CIVL 7/ D Boundary Value Problems - Axisymmetric Elements 1/8

Transcription

1 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts /8 xsymmtc poblms a somtms fd to as adally symmtc poblms. hy a gomtcally th-dmnsonal but mathmatcally only two-dmnsonal n th physcs of th poblm. In oth wo, th dpndnt vaabl s a functon of th coodnats and and not a functon of th of th angl u = u(, ). xsymmtc poblms a assocatd wth bo of voluton as ndcatd n th fgu blow h th-dmnsonal Laplacan opato n axsymmtc poblms ducs to u u u whch may b wttn as u u u h Posson bounday valu poblm s f u,, 0 n u g s on u su hs on n wh th sufac s th poton of th sufac wh th Dchlt typ bounday condtons a dfnd and s th potons wh th Numann o Robn bounday condtons a pscbd. h cospondng ngy functonal that wll sv as th bass fo a Rt fnt lmnt modl s u u Z u d d u f d d u uh 0 Du to th mathmatcal natu of th poblm, th analyss may b pfomd wthn a two-dmnsonal gon n th -plan whch s volvd about th -axs. h volvng gon dfnd th actual th-dmnsonal doman.

2 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts /8 h volvng gon dfnd th actual th-dmnsonal doman. h volvng gon dfnd th actual th-dmnsonal doman. Coss-scton and axal schmatc of th coaxal slot antnna Coss-scton and axal schmatc of th coaxal slot antnna Dsctaton - s usual th fst stp s dvlopng a fnt lmnt modl s th ctaton of th poblm gomty. s an ntoducton, w wll lmt ou cusson of th ctaton and fomulaton of th axsymmtc poblm to lna tangls.,,, Dsctaton - In tms of th ctaton, th functonal Z s now a sum of th ntgals ov ach lmnt n th doman, and th sum of sufac ntgals on th bounday sgmnts along u u Z u d d ' u uf d d ' uh 0

3 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts 3/8 Intpolaton - h smplst ntpolaton ov a staghtsdd th nod tangula lmnt s to assum th functon u(, ) s psntd by a lna plan. Intpolaton - Lnaly ntpolatd tangula lmnts psnt th vaaton of th dpndnt vaabl u ov an u, lmnt as u Lna psntaton of, u u u Lna psntaton of, u u u, u,, u,, u, Intpolaton - Lnaly ntpolatd tangula lmnts psnt th vaaton of th dpndnt vaabl u ov an lmnt as u, wh,, and a constant dtmnd by matchng th functon u wth th nodal valus of th lmnt u, u, u, Intpolaton - Solvng th th quatons fo,, and and substtutng bac nto th xpsson psntng th vaaton of u ov th lmnt sults n u, Nu N u N u wh a b c N,, 3 a b c wh,, and a pmutd cyclcally Intpolaton - h dtmnant of th coffcnts s Intpolaton - In matx notaton, th tbuton of th functon ov th lmnt s u, u N N u h lna tangula shap functons a llustatd blow wh s th aa of th lmnt. ny numbng schm that poc countclocws aound th lmnt s vald, fo xampl (,, ), (,, ), o (,, ). hs numbng convnton s mpotant and ncssay n od to comput a postv aa fo. N N N

4 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts /8 Intpolaton - h dvatvs of u ov th lmnt wth spct to both coodnats a u, N N u u Calculatng th dvatvs of th shap functons gvs N b b b b b 3 u, N N u u N c c c c c 3 b c Intpolaton - Obsvng th fom of th dvatv t s appant that th patal dvatvs of th functon u wll b constant ov a lna tangula lmnt. h a many poblms assocatd wth accuacy and convgnc fo ths typ of lmnt. In lastcty analyss, stss and stan a latd by a patal dffntal quaton, usng a lna tangula lmnt to cbd stss wll sult n a constant appoxmaton fo stan ov th lmnt. hfo, lmnts of ths typ a calld constant stan lmnts. Elmntal Fomulaton - h functonal fo th Posson quaton s u u Zu d u uf d uh ' ' 0 W can wt th functonal n th followng fom Z Z Zu Z Z 3 Elmntal Fomulaton Wh th componnts a dfnd as u u Z d d Z 3 u f d d Z Z u u h Elmntal Fomulaton - Evaluaton of Z u u u u Z d Rcall th fst dvatvs of u wth spct to and a u, N N u u u, N N u u Elmntal Fomulaton - Evaluaton of Z Rplacng th dvatvs wth th abov appoxmatons gvs N N N N Z u u u ud d u N N N N u u u d d N N N N d

5 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts 5/8 Elmntal Fomulaton - Evaluaton of Z h ntgals dfnd n a th lmntal stffnss matx. Fo th lna tangula lmnt w hav cussd th stffnss matx ducs to b b c c d d h sultng 3 x 3 lmntal stffnss matx contbuts to th global systm quatons at locatons cospondng to th lmnt no. Elmntal Fomulaton - Evaluaton of Z Z u In ths cas, th ntpolaton of u wth spct to and s usd to cb th bhavo along th bounday Z un Nu u N N u uau a NN h sultng s a x lmntal stffnss matx Elmntal Fomulaton - Evaluaton of Z 3 Z 3 u f d Substtutng th appoxmaton fo u nto th ntgal sults n Z3 u N f uf f Nfd h sultng s a 3 x lmntal load vcto Elmntal Fomulaton - Evaluaton of Z Z u h Substtutng th appoxmaton fo u nto th ntgal sults n Z u N h uh h Nh h sultng s a x lmntal load vcto Elmntal Fomulaton - In tms of th matx dfntons, th functonal may b wttn n th followng fom uu uau Zu, u, u3,, un uf ' uh wh th fst sum s ov th ach lmnt of aa cbng th doman and th scond sum s ov vy lmnt that has a sgmnt along th poton of th bounday. ssmbly - h assmbly s dnotd by th summaton n th matx quaton. h global matx fom of th fomulaton s ug KGuG Z ug FG ZuG K a F f h G G G G G G Z 0 u Z KG KG u F Ku F G u G G G

6 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts 6/8 Constants - h constants on th systm quatons a th focd bounday condtons u = g(s) on th sufac. hs condtons a appld to th systm quatons n a mann smla to that cussd fo on- and twodmnsonal poblms. Soluton - Dtals of th soluton of th smultanous quatons sultng fom axsymmtc bounday valu poblms a psntd n th two-dmnsonal scton of th nots. Computaton of Dvd Vaabls - In ths cas, th valus of th functon u a th pmay vaabls and u/ and u/ a consdd a sconday vaabl. h patal dvatvs a dtmnd by th followng xpssons u, u, u u N N b u u N N c u u Evaluaton of Matcs - Lna angula Elmnts Rcall th lmntal matcs hav th followng fom b b c c d d a NN f h Nfdd Nh hs ntgals a ssntally th sam as th tms dvlopd fo two-dmnsonal Posson s quatons usng lna tangula lmnt. h obvous dffnc s th coodnat whch appas n ach ntgal. Evaluaton of - Substtutng all th pcs of th tansfomaton n th tms gvs b b c c d d bb cc dd bb cc R R 3 h valu of R s th -coodnat of th cntod of th lna tangula lmnt. Evaluaton of f - In gnal, th ntgal f s f Nf xy, d Rplacng by th xact psntaton N, and assumng that th functon f may b appoxmatd by th lna ntpolaton N f, th lmnt matx f bcoms f N NN fd d

7 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts 7/8 Evaluaton of f - h fomula fo ntgatons of th typ s gvn wthout poof as a b c NNN d abc!!! a b c! hfo 6 f f 6 f symmtc 6 f Evaluaton of f - h fomula fo ntgatons of th typ s gvn wthout poof as a b c NNN d abc!!! a b c! If th functon f s a constant, f 0, th abov matx ducs to f 0 f h sultng s a 3 x lmntal load vcto Evaluaton of a a NN Snc a s valuatd along a sgmnt of th bounday, th ntpolaton functons duc to th on-dmnsonal countpats. N 0 N N 0 l a ld Evaluaton of a a NN s bfo, may b plac by th xact xpsson N and assumng that may b appoxmatd by a lna ntpolaton as N, thfo, a bcoms N 0 N N 0 l a NNN N Evaluaton of a - h ntgaton fomula fo th typ of ntgals s a b l NN I J ab!! a b! l [ a ] 3 l [ a ] 3 3 Evaluaton of a - h ntgaton fomula fo th typ of ntgals s a b l NN I J ab!! a b! l [ a ] 3 3 l [ a ] 3

8 CIVL 7/8 -D Bounday Valu Poblms - xsymmtc Elmnts 8/8 Evaluaton of h - Consd th ntgal Nh wh th ntgaton s along a bounday sgmnt of th lmnt. Snc, th ntgaton s computd along a sngl sd of th tangula lmnt, th ognal shap functons duc to N 0 N N h h 0 l h h h l d Evaluaton of h - Rplacng by th xact psntaton N, and assumng that th functon h may b appoxmatd by th lna ntpolaton N h, th lmnt matx h bcoms h NNN h N 0 N N h h 0 l h ld h h h l d Evaluaton of h - Wth ths assumpton th ntgal bcoms l 3 h h 3 h If th functon h s a constant, h 0, th abov matx ducs to lh 0 h 6 h sultng s a x lmntal load vcto Evaluaton of Matcs - Lna angula Elmnts Rcall th lmntal matcs hav th followng fom b b c c d d a NN f h Nfdd Nh It should b cla that any of th lmnts w hav cussd, (quadatc tangls, quadlatals, tc.) may b usd n conncton wth th axsymmtc functonal to dvlop a fnt lmnt modl. PROBLEM # - Us th axsymmtc fom of POIS36 to fnd th tmpatu tbuton n th poblm shown blow. Justfy you cmnaton of th poblm and psnt you soluton as a plot of sothmal lns at 0 o C ntvals. 6 cm xto sufac at 00 o C nto sufac at 0 o C End of xsymmtc Poblms cm nsulatd cm 8 cm