1 / 18

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:.

beau-berry
Télécharger la présentation

MECH593 Introduction to Finite Element Methods

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related