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MECH593 Introduction to Finite Element Methods

MECH593 Introduction to Finite Element Methods. Finite Element Analysis of 2D Problems Axisymmetric Problems Plate Bending. Axi-symmetric Problems. Definition:. A problem in which geometry, loadings, boundary conditions and materials are symmetric about one axis. Examples:.

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MECH593 Introduction to Finite Element Methods

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  1. MECH593 Introduction to Finite Element Methods Finite Element Analysis of 2D Problems Axisymmetric Problems Plate Bending

  2. Axi-symmetric Problems Definition: A problem in which geometry, loadings, boundary conditions and materials are symmetric about one axis. Examples:

  3. Axi-symmetric Analysis Cylindrical coordinates: • quantities depend on r and z only • 3-D problem 2-D problem

  4. Axi-symmetric Analysis

  5. Axi-symmetric Analysis – Single-Variable Problem Weak form: where

  6. Finite Element Model – Single-Variable Problem where Ritz method: Weak form where

  7. Single-Variable Problem – Heat Transfer Heat Transfer: Weak form where

  8. 3-Node Axi-symmetric Element 3 1 2

  9. 4-Node Axi-symmetric Element h 4 3 b 2 1 x a z r

  10. Single-Variable Problem – Example z T(r,L)= T0 Step 1: Discretization R T(R,z) = T0 L r T(r,0) = T0 Step 2: Element equation Heat generation: g = 107 w/m3

  11. Plate Bending

  12. Governing Equations of Classical Plates From force equilibrium --- Governing Equations for Classical Plates ----- (Distributed Transverse Loading) where Bending Stiffness (Flexural Rigidity) D = Eh3/12(1-n2)

  13. Strain Energy of Classical Plates

  14. Weak Form of Classical Plates Governing equation: (isotropic, steady) Weak form: Note: w is the deflection of the mid-plane and u is the weight function.

  15. Boundary Conditions of Classical Plates Essential Boundary Conditions ----- Natural Boundary Conditions ----- Examples: • Clamped : • Simply connected • free

  16. 4-Node Rectangular Plate Element Since the governing eq. is 4th order, at each node, there should 2 EBCs and 2 NBCs in each direction (but specify just 2 of them). For displacement-based finite element formulation, the DoFs should be on generalized displacements. In total, there are 3 DoFs per node: where

  17. Formulation of 4-Node Rectangular Plate Element Let Pascal’s Triangle ----- (incomplete 4th order polynomial)

  18. 3-Node Triangular Plate Element Let Pascal’s Triangle ----- (incomplete 3th order polynomial)

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