FINITE ELEMENT ANALYSIS OF
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1 FINIT LMNT NLYSIS OF D MODL PROBLM WITH SINGL VRIBL Fnt lmnt modl dvlopmnt of lnr D modl dffrntl qton nvolvng sngl dpndnt nknown govrnng qtons F modl dvlopmnt wk form. JN Rddy Modlqn D -
2 GOVRNING TION d d f d d d P t d n Ω L ondry pont P L L k lst dformton of r nnsltd r k L P k L Ht trnsfr n r JN Rddy Modlqn D -
3 FINIT LMNT PPROXIMTION ND FOR SKING SOLTION ON SB-INTRVLS. ppromton of t tl solton ovr t ntr domn rqrs gr-ordr ppromton. tl solton my dfnd y S-ntrvls s of dsontty of t dt. JN Rddy ppromton ovr s-ntrvls sntrvls llows lowr-ordr ppromton of t tl solton Modlqn D -
4 FINIT LMNT DISCRTIZTION ppromton ovr s-ntrvls sntrvls llows lowr-ordr ppromton of t tl solton typl lmnt gomtry nd fors nd fors or ts lmnt lngt JN Rddy Modlqn D -
5 Modlqn D - 5 JN Rddy w w d wf w d d d dw d d w d d w d wf w d d d dw d d w d wf w d d d dw d f d d d d w WK FORM OVR N LMNT WK FORM OVR N LMNT Ht/For npt d d d d Ht npt For ot
6 LINR ND BILINR FORMS ND TH VRITIONL PROBLM Wk Form dw d d d w wf dw d w d d B w l w B w d l w d w Blnr Form nd Lnr Form wf d w w Vrtonl Prolm: Fnd s tt olds for ll w w B JN Rddy dw d d w w d l w wf d w d w Modlqn D - 6
7 IVLNC BTWN MINIMM OF DRTIC FNCTIONL ND WK FORM drt Fntonl: I B - l d d Strn nrgy d d Vrtonl Prolm: Fnd s tt I s mnmm: f Work don y ppld fors δi B δ l δ for ll δ w s t sm s t wk form or t vrtonl prolm wt δ w JN Rddy Modlqn D - 7
8 Modlqn D - 8 JN Rddy FINIT LMNT MODL FINIT LMNT MODL j n j j Fnt lmnt modl { } { } j j j d f F d d d d d K F K ] [ Fnt lmnt ppromton to drvd ltr
9 Modlqn D - 9 JN Rddy PPROXIMTION FNCTIONS FOR PPROXIMTION FNCTIONS FOR LINR LMNT LINR LMNT
10 Modlqn D - JN Rddy LTRNT DRIVTION OF LTRNT DRIVTION OF PPROXIMTION FNCTIONS PPROXIMTION FNCTIONS Lnr lmnt Lnr lmnt j j δ j j f f - - nd nd α α α α ntrpolton fntons ltrnt drvton sng t ntrpolton proprty
11 LTRNT DRIVTION OF PPROXIMTION FNCTIONS drt lmnt ltrnt drvton sng t ntrpolton proprty JN Rddy α α nd nd.5. 5 nd α α α α Modlqn D -. 5
12 Modlqn D - JN Rddy NMRICL VLTION OF COFFICINTS NMRICL VLTION OF COFFICINTS for lmnt for lmnt-ws onstnt dt ws onstnt dt j j j d f F d d d d d K f f dt : For onstnt { } 6 f F K ] [ { } f F K ] [ Lnr lmnt: drt lmnt:
13 NMRICL XMPL - Prolm: Ws to dtrmn t dformton nd strsss n t tr mmrs of t strtr. Stl s s P Brss Rgd mmr onstrnd to mov orzontlly lmnm d d s 8ft..5 n. d. n. P l 6 ft s ps.875 n. 6ft. 6 ps 6 ps Solton: Not tt t govrnng qton s d d d d n lmnt JN Rddy Modlqn D -
14 NMRICL XMPL ontnd W s lnr lmnts to rprsnt t mmrs of t strtr nd ll t lmnts lmnt nods nd glol nods. lmnt nod nmrs Stl s s Brss P Rgd mmr onstrnd to mov orzontlly lmnm Glol nod nmrs T lmnt qtons for typl lmnt r f JN Rddy Modlqn D -
15 NMRICL XMPL ontnd T dsplmnt ondry ondtons prssd n trms of t glol dsplmnts r 5 T for qlrm ondtons r otnd y lookng t t qlrm of fors on t rgd r. W v P P P JN Rddy Modlqn D - 5
16 Modlqn D - 6 JN Rddy NMRICL XMPL NMRICL XMPL ontnd ontnd 6 T qlrm ondtons sggst tt w mst dd t sond qton of lmnt t sond qton of lmnt nd t frst qton of lmnt so tt w n rpl t sm of t s wt P. T lmnt qtons n trms of t glol dsplmnts r sng t ft tt w otn P
17 Modlqn D - 7 JN Rddy NMRICL XMPL NMRICL XMPL ontnd ontnd Ts w n ompt w rprsnts t longton n lmnts nd nd omprsson n lmnt. Ts omplts mpl on n ssttt t gvn dt to otn nmrl vls of t dsplmnts fors nd strsss n t mmrs. 7 W n tn dtrmn t mmr fors sng σ σ σ T mmr strsss r tn n omptd from
18 Modlqn D - 8 JN Rddy NMRICL XMPL NMRICL XMPL Prolm: Ws to dtrmn t nmrl solton of t dffrntl qton < < n d d Solton: W v t followng orrspondn omprd to t modl qton: d f f W ws to s ms of lnr lmnts to solv t prolm. T qtons of typl lmnt r 6
19 NMRICL XMPL ontnd W onsdr ms of lnr lmnts.5. T lmnt qtons r JN Rddy Modlqn D - 9
20 NMRICL XMPL ontnd T ondry ondtons r 5 T qlrm ondtons r 5 JN Rddy Modlqn D -
21 Modlqn D - JN Rddy NMRICL XMPL NMRICL XMPL ontnd ontnd T ssmld qtons r T ondnsd qtons for t nknown s nd s r
22 NMRICL XMPL ontnd Plot of t solton Plot of t drvtv of t solton.. Solton nlytl L FM Solton d/d nlytl L & FM & " & " * & & * & " " & * " *& Coordnt Coordnt JN Rddy Modlqn D -
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