Lctr 8; ntrodctor 2-dimnsional lastostatics; cont. (modifid 23--3)
ntrodctor 2-dimnsional lastostatics; cont. W will now contin or std of 2-dim. lastostatics, and focs on a somwhat mor adancd lmnt thn th CST-lmnt, naml th -nodd bilinar (and isoparamtric) lmnt. (, ) (, 3 3 ) n (, ) n 2 (, 2 2 ) 2 Howr, lt s first start with a qick riw of th CST-lmnt! 2 (modifid 23--3)
Riw- th CST-lmnt (, 3 3 ) W do not s an lmntwis/local coordinat sstm; onl th global on! n (, ) n 2 ( 2, 2 ) 2 Contrclockwis nmbring of th nods 3 (modifid 23--3)
Th CST-lmnt; cont. On th prios lctr w fond th following transformation diagram T V f C T F E d k K C D n n 2 2 (modifid 23--3)
Th CST-lmnt; cont. E 2 ( ) / 2 T V f C T F E d k K C D n n 2 2 5 (modifid 23--3)
Th CST-lmnt; cont. T V f C T F E d k K C D n n 2 2 6 (modifid 23--3)
Th CST-lmnt; cont. 2 T V 2 f 3 C 3 T 2 2 3 3 F Th rst of th diagram as sal! ow, what do w gt for or - nodd lmnt? E d k K C D n n 2 2 7 (modifid 23--3)
Th bilinar -nodd lmnt; cont. As for CST-lmnt! E T _ tda f C d T k K C F D n n 2 2 8 (modifid 23--3)
Th bilinar -nodd lmnt; cont. Sinc th strss/strain filds will not b constant in a -nodd lmnt, th strain nrg mst b writtn as an intgral. From this it follows (b MPE) that w gt an intgral in th transformation diagram! E T _ tda f d C T k K C F D n n 2 2 9 (modifid 23--3)
Th bilinar -nodd lmnt; cont. E 2 T _ tda 2 3 f d C 3 T k K C F D 2 2 3 3 n ot as, as in th CSTcas, to find th shap fnctions to! n 2 2 (modifid 23--3)
Th bilinar -nodd lmnt; cont. Th basic mthod to st p th shap fnctions for a -nodd lmnt is to introdc an additional lmntwis/local (so calld natral) coordinat sstm. Howr, bfor that, w will look a littl bit at th lowr lft part of th transformation diagram! E T _ tda f d C T k K C F D n n 2 2 (modifid 23--3)
Th bilinar -nodd lmnt; cont. Mor spcificall, lt s split th matri [ ε] in two parts (which is not against th law :) d n n 2 2 2 (modifid 23--3)
(modifid 23--3) 3 d W ha i.. Th bilinar -nodd lmnt; cont.
Th bilinar -nodd lmnt; cont. Th lowr lft part of th transformation diagram ma ths b rwrittn as shown blow. And wh is that fin, on ma ask? Wll, as w soon will s, it opns th wa to mak an adantag of an additionall introdcd natral coordinat sstm! Priosl w had [ ε] hr! d n n 2 2 (modifid 23--3)
Th bilinar -nodd lmnt; cont. Lt s now introdc th natral coordinat sstm according to th illstration, whr th smbol n in th transformation diagram indicats that an ntit/qantit dpnds on th natral coordinats. T n n n n n d n n 2 2 n n 2 2 5 (modifid 23--3)
Th bilinar -nodd lmnt; cont. ilinar shap fnctions ar as to st p in th natral coordinats! n n 2 T n n n n n d n n 2 2 6 (modifid 23--3)
Th bilinar -nodd lmnt; cont. 2 2 n n 2 T n n n n n d n n 2 2 7 (modifid 23--3)
Th bilinar -nodd lmnt; cont. 3 3 n n 2 T n n n n n d n n 2 2 8 (modifid 23--3)
Th bilinar -nodd lmnt; cont. n n 2 T n n n n n d n n 2 2 9 (modifid 23--3)
Th bilinar -nodd lmnt; cont. n n 2 As can b sn, w will for simplicit skip th ind n whn writing th natral shap fnctions! Howr, thr is no risk of confsing things, sinc w onl will st p th shap fnctions in th natral coordinats. n n 2 2 2 (modifid 23--3)
(modifid 23--3) 2 2 3 (,) (,) 2 (,) 3 (,) 3 3 2 2 3 2 3 2 3 2 3 2 (,) (,) W not Th bilinar -nodd lmnt; cont.
Th bilinar -nodd lmnt; cont. T n n n n n Plas obsr! Ths displacmnt componnts dpnd on th natral coordinats! d n n 2 2 22 (modifid 23--3)
(modifid 23--3) 23 d n n n n T n Th sb-matri fond in th transformation matri [Tn] is calld th Jacoban matri of th coordinat chang, and will b dnotd [J]! Th chain rl implis [Tn] Th bilinar -nodd lmnt; cont.
Th bilinar -nodd lmnt; cont. Ths, th strain-displacmnt matri [] is gin b th following prssion T n n n Sinc w alrad ar in fll control of [], [ n] and [n], it jst rmains to find th inrs of th transformation matri [Tn]! W ha T n n n n n d J Tn Tn J J 22 J J J 2 J 2 J J J J 22 2 J J 2 2 (modifid 23--3)
Th bilinar -nodd lmnt; cont. Ths, w mst fi [J]! T Tn T n n J n J n n n n n f w choos to intrpolat th coordinats in th sam wa as th displacmnts (so calld isoparamtr), w ha d n : nod and can find [J] b diffrntiating th prssion for {} in an appropriat wa- ALL DOE! 25 (modifid 23--3)
Th bilinar -nodd lmnt; cont. n ordr to calclat th lmnt stiffnss, w mst tak car of th intgral k T E tda This intgration (in th natral coordinats) is don nmricall b so calld Gass-qadratr, which basicall mans that w alat th intgrand at a nmbr of points, and mltipl with an associatd ara. Dpnding on lmnt tp, a crtain nmbr of alation points ar ndd, which for or -nodd lmnt i. t mst finall b notd that also th abo lmnt somtims bhas poorl, and that a highr ordr lmnt thn mst b sd. 26 (modifid 23--3)