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MECh300H Introduction to Finite Element Methods

MECh300H Introduction to Finite Element Methods. Finite Element Analysis (F.E.A.) of 1-D Problems. Historical Background . Hrenikoff, 1941 – “frame work method” Courant, 1943 – “piecewise polynomial interpolation” Turner, 1956 – derived stiffness matrice for truss, beam, etc

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MECh300H Introduction to Finite Element Methods

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

  2. Historical Background • Hrenikoff, 1941 – “frame work method” • Courant, 1943 – “piecewise polynomial interpolation” • Turner, 1956 – derived stiffness matrice for truss, beam, etc • Clough, 1960 – coined the term “finite element” Key Ideas: - frame work method piecewise polynomial approximation

  3. Axially Loaded Bar Review: Stress: Stress: Strain: Strain: Deformation: Deformation:

  4. Axially Loaded Bar Review: Stress: Strain: Deformation:

  5. Axially Loaded Bar – Governing Equations and Boundary Conditions • Differential Equation • Boundary Condition Types • prescribed displacement (essential BC) • prescribed force/derivative of displacement (natural BC)

  6. Axially Loaded Bar –Boundary Conditions • Examples • fixed end • simple support • free end

  7. Potential Energy • Elastic Potential Energy (PE) - Spring case Unstretched spring Stretched bar x - Axially loaded bar undeformed: deformed: - Elastic body

  8. Potential Energy • Work Potential (WE) f P f: distributed force over a line P: point force u: displacement B A • Total Potential Energy • Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable.

  9. Potential Energy + Rayleigh-Ritz Approach Example: f P B A Step 1: assume a displacement field f is shape function / basis function n is the order of approximation Step 2: calculate total potential energy

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

  11. Galerkin’s Method Example: f P B A Seek an approximation so In the Galerkin’s method, the weight function is chosen to be the same as the shape function.

  12. Galerkin’s Method Example: f P B A 3 2 1 1 2 3

  13. Finite Element Method – Piecewise Approximation u x u x

  14. FEM Formulation of Axially Loaded Bar – Governing Equations • Differential Equation • Weighted-Integral Formulation • Weak Form

  15. Approximation Methods – Finite Element Method Example: Step 1: Discretization Step 2: Weak form of one element P2 P1 x1 x2

  16. Approximation Methods – Finite Element Method Example (cont): Step 3: Choosing shape functions - linear shape functions x x x=0 x=-1 x=1 x2 x1 l

  17. Approximation Methods – Finite Element Method Example (cont): Step 4: Forming element equation E,A are constant Let , weak form becomes Let , weak form becomes

  18. Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 1: Element 1: Element 2: Element 3:

  19. Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Assembled System:

  20. Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 2: Element connectivity table global node index (I,J) local node (i,j)

  21. Approximation Methods – Finite Element Method Example (cont): Step 6: Imposing boundary conditions and forming condense system Condensed system:

  22. Approximation Methods – Finite Element Method Example (cont): Step 7: solution Step 8: post calculation

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

  24. Exercises – Linear Element Example 1: E = 100 GPa, A = 1 cm2

  25. Linear Formulation for Bar Element f(x) u2 u u1 x L = x2-x1 x= x2 x=x1 1 1 f1 f2 x=x2 x=x1

  26. Higher Order Formulation for Bar Element u2 u3 u u u u1 x x x 3 2 1 u4 u2 u3 u1 3 2 1 4 un u1 u4 u3 …………… u2 1 n …………… 2 3 4

  27. Natural Coordinates and Interpolation Functions x x x x x=-1 x=1 x=-1 x=1 2 1 3 1 2 x=-1 x=1 4 2 1 3 x=1 x=-1 x x=x1 x= x2 Natural (or Normal) Coordinate:

  28. Quadratic Formulation for Bar Element f3 f1 f2 x=1 x=0 x=-1

  29. Quadratic Formulation for Bar Element u2 f(x) u3 u1 P2 P1 P3 x=1 x=0 x=-1

  30. Exercises – Quadratic Element Example 2: E = 100 GPa, A1 = 1 cm2; A1 = 2 cm2

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

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