MECh300H Introduction to Finite Element Methods. Finite Element Analysis (F.E.A.) of 1-D Problems

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1 MECh300H Introduction to Finite Element Methods Finite Element Analysis (F.E.A.) of -D Problems

2 Historical Background Hrenikoff, 94 frame work method Courant, 943 piecewise polynomial interpolation Turner, 956 derived stiffness matrice for truss, beam, etc Clough, 960 coined the term finite element Key Ideas: - frame work method piecewise polynomial approimation

3 Aially Loaded Bar Review: Stress: Strain: Deformation: Stress: Strain: Deformation:

4 Aially Loaded Bar Review: Stress: Strain: Deformation:

5 Aially Loaded Bar Governing Equations and Boundary Differential Equation d d EA( ) du d Conditions f ( ) 0 Boundary Condition Types + 0 < prescribed displacement (essential BC) < L prescribed force/derivative of displacement (natural BC)

6 Aially Loaded Bar Boundary Conditions Eamples fied end simple support free end

7 Potential Energy Elastic Potential Energy (PE) - Spring case Unstretched spring PE 0 - Aially loaded bar undeformed: deformed: - Elastic body Stretched bar PE PE 0 L PE σεad PE σt εdv V 0 k

8 Work Potential (WE) A L WP u 0 Potential Energy f fd P u Total Potential Energy Π L B σε Ad 0 L 0 B u P fd P u f: distributed force over a line P: point force u: displacement Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium etremize the total potential energy. If the etremum condition is a minimum, the equilibrium state is stable. B

9 Potential Energy + Rayleigh-Ritz Eample: A f Approach Step : assume a displacement field u a φ ( ) i to n φ is shape function / basis function n is the order of approimation B i P i i Step : calculate total potential energy

10 Potential Energy + Rayleigh-Ritz Approach Eample: f P A B Step 3:select a i so that the total potential energy is minimum

11 Galerkin s Method Eample: d d u ( 0) EA( ) du d A du EA( ) + d 0 L P f ( ) 0 f B P Seek an approimation u ~ so d du~ wi EA f dv d ( ) d + ( ) 0 V u~ ( 0) 0 du~ EA( ) P d L In the Galerkin s method, the weight function is chosen to be the same as the shape function.

12 Galerkin s Method P f A B Eample: 0 ) ( ~ ) ( + dv f d du EA d d w V i 0 ~ ) ( ~ ) ( L i L L i i d du w EA fd w d d dw d du EA 3 3

13 Finite Element Method Piecewise Approimation u u

14 FEM Formulation of Aially Loaded Bar Governing Equations Differential Equation d d EA( ) du d + f ( ) 0 Weighted-Integral Formulation 0 < < L 0 L w d d EA( ) du d + f ( ) d 0 Weak Form L dw du du 0 EA( ) wf ( ) d w EA( ) d d d 0 L 0

15 Approimation Methods Finite Eample: Element Method Step : Discretization Step : Weak form of one element P P dw EA( ) d du d w( ) f ( ) d w( ) EA( ) du d 0 dw du EA( ) w( ) f ( ) d w d d ( ) P w( ) P 0

16 Approimation Methods Finite Eample (cont): Element Method Step 3: Choosing shape functions - linear shape functions u φ + u φu ξ l ξ ξ0 ξ ξ + ξ φ ; φ φ ; φ l l ( ) ( ξ + ) l ξ ; + l

17 Approimation Methods Finite Eample (cont): Element Method Step 4: Forming element equation E,A are constant Let w φ, weak form becomes u u EA d f d φp φ P 0 l l Let w φ, weak form becomes u u EA d f d φ P φp 0 l l EA EA u u φ l l φ f d + P EA EA φ u + u f d P l l φ + φ fd EA u P f P l + + u P f P φ fd

18 Approimation Methods Finite Eample (cont): Element Method Step 5: Assembling to form system equation Approach : Element : Element : Element 3: I I E A l I II II E A l II III E A III l I I I 0 0 u f P 0 0 I I I u f P III II II II 0 0 u f P II II + II 0 0 u f P III III + III 0 0 u f P III III III 0 0 u f P

19 Approimation Methods Finite Eample (cont): Element Method Step 5: Assembling to form system equation Assembled System: I I I I E A E A 0 0 I I l l I I I I II II II II E A E A E A E A u f I I P f P + 0 I I II II l l l l u f P I II I II f + f P + P II II II II III III III III + II III + II III E A E A E A E A u3 f3 P3 0 II II + III f + f P + P III l l l l u 4 f III III 4 P4 f P III III III III E A E A 0 0 III III l l

20 Approimation Methods Finite Eample (cont): Element Method Step 5: Assembling to form system equation Approach : Element connectivity table Element Element Element 3 k e ij K IJ local node (i,j) global node inde (I,J)

21 Approimation Methods Finite Eample (cont): Element Method Step 6: Imposing boundary conditions and forming condense system Condensed system: I I II II II II E A E A E A I + II II 0 l l l u f 0 II II II II III III III III E A E A E A E A + u3 f3 0 II II III III l l l l + III III III III u 4 f 4 P 0 E A E A III III l l

22 Approimation Methods Finite Eample (cont): Element Method Step 7: solution Step 8: post calculation u u φ + u φ du dφ dφ ε u u + d d d dφ dφ σ Eε Eu + Eu d d

23 Summary - Major Steps in FEM Discretization Derivation of element equation weak form construct form of approimation solution over one element derive finite element model Assembling putting elements together Imposing boundary conditions Solving equations Postcomputation

24 Eercises Linear Element Eample : E 00 GPa, A cm

25 Linear Formulation for Bar Element u f() u u P P L - P P + f f K K K K u u where K ij dφ dφ i j EA d K ji, fi d d ( φ ) i f d φ φ

26 Higher Order Formulation for Bar Element u u u u 3 3 u() uφ () + uφ () + u3φ 3() u u u u 3 u u() uφ ( ) + uφ ( ) + u3φ 3( ) + u4φ 4( ) u u() u u u 3 u n u φ ( ) + u φ ( ) + u φ ( ) + u φ ( ) + + u nφn u n ( )

27 Natural Coordinates and Interpolation Functions ξ- 0 l Natural (or Normal) Coordinate: ξ- ξ ξ ξ- ξ ξ 3 ξ φ ξ ξ ξ φ, ( ξ ), φ + l / ξ ξ + φ ( ξ + )( ξ ), φ 3 ( ξ + ) ξ ξ- ξ ξ φ ξ + ξ ξ ( ξ ), φ ( ξ + ) ξ ( ) 7 φ 6 ( ξ + ) ξ + ( ξ ), φ4 ( ξ + ) ξ ξ 9

28 Quadratic Formulation for Bar Element P P P 3 + f f f 3 K K K 3 K K K 3 K K K u u u 3 dφ dφ i j dφ dφ i j where Kij EA d EA dξ K d d dξ dξ l ji l and fi ( ϕi f ) d ( ϕi f ) dξ, i, j,, 3 φ φ φ 3 ξ- ξ0 ξ

29 Quadratic Formulation for Bar Element f() P P u u u 3 3 ξ- ξ0 ξ P 3 ( ξ ) ξ u( ξ ) uφ ( ξ ) + uφ( ξ ) + u3φ 3( ξ ) u u( ξ + )( ξ ) + u3 ξ φ ξ ( ξ ), + l / φ ( ξ + )( ξ ), φ 3 ( ξ + ) l dξ dξ d d l ξ ( ξ + ) ξ dφ dφ ξ dφ dφ 4ξ dφ3 dφ3 ξ +,, d l dξ l d l dξ l d l dξ l

30 Eercises Quadratic Element Eample : E 00 GPa, A cm ; A cm

31 Some Issues Non-constant cross section: Interior load point: Mied boundary condition: k

Introduction to Finite Element Method. Dr. Aamer Haque

Introduction to Finite Element Method 4 th Order Beam Equation Dr. Aamer Haque http://math.iit.edu/~ahaque6 ahaque7@iit.edu Illinois Institute of Technology July 1, 009 Outline Euler-Bernoulli Beams Assumptions