist o hich thr ar o shar strais ad o ora strais occr. Ths pas ar trd as pricipa pas ad th corrspodi strais ar ko as Pricipa strais. Th Pricipa strais ca b obtaid b first dtrii th thr ta prpdicar dirctios ao hich th ora strais ha statioar as. Hc for this prpos th ora strais i b Eqatio (.6b) ca b sd. i.. As th as of ad cha o ca t diffrt as for th strai. Thrfor to fid th ai or ii as of strai ar rqird to qat to ro if ad r a idpdt. Bt o of th dirctio cosis is ot idpdt sic th ar ratd b th ratio. No taki ad as idpdt ad diffrtiati ith rspct to ad t No diffrtiati ith rspct to ad for a tr t ( ) ( ) bstitti for ad fro Eqatio. t (.) Appid Easticit for Eirs T.G.ithara & L.GoidaRaj

12 Mod/Lsso oti th riht had prssio i th abo to qatios b ad (.a) Usi qatio (.a) ca obtai th as of ad hich dtri th dirctio ao hich th rati tsio is a tr. No tipi th first Eqatio b th scod b ad th third b ad addi th W t ( ) ( ) (.b) Hr Hc Eqatio (.b) ca b ritt as hich as that i Eqatio (.a) th as of ad dtri th dirctio ao hich th rati tsio is a tr ad aso th a of is qa to this tr. Hc Eqatio (.a) ca b ritt as ( ) ( ) oti ( ) Eqatio (.c) ca b ritt as th (.c) Appid Easticit for Eirs T.G.ithara & L.GoidaRaj

13 Mod/Lsso ( ) ( ) ( ) (.d) Th abo st of qatios is hooos i ad. I ordr to obtai a otriia sotio of th dirctios ad fro Eqatio (.d) th dtriat of th cofficits shod b ro. i.. ( ) ( ) ( ) Epadi th dtriat of th cofficits t J J J (.) hr J J J W ca aso rit as J J J 4 4 Hc th thr roots ( ) ( ) ad of th cbic Eqatio (.) ar ko as th pricipa strais ad J J ad J ar trd as first iariat scod iariat ad third iariat of strais rspcti. Iariats of trai Tsor Ths ar asi fod ot b tiii th prfct corrspodc of th copots of strai tsor ij ith thos of th strss tsor t ij. Th thr iariats of th strai ar: Appid Easticit for Eirs T.G.ithara & L.GoidaRaj

14 J J 4 ( ) J 4 ( ) Mod/Lsso (.) (.4) (.5).. OCTAHERAL TRAIN Th strais acti o a pa hich is qa icid to th thr coordiat as ar ko as octahdra strais. Th dirctio cosis of th ora to th octahdra pa ar. Th ora octahdra strai is: ( ) oct \ ( ) oct ( ) (.6) Rstat octahdra strai ( R ) oct Octahdra shar strai oct ( ) (.7) ( (.8) ) ( ) ( ) Appid Easticit for Eirs 4 T.G.ithara & L.GoidaRaj