The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems
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1 Swiss Federal Insiue of Technology Page 1 The Finie Elemen Mehod for he Analysis of Non-Linear and Dynamic Sysems Prof. Dr. Michael Havbro Faber Swiss Federal Insiue of Technology ETH Zurich, Swizerland Mehod of Finie Elemens II
2 Swiss Federal Insiue of Technology Page 2 Conens of Today's Lecure Moivaion, overview and organizaion of he course Inroducion o non-linear analysis Formulaion of he coninuum mechanics incremenal equaions of moion Mehod of Finie Elemens II
3 Swiss Federal Insiue of Technology Page 3 Moivaion, overview and organizaion of he course Moivaion In FEM 1 we learned abou he seady sae analysis of linear sysems however, he sysems we are dealing wih in srucural engineering are generally no seady sae and also no linear We mus be able o assess he need for a paricular ype of analysis and we mus be able o perform i Mehod of Finie Elemens II
4 Swiss Federal Insiue of Technology Page 4 Moivaion, overview and organizaion of he course Moivaion Wha kind of problems are no seady sae and linear? E.g. when he: maerial behaves non-linearly deformaions become big (p-δ effecs) loads vary fas compared o he eigenfrequencies of he srucure General feaure: Response becomes load pah dependen Mehod of Finie Elemens II
5 Swiss Federal Insiue of Technology Page 5 Moivaion, overview and organizaion of he course Moivaion Wha is he added value of being able o assess he non-linear non-seady sae response of srucures? E.g. assessing he: - srucural response of srucures o exreme evens (rock-fall, earhquake, hurricanes) - performance (failures and deformaions) of soils - verifying simple models Mehod of Finie Elemens II
6 Swiss Federal Insiue of Technology Page 6 Moivaion, overview and organizaion of he course Collapse Analysis of he World Trade Cener Mehod of Finie Elemens II
7 Swiss Federal Insiue of Technology Page 7 Moivaion, overview and organizaion of he course Collapse Analysis of he World Trade Cener Mehod of Finie Elemens II
8 Swiss Federal Insiue of Technology Page 8 Moivaion, overview and organizaion of he course Analysis of ulimae collapse capaciy of jacke srucure Mehod of Finie Elemens II
9 Swiss Federal Insiue of Technology Page 9 Moivaion, overview and organizaion of he course Analysis of ulimae collapse capaciy of jacke srucure Mehod of Finie Elemens II
10 Swiss Federal Insiue of Technology Page 10 Moivaion, overview and organizaion of he course Analysis of soil performance Mehod of Finie Elemens II
11 Swiss Federal Insiue of Technology Page 11 Moivaion, overview and organizaion of he course Analysis of bridge response Mehod of Finie Elemens II
12 Swiss Federal Insiue of Technology Page 12 Moivaion, overview and organizaion of he course Seady sae problems (Linear/Non-linear): The response of he sysem does no change over ime KU = R Propagaion problems (Linear/Non-linear): The response of he sysem changes over ime MU () + CU () + KU() = R() Eigenvalue problems: No unique soluion o he response of he sysem Av = λbv Mehod of Finie Elemens II
13 Swiss Federal Insiue of Technology Page 13 Moivaion, overview and organizaion of he course Organizaion The lecures will be given by: M. H. Faber Exercises will be organized/aended by: Jianjun Qin By appoinmen, HIL E13.1. Mehod of Finie Elemens II
14 Swiss Federal Insiue of Technology Page 14 Moivaion, overview and organizaion of he course Organizaion PowerPoin files wih he presenaions will be uploaded on our homepage one day in advance of he lecures hp:// The lecure as such will follow he book: "Finie Elemen Procedures" by K.J. Bahe, Prenice Hall, 1996 Mehod of Finie Elemens II
15 Swiss Federal Insiue of Technology Page 15 Moivaion, overview and organizaion of he course Overview Mehod of Finie Elemens II
16 Swiss Federal Insiue of Technology Page 16 Moivaion, overview and organizaion of he course Overview Mehod of Finie Elemens II
17 Swiss Federal Insiue of Technology Page 17 Moivaion, overview and organizaion of he course Overview Mehod of Finie Elemens II
18 Swiss Federal Insiue of Technology Page 18 Inroducion o non-linear analysis Previously we considered he soluion of he following linear and saic problem: KU = R for hese problems we have he convenien propery of lineariy, i.e: KU = λr, λ = 1 U = λu, λ 1 If his is no he case we are dealing wih a non-linear problem! Mehod of Finie Elemens II
19 Swiss Federal Insiue of Technology Page 19 Inroducion o non-linear analysis Previously we considered he soluion of he following linear and saic problem: KU = R we assumed: small displacemens when developing he siffness marix K and he load vecor R, because we performed all inegraions over he original elemen volume ha he B marix is consan independen of elemen displacemens he sress-srain marix C is consan boundary consrains are consan Mehod of Finie Elemens II
20 Swiss Federal Insiue of Technology Page 20 Inroducion o non-linear analysis Classificaion of non-linear analyses Type of analysis Descripion Typical formulaion used Maerially-nonlinear only Large displacemens, large roaions bu small srains Large displacemens, large roaions and large srains Mehod of Finie Elemens II Displacemens and roaions of fibers are large; bu fiber exensions and angle changes beween fibers are small; sress srain relaionship may be linear or non-linear Displacemens and roaions of fibers are large; fiber exensions and angle changes beween fibers may also be large; sress srain relaionship may be linear or non-linear Infiniesimal displacemens and srains; sress rain relaion is nonlinear Maeriallynonlinear-only (MNO) Toal Lagrange (TL) Updaed Lagrange (UL) Toal Lagrange (TL) Updaed Lagrange (UL) Sress and srain measures used Engineering srain and sress Second Piola- Kirchoff sress, Green-Lagrange srain Cauchy sress, Almansi srain Second Piola- Kirchoff sress, Green-Lagrange srain Cauchy sress, Logarihmic srain
21 Swiss Federal Insiue of Technology Page 21 Inroducion o non-linear analysis L Classificaion of non-linear analyses Δ P 2 σ = P / A ε = σ / E σ 1 E P 2 Δ= ε L ε < 0.04 ε L Linear elasic (infiniesimal displacemens) Mehod of Finie Elemens II
22 Swiss Federal Insiue of Technology Page 22 Inroducion o non-linear analysis L Classificaion of non-linear analyses Δ P 2 P 2 σ = P/ A σ Y σ σy ε = + E E ε < 0.04 T σ σ Y 1 E P / A 1 E T ε L Maerially nonlinear only (infiniesimal displacemens, bu nonlinear sress-srain relaion) Mehod of Finie Elemens II
23 Swiss Federal Insiue of Technology Page 23 Inroducion o non-linear analysis Classificaion of non-linear analyses Δ y y ε x L x L ε < 0.04 Δ= ε L Large displacemens and large roaions bu small srains (linear or nonlinear maerial behavior) Mehod of Finie Elemens II
24 Swiss Federal Insiue of Technology Page 24 Inroducion o non-linear analysis Classificaion of non-linear analyses Large displacemens, large roaions and large srains (linear or nonlinear maerial behavior) Mehod of Finie Elemens II
25 Swiss Federal Insiue of Technology Page 25 Inroducion o non-linear analysis Classificaion of non-linear analyses P 2 P 2 Δ Chang in boundary condiions Mehod of Finie Elemens II
26 Swiss Federal Insiue of Technology Page 26 Inroducion o non-linear analysis Example: Simple bar srucure R 4 3 Area Secion a L = 10cm a = 1cm 2 u Secion b L b = R 5cm σ σ Y 1 ε Y = E E T E = 10 N / cm E T ε 7 2 = 10 N / cm 5 2 σ Y :yield sress ε : yield srain Y Mehod of Finie Elemens II
27 Swiss Federal Insiue of Technology Page 27 Inroducion o non-linear analysis Example: Simple bar srucure Area Secion a L = 10cm Mehod of Finie Elemens II a = 1cm 2 u Secion b L b = u u εa =, εb = La Lb R+ σ A= σ A b σ ε = E ε = ε + Y a R 5cm (elasic region) σ σy E T (plasic region) Δ ε = σ σ Y R E 1 ε = Δσ (unloading) E E T 7 2 E = 10 N / cm 5 2 ET = 10 N / cm σ Y : yield sress ε : yield srain Y ε 2 4 6
28 Swiss Federal Insiue of Technology Page 28 Inroducion o non-linear analysis Example: Simple bar srucure Area Secion a L = 10cm a = 1cm 2 u Secion b L a = R 5cm σ σ Y R E 1 ε = E T E = 10 N / cm ET = 10 N / cm σ Y : yield sress ε : yield srain Y ε u u εa =, εb = L L a R+ σ A= σ A b T a σ ε = (elasic region) E σ σy ε = εy + (plasic region) E Δ ε = Δσ (unloading) E b Boh secions elasic 1 1 R R = EA u( + ) u = L L 310 σ a R 2 =, σb = 3A 3 a b R A 6 Mehod of Finie Elemens II
29 Swiss Federal Insiue of Technology Page 29 Inroducion o non-linear analysis Example: Simple bar srucure Area Secion a L = 10cm a Mehod of Finie Elemens II a = 1cm 2 u Secion b L b = R 5cm σ σ Y R E 1 ε = E T E = 10 N / cm ET = 10 N / cm σ Y : yield sress ε : yield srain Y ε secion b will be plasic when R = σ Y A σ u u 2 a = E, b ET( Y) Y L σ = a L ε σ b EA u ET A u R = + ETεYA+ σya La Lb R/ A+ ETεY σy R u = = E/ L + E/ L Secion a is elasic while secion b is plasic b 2 R u u εa =, εb = La Lb R+ σba= σaa σ ε = (elasic region) E σ σy ε = εy + (plasic region) ET Δσ Δ ε = (unloading) E u
30 Swiss Federal Insiue of Technology Page 30 Inroducion o non-linear analysis Wha did we learn from he example? The basic problem in general nonlinear analysis is o find a sae of equilibrium beween exernally applied loads and elemen nodal forces R F = 0 R = R + R + R B S C We mus achieve equilibrium for all ime seps when incremening he loading Very general approach F = F R I B τ dv m V ( m) = ( m) T ( m) ( m) includes implicily also dynamic analysis! Mehod of Finie Elemens II
31 Swiss Federal Insiue of Technology Page 31 Inroducion o non-linear analysis The basic approach in incremenal analysis is R F = 0 +Δ +Δ +Δ assuming ha R is independen of he deformaions we have F = F+ F +Δ We know he soluion F a ime and F is he incremen in he nodal poin forces corresponding o an incremen in he displacemens and sresses from ime o ime +Δ his we can approximae by F = KU Tangen siffness marix Mehod of Finie Elemens II F K = U
32 Swiss Federal Insiue of Technology Page 32 Inroducion o non-linear analysis The basic approach in incremenal analysis is We may now subsiue he angen siffness marix ino he equlibrium relaion KU = R F +Δ U = U+ U +Δ which gives us a scheme for he calculaion of he displacemens he exac displacemens a ime +Δ correspond o he applied loads a +Δ however we only deermined hese approximaely as we used a angen siffness marix hus we may have o ierae o find he soluion Mehod of Finie Elemens II
33 Swiss Federal Insiue of Technology Page 33 Inroducion o non-linear analysis The basic approach in incremenal analysis is We may use he Newon-Raphson ieraion scheme o find he equlibrium wihin each load incremen K Δ U = R F +Δ ( i 1) ( i) +Δ +Δ ( i 1) (ou of balance load vecor) U = U +ΔU +Δ () i +Δ ( i 1) () i wih iniial condiions U = U; K = K; F = F +Δ (0) +Δ (0) +Δ (0) Mehod of Finie Elemens II
34 Swiss Federal Insiue of Technology Page 34 Inroducion o non-linear analysis The basic approach in incremenal analysis is I may be expensive o calculae he angen siffness marix and, in he Modified Newon-Raphson ieraion scheme i is hus only calculaed in he beginning of each new load sep in he quasi-newon ieraion schemes he secan siffness marix is used insead of he angen marix Mehod of Finie Elemens II
35 Swiss Federal Insiue of Technology Page 35 Inroducion o non-linear analysis We look a he example again simple bar ( wo load seps) () i +Δ +Δ ( i 1) +Δ ( i 1) ( Ka + Kb) Δ u = R ( Fa Fb ) +Δ () i +Δ ( i 1) () i wih iniial condiions u = u; F = F F = F +Δ (0) +Δ (0) +Δ (0) a a b b K u = u +Δu a CA = ; Kb = L a = E if secion is elasic C = ET if secion is plasic CA L b Mehod of Finie Elemens II
36 Swiss Federal Insiue of Technology Page 36 Inroducion o non-linear analysis We look a he example again simple bar Load sep 1: = 1: ( K + K ) Δ u = R F F 0 0 (1) 1 1 (0) 1 (0) a b a b 2 10 Δ u = = ( + ) 10 5 Ieraion 1: ( i = 1) 4 (1) 3 u = u +Δ u = (1) 1 (0) (1) 3 1 (1) 1 (1) 4 εa = = La ε 1 (1) b F u Mehod of Finie Elemens II < ε (elasic secion!) 1 (1) u 3 = = < εy (elasic secion!) L b = ; F = (1) 3 1 (1) 4 a b 0 0 (2) 1 1 (1) 1 (1) ( Ka Kb) u R Fa Fb 0 Y Convergence in one ieraion! Δ = = u = `10
37 Swiss Federal Insiue of Technology Page 37 Inroducion o non-linear analysis We look a he example again simple bar Load sep 2: = 2 : ( K + K ) Δ u = R F F 1 1 (1) 2 2 (0) 2 (0) a b a b (4 10 ) ( ) ( ) Δ u = = ( + ) 10 5 Ieraion 1: ( i = 1) (1) 3 2 (1) 2 (0) (1) 2 2 (1) 3 a 2 u = u +Δ u = ε ε = < ε (elasic secion!) (1) 3 b Y = > ε (plasic secion!) Y F = ; F = ( E ( ε ε ) + σ ) A= (1) 4 1 (1) T 2 (1) 4 a b b Y Y ( K + K ) Δ u = R F F Δ u = ( 2) 2 2 (1) 2 (1) (2) 3 a b a b Mehod of Finie Elemens II
38 Swiss Federal Insiue of Technology Page 38 Inroducion o non-linear analysis We look a he example again simple bar i Δ u (i) 2 u (i) E E E E E E E E E E E E-02 Mehod of Finie Elemens II
39 Swiss Federal Insiue of Technology Page 39 The coninuum mechanics incremenal equaions The basic problem: We wan o esablish he soluion using an incremenal formulaion The equilibrium mus be esablished for he considered body in is curren configuraion In proceeding we adop a Lagrangian formulaion where we rack he movemen of all paricles of he body (locaed in a Caresian coordinae sysem) Anoher approach would be an Eulerian formulaion where he moion of maerial hrough a saionary conrol volume is considered Mehod of Finie Elemens II
40 Swiss Federal Insiue of Technology Page 40 The coninuum mechanics incremenal equaions The basic problem: x 3 δu δu δu δu 1 = 2 3 Configuraion corresponding o variaion in displacemens a +Δ δu u Configuraion a ime + Δ Surface area Volume +Δ V +Δ S x 2 Configuraion a ime 0 Surface area Volume 0 V 0 S Configuraion a ime Surface area Volume V S x (or x, x, x ) 0 + Δ Mehod of Finie Elemens II
41 Swiss Federal Insiue of Technology Page 41 The coninuum mechanics incremenal equaions The Lagrangian formulaion We express equilibrium of he body a ime +Δ using he principle of virual displacemens +Δ +Δ +Δ τδ ed V= R +Δ V +Δ ij Mehod of Finie Elemens II x 3 x (or x, x, x ) 0 + Δ δu1 δu = δu2 δu 3 Configuraion corresponding o variaion in displacemens a +Δ δu u Configuraion a ime +Δ +Δ Surface area S +Δ Volume V Configuraion a ime Surface area S Configuraion a ime 0 Volume V 0 Surface area S 0 Volume V x 2 +Δ τ : Caresian componens of he Cauchy sress ensor 1 δu δ u ( i j δ+δ eij = + ) = +Δ +Δ srain ensor corresponding o virual displacemens 2 xj xi δu : Componens of virual displacemen vecor imposed a ime +Δ i +Δ +Δ x : Caresian coordinae a ime +Δ i V : Volume a ime +Δ R= f δud V + f δu d S +Δ +Δ B +Δ +Δ S S +Δ i i i i +Δ +Δ V S f
42 Swiss Federal Insiue of Technology Page 42 The coninuum mechanics incremenal equaions The Lagrangian formulaion We express equilibrium of he body a ime +Δ using he principle of virual displacemens R= f δud V + f δu d S +Δ +Δ B +Δ +Δ S S +Δ i i i i +Δ +Δ V S where f +Δ B i +Δ S fi +Δ f δu : exernally applied forces per uni volume x 3 x (or x, x, x ) 0 + Δ : exernally applied surface racions per uni surface S : surface a ime +Δ +Δ : δu evaluaed a he surface S S i i f f δu1 δu = δu2 δu 3 Configuraion corresponding o variaion in displacemens a +Δ δu u Configuraion a ime +Δ +Δ Surface area S +Δ Volume V Configuraion a ime Surface area S Configuraion a ime 0 Volume V 0 Surface area S 0 Volume V x 2 Mehod of Finie Elemens II
43 Swiss Federal Insiue of Technology Page 43 The coninuum mechanics incremenal equaions The Lagrangian formulaion We recognize ha our derivaions from linear finie elemen heory are unchanged bu applied o he body in he configuraion a ime +Δ Mehod of Finie Elemens II
44 Swiss Federal Insiue of Technology Page 44 The coninuum mechanics incremenal equaions In he furher we inroduce an appropriae noaion: Coordinaes and displacemens are relaed as: x = x + u 0 i i i +Δ 0 +Δ i i i Incremens in displacemens are relaed as: +Δ i i i Reference configuraions are indexed as e.g.: +Δ S 0 i x = x + u u = u u f where he lower lef index indicaes he reference configuraion τ = τ +Δ +Δ ij +Δ ij Differeniaion is indexed as: u +Δ 0 +Δ i 0 m 0 ui, j=, 0 +Δ xm, n= +Δ xj xn x Mehod of Finie Elemens II
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