VSMN30 FINITA ELEMENTMETODEN - DUGGA

Transcription

1 VSMN3 FINITA ELEMENTMETODEN - DUGGA kl Maximum points: 4, Rquird points to pass: Assistanc: CALFEM manual and calculator Problm 1 ( 8p ) y 4 1. m x m Th isotropic two-dimnsional lasticity lmnt abov hav a thicknss of t =. m, a modulus of lasticity E = GPa and a Poisson s ratio of ν =.. Th lft dg is clampd and th othr dgs ar fr. Th lmnt is subjctd to a body load b T (x,y) = (, -4) kn/m 3. Th lmnt stiffnss matrix for this Mlosh lmnt is K = 1 9 a) Dtrmin th lmnt load vctor f l. b) Dtrmin th lmnt boundary vctor f b. c) Dtrmin th displacmnts u 1 u 8. d) Dtrmin all th strsss at th cntr of th lmnt. Hint: 9 1 =

2 Problm ( 8p ) Blow, suggstions of fiv lmnts with corrsponding approximation ar shown. Each lmnt rprsnts a thory that has bn studid during th cours. a) For -dimnsional hat flow, th compltnss rquirmnt stats that th tmpratur approximation must at last b abl to dscrib a constant tmpratur and a constant flux and it is writtn as: T ( x, y) α + α x + α y + possibly othr trms = 1 3 For ach of th fiv lmnts shown blow, writ th trms that ar rquird for fulfillmnt of th compltnss (as writtn for -dimnsional hat flow abov). b) For - dimnsional hat flow, th compatibility rquirmnt stats that th tmpratur must b continuous ovr th lmnt boundaris and it is writtn as: T continuous ovr th lmnt boundaris. For ach of th fiv thoris shown blow, giv th compatibility rquirmnt (as writtn abov). c) Dtrmin th lmnts blow if th convrgnc critrion is fulfilld and motivat (show) why.

3 Problm 3 ( 8p ) Th quilibrium quation for a dformabl body is givn by ~ T σ + b = whr b is th body forc vctor, σ is th strss vctor and ~ T x = y z y x z x z y Th kinmatic rlation is ε = u ~ whr u is th displacmnt vctor and ε is th strain vctor. Th constitutiv rlation is givn by σ = Dε In crtain situations a plan stat of strain or strss can b assumd which rsults in simplifications in th quations givn abov. a) For what kind of bodis is plan strain and plan strss, rspctivly, suitabl approximations? Giv xampls of both. b) What assumptions ar mad concrning th strain and strss vctor? Writ out th componnts of ε and σ for a plan strain and a plan strss cas in th xyplan. c) Rduc th quilibrium condition to yild th quilibrium condition for th two plan cass dscribd in b). d) Considr a plan strain cas in th xy-plan whr ε Τ =[.1,.,.45], E=1 Gpa and ν=.. Exprss σ zz in trms of ε.

4 Problm 4 ( 8p ).4. A B N h C b A stl cantilvr bam is loadd by a point load P=7 kn and a normal forc N= kn at its fr nd. Th bam cross-sction is rctangular with bxh=x8 mm. A wldd joint ABC runs through th bam at an angl as shown in th abov figur. a) Dtrmin th strss vctor σ at C. b) Dtrmin th traction vctor t at th surfac of th wldd joint at B and C. c) Dtrmin th traction vctor t at th boundary surfac at A. d) Dtrmin th magnitud of σ nn and σ nm at th surfac of th wldd joint at B. P Hint: n and m is th dirctions normal and tangntial to th surfac, rspctivly. σ b,max = M/W, whr M is th bnding momnt and W is th bnding rsistanc (bh /6) τ max = 3V/A, whr V is th shar forc and A is th cross-sction ara.

5 Problm 5 ( 8p ) In th cas of two-dimnsional hat flow w can writ th FE-quations as K a = f b + T f l = N A f l QtdA whr N is th shap functions, t is th thicknss and Q is th amount of hat supplid to th body pr unit volum and tim [J/m 3 s]. a) Vrify that th shap functions lft figur blow N 1, N and N 3 ar corrct for th lmnt in th N 1 = 1-ξ-η N = ξ N 3 = η b) Th us of isoparamtric lmnts implis that in ordr to valuat th FE-quations thy must b rformulatd. Calculat th Jacobian matrix J and th dtrminant dtj for th lmnt mapping shown blow. Is th mapping uniqu? Propos on way to chck th dtrminant in this spcific cas. c) Evaluat th load vctor f ( x, y) da = Axy f l by us of th intgral transformation Aξη f ( x( ξ, η), y( ξ, η))dt Jdξdη and by choosing on gauss point positiond at th cntr of th lmnt, i.. at ξ = η = 1/3. Paramtrs ar t =. m and Q = [J/m 3 s]. d) Dos th Gauss intgration provid an xact intgration in this spcific cas? η y 3 (,1 3 (,4) A xy 1 (,) A ξη (1,) ξ 1 (1,1) (5,) x Hint: Th Jacobian matrix J = x ξ y ξ x η y η

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