Chapter 5. Introduction. Introduction. Introduction. Finite Element Modelling. Finite Element Modelling

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1 Chaptr 5 wo-dimnsional problms using Constant Strain riangls (CS) Lctur Nots Dr Mohd Andi Univrsiti Malasia Prlis EN7 Finit Elmnt Analsis Introction wo-dimnsional init lmnt ormulation ollows th stps usd in th on-dimnsional problm. h displacmnt vctor u is givn b: u [ uv, ] whr u and v ar th and componnts o u. h strsss and strains: [,, ] [,, ] Introction Introction From Fig., th bod orc, traction vctor and lmntal volum ar givn b: = [, ] = [, ] dv = t da h strain-displacmnt rlations ar givn b: u v u v,, Strss-strain rlations: σ = Dε Figur Finit Elmnt Modlling Finit Elmnt Modlling h two-dimnsional rgion is dividd into straightsidd triangls. Nods and lmnts. Unilld rgions. Each nod - two dos. Figur h global displacmnt vctor: Q = [Q, Q,, QN] A tpical triangular lmnt: h lmnt displacmnt vctor: q = [q, q,, q] Figur q v q (, ) (, ) q 5 q (, ) (, ) u q q 5

2 Finit Elmnt Modlling From Fig., a tpical connctivit rprsntation is shown in abl. Elmnt numbr hr nods abl Constant-Strain riangl (CS) FEM uss th concpt o shap unctions or intrpolation. For CS, th shap unctions ar linar ovr th lmnt. From Fig., shap unction N is at nod and linarl rcs to 0 at nod and. Linar combination o ths shap unctions ar rprsnts a plan surac N + N + N rprsnts a plan at a hight o at nods,, and and it is paralll to th triangl. hus, th global shap unctions: N N N 8 Constant-Strain riangl (CS) h indpndnt shap unctions ar: whr ξ, η ar natural coordinat N N N Constant-Strain riangl (CS) h shap unction can b phsicall rprsntd b ara coordinat as shown in Fig.5. A A A N N N A A A Figur 9 Figur 5 0 Displacmnt Isoparamtric Rprsntation h displacmnt insid th lmnt can b writtn in trm o shap unctions: u Nq Nq Nq5 v Nq Nq Nq Or u ( q q5) ( q q5) q5 v ( q q) ( q q) q In matri orm: u = Nq Whr, N 0 N 0 N 0 N 0 N 0 N 0 N u v Shap unction Nodal coordinat u N q N q N q q q5 q q5 q5 q q q q q 5 v N q N q N q u v N N N N N N

3 Displacmnt Strain In th and coordinats: N N N Or N N N ( ) ( ) ( ) ( ) Or using th notation, ij = i j and ij = i j: Using th chain rul or partial drivativs o u, u u u u u u Evaluating th strains, in matri notation, u u u u Jacobian Matri Jacobian Matri h Jacobian o th transormation: J On taking th drivativ o and, d d d d d d d d d d d d J 5 d J d d d Jacobian Matri Jacobian Matri d J d d d h invrs o th Jacobian, J. J dt J J dt J d d d dt J d d d 7 dt J h magnitud o dt J is twic th ara o th triangl. A dt J 8

4 Jacobian Matri Strain-Displacmnt Rplacing u b th displacmnt v will lad to th sam rsult d d d dt J d d d dv dv dv d d d dv dt J dv dv d d d 9 h D strain-displacmnt rlations: u v ( q q5 ) ( q q5 ) ( q q ) ( q q ) dt J ( q q5 ) ( q q5 ) ( q q ) ( q q ) u v his quation can b writtn in th orm, q q q5 q q q dt J q q q q q5 q 0 Strain-Displacmnt Potntial-Enrg Approach h strain quation can b writtn in matri orm: ε = Bq Whr B is a () lmnt strain-displacmnt matri, B dt J h potntial nrg o th sstm is givn b: i i ε DεtdA tda tdl A u A u L u P Elmnt stinss. U ε DεtdA q B DBt ta q B DBqtdA = q B DBq = q k q i da q Potntial-Enrg Approach Potntial-Enrg Approach Whr th lmnt stinss matri, k tab DB hn, U q k q U = Q KQ h lmnt bod orc vctor. ta [,,,,, ] F raction orc. tl [,,, ] Point load. u P Q P Q P i i i i Global orc. F ( ) P h total potntial nrg. Q KQ Q F

5 Potntial-Enrg Approach E. 5. Summar t A,,,,, k t A B DB Evaluat th shap unction N, N and N at th intrior point P or th triangular lmnt shown blow (, 7) Elmnt bod orc vctor tl,,, Elmnt stinss matri (.5, ) P (.85,.8) (7,.5) Elmnt traction orc 5 P-5. Eampl Eampl