The Finite Element Method. Jerzy Podgórski

Transcription

1 Th Fnt Elmnt Mthod Jr Podgórs Novmbr 8

2 Introdcton Ths boo dals wth th s of th fnt lmnt mthod (FEM s an abbrvaton for th Fnt Elmnts Mthod or FEA for Fnt Elmnts Analss) to solv lnar problms of sold mchancs. W ar partclarl ntrstd n statc analss of bar strctrs (trsss frams) srfac strctrs (two-dmnsonal plats thr-dmnsonal plats shlls); lmnts that ar vr oftn sd n ngnrng strctrs. Obvosl thr ar man boos whch dscss ths problms for ampl th boos wrttn b th crators of FEM ncldng Bath (996) Znwc (97 994). In or opnon thr ar not nogh Polsh boos that ntrodc dffclt FEM problms n a smpl wa so that an ndrstandng of ts thortcal bass s possbl for popl who do not dal wth strctr mchancs on a dal bass. An ndrstandng of th FEM bass s ncssar for a contmporar dsgnr who has to s sts of comptr programms n th dsgn procss and thos calclatd modls ar st basd on th fnt lmnt mthod. Th ampl of a boo whch can b tratd as a manal s th boo b Raows and Kacpr (993). Ths boo rqrs som thortcal nowldg on th part of a radr. Th sam rfrs to th collctv boo dtd b Klbr (995) whch ntrodcs som othr comptr mthods sd n mchancs. Bt th problms prsntd n or boo ar hlpfl to ndrstand FEM and bsds t contans th ampls of rcss whch can b solvd b a radr wthot th s of a comptr. A good ampl of a FEM handboo prsnt on th Untd Kngdom mart s th boo wrttn b Ross (99). Hnc w hav dcdd to wrt a manal for ngnrs whch s as smpl as possbl (bt wthot trvalsng problms) n ordr to smplf th std and ndrstandng of FEM. Th contnt of ths boo s basd on lctrs whch hav bn gvn b on of th co-athors (J.P.) at th Faclt of Cvl Engnrng of th Tchncal Unvrst of bln snc 99. Howvr th contnt of ths boo has bn gratl broadnd and dpnd n comparson wth th lctrs. W hav also laboratd on man ampls smplfng th ndrstandng of dtald problms and algorthms of FEM. In ordr to std ths boo th radr shold hav basc nowldg of an scncs n partclar thos concrnng strngth of matrals and th thor of lastct. W assm that th radr s famlar wth trms sch as strss stran constttv rlatons (partclarl th gnralsd Hoo s law). Rfrncs gvn at th nd of th

3 boo laborat on ths topcs n dtal. Spcal attnton shold b pad to th boos on th thor of lastct wrttn b Fng (965) Tmoshno (966) and th boo on th strngth of matrals wrttn b Dląg t al. (999) Gawęc (998) Jastrębs t al. (985). Th std of FEM problms rqrs th s of matr algbra and w assm that th radr nows th bass of calcls. At th nd of th boo n Appnd on can fnd a short rvw of th most mportant nformaton concrnng matr algbra ncssar for radng ths boo. Knowldg of nmrcal mthods s not ssntal n ordr to ndrstand FEM bcas t s lnd wth comptr mplmntaton of algorthms. On th othr hand ths nformaton hlps whn sng rad-mad sts applng to FEM. Snc nmrcal mthods ar not alwas ncldd n th programm of th nvrst cors n mathmatcs w provd a rvw of mthods of storag of stffnss matrcs and of solvng larg sts of lnar qatons n Appnd. W also ncorag th radr to b famlar wth th boo wrttn b Gorg and (98) bcas t s partclarl dvotd to ths mthods. I wold l to than prof. Andr Garstc and dr. Wtold Kąol from Ponań Unvrst of Tchnolog who hav rvwd th manscrpt. W wold also l to than dr. Stvn Hard from Wals Unvrst at Swansa who has rad th boo to dntf rrors and mprov th clart of th matral. JP

4 Notaton a b - colmn matr vctors A B K two-dmnsonal matr ' K' vctors matrcs and scalars n th local coordnat sstm of an lmnt K X vctors matrcs and scalars n th global coordnat sstm as of th local coordnat sstm of an lmnt X Y Z as of th global coordnat sstm q lowr nd at vctors or matrcs dnots th nod nmbr q ppr nd at vctors or matrcs dnots th lmnt nmbr componnts of th local vctor n th local coordnat sstm X Y Z X Y componnts of th global vctor n th global coordnat sstm Z f F F F X Y Z X Y Z dsplacmnt vctor of nod forc vctor of nod p nodal dsplacmnt vctor of an lmnt F f F F componnts of th vctor ar sall dnotd b small lttrs st as a vctor cpt for th nodal forcs vctor whch s dnotd b captal lttrs n accordanc wth tradton. dt (A) stands for th dtrmnant of th matr A A T transpos of th matr A whch mans that f N N - nmbr of nods n a strctr N E - nmbr of lmnts n a strctr Elmnt nmbrs ar statd closr toth frst nod. T B A thn B A 3

5 N D - nmbr of dgrs of frdom of on nod N D - nmbr of dgrs of frdom of an lmnt N K - nmbr of dgrs of frdom of th whol strctr E - Yong s modls (modls of lastct) G - Krchhoff s modls (modls of lastct n shar) - Posson s rato - lngth of an lmnt V - volm of an lmnt A - cross-scton of a bar or th srfac ara of an lmnt J - nrtal momnt wth rgard to th as C - torsonal rsstanc charactrstcs 4

6 Introdcton to th Fnt Elmnt Mthod In ths chaptr w wll dscss basc concpts and algorthms of th fnt lmnt mthod. W wll also ncld ncssar nformaton rgardng sold mchancs. As w hav wrttn n th Introdcton w assm that th radr nows basc sss of mchancs of matrals and th thor of lastct thrfor th nformaton hr wll b onl a short srv and an ntrodcton to th matr notaton. Stabl rfrncs ar gvn at th nd of ths boo n partclar boos wrttn b Dlag t al. (999) Fng (965)Jastrbs t al. (985) Tmoshno and Goodr (96)... Th orgn and basc concpts of th Fnt Elmnt Mthod W can trac th bgnnngs of th fnt lmnt mthod to th 's and '3s of th th cntr whn athors l G.B.Man and H.Cross n th USA and A.Ostnfld n th Nthrlands mang s of fndngs prsntd n paprs wrttn b J. C. Mawll A. Castlano and O. Mohr proposng a nw mthod for solvng strctral mchancs problms whch s now nown as th dsplacmnt mthod. In th mddl of th th cntr J.Argrs P.C.Pattan S.Kls M.Trnr R.Clogh t al. accomplshd th gnralsaton of ths mthod. Th dd t on th bass of paprs wrttn b R.Coranta. In th '6s and '7s th fnt lmnt mthod was mprovd thans to th pblcatons b O.C.Znwc Y.K.Chng and R..Talor. Ths t has bcom a contmporar tool sd for solvng sss of sold mchancs tmpratr flows fld mchancs lctromagntc flds and othr sss. Th basc da of th fnt lmnt mthod (FEM) s to sarch for a solton to a compl problm (whch s wrttn n th form of a dffrntal qaton) b rplacng t wth a smplr and smlar on. It lads to th dscovr of an appromat solton th prcson of whch dpnds on th assmd appromaton mthods. In mchancs problms a solton gnrall conssts of dtrmnng dsplacmnts strans and strsss n a contnm. Ths sss appar n statcs and dnamcs of fram strctrs plats shlls and solds. Th qlbrm of a bod s sall wrttn n th form of a dffrntal qaton (or a st of dffrntal qatons) whch has to b ralsd wthn th bod and ts bondar condtons whch shold b ralsd on ts srfac. It s oftn vr dffclt or vn mpossbl to fnd act soltons. Th fnt lmnt mthod 5

7 proposs th followng wa of dtrmnng an appromat solton b Znwc (97): Th contnm s sparatd b magnar lns or srfacs nto a nmbr of fnt lmnts. Th lmnts ar assmd to b ntrconnctd at a dscrt nmbr of nodal ponts statd on thr bondars. Th dsplacmnts of ths nodal ponts wll b th basc nnown paramtrs of th problm. A st of fnctons s chosn to dfn nql th stat of dsplacmnt wthn ach fnt lmnt n trms of ts nodal dsplacmnts. Th dsplacmnt fnctons now dfn nql th stat of stran wthn an lmnt n trms of th nodal dsplacmnts. Ths strans togthr wth an ntal strans and th constttv proprts of th matral wll dfn th stat of strss throghot th lmnt and hnc also on ts bondars. Forcs concntratd at th nods (nodal forcs) whch dpnd on nodal dsplacmnts ar dtrmnd. Th rlatonshp btwn nodal forcs and dsplacmnts s dscrbd b th lmnt stffnss matr. A st of qlbrm qatons s wrttn for all nods hnc th problm bcoms on of solvng a st of algbracal qatons whch ar oftn lnar. Solvng sch a st of qatons wth stabl bondar condtons nabls th strans and strsss wthn lmnts to b calclatd. Th appromaton of th solton rqrs solvng man problms of whch th slcton of shap fnctons and dscrt sstms sm to b th most mportant ons. Th prson choosng a strctr modl (lastc plastc fram plat tc.) and a dscrt mthod shold hav consdrabl prnc. In th followng chaptrs w wll prsnt ncssar nformaton to smplf th wor of lss prncd srs of th fnt lmnt mthod... Basc assmptons and thorms of sold mchancs Hr w wll prsnt a fw basc assmptons and thorms of mchancs whch wll b sd n th sbsqnt chaptrs of ths boo. 6

8 ... Assmptons rgardng th lnar modl of a strctr In ths chaptr and som sbsqnt ons w wll b dalng wth lnar problms of mchancs. Ths mans that th procss of strctral dformaton can b wrttn b lnar dffrntal qatons. It nvolvs th followng consqncs: Dsplacmnts of strctr ponts whch appar drng dformaton ar small. nar dsplacmnts ar consdrabl smallr than th charactrstc dmnson of a strctr (for ampl th dflcton of a bam s a fw hndrd tms smallr than ts lngth) and angls of rotaton ar consdrabl smallr than on (for ampl a nodal angl of rotaton s smallr than. rad). Strans ar small. It nabls th rlatonshp btwn strans and dsplacmnts to b prssd wth th hlp of lnar qatons. Th matral s lnar lastc whch mans that t satsfs Hoo s law. It ma sm that sch lmts whch ar pt on both gomtr of a strctr and matral charactrstcs strongl rstrct th applcaton of th modl. In ffct ths lmts ar ralsd for man strctrs (th can rfr to most of thm) so th rang of sag of th modl s vr wd. Th radr shold now ths whn h procds wth th dscrpton of an ral problm n trms of mchancs qatons.... Strsss and strans W wll dnot componnts of th strss tnsor tradtonall (as t occrs n most boos on th fnt lmnt mthod). Ths mans that componnts of drct strss wll b dnotd b lttrs σ σ σ and componnts of shar strss b τ τ τ. Bcas of th smmtr of th strss tnsor (Fng (965) Tmoshno and Goodr (96)) w wll s onl s componnts whch whn prsntd n a colmn matr form th strss vctor: () Dnotng th componnts of th stran tnsor tradtonall w assm th followng dfntons: 7

9 () whrε ε ε ar th componnts of drct stran (nt longaton) and γ γ γ th componnts of shar stran (th ar th angls of th non-dlataton stran) ar th componnts of th dsplacmnt vctor n th Cartsan coordnat sstm. vctor: W wrt th componnts of stran n th form of a colmn matr - th stran ε (3) W smplf th calclaton of th ntrnal wor f w ta th componnts of th stran vctor γ (th angls of th volmtrc stran) nstad of sal tnsor dfntons: W V T T σ εdv ε DεdV V whr V mans th volm of a bod. (4)..3. Constttv qatons As w hav notd n or ntrodctor assmptons th rlatonshp btwn th componnts of th strss tnsor and th componnts of th stran tnsor (that s btwn σ and ε n or notaton) s prssd b th lnar qaton: σ= D ε (5) ε= D - σ (6) whr D s th sqar matr wth dmnsons 66 contanng th matral constants: 8

10 9 D (7) whr λ and μ ar th amé constants. Snc som othr matral constants l Yong s modls E and Posson s rato ν ar mor oftn sd n practc w prsnt th rlatonshps btwn thm and th amé constants b th followng formla: + E + E (8) Th amé constant μ s notd b th lttr G and s calld Krchhoff s (or shar) modls. Th nvrs matr D - wth th matral constants has an nsall smpl strctr whch s bst shown b mans of th constants E ν: ) ( ) ( ) ( E D (9) It shold b notd that matr D s smmtrcal whch mans that th dpndnc D = D T occrs. Ths dpndnc wll oftn b sd n convrsons...4. Plan strss In two-dmnsonal problms of thn plats th followng smplfcaton of th assmpton s: () whch lads to th plan strss crtron. If w pt Eqn. () nto Eqn. (5) tang nto consdraton data from Eqn. (7) w obtan:

11 () In plan strss th dmnsons of th strss and stran vctors and th matr of th matral constants ar rdcd b half and ths: σ ε () E D (3) E D (4)..5. Plan stran In problms rgardng dformatons of massv bldngs th plan stran crtron s oftn fond and t s prssd b th qatons: (5) Whn w nsrt th abov qatons nto Eqn. (6) tang also nto consdraton Eqn. (9) w gt th followng rlatons: (6) Ths s calld th plan stran. Aftr tang nto consdraton th abov Eqn. (5) and (6) w can notc that th rlatonshp btwn th rdcd strss and stran vctors Eqn. () lads to th followng matr of lastc constants: E D (7)

12 D (8) E..6. Eqlbrm qatons Th condton of qlbrm for a fd bod s satsfd whn th followng s qatons calld qlbrm qatons ta plac: n P M n whch can b wrttn as: n n P ; P ; P ; X n Y n M ; M ; M X n Y n Z Z (9) () whr P X P Y P Z ar th componnts of th forc P and M X M Y M Z ar th momnts of ths forc n rlaton to th as of a coordnat sstm and n s th nmbr of forcs. Whn a st of forcs s contand n for ampl th plan XY thn qlbrm Eqn. () ar rdcd to th followng thr qatons: n P ; P ; M () X n Y n Z..7. Th prncpl of vrtal wor Eqlbrm Eqn. (9) dfn condtons for a st of forcs actng on a rgd bod. In th cas of an lastc bod whch dforms d to forcs actng on t w hav to dtrmn condtons for trnal forcs as wll. Ths can b don b sng th prncpl of vrtal wor whch sas that th trnal wor d to vrtal dsplacmnts s qvalnt to th ncras of th potntal nrg of th ntrnal forcs: n P () E

13 whr s th vctor of th vrtal dsplacmnt at th pont th dot mans th scalar prodct of th vctor of th forc P and th vctor of th vrtal dsplacmnt s th potntal nrg of ntrnal forcs: E T σ ε V T dv ε σ dv V E σ - In Eqn. (3) ε dnots th stran vctor whch rslts from th vrtal dsplacmnt. Th vrtal dsplacmnt mst satsf th followng condtons from Nowac (976): of ths boo. obtan: n P t shold b ndpndnt of forcs actng on a sold t shold b consstnt wth th constrants so that t s nmatcall allowabl t shold b ndpndnt of tm. Eqn. () wll b sd man tms n dffrnt forms n th sbsqnt chaptrs..8. Clapron s thorm Changng vrtal dsplacmnts nto th ral ons n Eqn. () and (3) w V (3) T T σ ε dv ε σ dv (4) V Th abov qaton prsss th contnt of Clapron s thorm whch sas that for th lastc bod n qlbrm th wor of trnal forcs s qal to th potntal nrg of ntrnal forcs (lastc nrg). Morovr th lastc bod has to satsf th condtons dscrbd b Gawęc (998) and Jastrębs t al. (985): matral of whch th bod s composd racts accordng to Hoo s law bod dos not possss th bondar condtons whch dpnd on th dformaton of a strctr bod tmpratr s constant thr ar no ntal strsss and strans. Bods whch satsf ths condtons ar calld Clapron s bods.

14 ..9. Th Btt rcprocal thorm of wor and th Mawll rcprocal thorm of dsplacmnts. t s nsrt th constttv rlaton nto Eqn. () prssng th prncpl of vrtal wor Eqn. (5). Ths w obtan: n P T T σ ε dv Dε ε dv ε V V V T Dε dv In th abov qatons w hav mad s of th smmtr of th matr of lastc constants D=D T. Blow w wll appl th prncpl of vrtal wor n a dffrnt wa naml w attach vrtal loads (a st of forcs (5) P ) actng at th sam nods as th actal loads bt of a dffrnt val and drcton. Th wor don b ths forcs for th actal dsplacmnt s qal to: n P T T σ ε dv Dε ε dv ε V V V T Dε dv Th rght hand sds of Eqn.(5) and (6) ar dntcal whch can b smpl chcd b drct calclatons. Hnc w obtan th qaton: n n P P (7) whch prsss th rcprocal thorm of wor formlatd b E.Btt n th nntnth cntr. Ths thorm can b wrttn as follows (Nowac (976)): Th st of forcs P dos th sam wor at th dsplacmnts ndcd b th st of forcs P as th st of forcs P dos at dsplacmnts ndcd b forcs P. w obtan: a a If w brng down both sts of forcs to sngl nt forcs actng at th pont a (8) a a Ths rlatonshp s calld th rcprocal thorm of dsplacmnts and was formlatd b J.C.Mawll n 864. (6) 3

15 .3. Algorthm of th Fnt Elmnt Mthod Th fnt lmnt mthod as a comptr mthod s charactrsd b a strctl dfnd and smpl algorthm. W wll show th most mportant stags of ths algorthm. Som of thm wll b dscssd n dtal n frthr parts of ths boo. A. Dscrtaton At ths stag th dvson of a strctr nto fnt lmnts s don. In th cas of framwors t s oftn obvos snc vr straght sgmnt of a bar bcoms an lmnt. In th cas of D srfacs w dvd thr ara nto tranglar and/or qadranglar lmnts and n th cas of solds w dvd thm nto ttrahdral and hahdronal (brc) lmnts. At ths stag w dcd abot ponts of lmnts contacts gv coordnats of th nods and stat th mannr of conncton btwn nods and lmnts. B. Calclaton of lmnt stffnss matrcs On th bass of matral proprts and topologcal data gvn n th frst stag matrcs prssng rlatonshps btwn nodal forcs and nodal dsplacmnts of an lmnt ar formd. C. Aggrgaton (constrcton) of a global stffnss matr Now lmnt stffnss matrcs ar dvdd nto blocs whch mrg nto a global stffnss matr for whch th nformaton abot constrcton topolog s sd. Modfcatons tang nto consdraton bondar condtons ar oftn ntrodcd nto th global matr at that stag. D. Constrcton of a global loads vctor Hr w calclat load vctors of lmnts whch aftr bng dvdd nto blocs ar nsrtd nto th global vctor of nodal loads. Whn th global vctor s blt thn ts componnts shold b modfd wth rgard to bondar condtons. E. Solton of a st of qatons At ths stag a st of lnar qatons wll b solvd. In ffct w wll obtan th nodal dsplacmnts of a strctr. F. Calclaton of ntrnal forcs and ractons If w obtan dsplacmnts w can thn calclat strans strsss and ntrnal forcs n a strctr. Aftr havng calclatd lmnt nodal forcs ractons at constrants (spports) of th constrcton can also b calclatd. 4

16 Th sstms of FEM sall hav a modlar strctr. Indvdal stags of th algorthm ar solvd b spcalsd modls of th sstm. Th frst stag (A) complmntd b dfnng matral proprts and dscrbng constrcton loads s calld a prprocssor. In old sstms ths stag dpndd on manal craton of a data fl (npt fl). At prsnt sch a staton occrs vr rarl bcas manall npttng data for th tpcal problm of FEM (covrng a fw thosands of nods) s vr hard wor. Contmporar prprocssors ar sall graphc programms qppd wth tools smplfng th gnraton of lmnt mshs. Stags (B) (C) (D) and (E) ar sall prformd b th modl calld a procssor. Apart from th opratons mntond abov th procssor oftn dals wth a stabl arrangmnt of qatons n ordr to rdc th amont of mmor for th storag of th stffnss matr and to acclrat th procss of solvng sstms of qatons. Th sth stag (F) complmntd b graphcal otpt s ndrtan b a postprocssor. A larg amont of rslts that w gt aftr solvng an sstm of qatons and calclatng ntrnal forcs s vr dffclt to ntrprt wthot sng graphcal tchnqs. Contmporar FEM sstms ar qppd wth a graphc postprocssor prodcng color maps of strsss dsplacmnts and othr paramtrs whch smplf analss. Althogh vsal tchnqs ar strongl lnd wth th fnt lmnt mthod th ar not a part of ths cors; hnc th ar not dscrbd n ths boo. W wll concntrat on th procssor and parts of th postprocssor..3.. Craton of lmnt stffnss matrcs As w hav alrad notd n ths chaptr (pont.) w assm that aftr havng dvdd th strctr nto fnt lmnts ths lmnts ar onl n contact wth ach othr at nods. It wll b convnnt f w magn a nod as a matral partcl movng drng th dformaton procss casd b trnal loads affctng th strctr (forcs tmpratrs tc.). W can dscrb th movmnt of a nod b gvng th componnts of th dsplacmnt vctors. W wll b ntrstd n dffrnt tps of moton accordng to th lmnt tp. In som cass th wll b dsplacmnts (n trss lmnts two-dmnsonal plan lmnts solds) n othr cass thr ar rotatons (n bams frams plats shlls). All ncssar componnts of a nodal dsplacmnt crat 5

17 th sstm of paramtrs calld dgrs of frdom. W wll mar th nmbr of dgrs of frdom as N D. In Tabl thr s nformaton on th nmbr of dgrs of frdom for nods of tpcal ngnrng strctrs. Dgrs of frdom ar gvn as componnts of dsplacmnt vctors n th Cartsan coordnat sstm. Tabl. Th dgr tps of frdom for lmnts. Tp of strctr Nmbr of dgrs of frdom Dsplacmnts Rotatons N D φ φ φ plan trss spac trss 3 plan fram 3 spac fram 6 grllwor 3 two-dmnsonal plat 3 shll 6 sold (brc) 3 Fgr. Th plan fnt lmnt wth for nods. 6

18 7 t s magn som qadrlatral lmnt (for convnnc w wll ta a plan lmnt whch s as to draw) havng nmbr (Fgr ). Th nods of ths lmnt ar locall nmbrd: l and th hav thr global nmbrs rspctvl: n n n n l. Nodal coordnats ar alwas gvn n th global coordnat sstm XY bt for convnnc w s an local coordnat sstm whl formng an lmnt stffnss matr. Th local coordnat sstm s chosn at random. W grop nodal dsplacmnts n th dsplacmnt vctor: Y X Y X Y X ly lx l (9) Th st of all nodal dsplacmnts of an lmnt forms th vctor of nodal dsplacmnts of an lmnt: ly lx Y X Y X Y X l (3) Th dsplacmnt of a crtan pont m wthn th lmnt s wrttn n th form of th vctor: ) ( ) ( ) ( Y X Y X Y X Y X (3) If th componnts of vctors ar dfnd n a local coordnat sstm thn w wll dnot thm as th sgn ' (prm) for nstanc: ) ( ) ( ) ( (3a) Smlar notaton can b sd n Eqn. (9) and (3) bt for th tm bng w wll s onl global rlatonshps for convnnc. Now w assm that th dsplacmnt of som pont m dpnds on nodal dsplacmnts of an lmnt: N ) ( ) ( (3)

19 whr N() s th matr componnt whch dpnds on th coordnats of a pont. Th dmnsons of th matr N() dpnd on lmnt tp. Th nmbr of rows of th matr N() s qal to th nmbr of dgrs of frdom of th pont m and th nmbr of colmns rprsnts th nmbr of dgrs of frdom of th lmnt. In or ampl whr th pont has two dgrs of frdom and th lmnt has 4=8 dgrs of frdom th matr N() has two rows and ght colmns. Ths t wll b convnnt to prsnt Eqn. (3) n a dvlopd form: ( ) N ( ) N ( ) N ( ) l N ( ) l (3a) whr matrcs N ()... N l () ar qadratc matrcs contanng fnctons whch show th nflnc of th dsplacmnts of nods... l on th dsplacmnt of th pont m. In th fnt lmnt mthod ths fnctons ar nown as shap fnctons or dsplacmnt fnctons and th ar vr mportant for th formlaton of FEM qatons. Matrcs N ()... N l () ar calld matrcs of shap fnctons of nods... l and th matr N s th matr of shap fnctons of an lmnt. It s obvos that shap fnctons shold flfl som condtons to b sfl for th appromaton of th fld of an lmnt dsplacmnt. If w magn that th pont m s at a nod thn ts dsplacmnts shold b qal to th dsplacmnts of ths nod bt th dsplacmnts of othr nods shold not hav an nflnc on thm (Fgr ). Fgr. Th dformaton of th lmnt srfac whos th nod s dsplacd b a nt n th drcton prpndclar to ths lmnt. 8

20 N p Ths condton can b prssd n th followng wa: q q pq (33) whr pq s Kronncr s dlta: pq - whn - whn and p and q rprsnt an local nmbr of nods... l. p=q p q Condtons of tp Eqn. (33) allow s to dtrmn th coffcnts of shap fnctons. W wll consdr som othr condtons whch hav to b flflld b fnctons N p () n latr parts of ths chaptr. stran vctor: Sbstttng Eqn. (3) for Eqn. (5) w calclat th componnts of th lmnt ε DN (34) whr D s th matr wth dmnsons 3 N D for both plan strss and plan stran or 6 N D for thr-dmnsonal problms (N D s th nmbr of dgrs of frdom of a nod) contanng dffrntal oprators comng from th dfnton of stran Eqn. () For a two-dmnsonal problm N D = and th matr of dffrntal oprators has th followng form: D (35) whr smbol rspct to. sgnfs dffrntaton wth rspct to : = and wth W assm th notatons: B D N (36) and consstntl B D N : B D N. l l (37) Th smplf frthr transformatons. 9

21 prsntd as: Aftr tang nto consdraton ths notatons rlaton Eqn. (34) can b ε B (38) Th matr B() has dmnsons 3 n D (or 6 n D for thr-dmnsonal problms of strss). For a qadrlatral lmnt n a two-dmnsonal problm matr B() has dmnsons 3 8. As wth matr N() w now smlarl dvd th matr B() nto blocs: B ( ) B ( ) B ( ) B ( ) B ( ) (39) l Matrcs B... B l ar matrcs contanng stran shap fnctons of nods... l and B () s th matr contanng stran shap fnctons of th lmnt. Hr w rplac ractons btwn nods and lmnts b concntratd forcs. Th schm of ths ractons s shown n Fgr 3. Fgr 3.Th arrangmnt of nodal forcs ovr th lmnt. Now w collct th componnts of nodal forcs nto th nodal forc vctor: FX F F X X FlX f f F f f l Y FY F (4) Y FlY and th forcs actng on an lmnt nto th nodal forc vctor of an lmnt:

22 FX FY f FX f F Y f (4) f FX fl FY F lx FlY t s loo for th rlatonshp btwn nodal forcs f and nodal dsplacmnts. W appl th prncpl of vrtal wor Eqn. () tratng th nodal forcs as th trnal loads on an lmnt. Th lmnt s loadd both on ts nsd and bondar and w dnot th load whch dpnds on th coordnats of a pont as follows: q q (4) q W dvd constttv Eqn. (5) (or for nstanc Eqn. () and (3) for plan strss) nto parts n ordr to consdr ntal strans and strsss: ε o σo σ D ε (43) whr ε o s th ntal stran vctor (for ampl casd b tmpratr loads) and σ o s th ntal strss vctor (g. rsdal strsss). Now w r-wrt Eqn. () prssng th qalt of trnal and ntrnal wor for th lmnt n qlbrm: T T f q da A T ε σ dv (44) V Th lft hand sd of ths qaton rprsnts trnal wor whl th rght hand sd dnots ntrnal wor for ths lmnt A rprsnts th srfac of an lmnt and Vs ts volm. W s Eqn. (3) (38) and (43) n th abov qaton: T T T f N qd B DB f A A ε o σ o dv (45) V Aftr th transformaton w obtan ts fnal form as follows: K fq f f (46) o o whr th followng vals hav bn notd:

23 f q N A - nodal forcs vctor d to trnal loads: T q da - nodal forcs vctor d to ntal stran: T Dε dv o o (48) V f f K o B - odal forcs vctor d to ntal strss: B V T σ dv o - lmnt stffnss matr: B V T DB dv (47) (49) Ths calclatd nodal forc vctors contan forcs actng on th lmnt. Th shold b mard wth th ngatv sgn whn formng qlbrm qatons. Th matr K can b dvdd nto a bloc of qadratc matrcs th nflnc of th dsplacmnt of th nod q on th forcs at th nod p: K pq B V p T DB dv q K pq (5) dscrbng Thr ar 4 4=6 blocs n th stffnss matr of th lmnt wth for nods (Fgr 3). Snc th stffnss matr s smmtrcal t mans that K T (5) K whch coms from Eqn. (5) and t s a smpl consqnc of th Btt rcprocal thorm of wor; thn blocs qp T pq K pq hav to rals th condtons: K K (5) Eqn. (5) or (5) rprsnts a stp n formlatng qlbrm qatons of th strctr bt th stffnss matr has not alwas bn dtrmnd ths wa. For smpl lmnts sch as a trss lmnt or a fram lmnt som othr was (somtms smplr) of obtanng rlaton Eqn. (46) st. W wll show ths n nt chaptrs. If all transformatons ladng to Eqn. (5) hav bn don n th local coordnat sstm () thn th rsltng stffnss matr shold b transformd to th global coordnat sstm (XYZ). Ths transformaton s achvd b mltplng th matr

24 K ' (sgn prm dnots a matr n th local coordnat sstm) b th transformaton matr of th lmnt. Th dtald strctr of ths matrcs s laboratd on n Chaptrs 3 and 4 hr w smpl llstrat th transformaton: K R K R T (53) R whr R R (54) R R... R - transformaton matrcs of nods.... Th transformaton matrcs of th nods contan cosns of angls btwn th as of th global and local coordnat sstms: CX CY CZ R C X C Y C Z (55) C X CY CZ whr for nstanc C cos Y Y tc. α Y s th angl btwn th as of th local coordnat sstm and th Y as of th global sstm..3.. Aggrgaton (constrcton) of a global stffnss matr Rlaton Eqn. (46) allows s to wrt qlbrm qatons of a nod n th form contanng nodal dsplacmnts as nnown. t s magn a nod as an ndpndnt part of a constrcton and dsconnct lmnts from nods n ordr to show nodal forcs (Fgr 4). 3

25 nods. Fgr 4. Th snss of forcs rprsntng th ntracton btwn lmnts and E n F X W wrt a st of qlbrm qatons of th nod n th scalar form: E n F Y E n F Z (56a) For th nods wth rotatonal dgrs of frdom th qlbrm qatons of momnts wll b ncssar: E M X n E M Y n E M Z n (56b) In Eqn. (56) smmaton s rqrd for all lmnts connctd to th nod hnc ndcs... E n ar nmbrs of lmnts connctd to th nod E n s th nmbr of lmnts connctd to th nod n. W nsrt rlatonshp Eqn. (46) nto Eqn. (56) rmmbrng to chang th sgn of th nodal forcs comng from th chang of sns of th forcs actng on th lmnt and nod (Fgr 4): En f (57) n 4

26 In ths qaton smbol f n dfns onl ths componnts of vctor act on th nod n. W convrt ths qaton nto a mor convnnt form: f whch En K n n p n (58) whr p fq f f s th vctor of th nodal forcs d to trnal loads ntal strans and strsss. o o Arrangng qatons for vr nod of th strctr smlar to Eqn. (56) w obtan a st of qatons whch allow s to calclat nodal dsplacmnts for ths strctr. Snc smmaton s don for th lmnts n Eqn. (56) (th forc vctors whch blong to ths nod) formaton of a st of qatons basd on th qlbrm of sccssv nods s not ffctv. Ordrng nods and dgrs of frdom s ncssar for ths opraton. So far w hav sd local nmbrs for nods of lmnts l.. bt ntrodcng global nmraton of nods s ncssar whl bldng th global st of qatons. t n stand for a global nmbr of th nod rprsntd b th local nmbr and lt s p b a global nmbr of dgrs of frdom rprsntd b th local nmbr p. Now w form a rctanglar matr of connctons of th lmnt A. Th nmbr of rows of th matr A s qal to th global nmbr of dgrs of frdom of th strctr N th nmbr of colmns s qal to th nmbr of dgrs of frdom of th lmnt N D. Most componnts of th matr A ar qal to ro apart from th componnts havng th val of whch ar statd n rows s p and colmns p. Hnc th strctr of th matr A contans nformaton abot connctons btwn th lmnt and nods or bng mor act abot th rlatonshp btwn th dgr of frdom of th lmnt and th global dgr of frdom of th strctr. Th formaton of th conncton matr can b most asl stdd on th followng ampl. Fgr 5prsnts a plat dvdd nto fv tranglar lmnts. Th plat has s nods nmbrd from to 6 vr lmnt has a local notaton of nods. Tabl shows global nmraton of dgrs of frdom of a two-dmnsonal lmnt of th plat. 5

27 lmnts. Fgr 5. Th mplar dscrtsaton of th D mmbran nto fv fnt Tabl. Th global nmbrs of dgrs of frdom for nods n th plat (Fgr 5). Nod nmbr Global nmbrs of dgrs of frdom of nods n nx ny

28 Tabl 3.Th global nmbrs of dgrs of frdom for lmnts n th plat (Fgr 5). Elmnt nmbr Global nmbrs of dgrs of frdom of lmnt s p allocaton vctor X Y X Y X Y Tabl 3shows th dpndnc btwn local and global dgrs of frdom. Hnc th conncton matr cratd for lmnt No 3 wll hav th followng form: whr all ro lmnts ar nglctd for clart. Mltplng th nodal forc vctor of an lmnt b th conncton matr cass th transfr of stabl blocs of th local vctor to th global vctor. Now smpl addton of ths vctors s possbl: 7

29 N E N E N E A f A K A p (59) Hr t s ncssar to prss th nodal dsplacmnt vctor of lmnts b mans of th global vctor: T A whch shold b pt nto Eqn. (59). Fnall w obtan th sstm of qatons n th form: N E A K N T A E or n a shortr form A p (6) K p. (6) vctor N Matr E K A K A N E T s calld th global stffnss matr of a strctr p A p s th global vctor of nodal forcs of th strctr th vctor contanng th dsplacmnt of all nods s th global dsplacmnt vctor. A smlar mthod of aggrgaton s dscrbd n th boo wrttn b Raows and Kacpr (993) whr matr A T s calld th conncton matr. Th mthod of aggrgaton sng th conncton matr s not stabl for comptr mplmntaton bcas t ss th bg matr A. It s mor ffctv to plot nformaton whch s contand n allocaton vctors. Vctors for th prvos ampl ar ncldd n Tabl 3. Th aggrgaton mthod sng allocaton vctors wll b prsntd n th scond chaptr n sctons dvotd to bldng th stffnss matr of a trss Rmars rgardng th shap fnctons of an lmnt Fnctons appromatng th dsplacmnt fld wthn lmnts whch ar n fact shap fnctons dscrbd n Sc..3. cannot b chosn n frl. Th shold flfl som condtons whch dcd abot th qalt of ths fnctons or thr sflnss for appromaton of dsplacmnts strans and strsss. W qot ths crtra aftr Znwc (97). A. Crtra of rgd bod movmnts 8

30 Th dsplacmnt fncton chosn shold b n sch a wa that t shold not prmt stranng of an lmnt to occr whn th nodal dsplacmnts ar casd b a rgd bod dsplacmnt. B. Crtron of stran stablt Th shap fncton shold nabl th constant fld of strans n an lmnt to appar. C. Crtron of stran agrmnt Th dsplacmnt fnctons shold b so chosn that th strans at th ntrfac btwn lmnts ar fnt. Crtra (A) and (B) sm to b obvos. Snc som componnts of stran (or strss) can b ro thn appromaton fnctons shold b abl to rprodc ths problms. Constant and lnar parts of polnomals whch w oftn s to bld a shap strctr assr ralsaton of condtons (A) and (B). Crtron (B) s th gnralsaton of crtron (A) and t was formlatd b Bal Chng Irons and Znwc (97 994) n 965. Crtron (C) rqrs that shap fnctons shold assr contnt of drvatvs to th dgr whch s lowr b on than dffrntal oprators bng n th matr D (comp. Eqn. (34)). W plan ths sng th followng ampl. In th twodmnsonal problm of a plat th strans ar dfnd b th frst drvaton of th dsplacmnt fncton (comp. Eqn. (34) and (35)) bcas th dsplacmnt fld has to b contnos on th bondar btwn lmnts and dsplacmnts fnctons hav to b of class C. For plat lmnts th crvatrs gvn b th scond ordr drvatvs ta th rol of dsplacmnts (comp. chaptr 7). Hnc th dsplacmnt fncton of a plat shold assr contnt both of th srfacs of a plat dflcton and ts frst drvatons nsd and on th bondars btwn lmnts. Thn th dsplacmnt fld shold b contnos and smooth wthn th plat. Ths fnctons ar sad to b of class C. Crtra (A) and (B) hav to b ralsd crtron (C) dos not. For nstanc th shap fncton of plat lmnts dos not oftn achv th condton of contnt (contnt of th frst drvatons on bondars of lmnts). If all crtra ar ralsd thn w sa that th dscrbd lmnts ar adst ons If onl crtra (A) and (B) ar achvd thn lmnts ar calld not adst ons. Th rslt of applng adst and not adst lmnts to dscrtaton of a strctr s prsntd nfgr 6. Th convrgnc of rslts obtand wth th hlp of 9

31 dsplacmnt th dffrnt tps of lmnts whch ar sd for dscrtaton of a qadratc plat s shown n th sam fgr. act rslt non adst lmnts adst lmnts nmbr of lmnts Fgr 6. Th prcson of calclatons for ncompatbl and non-ncompatbl lmnts dpndng on th nmbr of lmnts. Apart from th thr lstd crtra w can also add som othrs whch dtrmn th choc of appromaton polnomals. Ths choc shold assr sotrop wth rspct to as of a coordnat sstm. W wll show ths sng th ampl of bldng shap fnctons of plat lmnts (two- and thr-dmnsonal problms). If w prsnt appromaton polnomals n th form of Pascal s trangl thn th choc of part of ths trangl shold b smmtrcal n wth rspct to ts as. It s shown nfgr 7. Fgr 7. Pascal s trangl. 3

32 W can s Hrmtt (dscrbd n chaptr 4 of ths boo) and agrang polnomals (Znwc (97)) bt w alwas hav to mantan th condton of sotrop. Thr s a long lst of rfrncs as far as shap fnctons ar concrnd bt w rcommnd th followng boos: Bath (996) Raows and Kacpr (993) Rao (98) Znwc and Talor (994). 3

33 D trss strctrs D trsss ar on of th most common tps of strctrs. Th strctr of a trss s conomc snc th rato of th strctr wght to forcs carrd b ths strctr s prssd as a small nmbr. Accordng to assmptons loads (concntratd forcs) wll act on nods onl (tmpratr loads ar an cpton hr) and conncton bars wll b ond wth nods n an artclatd wa. Althogh most strctrs whch hav bn blt latl ar trsss wth rgd nods (th ar bascall fram constrctons whch ar prsntd n Chaptr 4) mthods of solvng problms n trss statcs wth artclatd onts ar stll vr mportant n ngnrng practc. Th sstm of a plan trss wth an artclatd ont s th smplst ampl of an constrcton showng th da of th fnt lmnt mthod wthot mplong an complcatd dtals. Thogh th strctr of th mthod s vr smpl most notons algorthms and rlatons connctd wth th FEM algorthm wll b rlvant n dscssons of mor compl strctrs..4. Basc rlatons and notatons W assm that th bar of a plan trss (w wll also call t an lmnt) s straght and homognos (t mans that t s mad from a homognos matral wthot fractrs and hols and has a constant cross scton) and t ons nods (th frst nod) and (th last nod). Notatons for ths nods ( ) ar local notatons whch ar th sam for all bars and th ar to dfn lmnt orntaton. On th othr hand strctr nods also hav global nmbrs whch allow s to dntf thm. Global nmbrs ar mard as n (th global nmbr of th frst nod) and n (th global nmbr of th last nod). Th nod of a plan trss can mov on th plan XY onl. In mchancs t mans that th nod has two dgrs of frdom bcas n ordr to dtrmn ts locaton drng ts moton t shold b gvn two coordnats. Th staton of th nod of a rgd strctr wll b dtrmnd b ntal coordnats X Y wth rspct to th coordnat sstm whch wll b sd for th dscrpton of th whol strctr. W sa that ths sstm s global and ts as wll b dnotd b captal lttrs X Y. Th locaton of th nod aftr ts dformaton casd b loads s dtrmnd b two componnts of th dsplacmnt vctor of nods X and Y. Ths mthod of dscrpton of th strctr movmnt s calld th angrang dscrpton n mchancs. Th dscrpton of som dpndnc btwn forcs and lmnt dsplacmnts bcoms mch smplr whn w ntrodc a local coordnat sstm 3

34 whch wll b dnotd b small lttrs. Th as of th sstm ovrlaps th as of th bar and has ts bgnnng at th frst nod of an lmnt whl th as s prpndclar to th as and s drctd n sch a wa that th Z as of th global coordnat sstm and as of th local sstm hav th sam sns and drcton. Bcas w accpt that both coordnat sstms ar rght-torson w can obtan th as b rotatng th as clocws throgh th angl π/. Fgr 8.Nodal forcs and dsplacmnts for th D trss lmnt: a) n th global coordnat sstm; b) n th local coordnat sstm. Th most mportant notatons drctons as wll as snss of vctors and th coordnat sstms ar shown n Fgr 8. Nodal dsplacmnts and forcs of lmnts ar wrttn as colmn matrcs whch w wll call vctors. Th nodal dsplacmnt vctor of th frst nod and th last nod n th local coordnat sstm:. (6) 33

35 Th nodal dsplacmnt vctor of th lmnt n th local coordnat sstm: Th nodal forcs vctor of th frst nod and th last nod n th local coordnat sstm: F f ' F F f ' F (63) (64) Th nodal forcs vctor of th lmnt n th local coordnat sstm: f ' f ' f ' F F F F. (65).5. Th lmnt stffnss matr of a plan trss n th local coordnat sstm W loo for th rlaton btwn nodal forc vctors and nodal dsplacmnt vctors (comp. Chaptr ) whch s ncssar to prss qlbrm qatons dpndng on th nodal dsplacmnts. K' ' f ' Th gnral mthod of bldng sch a rlatonshp conssts of sng th prncpl of vrtal wor (comp. Chaptr ) bt n ths cas w wll appl dffrnt approach. W wll s th qlbrm condtons n thr basc forms whch s possbl n th cas of bar lmnts. rlatons: Eqlbrm qatons for th lmnt (Fgr 8) lad to th followng F F F (66) F F F (67) M F and w obtan 34

36 F ; F F ; F. (68) Snc th st of thr qlbrm Eqn. (67) or (68) contans for nnown paramtrs ths problm s statcall ndtrmnat. Th arrangmnt of an addtonal qaton s ncssar n ordr to ma th dtrmnaton of nodal forcs possbl. Ths qaton oght to s th rlaton btwn nodal dsplacmnts of an lmnt and ts ntrnal forcs. Hoo s law wrttn for a smpl cas of aal tnson of a straght and homognos bar contans ths rlatons (Fgr 9): N E A (69) whr N s th aal forc n th bar (th postv val of an aal forc alwas mans tnson) s th bar lngth Δ sgnfs ncrmnt of th bar lngth d to th bar tnson casd b th forc N; E s Yong s modls of th matral from whch th bar s mad; A s th ara of th bar cross scton. Fgr 9. Th bar strchd along ts own as wth th notaton sd n Eqn. (69). Comparng Fgr 8 and Fgr 9w can obsrv smpl rlatons btwn nodal forcs actng on th bar that s F F (Fgr 8) and th aal forc N (Fgr 9): F N ; F N. (7) dntcall. As t s shown abov ths rlatons satsf th thrd qlbrm Eqn. (68) Th ncrmnt of th bar lngth d to tnson rslts from aal dsplacmnts of th bar ndngs: (7) whch aftr nsrtng nto Eqn. (69) lads to th rlaton: N EA. (7) 35

37 Tang nto consdraton th rlatonshp btwn th aal forc of th lmnt and nodal forcs Eqn. (7) wth rspct to Eqn. (7) w obtan: F EA EA ; F F ; (73a) F. (73b) Th rsltng rlatons ar th sarchd rlatons Eqn. (66) btwn th nodal forcs and nodal dsplacmnts of th trss lmnt. W wll wrt thm on mor tm n a dffrnt form: EA EA F = F. EA EA F (74) F qaton: Aftr consdrng notatons Eqn. (63) (65) and (66) th abov form lads to th K' = EA EA EA EA (75) whch dfns a matr Kʹ. Ths matr wll b calld th lmnt stffnss matr of a plan trss. Th matr n th form of Eqn. (75) prsss rlatonshps btwn th vctor ʹ and th nodal forc vctor of an lmnt Th stffnss matr Kʹ can b smplfd to: f n th local coordnat sstm. K' J' J' J' J' whr J' s th sqar matr dfnd n th followng wa: (76) 36

38 J' EA (77).6. Coordnat sstm rotaton Th form of th lmnt stffnss matr dtrmnd n th local coordnat sstm wll not b convnnt n frthr consdratons for whch w wll s matrcs of dffrnt lmnts. Th most convnnt mthod s transformng all matrcs to th form whch s dfnd n on common coordnat sstm. Sch a sstm wll b calld th global coordnat sstm. It can b th sstm of a crtan tp: cartsan polar or crvlnar. Th cartsan coordnat sstm s th most convnnt sstm for a trss. Nodal coordnats of a strctr ar sall gvn n th global coordnat sstm. Now w convrt th lmnt stffnss matr to th global sstm. W start th transformatons b fndng rlatonshps for a sngl nod: cos X sn Y or n matr form: X Y c s sn cos s c whr c cos and s sn. (78) (79) Fgr. Dsplacmnt vctor componnts n th global and local coordnat sstms rotatd throgh th α angl. Dnotng 37

39 X Y and tang nto consdraton notaton Eqn. (6) w obtan: (8) R (8) whr c s R (8) s c s th transformaton matr of th vctor from th local sstm to th global on. A rvrs rlaton wll b rqrd: ' R whr R R I R s th nvrs matr of R ; t mans that t has sch a proprt that whr I s th dntt matr (83) (84) I (85). Th matr R l othr transformaton matrcs has th proprt that R T R (86) t mans that R s th orthogonalt matr (th dtrmnant of ths matr s qal to T.. dt(r )=; dtr R b mang a drct calclaton: R c s s c c s ). W can asl chc th proprt Eqn. (86) of th matr s c c s sc cs cs sc c s T R I Th transformaton matr contans th blocs of th nodal transformaton matr: R R R (87) whr R s th transformaton matr of th frst nod R s th transformaton matr of th last nod and s th part of th matr contanng ro vals. Th transformaton matrcs R and R ar sall dntcal (for straght lmnts) bcas 38.

40 rotaton angls of th vctor of nods and ar qal. Snc th trss lmnts ar straght w can wrt R =R. Fnall th rlatonshps btwn th nodal dsplacmnt vctor of th lmnt prssd n th local sstm and th sam vctor n th global sstm hav th form: R ' (88) T R ' (89) Th rlatonshp btwn th nodal forc vctor of an lmnt n th local sstm and th sam vctor n th global sstm s dntcal to th rlatonshp that w hav obtand n th qatons dscrbng dsplacmnts f R f ' and (9) T R f f' (9) f f R f ' (9) T R f '. (93).7. Th lmnt stffnss matr n th global coordnat sstm Mltplng Eqn. (66) b th transformaton of th matr rlaton (89) for T R R R K' f' w obtan R and sbstttng (94) On th bass of rlaton Eqn. (9) th rght hand sd of ths qaton s qal to f so f w ntrodc th notaton K R K' R T (95) w obtan f K (96) It s th rqrd rlatonshp btwn nodal forcs and dsplacmnts of th lmnt n th global coordnat sstm. If w prform th mltplcaton n Eqn. (95) w obtan 39

41 J J K (97) J J EA c sc whr J. (98) sc s W can chang form Eqn. (98) of th matr J nto th qvalnt on n whch trgonomtrc fnctons do not st. t s not that X Y c cos and s sn. (99) Aftr nsrtng ths rlatons nto Eqn. (98) w obtan J EA 3 () X X Y X Y Y.8. Nodal qlbrm qatons and aggrgaton of a stffnss matr Rplacng stng bars (lmnts) of a trss b nodal forcs w obtan a grop of nods whch can b tratd as matral partcls wth two dgrs of frdom. Ths nods ar loadd wth concntratd forcs comng from lmnts or trnal loads. Th qlbrm condtons for sch a nod ar as follows: P F P X E n nx nx P F P Y E n ny ny whr w hav dnotd th lmnt nmbrd actng on a nod n () F nx - componnt n th drcton X of nodal forcs from PnX - componnt n th drcton X of th trnal forcs actng on th nod n E n - nmbr of lmnts ond to th nod n. 4

42 Fgr. Nodal and trnal forcs actng on th trss nod. Now w transform th st of Eqn. () to th form contanng nodal dsplacmnts: K K K K p () n n n N n n n 4

43 In Eqn. () sgnfs th global vctor of nodal dsplacmnts of a strctr N n PnX p n s th vctor of trnal forcs actng on th nod n PnY matrcs K n ar qadratc matrcs wth dmnsons dtrmnd as follows: whr =n - f f En E n nn J K (3) - ar nmbrs of th lmnts ond at nod n K n and nods and n ar not drctl connctd b an lmnts thn K n and nods and n ar connctd b som lmnt wth a nmbr thn n J J sgnfs th bloc of a stffnss matr of th lmnt (comp. Eqn. (98)). Arrangng qlbrm Eqn. () for all nods of a strctr w obtan th fnal form of qatons allowng dtrmnaton of nodal dsplacmnts of th trss: n K K K n K Nn p K K K n K Nn p = K n K n K nn K nnn n p n K Nn K Nn K N n n K NnNn Nn p Nn or K p (4) 4

44 Th matr K of th st of Eqn. (4) s th global stffnss matr of th strctr th vctor s th global vctor of nodal dsplacmnts of th strctr and th vctor p s th global vctor of nodal forcs of th strctr. Carfl nmbrng of th nods can allow K to th bandd matr whch s charactrsd b a fact that non-ro componnts appar on th man dagonal and closl to t. Th matr K s a smmtrc matr whch mans that ts componnts satsf qatons: K K or T K K (5) whch rslt from th prncpl of vrtal wor (comp. Chaptr ). Componnts K nn whch ar on th man dagonal ar alwas postv K nn (6) whch s a drct conclson drawn from dfntons Eqn. (98) and(3). Global nmbr of th frst nod of an lmnt Global nmbr of th last nod of an lmnt Nod nmbrs.... n.... n.. N n : : n K= : : n : K n n K n n K nn K n n + + N n Fgr. Th J stffnss -J matr aggrgaton schm. K = - J J

45 Th ro componnt K nn dmonstrats gomtrc changablt of a strctr and shold b rmovd b a stabl chang of a gomtrc schm. Th matr K prsntd n Eqn. (4) s a snglar matr (t mans dt K=) hnc th st of Eqn. (4) cannot b solvd wthot modfng t. Ths modfcaton wll dpnd on th consdraton of bondar condtons. W wll consdr ths problm n th nt scton. Th procss of bldng th global stffnss matr s calld aggrgaton of a matr. It can b don b mans of th mthod dscrbd n Chaptr dmandng formaton of conncton matrcs. Snc ths matrcs ar larg thn thr s s not convnnt and th ar rarl sd n comptr mplmntaton of th FEM algorthm. Th mthod of smmaton of blocs shown b Eqn. () and (3) s mch smplr. Th form of matr Eqn.() and (3) ma sm to b complcatd bt n fact w hav vr smpl opratons of nsrton of blocs hr. Ths mthod s bst shown n Fgr. + sgns locatd at arrows pontng to th plac of locaton of blocs K man that blocs J shold b addd to th stng contnts of clls of matrcs n n K n n or K and blocs J lng bond th dagonal shold b addd to clls K n n or K n n. In th cas of a trss whr nods ar sall ond b on lmnt blocs lng bond th man dagonal contan onl a sngl matr J. Bt blocs lng on th man dagonal nod n. K n n contan sms of as man matrcs J as lmnts ond wth th.9. Bondar condtons As t was notd n th prvos scton of ths Chaptr that th global stffnss matr of a strctr s most oftn a snglar matr drctl aftr th aggrgaton. It mans that th dtrmnant of ths matr s qal to ro. Bcas th st of Eqn. (4) has to hav onl on solton for statc problms w hav to modf th global stffnss matr. It shold b don n sch a wa that th solton of th st of lnar Eqn. (4) s possbl. Th rason for th snglart of th matr K s th lac of nformaton abot spports of th constrcton ths w nd to dfn what th spport of th nod s. For trsss thr ar two tps of spports possbl: an artclatd spport and an artclatd movabl spport. Th artclatd spport (shown n Fgr 3a) prvnts movmnts of a nod n an drcton whch mans: 44

46 rx ry. (7) Th movmnt of th spport nod r cass ractons n two componnts: R X and R Y (Fgr 3a) whch contract th movmnt of th nod r. W sa that ths spport assrs fr spport of a nod. Fgr 3. Plan trss spport tps. Th nt spport shown n Fgr 3b s calld an artclatd movabl spport and t prvnts movmnts of a nod along on ln onl bt t allows movmnt of a nod n prpndclar drcton wth rspct to ths ln. Th racton occrrng n th artclatd movabl spport can hav th drcton of ths ln onl (Fgr 3bcd). Th spport can appar n a fw forms two most oftn occrrng varants (shown n Fgr 3bc) gv vr smpl spport condtons: - spport wth th possblt of movmnt along th Y as of th global coordnat sstm (Fgr 3b) rx (8) - spport wth th possblt of movmnt along th X as of th global coordnat sstm (Fgr 3c) ry. (9) 45

47 Th thrd varant of a movabl spport cass problms whn dscrbng th bondar condtons bcas th drcton of th racton of ths spport (Fgr 3d) s not paralll to an as of th global coordnat sstm. It s mportant bcas qlbrm Eqn. () ladng to Eqn. (4) wr wrttn n th global coordnat sstm. In th cas of a spport wth movmnt not paralll to an as of th global coordnat sstm (w wll call sch spports sw spports) w hav to wrt th bondar condtons n th sstm '' connctd wth th spport. Th sstm '' s rotatd wth rspct to th global sstm b an angl α' (Fgr 3d). W wll plan th transformaton mthod for a st of qatons at a spport nod to th local sstm n th nt scton. Now w wll focs on dscrbng th bondar condton. W wrt th condton of absnc of a movmnt along th ' as analogosl as n Eqn. (9): r'. () Eqn. ((7)...() dscrbng th bondar condtons gv s th vals of dsplacmnts at spport nods. Hnc som qatons of st Eqn. (4) shold b rmovd bcas th contan nnown forcs actng on spport nods (constrant ractons). Ths qatons can b rplacd b qatons of bondar condtons (for ampl Eqn. (7)). It s sall don b modfng som Eqn. (4). t m b th global nmbr of th dgr of frdom whch s lmnatd b th bondar condton: thn w modf th row wth th nmbr m n th global m stffnss matr K rplacng t b a row contanng ros and th val n th colmn m: K K K m K Nn K K K m K Nn P P = m K Nn K Nn K N m n K NnNn Nn P Nn or o r K p. () Th nodal load vctor p shold b modfd so that qaton m contans ro on th rght sd. Th modfd matrcs ar mard n Eqn. () b a sprscrpt r. 46

48 K m = whn Ths changs n th stffnss matr dstrb th smmtr bcas K m bt m (comp. Eqn. ()). Th absnc of smmtr n th stffnss matr dos not prvnt th solvng of th qlbrm Eqn. (4) bt t consdrabl loads th comptr mmor storng coffcnts K thr n th cor mmor (RAM) or trnal spac (ds) whch lngthns th solton tm for a st of qatons (comp. Appnd ). Ths lt s tr to rstor th smmtr of th matr K o (Eqn. ()). t s not that th trms locatd n th colmn wth th nmbr m ar mltpld b th ro val of th dsplacmnt m. Hnc w can nsrt ros nstad of coffcnts n th colmn m (cpt for on coffcnt n th row m whch has to b qal to ). If w modf th stffnss matr n that wa th solton of or problm wll b th sam and th matr wll b a smmtrc on: K K K Nn K K K Nn K r () K Nn K Nn K NnNn r K p Fnall w solv th problm: whr th matr r (3) r r K s smmtrcal and s not snglar whch mans that dt K f w hav proprl chosn th bondar condtons. On th bass of th thorm abot th postv val of a stran nrg (comp. Eqn. (45) Chaptr ) w can concld that th matr r K has to b postv-dtrmnant thn dt K r. (4) Hnc th st of Eqn. (3) has on solton. In small fnt lmnt sstms (programs) th matr r K s sall lft n th form notd n Eqn. (). arg and compl sstms sd to solv problms dscrbd b man thosands of qatons sall rmov rows and colmns contanng ros 47

49 from th matr r r K and vctorp. Ths s don to rdc th dmnsons of a solvd problm. Ths mthod of modfcaton of th matr r K rqrs r-nmbrng of dgrs of frdom of a strctr. Bcas t s not strctl ond wth FEM and s connctd wth th comptr mplmntaton of th FEM algorthm w wll not dscrb t hr... Transformaton of th stffnss matr for a sw spport Now w ar planng was of transformng an lmnt stffnss matr ond to a spport nod b mans of a sw spport (Fgr 3d). W choos th coordnat sstm '' n sch a wa that th drcton of a spport racton covrs th ' as and th movmnt wll b paralll to th ' as (an altrnatv choc of th local coordnat sstm s obvosl possbl). Th ' as s rotatd wth rspct to th X as of th global sstm b th angl α' whch w wll dm to b postv whn th rotaton from th X as to th ' as s antclocws. Th postv angl α' s shown n Fgr 3d. If w wrt qlbrm qatons for th spport nod r n th sstm '' thn th bondar condton of ths spport s dtrmnd b Eqn. (). t s tr to prform th ncssar transformaton. W ma s of rlatons Eqn. (8) and (83) whch srvd s n Sc..3 to pass from th local sstm of an lmnt to th global on. Thn w prss th nodal forcs vctor at th nod r as follows: F F r' r' c' s' s' F c' F rx ry or n an abbrvatd form: r T R' r fr f'. (5) Nt w transform th nodal dsplacmnts vctor of th spport nod from th local sstm to th global on as follows: rx ry c' s' or n a clos form: s' c ' r' r' 48

50 49 r r r R. (6) In Eqn. (5) and (6) w hav mard c s s c r R cos c sn s and T r R s th transpos of th matr r R. t s assm that an lmnt ons nods r and r spportd b sw spports whch ar rotatd b angls and (Fgr 4). Thn w wrt qlbrm qatons for nods r and r n th local coordnat sstm at th nod r and at th nod r. Th transformaton of nodal forcs vctors and nodal dsplacmnts vctors of th lmnt s as follows: - for a nodal forcs vctor f R f T ' ' (7) or n a dvlopd form r r r r r r f f R R f f T T ' ' ' ' - for th nodal dsplacmnts vctor R ' ' (8) or r r r r r r R R.

51 Fgr 4. Th bar wth sw spports. Insrtng rlatonshp Eqn. (8) nto (96) and th rslt nto(7) w gt th qaton transformng th stffnss matr of th lmnt from th global coordnat sstm to th spport coordnat sstm: T R' K R' ' f' (9) W smplf ths qaton to th form: f ' K' ' () n whch w ma s of th sbsttton: T R' K R' K' () dfnng th lmnt matr n th spport coordnat sstm. On of angls α' (Fgr 4) s most oftn qal to ro bcas t rarl happns that a trss bar ons two spport nods spportd b a sw spport. Th transformaton matr of a ro angl s a nt matr. Bcas (c'= s'=) thn th lmnt transformaton matr s smplfd to th form: R' R' r I () whn th scond nod s dscrbd n th global sstm bt w transform forcs and dsplacmnts at th frst nod r and 5

52 R' I R' r (3) whn th transformaton concrns th last nod r onl. As t has bn shown that th stnc of sw spports complcats th smpl FEM algorthm prsntd n Chaptr bcas t rqrs addtonal transformatons of lmnt stffnss matrcs bfor th aggrgaton of th global matr s don. Thr ar som othr smplr thogh appromat mthods of solvng ths problm and th wll b dscssd n th nt scton concrnng bondar lmnts... Elastc spports and bondar lmnts Not all nds of spports appld to spport trsss can b dscrbd b th bondar condtons of tps Eqn. (8) (9) and(). Thr ar flbl spports whch hav dsplacmnts connctd wth a spport racton for nstanc th lnar rlaton of th followng tp: R R h rx rx rx h ry ry ry (4) whr h rx s th spport stffnss n th drcton of th X as and h ry s th spport stffnss n th drcton of th Y as. Th lnar sprng shown n Fgr 5s a good modl of ths tp of spport. Fgr 5. Th lastc spport modl. If w trat ractons R rx and R ry actng on th nod spportd lastcall as trnal forcs thn w obtan th nodal forcs vctor contanng nnown dsplacmnts rx ry : 5

53 P X P X P Y P Y P X P X P Y P Y p= = R rx r hrxrx (5) R ry hryry P N X n N n P N X n P N Y n P N Y n Th vctor p cannot b absoltl sd as th rght hand sd of Eqn. (4) n whch nnown vals of nodal dsplacmnts shold b on th lft hand sd of th qaton. Now w ar transformng th vctor p dscrbd b Eqn. (5) n sch a wa that nodal ractons of th lastc nod r wll b movd to th lft hand sd of th qlbrm qaton: s K p whr r strctr and (6) s K s th stffnss matr contanng nformaton abot lastc spports of th r p s th nodal forcs vctor n whch th bondar condtons wrttn n Eqn. () (w can trat th lastc spports as fd ons aftr transfrrng th rlatons whch dscrbd thm to th lft hand sd of th qaton) ar consdrd. Th matr s K s wrttn b th qaton: 5

54 K K K m K ( m ) K Nn K K K m K ( m ) K Nn K s K m K m K mm h rx K m( m ) K mnn r (7) K ( m ) K ( m) K ( m ) m K ( m )( m ) ry h K ( m ) N n N n K Nn K Nn K N m n K N ( n m ) K NnNn whr m s th global nmbr of th frst dgr of frdom of th nod r. Wth standard nmbrng m=(r-)n D + whrn D s th nmbr of dgrs of frdom of th nod. For a D trss N D = th nmbr of th frst dgr of frdom of th nod r s qal to m=r-. At ths stag th modfd matr s K contans th stffnss of lastc spports whch ar addd to th trms comng from th trss lmnt of a strctr. Ths sms ar locatd on th man dagonal of th matr n rows dscrbng th qlbrm of th nod r. Sch an ntrprtaton of lastc spports lads to a convnnt althogh smplstc wa of consdrng fd spports. W sbsttt thm for lastc spports wth vr larg stffnss for ampl H= 3 onto th man dagonal. Ths mthod was formlatd b Irons and Ahmad (98) who mltpls trms lng n a stabl row on th dagonal of th matr K b nmbrs of th ordr of 6. Aftr nsrtng a hgh val onto th dagonal t s rrlvant to nsrt ros both n rows and colmns of th matr K as wll as rows of th vctor of th rght hand sd p. It s vr mportant for larg stffnss matrcs whch ar oftn stord n strctrs of data dffrnt from qadratc tabls (comp. Appnd ). Th smplct of ths mthod nsrs that t s commonl sd n th comptr mplmntaton of th FEM algorthm nstad of th act mthod dscrbd n Sc..6. Elastc spports also sggst th s of a spcal spport lmnt whch cold sbsttt an lastc constrants and fd spports (whch shold b tratd as lastc spports wth larg stffnss). Ths spport lmnt rotatd b an angl α wth rspct to th global coordnat sstm s shown nfgr 6. 53

55 Fgr 6Th bondar lmnt schm. W can asl obtan th stffnss matr of sch an lmnt from th matr of an ordnar trss lmnt dscrbd b Eqn. (75) n th local coordnat sstm or Eqn. (97) n th global sstm. W do t n sch a wa that w sbsttt th stffnss of a bar EA/ for th stffnss of th lastc bondar lmnt b. In gnral th nod o of ths lmnt s alwas fd so w can rmov t from th st of qatons whch allows s to trat th bondar lmnt as an lmnt wth two dgrs of frdom: K b b c sc sc s whr smlarl to Eqn. (78) c cos s sn. (8) Whn w want to sbsttt th fd spport for ths lmnt w accpt b =H. Th val of H dpnds on th comptr sstm n whch th program wll b startd and most of all t dpnds on th tp of ral nmbrs. W can ta for ampl H= 3 as rfrnc for man sstms... Th nodal loads vctor wth tmpratr load As w hav alrad notd n th ntrodcton to ths Chaptr trss loads whch act on lmnts and do not act on nods drctl ar tmpratr loads. Now w wll show how w can rplac ths load b nown loads that s concntratd forcs actng on th nods of a strctr. 54

56 Fgr 7. Th lmnt tnson casd b tmpratr. As w now th ncras n tmpratr of an lmnt cass t to lngthn whch wth th assmpton of a stad ncras n th tmpratr of th whol bar s dscrbd b th qaton: t tto (9) whr α t s th coffcnt of thrmal panson of th matral from whch th lmnt s mad Δt o stands for an ncrmnt of tmpratr n th mddl fbrs (onng cntrs of gravt of cross sctons of an lmnt). W assm a stad ncras n tmpratr n th whol scton and homognt of th matral. If w accpt that th lmnt has no frdom to grow bt s lmtd b fd nods w obtan an aal forc whch s st p wthn th lmnt: N tda E tda E ttoda E t t A A A A o (3) whr E s Yong s modls of th matral and A sgnfs th srfac ara of th cross scton of th lmnt. Th nodal forcs vctor of th lmnt d to th tmpratr wrttn n th local coordnat sstm s qal to: t f ' EA t to (3) aftr transformaton to th global sstm wth th hlp of rlaton Eqn. (9)w obtan c t s f EA tto (3) c s 55

57 whr c cos s sn α - s th angl dtrmnng a slop of th loadd lmnt wth rspct to th global coordnat sstm. Snc forcs actng on th nods ar ncssar for th qlbrm qatons and as t s nown th ar of oppost drcton to othr forcs actng on lmnts thn w sbtract thm from othr forcs whl bldng th global nodal forcs vctor. Ths s shown nfgr 8. P X P Y P X P Y P n X P n Y n Global nmbr of th frst nod of an lmnt f t = P X t P Y t P X t p= P n X P n Y n Global nmbr of th last nod of an lmnt P Y t P N X n Nn P N Y n P P t X t Y t EA t cos P EA t cos t o t EA t sn P EA t sn t o X Y t t o o Fgr 8. Th tmpratr load ncldd nto th nodal load vctor. 56

58 .3. Th gomtrc load on a trss Th fnal tp of trss load whch w wll dscrb s th gomtrc load (forcd dsplacmnts of nods). W assm that th nod r s dsplacd b th vctor d (Fgr 9). It s ncssar to appl forcs to th nod to cas ths dsplacmnt. Vals of ths forcs ar not nown whras w now componnts of th dsplacmnt of th nod r: rx d X ry d Y (33) whr d X d Y ar th componnts of th vctor of th forcd dsplacmnt d. Fgr 9. Th schm of th gomtrc load actng on th trss. Eqn. (33)s l th nown qatons of th bondar condtons (8)and (9)bt wth on dffrnc hr w hav obtand nonhomognos qatons. It changs th procdr of smmtrsaton of th stffnss matr. Prvosl w nsrtd ros nto stabl colmns of th matr K whch dd not ndc an consqncs bcas ths matr was mltpld b ro vals of dsplacmnts of th spport nods. At ths tm w hav to p th componnts of th matr occrrng n ths colmn bcas th ar mltpld b gvn dsplacmnts (comp. Eqn. (33)) and th ar sall not qal to ro. Hnc transformatons of th stffnss matr K and nodal loads vctor p ladng to th consdraton of th gomtrc load shold loo as follows: W form vctors rx and ry whch ar stabl colmns of th matr K ond wth th dsplacmnts of th nod r. rx s th colmn wth a nmbr qal to th dsplacmnt global nmbr rx and ry s th colmn wth a nmbr qal to th dsplacmnt global nmbr ry. 57

59 W mov th nodal forcs d to th nown dsplacmnts d X and d Y to th rght hand sd of th st of qatons: d p p d d rx X ry Y. (34) W consdr bondar condtons n th standard wa as n Sc..6. Howvr thr s on dffrnc w pt nown vals nto th rows of th rght hand sd vctor d p. Ths rows hav th global nmbrs qvalnt to th dgrs of frdom rx and ry. r K p whr Aftr mang th abov transformatons th followng st of qatons rss: rd (35) r K s th stffnss matr whch s modfd b th standard consdraton of th bondar condtons as n Eqn. () and Eqn. (34)aftr nsrtng nown vals of dsplacmnts: rd d p s th modfd vctor p dtrmnd b P rx d X P ry d Y. P X P Y P X P Y : : : : : : P rx n r d X n r p d = P ry - rx d X ry d Y p rd = d Y : : : : : : : : : : : : : : : : : : P NnX N n N n P NnY Fgr a. Prparng th gomtrc load vctor. 58

60 .... n r N n : n r K= : : : N n rx ry Fgr b. Th gomtrc load ncldd nto th nodal load vctor..4. Spport ractons ntrnal forcs and strsss n lmnts Aftr aggrgaton of th stffnss matr consdraton of th bondar condtons and bldng th nodal forcs vctor w obtan th st of lnar qatons n forms Eqn. (3)(6) or (35)wth a postvl dtrmnd smmtrc matr. Mthods of solvng sch qatons ar dscrbd n Appnd. Th solton of th st of qatons s th nodal dsplacmnts vctor of a strctr. Knowng nodal dsplacmnts allows s to dtrmn control sms of nods and spport ractons n th spport nods n a vr smpl wa. And thn w ma s of Eqn. (4) n whch th matr K dos not contan an nformaton abot th spport constrants. r K p. (36) Th vctor of ractons r shold contan ros at fr nods and vals of ractons at spport nods. If w assm th occrrnc of th local coordnat sstm n som nods (th sw spports) thn th componnts of ractons wll b prssd n th local coordnat sstm. Snc nmrcal rrors rsltng from approachng vals of nmbrs stord n th comptr mmor ncras drng th solton procss th control sms ar rarl qal to ro and th ar most oftn small nmbrs for ampl th ordr of -. Componnts of th global dsplacmnts vctor nabl th bldng of global dsplacmnts vctors for th lmnts (Fgr ). Snc th componnts of th vctor ar not alwas wrttn n th global coordnat sstm (th sw spports) thn t can happn that som componnts of 59

61 th vctor ar prssd n th global sstm and othrs ar prssd n th local coordnat sstm. To smplf frthr dscsson w standards th dscrpton of th vctor brngng down th componnts to th global coordnat sstm b tang advantag of Eqn. (8). It shold b notd that t s onl ncssar for lmnts ond to a nod whch s spportd b a sw spport. X Y X Y n X n Y n Global nmbr of th frst nod of an lmnt = X Y X = n X n Global nmbr of th last of an lmnt Y n Y Nn X N Y n N n Fgr. Th gomtrc load ncldd nto th global stffnss matr. Nodal dsplacmnts of an lmnt allow th ntrnal forc N n a trss lmnt to b calclatd qt asl. W can thr ma s of Eqn. (7) whch rqrs nowldg of dsplacmnts n th local coordnat sstm of th lmnt or on th bass of Eqn. (7) (74)and (93)w sarch th rlatonshp: 6

62 X X Y Y N EA c s whr smlarl to Eqn. (79) c cos and s sn. (37) Strsss n th trss lmnt assmng that th bar s homognos ar th aal strsss onl whch can b calclatd sng a smpl rlatonshp: N E A c s. X X Y Y (38) If th lmnt s loadd wth a tmpratr gradnt thn th corrcton comng from thrmal panson of th matral shown n Eqn. (37) and (38)shold b tan nto consdraton: E t E X X Y Y t o c s t (39) and EA N A c s t X X Y Y t o. (4) Th calclaton of dsplacmnts constrand ractons and ntrnal forcs n th lmnt complts th statc analss of th trss.. 3D trss strctrs Althogh 3D trss strctrs hav bn arond for a long tm (comp. Tmoshno and Goodr (96)) th hav bn sd vr rarl ntl now. Th ar partclarl dffclt to solv. Thogh a srs mthod smplfng th calclaton of ntrnal forcs (th mthod of nodal qlbrm and ts graphc varant - Crmona s mthod and th mthod of sctons - Rttr s mthod tc. ) has bn dvsd for statcall dtrmnd plan trsss n cas of spac trsss onl th mthod of nodal qlbrm has rmand. arg sts of qatons whch ar gnratd b ths mthod for spac trsss hav dscoragd ngnrs from dsgnng ths tp of strctr. 3D strctrs loong l trsss n fact ar sldom trsss. For nstanc th famos Effl s towr or spport colmns of ovrhad powr lns masts (n partclar wth th qadranglar crosss) ar most oftn spac frams bcas th p thr gomtrc stablt thans to bnt lmnts whch do not st n classcal trsss. Both th s of comptrs and nw mthods of statcs analss of a strctr mang s of nw 6

63 tchncal possblts (th fnt lmnt mthod s on of th man mthods among thm) hav nabld consdrabl progrss n dsgnng spac trsss. On of th most poplar ss of ths strctrs s n strctral roofs. Eampls of spac trsss ar prsntd n Fgr 3. a) th spac strctr b) th towr Fgr 3. Th ampl of 3D trsss... Notaton and basc rlatons Th nod of a spac trss has thr dgrs of frdom bcas n ordr to dscrb ts movmnt w hav to gv thr componnts of a dsplacmnt vctor. Th dsplacmnt vctor and forcs actng on an lmnt of th spac trss ar shown nfgr 4. As n Chaptr componnts of forcs and dsplacmnts vctor ar collctd n colmn matrcs whch wll b calld vctors; nodal dsplacmnts vctor of th frst nod n th global coordnat sstm: X Y Z (4) 6

64 th sam vctor n th local coordnat sstm: ' (4) vctor of nodal forcs actng at th frst nod of an lmnt wrttn n th global sstm: f F F F X Y Z (43) and n th local sstm: F f ' F F (44). Th abov vctors form forcs and dsplacmnts vctors of an lmnt: vctor of th nodal dsplacmnts of an lmnt wth th nod (th frst on) and (th last on) s wrttn n th global coordnat sstm as follows: X Y Z X Y Z (45) ts dscrpton n th local sstm s: ' ' '. (46) vctor of th nodal forcs of an lmnt n th global sstm 63

65 f f f F F F F F F X Y Z X Y Z (47) and n th local sstm f ' f ' f ' F F F F F F. (48) Intrprtaton and manngs of th smbols sd hr can b fond nfgr 4. 64

66 Fgr 4. Nodal forcs and dsplacmnts for th 3D trss lmnt: a) n th global coordnat sstm; b) n th lmnt local coordnat sstm... Th lmnt stffnss matr of a spac trss Th rlatonshp btwn nodal forcs and nodal dsplacmnts for a spac trss s dntcal to that for a plan trss f w anals t n th local coordnat sstm. Obvosl th thrd forc s F or F bt th qlbrm qaton of momnts wth rspct to th as rslts n th ro val of ths forc: F F F F F aftr consdrng q. f F F F F (49) 65

67 aftr consdrng q. F F F F M M F F M F F. Th rlatonshp btwn an aal forc and dsplacmnts whch s dntcal to th rlaton prsntd n Chaptr (comp. Eqn. (7)) allows s to prss th sarchd dpndnc as follows: f ' K' ' whr K' J' J' J' J' (5) (5a) J' EA (5b). Th transformaton of ths qatons from th local sstm to th global on wll b don analogosl to th transformaton prformd n cas of a D trss (Eqn. (9) (95) (96)). In ordr to complt th transformaton of th lmnt stffnss matr to th global sstm w nd th rotaton matr of a nod R and thn w can dtrmn componnts of th matr J smlar to thos dscrbd b Eqn.(98). 66

68 sstm. Fgr 5. Th trss lmnt arrangmnt wth rgard to th global coordnat Snc th locaton of th and as of th local sstm s not ssntal for trss lmnts w wll choos th drcton of th as n sch a wa that t wll alwas b paralll to th XY plan of th global sstm bt for bars paralll to th Z as thr wll b an addtonal assmpton that th as s paralll to th Y as (comp.fgr 5). Th rotaton from th local coordnat sstm to th global on wll b composd of two ntrmdat rotatons. Frst w rotat th sstm to th ntrmdat sstm '''''' slctd so that th '' as s paralll to th XY plan and nt w rotat th sstm '''''' b an angl γ so that th '' and X as ar paralll. Th frst rotaton arond th as gvs th followng rslt: '' '' '' c s s c or n a shortr form R (5) whr c cos Z s sn X X X Y Y Y Z Z Z X Y Z. 67

69 Th scond rotaton arond th as lads th qatons to th global sstm: X Y Z c s s c '' '' '' or n a shortr form R (53) X whr c cos Y s sn whn ''= w assm γ= hnc c and s. Th composton of both rotatons whch mans pttng Eqn. (5) nto(5) gvs th sarchd rotaton matr of a nod R R ' (54) whr R R R. Aftr mltplng matrcs R R w obtan th fnal form of th rotaton matr R : c c s c s R s c c s s s. (55) W calclat th transformaton of th bloc J of th lmnt stffnss matr of th spac trss from th local coordnat sstm to th global on as n Chaptr (comp. th smlar transformaton of th stffnss matr Eqn. (95)) J R J' R T. Insrtng rlatons Eqn. (5b) and (54) nto th abov qaton w obtan: c c c s c s c s c (56) c c s EA J c s cc. (57) c c s s c s s Aftr th ntrodcton of a convnnt notaton: C X X C Y Y C Z Z (58) 68

70 whch ar calld drcton cosns of an lmnt w obtan a vr smpl form of th bloc J of th stffnss matr: C X CXCY CXCZ EA J CXCY CY CYCZ. (59) CX CZ CYCZ CZ Rlaton Eqn. (58) obtand aftr bng nsrtd nto Eqn. (5a) gvs s th lmnt stffnss matr for th spac trss n th global coordnat sstm..3. Th vctor of tmpratr loads for an lmnt of 3D trss Snc formng a loads vctor of a trss for concntratd forcs s dntcal to formng t for a D trss w wll also not dscss th vctor p. On th othr hand w wll dscss th vctor of nodal forcs d to a tmpratr load. Componnts of ths vctor n th local coordnat sstm ar dntcal (apart from th corrcton n rfrnc to th thrd componnt of th vctor!) to th componnts of th vctor for a plan trss Eqn.(3). t f EA tto. (6) Th transformaton to th global sstm procds n agrmnt wth Eqn. (9) n th followng wa: f t t whr R R f ' (6) R s th lmnt rotaton matr: R R. Eqn. (54). (6) Snc a trss lmnt s straght R =R whr th matr R s dfnd b Aftr nsrtng Eqn. (54) nto (6) and mltplng thm w obtan 69

71 c c s c s t f EA tto (63) c c s c s or n anothr form: C X CY C t Z f EA tto. (64) C X C Y CZ trss. Th rmanng procdr s dntcal to th on mplod n cas of a plan.4. Th bondar lmnt In Chaptr w pland wdl dffrnt tps of bondar condtons and also lastc bondar lmnts. Snc th ar vr sfl lmnts for modllng man dffrnt bondar condtons w wll pa mor attnton to thm n ths chaptr concntratng on dffrncs btwn plan and spac trsss. W wll dscss th most gnral lastc lmnt wth stffnss b droppng wth rspct to as of th global sstm wth th angls α X α Y α Z whos drcton cosns ar qal to c X cos X c Y cos Y c Z cos Z. (65) Th stffnss matr of ths lmnt n th local sstm s analogos to th matr stffnss of an ordnar trss lmnt bt ths lmnt has thr dgrs of frdom so th stffnss matr contans onl on bloc J (Eqn. (5b) K' b b (66). Transformng ths lmnt to th global coordnat sstm w obtan a matr whch s vr smlar to th on obtand n Chaptr for a plan trss: 7

72 K b b c c c c c c c c c c cx cx cy cx cz X Y Y Y Z X Z Y Z Z (67) Bondar lmnts can form for ampl an lmnt wth thr dffrnt tps of stffnss paralll to as of th local sstm : K' b. (68) Th transformaton of ths matr to th global sstm s analogos to th transformaton of th bloc J Eqn.(56) dscssd arlr. W do not gv th rslt of ths transformaton hr lavng ts cton as an rcs for th radr..5. Strsss and Intrnal forcs As n Sc.. of Chaptr w prsnt hr qatons to calclat strsss and ntrnal forcs n an lmnt: E t E t o t or n anothr form: (69) E ' E t to (7). E Th transformaton of th vctor R T E tto to th global sstm gvs th rlatonshp: (7) whch aftr mltplcaton gvs componnts of drct strss n an lmnt as follows: E T T c c R T tt T whr c s th vctor of lmnt drcton cosns: c c c o c (58). X Y Z (7) Calclatng th normal forc conssts of ntgratng strsss on th srfac of a cross scton wth an assmpton of homognt of th strss fld (as n Chaptr ) N A EA T T c c R T t t o. (73) 7

73 Th rmanng spport ractons ar calclatd wth th hlp of Eqn. (36). W do t actl n th sam wa as t has bn don for th D trss so w wll not dscrb hr th abov problm for a spac trss n dtal. 3. D fram sstms Th corrct choc of modl for a strctr s vr mportant for qalt and actnss of th rslts obtand. Th choc of fram or trss (for ampl a trss wth fd nods) s oftn sbctv and t dpnds on prnc and ntton of th analst. In ths chaptr w wll prsnt th followng modl of a bar strctr - a D fram whch gvs mor possblts of modllng ral strctrs. Th lmnt of a D fram s mor gnral than a trss lmnt prsntd n Chaptr bcas wth hlp of ths lmnt w can also modl dal trss strctrs (artclatd conncton of lmnts at nods). W can smpl sa that a fram s a strctr whos bars can b bnt whl trss lmnts can b onl comprssd and strtchd. It has th followng consqncs: bar (an lmnt) of a fram can b loadd btwn nods modllng of dffrnt tps of loads s possbl for ampl: concntratd forcs concntratd momnts dstrbtd loads conncton of an lmnt wth a nod can b a fd conncton provdd that th rotaton of a nod and of a nodal scton of th lmnt ar dntcal or t can b an artclatd conncton whn ndpndnt rotatons of a nod and a nodal scton ar possbl nod of a D fram has thr dgrs of frdom whch mans that w hav to now two componnts of a translaton vctor: X Y and th rotaton angl φ Z n ordr to dtrmn th locaton of ths nod. In th cas of plan frams w wll nglct nd Z of rotaton angls n or notaton bcas all rotaton angls on th plan XY (whch w wll s to dscrb th strctr) ar rotatons wth rspct to th Z as. t s assm that a fram lmnt s straght and homognos whch mans that t s mad from a homognos matral and has a constant cross scton. Th vw of a fram lmnt drctons snss of nodal dsplacmnts and forcs whch w wll consdr as postv ar prsntd n Fgr 6. 7

74 Fgr 6Nodal loads and dsplacmnts for th plan fram lmnt: a) n th global coordnat sstm; b) n th lmnt local coordnat sstm. 3.. Th lmnt stffnss matr for a D fram W grop nodal dsplacmnts and forcs shown n Fgr 6ab n colmn matrcs st as w dd prvosl n Chaptrs and 3. Th ar calld vctors: dsplacmnt vctor of th frst nod and th last nod n th local sstm (Fgr 6b) ' '. (74) nodal forcs vctor n th local coordnat sstm 73

75 74 f ' F F M f ' F F M. (75) lmnt dsplacmnt vctor n th local coordnat sstm ' ' '. (76) lmnt forcs vctor n th local coordnat sstm f f f ' ' ' F F M F F M. (77) W can also dscrb all th vctors formlatd abov n th global sstm: Y X Y X (78) f X Y F F M f X Y F F M (79) Y X Y X (8)

76 f f f FX F Y M FX F Y M. (8) As n th prvos chaptrs th rlatonshp btwn nodal forcs and nodal dsplacmnts wll b of grat mportanc. Ths rlaton (analogos to Eqn. (66) for a trss) n th local coordnat sstm has th form: K' ' f ' (8) and n th global sstm K f. (83) At th momnt w wll concntrat on sarchng for th stffnss matr th local coordnat sstm and nt ts transformaton to th global sstm. K n Eqlbrm qatons of th lmnt prsntd n Fgr 6b lad to th followng rlatons btwn nodal forcs: F F F F F ; F F F ; (84) M M th vctor M F. It has bn shown that thr qatons ar nabl to calclat s componnts of f. Th dscsson concrnng lmnt strans wll provd ths mssng qatons. Th dformaton casd b th aal forcs F and F s dntcal to th dformaton of a trss lmnt hnc w ta advantag of prvosl dtrmnd dpndnc Eqn. (7) and Eqn. (73a). W wll obtan th rmanng qatons whn w consdr th flral dformaton of an lmnt and th rlatonshp btwn sharng forcs and bndng momnts. Th wll-nown rlatonshp btwn crvatr and bndng momnt s (comp. Jastrębs t al. (985)): 75

77 d d d d 3 M EJ (85) whr ρ rprsnts th rads of a crvatr E s Yong s modls of a matral J s th momnt of nrta of an lmnt cross scton (comp.fgr 6a). Fgr 7. Th sgmnt of th bar lmnt wth ntrnal forcs. qaton: T( ) Th qlbrm of on scton of a bar n bndng (Fgr 7) gvs th dm d ( ). (86) Snc w ar dalng wth lnar strctrs wth small dflctons w assm d whch smplfs Eqn. (85) to th wll-nown form: d d M d ( ) EJ. (87) Th oppost sgn of th rght hand sd of Eqn. (85) and (87) (comp. Jastrębs t al. (985)) to th on that w hav sall assmd coms from th sns of th as of th local coordnat sstm whch s orntatd antclocws n or assmptons. Dffrntatng ths qaton twc w obtan th rlatonshp (comp. Jastrębs t al. (985) Nowac (976)): 76

78 4 d q 4 d ( ) (88) EJ whr q () dnots th dstrbtd load whch s prpndclar to th as of an lmnt. Hr th lmnt s fr from nodal loads ths q. Fnall w obtan th st of dffrntal qatons: 4 d a) 4 d b) c) d M ( ) d EJ 3 d T( ). 3 d EJ (89) Aftr ntgratng rlatons Eqn. (89a) w obtan th followng qatons: bndng ln of th fram lmnt: 3 ( ) C C C3 C4 6 bndng momnt: M( ) EJ C C (9) (9) sharng forc: T( ) EJ C (9) whr C... C 4 ar ntgraton constants whch shold b dtrmnd on th bass of bondar condtons. W hav for bondar condtons: at nod = : ( ) d d at nod =: (93) ( ) (94) 77

79 d d. Aftr nsrtng ths condtons nto Eqn. (9) w obtan th followng vals of th ntgraton constants: 6 C C 4 6 (95) C3 C4. Hnc aftr pttng th abov qatons nto Eqn. (9) (9)and consdrng th snss of both nodal and bndng momnts (comp. Fgr 6 andfgr 7) w obtan M EJ M ( ) 4 6 M F EJ M ( ) 4 6 EJ T( ) 6 6 (96) F EJ T ( ) 6 6. Fnall tablatng Eqn. (73a) and (96)n a stabl sqnc w obtan th stffnss matr: 78

80 K' = EA EA EJ EJ EJ EJ EJ EJ 6 EJ 4 EJ 6 (97) EA EA EJ EJ EJ 3 6 EJ 3 6 EJ EJ 6 EJ EJ 6 4 Th rlatonshps dscrbd b Eqn. (96) ar calld transformaton formla of th dsplacmnt mthod n strctral mchancs (n som othr form) (comp. Nowac (976)). 3.. Transformaton of th stffnss matr from th global coordnat sstm to th local on Th transfr of th matr K' to th global coordnat sstm s don accordng to rls analogos to th rls dscrbd b Eqn. (75) n Sc..4. In ordr to obtan th transformaton matr of an lmnt w nd R that s th transformaton matr from th local sstm to th global on for th nod. Snc th thrd dgr of frdom of fram nods s a rotaton wth rspct to th as whch dos not chang ts locaton bcas t s alwas prpndclar to th plan th rotaton wll b th sam as for a trss lmnt: X cos sn Y sn cos Z or n th matr form X c s Y s c R 79

81 c s whr R s c (98) In accordanc wth th assmpton accptd n th ntrodcton that th fram lmnt s straght th transformaton matr of th nod s dntcal to R whch lads to th fnal form of th lmnt stffnss matr: R c s s c c s s c. (99) Aftr mltplng matrcs dscrbd b Eqn. (95) w obtan th stffnss matr of a fram lmnt n th global coordnat sstm. Unfortnatl ts form s rathr compl: φ φ c s sc -6s c s sc -6s F sc s c 6c sc s 6c F c EJ K -6s 6c 4-6s -6c M () c s sc -6s c s sc 6s F sc s c -6c sc s c -6c F -6s 6c 6s -6c 4 M J cos A c s sn 3.3. Statc rdcton of th stffnss matr Fram lmnts ar not alwas ond at a nod nsrng th agrmnt of all nodal dsplacmnts and dsplacmnts n th bar scton at ths nod. Artclatd onts shown n Fgr 8ar ampls of sch ncomplt connctons. 8

82 Fgr 8. Th lmnt ont schm wth on lmnt abl to rotat (an artclatd ont). At ths ont th angl of th nodal rotaton dos not nflnc th rotaton of th lmnt scton of a nod. Th lattr can rotat ndpndntl of th nod (th lmnt n Fgr 8). W dtrmn th nnown angl of th rotaton of sch an lmnt sng an addtonal qaton whch s gvn b th qlbrm condton of momnts n a ont. Hnc w can rdc th nmbr of dgrs of frdom of th lmnt bcas th addtonal qlbrm condton allows s to lmnat on dsplacmnt from th st of qatons. W wll show th wa to lmnat th dgr of frdom sng th ampl of two tps of connctons of an lmnt wth a nod. Eampl No - artclatd conncton (Fgr 9). Fgr 9. Th lmnt wth th artclatd nod ont. Addtonal qlbrm condton of a scton at th nod : M = () lads aftr consdrng Eqn. (75) (8)and(97) to condtons: EJ () and ths w calclat th rqrd val of th rotaton angl of th scton at th nod : 8

83 (3) Aftr pttng ths rslt nto Eqn. (8) and tang nto consdraton matr (97) w obtan: EJ EJ F (4) EJ F EJ EJ M EJ and hnc th nw stffnss matr of an lmnt wth th ont at th nod :

84 EA EA K' ( 3 ) EJ 3 3 EJ 3 3 EJ 3. EA EJ 3 3 EJ 3 EA EJ 3 3 EJ 3 EJ 3 3 EJ (5) Sprscrpts (3) n th notaton of th stffnss matr ndcat that th thrd dgr of frdom s lmnatd at th frst nod. Eampl No - movabl conncton ( Fgr 3) Fgr 3. Th lmnt wth th nod translaton possblt. : Hr th addtonal condton s th dsapparanc of th aal forc at th nod F = (6) whch aftr analogos transformatons lads to th qaton: F = (7) and t dos not chang th rlatons for th rmanng nodal forcs. Th stffnss matr of sch an lmnt tas th followng form: 83

85 EJ 3 EJ 6 EJ 3 EJ 6 K' ( ) EJ EJ 6 EJ 4 EJ 6. = (8) EJ 3 EJ 6 EJ 3 EJ 6 EJ 6 EJ EJ 6 EJ 4 Sprscrpts () n th notaton of ths matr ndcat that th frst dgr of frdom at th last nod of an lmnt has bn lmnatd. Th abov procss s calld th statc rdcton of a stffnss matr. Now w wll gv th matr notaton of an opraton ladng to a rdcd stffnss matr. For th sa of smplct w assm that th last dgr of frdom of an lmnt s th lmnatd dgr of frdom. Nodal forcs nodal dsplacmnts vctors and th stffnss matr ar dvdd nto blocs: f = K K (9) f K K whr accordng to th smmtr of th matr w hav T K K T K K K s th matr wth dmnson and ths t s a scalar th blocs f and ar also scalars. Th rslts of th mltplcaton of matr blocs Eqn. (9)ar: a) f K K b) f K K scalar. () From Eqn. (b) w calclat 84

86 K K () and aftr nsrtng ths rlaton nto (a) w obtan f K K K K () or f K'' (3) whr K' ' K K K K (4) T s th condnsd lmnt stffnss matr. Vctor f of an lmnt load stll rmans to b dtrmnd. W obtan t b composng both th load vctor th vctor o f of an lmnt wth rgd connctons wth nods and f of th load casd b dsplacmnts of nods fr from constrants f f f f f f o f o o f f. (5) Snc o f f f thn o o o (6) f f K K (7) and hnc o o K f bcas othr dsplacmnts contand n ar qal to ro. Fnall w obtan o o o o f f K K f. (8) (9) In ths wa w can lmnat an dgr of frdom bt t rqrs som mor compl transformatons. W lav ths problm to b solvd b th radr. 85

87 Mr Mr Rr X Mr Rr Y Rr Y 3.4. Bondar condtons of plan fram strctrs Spports for plan frams ncld artclatd and fd spports all lstd n Chaptr. Th lattr ons prvnt th rotaton of a spport nod. Smbolc notaton of ths spports and th bondar condtons dscrbng thm ar shown n Fgr 3. Consdrng bondar condtons rqrs th modfcaton of a global stffnss matr of a strctr and t s don dntcall as for a plan trss (Sc..6) ths w wll not dscrb th wa of modfng ths matr hr. A whol rang of othr spports sch as movabl sw spports and lastc spports consdrd analogosl to spports of trsss dscrbd n Chaptr s also possbl. As a gnral mthod of consdraton of non-tpcal spports w propos to consdr th s of stabl bondar lmnts nstad of ths spports. W wll dscss ths n th nt scton. Rr X a) rgd spport Y r = r X = r Y r = X b) rgd-movabl spport (a dsplacmnt n th drcton of th global X as) Y r r = r Y = X c) rgd movabl spport (a dsplacmnt n th drcton of th global Y as) Y r = r X = r X Fgr 3. Plan fram spport tps. 86

88 3.5. Bondar lmnts of D frams Introdcng a bondar lmnt s a convnnt wa to avod problms connctd wth th consdraton of dffrnt non-tpcal bondar condtons. It allows n fact s to modl fd and fd-movabl spports wth appromat actnss and to sbsttt lastc spports. Now w wll prsnt a sngl lastc spport nclnd at som angl. Th schm of ths lmnt and notatons sd ar shown nfgr 3. Fgr 3. Th plan fram lastc spport schm. Stffnss of sprngs: h r and h r ar forcs whch shold b appld to thr nds n ordr to ndc ntar tnsons. Rotaton stffnss of a spport g r s a momnt ncssar to ndc th rotaton of th nod r qal to on radan. form: K b Th stffnss matr of sch an lmnt n th local coordnat sstm has th hr hr gr. () Its transformaton to th global sstm s don analogosl to th cas of normal fram or trss lmnts cpt that t concrns on nod onl Eqn. (95). Th rotaton matr s gvn b Eqn. (88). Hnc w can wrt th qaton transformng th matr K' b to th global sstm: K b b T RrK' Rr. () Aftr tang nto consdraton Eqn. (88)and ()w obtan 87

89 K b whr c h s h sc h h sch h c h s h r r r r r r r r s sn c cos. gr () If w modl flbl spports w oght to assm hgh stffnss of a stabl sprng. In most cass stffnss of th ordr of 3 assrs smlart btwn rslts obtand wth ths mthod and th rslts obtand wth th act mthod Intrnal forcs d to a statc load Th vart of loads whch can act on a fram strctr s consdrabl gratr than t was n th cas of a trss. Fram lmnts can b affctd b concntratd (forcs momnts) dstrbtd (prssr momnt loads) and tmpratr loads. Th formlaton of qlbrm qatons rqrs sbsttton of ntrnod loads for an qvalnt st of concntratd forcs and momnts actng on nods. Th wa of rdcng ths loads wll b th sbct of or dscsson n ths scton. Eqn. (9) and (95)dfn dsplacmnts of an lmnt bndng n th drcton of th as of th global sstm. Aftr addng th qatons dscrbng th dsplacmnts n an aal drcton w obtan rlatons dfnng th dsplacmnts vctor for an pont btwn nods N (3) whr N s th rctanglar matr of shap fnctons. It contans two blocs: N () - matr of th shap fnctons for th frst nod and N () - matr of th shap fnctons for th last nod. N N N. W can obtan both matrcs from Eqn. (9) and (7): (4) N 3 5 (5)

90 6 - ξ ( ξ) - ξ( 3ξ) - + 6ξ N whr non-dmnsonal dsplacmnt fnctons ( =... 6)) and thr drvatvs ar srvd n Tabl 4. Th convnnt non-dmnsonal coordnat / s ntrodcd hr. t s consdr now th bar (an lmnt) of a plan fram loadd wth statc loads (Fgr 33). Fgr 33. Th plan fram lmnt loadd wth statc loads. W wll fnd nodal forcs qlbrm. W wll s th prncpl of vrtal wor hr: n T f ' f b mang s of condtons of lmnt (6a) whr n s th wor of nodal forcs q q m whr s th wor of trnal forcs (statc loads). o d (6b) Tabl 4.Non-dmnsonal dsplacmnt fnctons. 89

91 ω ω graph Nr ωʹ ωʹ graph ωʺ ωʺ graph ξ ξ 3 3ξ + ξ 3-6ξ( ξ) ξ 4 ξ (3 ξ) 6ξ( ξ) 6 ξ 5 ξ( - ξ + ξ 3 ) - 4ξ + 3ξ ξ Concntratd forcs and momnts can also b analsd b dscrbng thm n th followng wa: q( ) ( o) P M o ( o ) M o (7) whr δ( o ) s Drac s dlta dfnd as (comp. Nowac (979)) ( o ) whl o ; ( o ) whl o ; (8) ( o ) whl o 9

92 and ( ) d. o Th lmnt qlbrm s mantan whn n + = whch mans T T f q whr q() s th vctor of trnal loads: d (9) q q q. (3) mo Pttng th prsson dscrbng th lmnt dsplacmnts vctor Eqn. (3)nto (9)w obtan rlatons: T f T f T q N d (3) Nq d (3) whch nabls s to rplac loads actng on lmnts b loads actng on nods. It shold b notd hr that thr ar forcs actng on th nods n th qlbrm qatons and that ths forcs act aganst thos actng on th lmnt (comp. Fgr. 8) ths th shold b sbtractd from th nodal forcs vctor of th strctr. W chc th ffctvnss of Eqn. (3) for thr smpl ampls whn:. th load wth a concntratd forc s appld to th cntr of an lmnt. th load wth a concntratd momnt 3. th dstrbtd load whch s constant for th whol lmnt. 9

93 9 Eampl No. Fgr 34. Th fram lmnt loadd wth a concntratd forc. W ntrodc a non-dmnsonal coordnat / to ma th calclatons mor convnnt and wrt th concntratd forc as follows:.5 P q and aftr pttng t nto Eqn. (3) w obtan d P d P T T N N f /8 / /8 / P P whch mans that F P F P M 8 F P F P M 8.

94 93 Eampl No. Fgr 35. Th fram lmnt loadd wth a concntratd momnt. W wrt th concntratd momnt appld to th cntr of an lmnt b sng Drac s dlta: q 5 M o. Aftr nsrtng th load vctor nto Eqn. (3) w obtan d M d M o o /.5.5 /.5.5 T T N N f M o whch mans that F o M F 3 o M M 4 F o M F 3 o M M 4.

95 Eampl No 3. Fgr 36. Th fram lmnt loadd wth a nforml dstrbtd load. Th contnos load nforml dstrbtd on th whol lngth of an lmnt gvs a load vctor: q f ' q q o. o Aftr nsrtng th vctor q nto Eqn. (3) w obtan th qaton: whch mans that / / d qo / / F F qo M qo F F qo M qo Forcs casd b a tmpratr load Th acton of a tmpratr on fram lmnts can cas flon. Ths happns whn th tmpratr fld s not homognos n th cross scton. In th cas of a trss th flon of bars dd not cas ncrasng nodal forcs bcas trss lmnts ar connctd b mans of ontd nods. Bars of fram strctrs can ma a nod 94

96 rotat hnc w hav to dtrmn forcs at th nod n th lmnt ndrgong th acton of th non-nform tmpratr fld. Fgr 37. Th tmpratr dstrbton n th lmnt cross scton. t s consdr an lmnt of whch th ppr fbrs ar affctd b an ncras n a tmpratr Δt g and th lowr fbrs ar affctd b an ncras n a tmpratr Δt d (Fgr 37). Th tmpratr fld can b wrttn as follows: t to h t h (33) whr t t t o h d g g d s th ncras n th tmpratr of th mddl fbrs t t t s th dffrnc of tmpratrs btwn trm fbrs h s th h g d hght of th cross scton d s th dstanc btwn th cntr of gravt and th lowr fbrs g s th dstanc btwn th cntr of gravt and th ppr fbrs. Strans of th lmnt fbrs ndcd b th tmpratr fld ar qal to t t t t t t o h h whr α t s th panson coffcnt of th matral. If bars cannot dform frl thn strsss rs nsd thm: E Et t t t o h h whch th ntrnal forcs rslt from: (34) (35) N t h da te to da da. (36) h A A A 95

97 Snc th scond ntgral occrrng n Eqn. (36) s th statc momnt wth rgard to th as whch crosss th cntr of gravt ths momnt has to b qal to ro. Ths w obtan N t t EA (37) t o l n th cas of a trss lmnt. Th scond ntrnal forc casd b tmpratr strsss s th bndng momnt: M t da E t A t h t o da da. (38) h A A Th frst ntgral n th abov qaton has to b qal to ro smlarl to Eqn. (36) and th scond on s th momnt of nrta of th cross scton calclatd wth rgard to th mddl as. Ths w can wrt an qaton dscrbng th bndng momnt d to tmpratr strsss as M t whr t h t h EJ (39) J A da s th momnt of nrta of th lmnt scton wth rgard to th as crossng th cntr of gravt of th scton. W calclat forcs at nods mang s of th prncpl of vrtal wor st as w dd n Sc.4.6: T t f whr t t T t d (4) t N t (4) Mt s th vctor of th ntrnal forcs ndcd b a tmpratr. Th ro val of th prsson n th scond row of th vctor coms from th fact that th tmpratr dos not cas sharng forcs n th lmnts ε() s th vctor of dsplacmnts gradnts: 96

98 d d d ε B d d d ( ) (4) B B B B B B s th matr of drvatvs of shap fnctons:. On th bass of Eqn. (5) w calclat ' whr ntabl 4. ' 3' 5' 3'' 5'' 4 ' 6 ' 4 '' 6 '' (43) (44) ( =... 6)) ar nondmnsonal fnctons gvn On th bass of Eqn. (4) w calclat componnts of th nodal forcs vctor: f t B T t d. (45) t Aftr nsrtng matr Eqn. (44)nto Eqn. (45) w obtan 97

99 d t h J d t h J d t A d t h J d t h J d t A E h h o h h o t t f (46) whr and ar non-dmnsonal coordnats at both th bgnnng and nd of th acton ntrval of th tmpratr load (Fgr 38). Fgr 38. Th tmpratr loadd fram lmnt. In th cas whn th tmpratr load s constant and occrs along th whol lngth of th lmnt w obtan th followng qaton from Eqn. (5): h t J A h t t J t A E h o h o t t f. (47) Both Eqn. (5)and (6) dscrb ntrnal forcs actng on th lmnt. So whn w form th load vctor of a strctr w shold sbtract componnts of ths vctor from stabl componnts of th global vctor.

100 Statcs of a 3D fram sstm A thr-dmnsonal fram strctr s th most gnral tp of bar strctrs. Elmnts of a spac fram can srv for modllng of all th prvosl dscrbd strctrs (D and 3D trsss D frams) and som othrs sch as grllwors bams bron n a plan and loadd prpndclarl to ts plan tc. A fw ampls of strctrs whch cannot b modlld b lmnts prsntd so far bt can onl b modlld wth th hlp of 3D fram lmnts ar prsntd n Fgr 39. Fgr 39. Th 3D fram ampls Th lmnt stffnss matr of a 3D fram An nod of a spac strctr has s dgrs of frdom whch mans that t can sbmt to thr ndpndnt dsplacmnts and thr rotatons. Hnc a fram lmnt has twlv dgrs of frdom. Componnts of both nodal forcs and 99

101 dsplacmnts of th fram lmnt ar shown n Fgr 4. Th local coordnat sstm has to b chosn n sch a wa that as and ar th prncpal as of a cross scton bcas t smplfs th dscsson of a bndng of problm. Bndng of sch an lmnt can b analsd as two ndpndnt phnomna of bndng n plans and. Fgr 4. Nodal loads and dsplacmnts for th 3D fram lmnt n th lmnt local coordnat sstm. Hr w wll prsnt nodal dsplacmnts and forcs smlarl that s n th form of vctors (colmn matrcs). Th nodal dsplacmnt vctor of an lmnt n th local sstm s: ' ' ' whr (48)

102 (49) ' s th dsplacmnt vctor of th nod n th local coordnat sstm dsplacmnt vctor of th nod n th local coordnat sstm. Th nodal forc vctor of an lmnt n th local sstm s ' s th f ' f ' f ' whr (5) F F F f ' M M M F F F f ' M M M f ' s th forc vctor of th nod n th local coordnat sstm of th nod n th local coordnat sstm. th form: (5) f ' s th forc vctor As sal w loo for th rlatonshp btwn nodal forcs and dsplacmnts n f ' K' ' (5) whr th stffnss matr K s a sqar and smmtrc matr wth dmnsons. Most componnts of ths matr can b calclatd on th bass of th rslts obtand for a D fram n Chaptr 4. Snc th bndng n prncpal plans of th cross scton s ndpndnt w wll splt th dformaton of th lmnt of a thrdmnsonal fram nto a fw smplr form: aal tnson whch s dntcal to that n a trss bndng n th plan whch s smlar to th stats of a D fram; modfcatons concrn th sgns of ntrnal forcs torson.

103 Torson of a fram lmnt s a stat whch has not bn dscrbd so far. Th dpndnc btwn a nodal torson momnt and a torson angl of an lmnt s qt smpl (comp. Jastrębs t al. (985)) and rsmbls th rlaton btwn an aal forc and an lmnt tnson: whr M GC (53) s th ncras n th torson angl d to th torson momnt M E G - s Krchhoff s modls and C s th torsonal rsstanc charactrstcs. ( ) Th constant C has th dmnson of a momnt of nrta and s qal to th polar momnt of nrta for crclar-smmtrc sctons (comp. Jastrębs t al. (985)) bt for othr sctons t shold b calclatd b s of qt compl mthods (comp. Tmoshno and Goodr (96)). Th calclaton mthod of ths constant for a fw poplar cross sctons n ngnrng practc s gvn n Appnd 3. Eqn. (53) allows s to wrt th rlaton btwn th nodal rotatons arond th as and nodal torson momnts: M M GC GC. (54) Th abov qatons ar th sarchd rlaton whch allows s to wrt th lmnt stffnss matr. Snss of nodal forcs casd b ntar nodal dsplacmnts whch allow s to dtrmn sgns of th prssons of th stffnss matr ar shown n Fgr 4.

104 Fgr 4. Th sgns of th nodal forc vctor casd b ntar nodal dsplacmnts. 3

105 Th lmnt stffnss matr s prsntd b qaton (55) 3.9. Tranformaton of th stffnss matr to th global coordnat sstm Th lmnt stffnss matr shold b transformd to th global sstm. Th transformaton mthod of th matr of a fram lmnt s analogos to th transformaton of an lmnt of a 3D trss prsntd n Chaptr 3 bt th thrd rotaton 4

106 arond th as of th local sstm s ncssar n ordr to lad as and to th poston of th prncpal cntral as of nrta of an lmnt cross scton. Sch a choc of local as s vr mportant for bldng th stffnss matr whch has bn notd at th bgnnng of ths chaptr. Th locaton of an lmnt n spac appld tps of coordnat sstms and rotaton angls notatons ar prsntd nfgr 4. Fgr 4. Th fram lmnt arrangmnt wth rgard to th global coordnat sstm and th notaton of basc vctors. In Fgr 4th followng notatons ar sd: as basc vctors of as of th local coordnat sstm and E X E Y E Z as basc vctors of as of th global coordnat sstm. Th wll b hlpfl n sbsqnt transformatons Us of th rotaton angl α for bldng th transformaton matr Now w prform th transformaton of a crtan dsplacmnt vctor ' from th local sstm to th global on b th composton of thr rotatons: R R R ' (56) whr R c s (57) s c 5

107 s th rotaton matr arond th as b an angl α c R s s c s th rotaton matr arond th ' as b an angl β (58) c s (59) R s c s th rotaton matr arond th '' as b an angl γ. In Eqn. (57) (58)and (59)w hav c cos s sn c cos s sn c cos and s sn. Eqn. (56) can b wrttn n a smplr wa: R ' (6) whr R R R R s th transformaton matr and th nvrs rlaton s: T R (6) whr R T R T R T R T. Wth ths mthod of transformaton fnctons of angls γ and β can b dtrmnd on th bass of nodal coordnats of an lmnt (th dpnd on drcton cosns of an lmnt - comp. Sc.3.) and th angl α s an addtonal paramtr whch has to b gvn for all lmnts Us of a drcton vctor Hr w wll prsnt anothr wa of dtrmnng th transformaton matr. t an addtonal paramtr dtrmnng an lmnt b a drcton vctor (Fgr 4) whch s locatd on th as of th local sstm and ts modls s qal to nt (sch a vctor s calld a basc vctor or a vrsor of an as). Hnc w hav: vctor of th as of th local sstm dtrmnd on th bass of lmnt coordnats (ts componnts ar drcton cosns of th lmnt) X Y Z X Y Z (6) 6

108 gvn drcton vctor of th lmnt X Y Z (63). W loo for th thrd basc vctor whch allows s to wrt th transformaton of an vctor from th local coordnat sstm to th global on XYZ. Snc th sstm s th rght cartsan coordnat sstm thn th vrsors of ths sstm ar orthogonal. Ths w can wrt (64) and from hr w calclat X Y Z (65) whr X Y Y Z Z Y X X Z Z Z X X Y Y (66) ar th coordnats of th vrsor of th local as wth rgard to th global coordnat sstm. Snc an vctor can b prsntd as a sm of prodcts of ts coordnats and vrsors thn w obtan: E E E X X Y Y Z Z X X Y Y Z Z E E E E E E X X X X or lss Z Z Z Z X X Y Y Z Z E E E Y Y Y Y (67) R ' (68) whr R s th rotaton matr of a nod 7

109 X X X R Y Y Y Z Z Z (69) Us of a drcton pont Th ncsst to gv th drcton vctor n th form Eqn. (63)oftn cass dffclts drng data npt. Hr w prsnt on of th possblts of smplfng th wa of passng th drcton of an lmnt as whch s sd n th Atods Smlaton Mchancal (AGOR) sstm. Th 3D fram lmnt s dtrmnd b thr ponts ( - th frst nod - th last nod - th drcton nod). Th ponts dtrmn a plan n th thr dmnsonal spac. Th as of th local coordnat sstm s n ths plan. Th as s dtrmnd b th ln passng throgh ponts. W fnd coordnats of vrsors for sch a dfnton of drctons of th local as. t X Y Z dnot coordnats of th pont n th global sstm. If analog w dnot coordnats of ponts and thn th lmnt coordnats n th global sstm ar qal to X X X Y Y Y Z Z Z (7) X Y Z and from hr w calclat th componnts of vctor : X X Y Y Z Z. (7) W form th vctor v connctng th pont and th drcton pont (Fgr 43): X X v Y Y. (7) Z Z 8

110 Fgr 43. Th drcton pont applcaton to dtrmn th as of th local coordnat sstm. Th vctor prodct of th vctors and v gv a vctor whch s prpndclar to th plan. Ths vctor wll b th vrsor : w v (73) w X v Y Y v Z Z w Y v X X v Z Z w Z v (74) w w w w X Y Z X X v Y Y X wx w Y w Y w Z w Z w. (75) Now w obtan th coordnats of th vrsor from th vctor prodct of th vrsor b : X Y Y (76) Z Z Y X X Z Z Z X X Y Y. (77) On th bass of rslts Eqn. (7) (75)and (77) w can form th transformaton matr R as n Eqn. (69) Th transformaton matr of an lmnt Now w bld th transformaton matr of an lmnt. Nodal dsplacmnt vctors and nodal forc vctors hav bn gropd so that w can dvd thm nto blocs contanng thr dsplacmnts or rotatons and thr forcs or momnts rspctvl. Aftr ths opraton w can transform vr bloc ndpndntl R R R R R (78) 9

111 whr R s th rotaton matr of th nod and R s th rotaton matr of th nod. Snc th lmnt s straght as t was n prvos cass (D and 3D trsss D fram) w assm R = R. W obtan th transformaton of th stffnss matr to th global sstm b mltplng matrcs dntcall as n Eqn. (95). K R K R T (79) whr R s dtrmnd b Eqn. (78). Th form of th matr global sstm so w wll not gv t. 3.. Bondar condtons for a 3D fram K s too compl n th Bondar condtons stng n 3D fram spports ar vr smlar to condtons dscrbd for two-dmnsonal frams. Dffrncs concrnng dgrs of frdom whch do not st n plan frams ar obvos. W laborat onl thos bondar condtons whch dscrb fram spports of spac strctrs (Fgr 44) and whch ar most oftn appld. a) rgd spport Z r Y = r X = r Y = r Z r X = r Y = r Z = X Y b) lnar movabl spport (along th X as) Z X Z r Y = r Y = r Z r = r X = Y = r Z X

112 d) clndrcal ont (rotaton arond th Y as) Z Z Y Y r = r X = r Y = r Z r X = r Z = X ) movabl plan spport (a dsplacmnt on th XY plan) X Z r Y = r Z X Fgr 44. 3D fram spport tps. Modfcaton of th global stffnss matr (comp. pont.6) s th wa of consdrng bondar condtons st as w hav don n rfrnc to prvosl dscrbd strctrs. 3.. Bondar lmnts A choc of spports to b sd n a spac strctr ncrass f w add lastc constrants and sw spports.

113 As n prvos chaptrs w propos to s lastc and fd bondar lmnts for modllng ths constrants. In fact w can s a sngl lmnt dscrbd n Chaptrs or 3 of whch w can compos a mor compl spport bt for convnnc w wll show hr th s of th matr of a vrsatl lastc lmnt wth s dgrs of frdom: K' b hrx hry hrz grx gry grz (8) whr h rx h ry h rz ar sprng rats and g rx g ry g rz ar flral (or torson) stffnss of sprngs. Th transformaton of ths matr to th global sstm s smlar to th on prsntd n Chaptr 4 (Eqn. ()). Snc ractons of or lmnts ar contand n two ndpndnt vctors: th vctor of spport forcs and th vctor of spport momnts thn th transformaton matr has th form: R b R r R (8) r whr R r s th rotaton matr of th nod gvn b Eqn. (69). Aftr th mltplcaton w obtan th stffnss matr of th bondar lmnt n th global coordnat sstm: K b R b K b H b T R G (8) whr H s th stffnss matr for a movmnt and G s th stffnss matr for a rotaton: X hrx X hry X hrz H YhrX YhrY YhrZ h h h Z rx Z ry Z rz. (83) It s as to obtan th matr G from th matr H changng th stffnss of tnson of sprngs h rx h ry h rz nto th stffnss of bndng sprngs g rx g ry g rz.

114 4. Two-dmnsonal lmnts Strctrs dscssd n th prvos chaptrs wr modlld b mans of bar strctrs whos qlbrm qatons as wll as thr gomtrcal rlatonshps ar dscrbd wth th hlp of dffrntal qlbrm qatons and whos ndpndnt varabl s masrd along th bar as. Ths rathr smpl strctr lts s gt famlar wth th ssnc of th FEM and convncs th radr that ths mthod s ffcnt n solvng vr compl and tndd problms n strctral mchancs. Now w wll dscss srfac strctrs sch as D lmnts plat and shll for whch dsplacmnts strans ntrnal forcs ar th fnctons of two ndpndnt coordnats. As a rslt qlbrm qatons ar partal dffrntal qatons mch mor dffclt to b solvd than ordnar qatons. Dffrntal qlbrm qatons for bar strctrs ar smpl nogh to b ntgratd. Thr act rslts can b sd as lmnt shap fnctons. Th staton s qt dffrnt for srfac strctrs. Partal dffrntal qatons dscrbng th qlbrm of thos strctrs hav nq soltons onl for vr smpl problms. Soltons obtand b sng th appromaton mthod (for ampl b panson n a srs) ar vr laboros and th rqr a lot of wor and thrfor a comptr has to b sd n ordr to solv a st of qatons and sm srs. In sch a staton a nmrcal mthod whch assms som smplfcaton at th stag of formaton of lmnt qlbrm qatons appars to b mor ffctv. That s wh th fnt lmnt mthod has broght so man sgnfcant rslts to contnm mchancs. It can b asl notcd n th ampl of a two-dmnsonal lmnt whch s th cas of th smplst contnm. Th D lmnt (slab lmnt) can b dfnd as a sold of whch on dmnson (thcnss) s consdrabl smallr than th two othrs and whos mddl plan (th srfac paralll to both trnal srfacs of an lmnt) s a plan (Fgr 45). A plat lmnt has also sch a shap bt th slab lmnt dffrs from a plat th wa t s loadd. Th slab lmnt can b loadd onl wth th load actng n ts plan and b th tmpratr dpndnt pon th and coordnats. On th othr hand th plat can b loadd wth a forc prpndclar to ts srfac or an tmpratr fld. Plat lmnts wll b dscssd n th followng chaptr. 3

115 Z Y a) q( X Y) Y h Z X >>h X Y Z b) X Fgr 45. Th mplar applcaton of a slab D lmnt. 4.. Plan strss and stran Whn trnal srfacs of a D lmnt ar fr and ths lmnt s thn nogh w can assm that n rfrnc to th whol thcnss of th lmnt. Thn t s sad that ths s a plan strss problm. Th thnnr th D lmnt (comp. Nowac (979) Tmoshno and Goodr (96)) th bttr th appromaton s. Hnc onl th componnts of strss shown n Fgr 46 ar non-ro. 4

116 Fgr 46. Strss tnsor componnts n plan strss. and Wth rgard to th smmtr of a strss tnsor componnts of shar strss ar qal ths w hav thr ndpndnt componnts of strss whch w compos n th strss vctor: σ. (84) A compltl dffrnt cas occrs whn th componnt Z n Fgr 45b s vr sgnfcant that s h<< X Y Z and th spport and load condtons ar constant along th as whch s prpndclar to th lmnt. Th strctr satsfng ths condtons can also b analsd b applng plan stat whch n fact s plan stran. Snc th cross dmnson of th strctr shown n Fgr 45b prvnts th strctr dformaton n th drcton prpndclar to th cross scton th thn lar ct ot from ths strctr s n th stat dscrbd b th qaton:. (85) coms from th abov qatons bt th frst qaton allows to calclat th componnt on th bass of two othr componnts of a drct strss. Ths w hav (86) whch allows to lmt th nmbr of sarchd componnts of th strss vctor to thr componnts gvn n Eqn. (84). 5

117 W also grop ndpndnt componnts of th stran tnsor n a colmn matr whch w hav calld a stran vctor: ε. (87) Thr s a rlatonshp btwn vctors σ and ε dscrbd b constttv qatons whos form dpnds on th modl of th matral whch th strctr s mad of. In ths boo w dal onl wth lastc sotropc matrals whch ob Hoo s law. Hnc w can wrt th constttv qaton as follows: σ = D ε (88) whr D s a sqar matr contanng matral lastc constants dscrbd n Chaptr. For plan strss th matr Dhas th form wrttn b Eqn. (3). Plan stran rqrs anothr matr for lastc constants whch s dscrbd b Eqn. (7). 4.. Gomtrc rlatonshps A crtan pont can mov onl on th plan drng th dformaton procss and thn th dsplacmnt vctor of ths pont () has two componnts: ( ) ( ) ( ) (89). Som nown rlatons st (Tmoschno and Goodr (96)) btwn th componnts of dsplacmnt and stran vctors: (9) whch can b prsntd n th form: ε =D () (9) whr D s th matr of dffrntal oprators Eqn. (35) Th stffnss matr of an lastc lmnt t s dvd a contnm nto fnt lmnts. W wll dscss onl a tranglar D lmnt n ths boo and w wll choos sch lmnts drng dscrtaton (Fgr 47). 6

118 Fgr 47. Nodal forcs and dsplacmnts for th D lmnt n th global coordnat sstm. Accordng to assmpton Eqn. (89)t s sn that vr nod of an lmnt has two dgrs of frdom and all nodal forcs hav two componnts. Th local coordnat sstm s chosn n sch a wa that ts as ar paralll to th as of th global coordnat sstm. Hnc dstngshng componnts of local and global vctors and matrcs s nsgnfcant. Now w grop nodal dsplacmnts and forcs n th vctors of: nodal and lmnt dsplacmnts nodal and lmnt forcs (9) F F f F F F f f F f F F f F f. (93) F f F F 7

119 Snc w loo for th dpndnc btwn nodal dsplacmnt and nodal forcs vctors of an lmnt w appl th prncpl of vrtal wor (comp. Chaptr ) whch rqrs gvng th rlaton btwn dsplacmnts of ponts lng wthn th lmnt and dsplacmnts of nods. Accptng rrors comng from appromaton w assm that ths rlatonshp can b wrttn b th fncton of two varabls: ( ) N ( ) N ( ) N ( ) and ( ) N ( ) N ( ) N ( ) (94) or th gnral matr form: ) ( ) N ( (95) whr N () s th matr of shap fnctons of th lmnt: N ( ) N ( ) I N ( ) I N ( ) I (96) and N () N () N () ar th shap fnctons for nods. t s now assm th smplst of all possbl forms of th shap fncton for th nod N ( ) a b c (97) whr a b c ar constants whch w dtrmn on th bass of consstnc condtons N ( ) N ( ) N ( ). (98) Aftr nsrtng ths condtons nto Eqn. (97) w obtan th st of qatons: a b c whch aftr bng solvd gv th vals of coffcnts of th shap fncton. Eqn. (99) can also b wrttn n th gnral form: (99) Mα = δ whr δ (3) 3 whch aftr modfcaton dpndng on th chang of nto (or ) allows s to dtrmn th coffcnts of th shap fnctons for th sbsqnt nods. δ mans th Kroncr s dlta n ths qaton. 8

120 9 W solv th st of Eqn. (99) b th Cramr mthod W dt M W a W b W c thn W W a a W W b b W W c c. (3) Smlarl f w chang th nd nto and w fnd δ W a W b W c a W W a b W W b c W W c. (3) Fnall w hav

121 δ a W b W c W W W a a W W b b W W c c. (33) for nod. Constants a a a ar nsgnfcant for frthr transformatons (bcas th ar connctd wth th rgd moton of a D lmnt) and th can b nglctd whn solvng th st of Eqn. (3). Aftr dtrmnng th shap fnctons of th lmnt lt s com bac to ts strans. W nsrt Eqn. (95)n (9): ε=d B N ) ( ) ( (34) obtanng th dpndnc btwn th nodal dsplacmnts of th lmnt and ts strans. Th matr B n Eqn. (34) s calld a gomtrc matr and t can b prssd as follows: B B B B ( ) ( ) ( ) ( ) whr n B D n n n n n b c c b ) ( N (35) s th gomtrc matr of an nod n. Ths w hav all componnts whch ar ncssar to wrt an lmnt qlbrm qaton. W appl th prncpl of vrtal wor whch sas that th

122 trnal wor (don b trnal forcs - hr nodal forcs) has to b qal to ntrnal wor (don b strss) of a D lmnt: T f V T ε σ dv. (36) W transform ths qaton frst sbstttng th constttv rlaton Eqn. (88) for δ and nt sbstttng gomtrc rlatons (34)for ε: T T T T f B DB dv B DB dv V V. (37) In ths qaton th nodal dsplacmnt vctors of th lmnt bng ndpndnt of varabls and ar tan to th front and bac of th ntgral. Eqn. (37) can b solvd ndpndntl of lmnt dsplacmnts onl whn f DB d V B V T (38) whch aftr comparson wth th nown rlaton was rfrrd to n all prvos chaptrs of ths boo: f K gvs s th qaton dtrmnng coffcnts of th lmnt stffnss matr: K B V T DB dv. (39) Bldng th lmnt stffnss matr can b consdrabl as f w not that ths matr dvds nto blocs: K K K K K K K K K K n whch an of thm for ampl K can b calclatd from th qaton: (3) K B V T DB dv (3) and othrs comng from analogos qatons formd aftr stabl changs of ndcs hav bn mad. Th nsrton of th gomtrc matrcs B and B gvn b Eqn. (35) and th matr D gvn b Eqn. (3) nto (3)rslts n

123 K T T B DB dv B DB Ab V EAb bb cc b c b c b c b c cc bb whr A s th srfac of a slab lmnt and b s th ts thcnss. Th abov matr s th stffnss matr for plan strss. (3) Not that matrcs B B and D do not contan componnts dpndnt on varabls ths w can ta thm otsd th ntgral. W obtan th bloc of th stffnss matr for plan stran accptng th matr of matral constants accordng to Eqn. (7): EAb b b c c b c b c b c b c cc bb. K (33) Snc th local coordnat sstm s assmd n sch a wa that ts as ar paralll to th global coordnat sstm thn w do not hav to transform th stffnss matr Elmnt stran and strss W also calclat lmnt strans. Th ar gvn b Eqn. (34) and tang nto consdraton Eqn. (35) w hav b n bn n n n c nn bnn n. (34) n W s that componnts of th stran vctor ar constant wthn th lmnt whch s th consqnc of th assmpton of lnar shap fnctons. Ths lmnt s calld CST (constant stran trangl). W dtrmn lmnt strsss from th constttv Eqn. (88) and Eqn. (3) or (7) accordng to th nd of varant that w dal wth. It s obvos that strans st as strsss ar constant wthn th CST lmnt A Nodal forc vctor for a dstrbtd load oads on slab lmnts can b tratd as loads on plan trsss whch mans that th can b appld to th nods of a strctr. Bt f a dstrbtd load actng on

124 th bondar of an lmnt s gvn thn t shold b convrtd to concntratd forcs actng on th nods of an lmnt (Fgr 48). Fgr 48. Nodal forcs rprsntng contnos loads. Smlarl as n prvos chaptrs w appl th prncpal of vrtal wor gvng th followng qlbrm qaton for ths cas: T T f q d (35) whr (ξ) contans fnctons dscrbng th dsplacmnt of th loadd dg and q q contans fnctons dscrbng th load on th dg s th lngth of th q dg - ξ s th non-dmnsonal coordnat tang ro val at th nod and val at th nod. Snc w assm lnar shap fnctons for th lmnt thn w wrt th vctor (ξ) as follows: N (36) whr N s th matr of shap fnctons for dsplacmnts of th bondar. o o o N N I N I N whr N ( ) N o ( ) o (37) or n th dvlopd form N. (38) 3

125 f obtan f Aftr nsrtng rlaton Eqn. (36)nto Eqn. (35) w obtan T N q d (39) Aftr tang nto consdraton th shap fnctons dscrbd b Eqn. (38) w q q q q d. (3) For ampl lt s calclat th nodal forc vctor d to th lnar dstrbtd load on th dg - of val q q - at th nod and q q - at th nod. W wrt sch a load wth th hlp of a non-dmnsonal coordnat ξ: q q q q q and aftr nsrtng th abov qaton nto Eqn. (3) w obtan (3) f q d q d q d q d q d q d q d q d whch aftr ntgraton gvs (3) 4

126 f 6 q q q q q q q q. For a partclar cas whn th load s constant and qal to bass of Eqn. (33) w obtan q ( ) q q o o (33) on th f qo q o qo qo. (34) It shold b rmmbrd that th calclatd forcs ar forcs actng on th lmnt. W obtan th ncssar nodal forcs changng th sns of vctors whch mans: p f (35) whr p s th nodal forc vctor for th nods tochng th lmnt A Nodal forc vctor d to a tmpratr load As n th prvos scton w appl th prncpal of vrtal wor to calclat altrnatv nodal forcs rplacng a tmpratr load. In accordanc wth th fatrs of a CST lmnt w wll ta nto consdraton onl a constant tmpratr fld wthn th lmnt. Th stabl qaton of vrtal wor has th form: T t T f ε σt d V T V ε Dεt dv (36) V whr σ t s th strss fld n th lmnt whch s casd b th tmpratr and ε t s th stran of th lmnt casd b th chang of a tmpratr. Assmng sotrop of a D lmnt w obtan 5

127 6 ε t = t t (37) Aftr nsrtng gomtrc rlaton Eqn. (34) nto Eqn. (36) w obtan D B D B f T T t t t tab d t V V. (38) For a plan strss problm ths qaton s smplfd to th followng rlaton: t t c b c b c b teab PSN f (39) whr b... c ar coffcnts of shap fnctons of th CST lmnt. Plan stran gvs a slghtl dffrnt nodal forc vctor: t t c b c b c b teab PSO f. (33) As n prvos sctons w shold chang th sgns of componnts of nodal forcs bfor applng thm to th nods: p f t t. (33) W calclat strsss n th lmnt ndrgong th acton of a tmpratr tang nto consdraton strans casd b th thrmal panson of th lmnt: σ t = D(ε - ε t ) = t t B D. (33)

128 4.7. Bondar condtons of a D lmnt Bondar condtons of a two-dmnsonal strctr can b tratd analogosl to th condtons n a plan trss bcas th nods of both sstms hav two dgrs of frdom on th XY plan. Hnc w hav: fd spports (at th nod r nfgr 49) and spports whch can mov along th X as (at th nod r ) nt spports whch can mov along th Y as (at th nod r 4 ) or sw spports (at th nod r 3 ). Th bondar condtons for ths spports ar as follows: nod r : r X nod r : r Y nod r 4 : r 4 X ry for nodr 3 whr constrants ar not consstnt wth th as of th global coordnat sstm w propos th s of bondar lmnts dscrbd n Chaptr. and spports. Fgr 49. Th D lmnt (slab lmnt) schm dvdd nto fnt lmnts 7

129 5. Statcs of plats Plats ar on of th most commonl sd lmnts n strctrs. Th can b fond n almost vr bldng or mchancal strctr. Th gomtrc shap of a plat can b dfnd smlarl to a D lmnt (Chaptr 6) bt th dffr n th wa of loadng. Plats ar loadd wth normal loads to thr srfacs whch cas bndng. Bndng s not prsnt n th cas of th dformaton of th D lmnt. Analtcal mthods of dtrmnng both dflctons and ntrnal forcs wr dscrbd b Elr Brnoll Grman agrang Posson and spcall b Navr n paprs whch appard at th nd of th 8 th cntr dscrbd b Rao (98). tratr dvotd to th thor of plats s nsall rch th boos of Kącows (98) Nowac (979) Tmoshno and Wonows-Krgr (96) ar rcommndd to ntrstd radrs. Man mportant statcs and dnamcs problms of plats wr solvd b analtcal mthods (manl b th mthod of th Forr srs) bt th ar naccrat both n th cas of problms wth compl bondar condtons and complcatd shaps of plats. Howvr th fnt lmnt mthod has provd to b nvrsal and althogh t gvs appromat soltons th ar prcs nogh for practcal applcatons. 5.. Basc assmptons and qatons of th classc thor of plats W assm that ths plats th assmptons of th classc thor of thn plats (Tmoshno and Wonows-Krgr (96)): a) thcnss of a plat s small n comparson wth ts othr dmnsons; b) dflctons of plats ar small n comparson wth ts thcnss; c) mddl plan dos not ndrgo lngthnng (or shortnng); d) ponts lng on th lns whch ar prpndclar to th mddl plan bfor ts dformaton l on ths lns aftr th dformaton; ) componnts of strss whch ar prpndclar to th plan of th plat can b nglctd. From pont d) of th abov assmptons t follows that th dsplacmnt of ponts lng wthn th plat vars lnarl wth ts thcnss (Fgr 5): w w w( ). (333) 8

130 Fgr 5. Th bar sgmnt dformaton schm. Ths stans ar prssd b th rlatons: w w w. (334) Th stran vctor can b prsntd n th form: ε = - w() (335) whr vctor s th vctor of dffrntal oprators: =. t s assm that thr s a plan strss condton n th plat so th strss vctor can b dtrmnd as follows: σ = D ε= D w() (336) whr D s th matr of matral constants dtrmnd for plan strss (Eqn. (3)). Now w ntrodc n th prsson of ntrnal forcs (momnts and sharng forcs Fgr 5) M Q h/ d M h/ h/ h/ d Q h/ h/ h/ h/ d. d M h/ h/ d (337) a) strsss 9

131 mddl plan h b) ntrnal forcs M q () Q M M d Q + Q M + d M d d M Q Q + Q d M M + d M + M + M M d d plat lmnt. Fgr 5. Th dstrbton of strsss trnal loads and ntrnal forcs n th Th qlbrm of an nfntsmal plat lmnt shown n Fgr 5b lads to th st of qatons: Q Q q( ) M M Q (338) M M Q. Aftr ntgraton Eqn. (337) tang nto consdraton Eqn. (336) w obtan 3

132 w w M D w w M D (339) w M D whr D dnots th plat stffnss dfnd b th qaton Eh D 3. From th last two Eqn. (338) w obtan rlatons for th sharng forcs: (34) 3 3 w w Q D 3 (34) 3 3 w w Q D 3. Insrtng th abov qaton dscrbng sharng forcs nto th frst Eqn. (338) w obtan 4 4 w 4 4 w w q ( ) D. 4 (34) It s a bharmonc partal dffrntal qaton whch shold b satsfd b th fncton of dflcton w() wthn th plat. Th followng bondar condtons shold b ralsd at th dgs of th plat: w a) w = - on th fd dg n w b) w = - on th fr spportd dg n c) M n = V n = - on th fr dg. In th abov qatons n dfns th drcton of th ln whch s prpndclar to th dg and V n s th rdcd forc ntrodcd b Krchhoff n 85 dscrbd b Tmoshno and Wonows-Krgr (96). Ths forc ons th nflnc of th torson momnt M ns and th sharng forc Q n on th fr dg Fgr 5b: 3

133 3 M ns w V n Qn D 3 s n ns 3 w (343) whr n dscrbs th drcton of th ln whch s prpndclar to th dg and s s th drcton of th ln whch s paralll to th dg of th plat. Th modfcaton of th bondar condtons s ncssar hr bcas th forth ordr Eqn. (34) cannot b solvd for thr bondar condtons comng from th rqrmnt of ro strss on th fr dg: M ns = M n = Q n =. 5.. A fnt tranglar lmnt of a thn plat Now w show th wa of bldng th stffnss matr of a tranglar lmnt of a thn plat (Fgr 5). a) nodal dsplacmnts b) nodal forcs Fgr 5. Nodal loads and dsplacmnts for th plat lmnt n th local coordnat sstm. W also ntrodc a fw convnnt notatons: w() stands for th fncton of dsplacmnt of th mddl plan of an lmnt; w s th rotaton angl of th lmnt abot th as; w s a rotaton angl of th lmnt abot th as. As sn n Fgr 5 th nod of a plat lmnt has thr dgrs of frdom. Hnc nodal dsplacmnt vctors of th lmnt n th local sstm can b wrttn as follows: 3

134 33 w ' w ' w ' (344) and an lmnt dsplacmnt vctor: ' ' ' '. (345) Drctons of both nodal dsplacmnts and forcs (Fgr 5b) ar th sam so th nodal forcs vctors hav a smlar notaton: f ' Q M M f ' Q M M f ' Q M M. (346) Hnc w wrt th nodal forc vctor of th lmnt as follows: f f f f ' ' ' '. (347) W appromat th srfac of th dformd lmnt b th polnomal of th thrd ordr proposd b J..Tochr n 96: ) ( a a a a a a a a a w T a (348) whr = 3 3 a= a a a a a a a a a.

135 34 W dtrmn th coffcnts a... a 9 of th fncton w() from th bondar condtons at th nods : w w ) ( ) ( ) ( w w ) ( ) ( ) ( w w ) ( ) ( ) (. (349) Aftr calclatng th rotaton angls w obtan ) ( ) ( a a a a a w ) ( 3 ) ( a a a a a w. (35) Now w nsrt Eqn. (348) and (35) nto bondar condtons (349) obtanng: Ma ' (35) whr M s th sqar matr dpndnt on nodal coordnats of th lmnt.

136 a a a 3 a 4 a 5 a 6 a 7 a 8 a 9 w φ - φ 3 w M= φ (35) φ 3 3 w 3 φ φ W can prsnt th solton of Eqn. (35) as follows: a M ' (353) whr M - s th nvrs matr of M. Th solton of M - s possbl whn dt M (comp. Appnd ) whch s not alwas th cas n or problm bcas dt M qaton 5 5 (354) It mans that n cass whn th nod of th lmnt s on th ln dscrbd b thn th matr M s snglar. Ths th problm s solvd b changng th local coordnat sstm. Now w calclat a stran vctor dtrmnd b Eqn. (335). ε = w() = T M whr B M (355) B = T s a rctanglar matr of whch componnts ar qal to: 6 B 6 4( ). Comparng Eqn. (355) wth th dfnton of th gomtrc matr b Eqn. (36) and (38) w obtan (356) B dscrbd 35

137 36 B B M. (357) Hnc w can ma s of th dfnton of th stffnss matr contand n Eqn. (5): A A / / ' M DB B M DB B K d d d h h T T T V V A A 3 M DB B M d Eh T T. (358) Aftr dnotng th ntgraton n th abov qaton b K and applng th dfnton of plat stffnss w hav ' M K M K T D. (359) Aftr calclatng th matr mltplcaton nsd th ntgraton n Eqn. (358) w hav A SdA K (36) whr S. Whl calclatng th ntgraton of fnctons tng n Eqn. (36) th followng rlatons ar hlpfl: d A A d 6 A A (36)

138 A A A A d 6 A da da 4 da. Matr Eqn. (358) s dtrmnd n th local coordnat sstm. W hav to transform t to th global coordnat sstm n accordanc wth rlaton Eqn. (53): K R K R T. Th rotaton matr of an lmnt R s qal to: R R R R (36) whr R R R ar th transformaton matrcs of nods. If w s th sam coordnat sstms for all nods (t has bn don n ths chaptr) thn w can s onl on transformaton matr: R = R R = R R c s s c whr (363) c cos s sn and α s th angl btwn th X as of th global sstm and th as of th local sstm (Fgr 53). Val n th frst row of th matr R s th consqnc of a fact that as Z and ar paralll. 37

139 Fgr 53. Th plat arrangmnt n th global coordnat sstm. Th tranglar lmnt for whch th matr stffnss has bn obtand has a convnnt fatr. Naml t allows s to dscrt plats of an shap wthot an dffclt. Ths lmnt ond wth a D tranglar lmnt can b sd as a shll lmnt (comp. Raows and Kacpr (993)). Elmnts of an othr shaps (rctanglar or qadrlatral) ar prsntd n th boos wrttn b Bath (996) Raows and Kacpr (993) Rao (98) or Znwc (97 994) A tranglar lmnt of a thn shll As t has bn notd at th prvos pont an lmnt contanng D tranglar and plat lmnts can b sd as a shll lmnt. Appromatng a crvd srfac (whch s th mddl srfac of a shll) wth th hlp of plat lmnts rmnds th smplfcaton w appl to approach th arc wth th hlp of a bron ln. W nttvl fl that th smallr th crv ln sgmnts ar th bttr th rplac th crv as of th arc (Fgr 54). Smlarl th smallr th plan shll lmnt dmnsons and th smallr β angls (comp. Fgr 54) of nghborng lmnts ar th bttr ths lmnt dscrbs dsplacmnts and ntrnal forcs n th strctr. Dtald calclatons and prmnts confrm th corrctnss of ths appromaton (comp. Znwc (97)). 38

140 Fgr 54. Th mplar shll dvson nto fnt lmnts. Connctng dsplacmnt and ntrnal forc vctors of th tranglar lmnts dscrbd b Eqn. (9) (344) (96) and (346) w obtan shll lmnt nods possssng fv dgrs of frdom: ' F F f ' F M M. (364) In Eqn. (364) and F dnot rspctvl a nodal dsplacmnt and a forc paralll to th as of th local coordnat sstm. In Eqn. (344) and (346) ths vals ar mard wth w and Q (comp. Fgr 5 and Fgr 55). 39

141 a) nodal dsplacmnts b) nodal forcs D lmnt plat lmnt spplmntar angl and momnt Fgr 55. Th shll lmnt composton of a D and plat lmnts. shll lmnt Smplfng th dscrpton of a nod movmnt b dsrgardng th rotaton arond th as prpndclar to th lmnt lads to th snglart of th shll stffnss matr modlld b th lmnts mntond bfor. Ths dffclt s solvd b assmng thr componnts of th rotaton and momnt vctors whch rqrs th valaton of th plat lmnt torsonal stffnss. Snc th torsonal stffnss s not mportant n shll statcs and dnamcs problms th fcttos val of ths stffnss s oftn assmd (comp. Znwc (97)). Hnc th dpndnc btwn th torsonal momnts and angls can b prsntd as a varabl ndpndnt of othr nodal forcs and dsplacmnts of an lmnt: M M M EhA (365) In th abov rlatonshp sggstd b Znwc (97) E s Yong s modls h s th lmnt thcnss A s th ara of a cross scton and α dnots an ndmnsonal coffcnt whch s so small that t dos not hav an sgnfcant nflnc on th solton of a st of qatons. Th val of ths coffcnt s assmd wthn th rang.. Znwc (97 994) sggsts tang th val qal to.3 or lss. 4

142 Srvng th dscrbd matrcs w obtan th stffnss matr of th tranglar shll lmnt nods havng s dgrs of frdom: f ' K' ' or f ' K' K' K' f ' K' K' K' f ' K' K' K' ' ' ' whr f and dnot fll vctors of nodal forcs and dsplacmnts: (366) (367) ' F F F f ' M M M. (368) Evr bloc of th stffnss matr n Eqn. (367) conssts of D lmnt plat and torsonal parts (Eqn. (365)) F F t K F = M p K φ (369) M φ M a s φ a s a s whr a s =αeha dscrbs th fcttos torsonal stffnss stng n Eqn. (365) t K s th stffnss matr bloc for th plat lmnt (Eqn. (359)). Transformaton of ths matr to th global coordnat sstm can b don n th wa dscrbd n Chaptr 5 (p.5.. p.5..3) n whch w prsnt th transformaton of th stfnss matr of a 3D fram lmnt wth nods havng s dgrs of frdom st as th nods of a shll lmnt. Th mthod of obtanng th componnts of th rotat matr dscrbd at pont 5..3 s stabl for th tranglar shll lmnt whos and 4

143 nods dtrmn th drcton of th local as and th thrd nod can b a drctonal pont. Th shll lmnt dscrbd abov s th smplst lmnt whch nabls s to solv an shll statcs problm. Thr crtanl ar mor compl lmnts both plan and spac lmnts wth at last for nods dscrbd n boos dvotd to ths sbct as Irons and Ahmad (98) Raows and Kacpr (993) Rao (98) Znwc (97 994). W mst rmmbr abot th possblt of sgnfcant smplfcaton of a shll lmnt dscrpton n cas of asmmtrc strctrs. It s also possbl to s con or crvlnar lmnts wth nods havng thr dgrs of frdom (Raows and Kacpr (993) Znwc (97 994)). 6. Brc lmnts Brc s th thr-dmnsonal lmnt whch can b dfnd as a bod to whch all dmnsons ar of th sam ordr. Th shap of th bod and th load s an of avalabl. Wth brc lmnts fll 3D sold constrctons can b modld. Brc lmnts can rplac vr othr tp of lmnt l fram shll and plat lmnts. A sold modl s dvdd nto brc lmnts and th gomtrc shap of th lmnts can b ttrahdron hahdron or prsm wth trangl bas whch mans that a brc lmnt s bld of trangls and qadrlatral. Tpcal 3D shaps of lmnts ar shown n Fgr 56. a) b) c) Fgr 56. Thr-dmnsonal shaps of brc lmnts. a) for-nods (ttrahdron) b) s-nods c) ght-nods (hahdron). In ths chaptr w wll show how th ttrahdron s rgardd n Fnt Elmnt Mthod. In Fgr 57 o can s how dsplacmnts and nodal forcs ar locatd. 4

144 Fgr 57. Nodal forcs and dsplacmnts for th 3D lmnt n th global coordnat sstm. As t can b sn n ach nod thr ar thr dsplacmnts and forcs n all global drctons bt thr ar no rotatons and momnts. Vctor of movmnts of lmnt nods and nodal forcs can b wrttn as follows: l l l F F F F F F f (37) F F F Fl Fl F l 6.. Rlaton btwn stran strss and dsplacmnts Th brc wors n a spatal stat of strss. In Fgr 46 componnts of strss tnsor ar shown. Thr ar thr componnts of normal strsss σ σ σ and s shr strsss bt accordng to th smmtr of a strss tnsor componnts of shar strss w hav τ = τ τ = τ and τ = τ ths w hav s ndpndnt componnts of strss whch ar composd n th strss vctor: 43

145 44 σ (37) Fgr 58. Strss tnsor componnts. Bcas of th fact that th brc wors n thr-dmnsonal stat of strss and stran th stran vctor s smlar to strss vctor. Rlatonshps btwn th componnts of dsplacmnt and stran vctors ar smlar to thos for D lmnts: ε (37) Th rlatonshp btwn vctors σ and εs l n two-dmnsonal lmnts dscrbd b constttv qatons.for lastc sotropc matrals th constttv qaton s shown blow:

146 45 σ = D ε (373) whr D s th sqar matr wth dmnsons 66 contanng th matral constants dscrbd n Chaptr shown n Eqn. (7). 6.. Stffnss matr of 3D lmnt Th stffnss matr s th sam as for all prvos chaptrs: f K whr K has th sam formla as n Chaptr 6: V dv DB B K T. (374) In cas of brc lmnt B has followng formla: B DN (375) whr N s matr of shap fncton (whch wll b dscrbd latr) Ds a matr of dffrntal oprators: D. (376) 6.3. Shap fncton of 3D lmnts Dsplacmnt of vr pont of brc lmnt n thr-dmnsonal coordnat sstm can b wrttn as a fncton: a a a a 4 3 ) ( b b b b 4 3 ) ( c c c c 4 3 ) (. (377)

147 46 Ths ar dsplacmnts n vr of thr dmnsons locatd n an pont wth coordnats ( ). All fnctons ar lnar. Eqn. (377) can b wrttn as a fncton dpndng on dsplacmnts n vr of for nods n ttrahdral lmnt. Dsgnatons ar rlatd to Fgr 57: l l N N N N ) ( ) ( ) ( ) ( ) ( l l N N N N ) ( ) ( ) ( ) ( ) ( l l N N N N ) ( ) ( ) ( ) ( ) (. (378) whr: d c b a N ) ( d c b a N ) ( d c b a N ) ( d c b a N l l l l l ) (. (379) Eqn. (378) can b wrttn n matr form: N ) ( (38) whr ( ) s a dsplacmnt vctor of an pont locatd nsd th brc lmnt s nodal dsplacmnt vctor (Eqn. (37)) and N s stffnss fncton matr. ) ( ) ( ) ( ) ( (38) l l l N N N N N N N N N N N N N. (38) Shap fncton for lmnt of ght nods can b wrttn for an nod as bllow: ) ζ )( η )( ξ ( 8 o o o N (383) whr. l l l (384)

148 47 Ths shap fncton s basd on agrangan ntrpolaton for th thr varabls of fncton passng throgh two ponts. Dsgnatons for ght-nod lmnt ar shown n Fgr 59. Fgr 59. Dsplacmnts for cbc brc lmnt. Now whn w hav th shap fncton w can rtrn Eqn. (375) whr th gomtrc matr for nods B can b prssd for for-nod lmnt: N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N l l l l l l l l l B DN (385) B sbstttng Eqn. (379) nto Eqn. (385) w can obtan a smplfd form: l l l l l l l l l b d b d b d b d c d c d c d c d b c b c b c b c d d d d c c c c b b b b B. (386)

149 Stran and strss n lmnt on ttrahdron ampl Now aftr nowng th gomtrc matr for ttrahdral lmnt th stran vctor can b obtand from th followng qaton: B N ε ) ( D (387) whch s l l l l l l l l l b d b d b d b d c d c d c d c d b c b c b c b c d d d d c c c c b b b b (388) whr th dsplacmnt vctor ʹ s dscrbd b Eqn. (37) Knowng th stran vctor t can b sd n Eqn. (88) to obtan th strss vctor: E ) ( ) ( ) ( ) )( ( ) ( (389) 6.5. Rls of FEM msh formaton for 3D brc modls Emplar of th 3D modls ar shown n Fgr 6.

150 Fgr 6. Eampl of th3d brc lmnts s. Thr ar som rls that ar nd to b followd: FEM grd mst ta nto accont th shap of th strctr n a hol n th modl nods mst b locatd so that thr s no possblt to crat a FEM lmnt bt to form an mpt ara msh nods mst b locatd n locaton of concntratd loads msh nods mst b locatd n th bondar condton ponts th dgs of th grd mst b locatd on th bordr btwn parts of lmnts mad of dffrnt matrals f th ob s smmtrcal th msh also shold b smmtrcal. FEM msh shold b concntratd n aras of hgh strss concntraton and n aras of rapd chang n strss (hgh val of gradnt). Sch aras ar locatd: at th cornrs arond th ponts of applcaton of concntratd forcs arond th spports If th componnt s narrow thn n cross scton thr shold b at last for blts of lmnts. Onl thn th wll b abl to dscrb th chang n strss n th cross scton. 49