UNIT – III FINITE ELEMENT METHOD

1. UNIT – IIIFINITE ELEMENT METHOD Presented by, G.Bairavi AP/civil

2. PLANE STRESS AND PLANE STRAIN 2

3. Plane stress 3

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19. Formulation of Two-Dimensional Elasticity Problems 19

20. Simplified Elasticity Formulations The General System of Elasticity Field Equationsof 15 Equations for 15 Unknowns Is Very Difficultto Solve for Most Meaningful Problems, and So Modified Formulations Have Been Developed. Stress Formulation Eliminate the displacements and strains from the general system of equations. This generates a system of six equations and for the six unknown stress components. Displacement Formulation Eliminate the stresses and strains from the general system of equations. This generates a system of three equations for the three unknown displacement components. 20

21. Solution to Elasticity Problems x F(z) G(x,y) z y Even Using Displacement and Stress FormulationsThree-Dimensional Problems Are Difficult to Solve! So Most Solutions Are Developed for Two-Dimensional Problems 21

22. Two and Three Dimensional Problems Two-Dimensional Three-Dimensional x x y y z z z Spherical Cavity y x 22

23. Two-Dimensional Formulation y y 2h R z x x R z Plane Stress Plane Strain << other dimensions 23

24. Examples of Plane Strain Problems P z x y y x z Long CylindersUnder Uniform Loading Semi-Infinite Regions Under Uniform Loadings 24

25. Examples of Plane Stress Problems Thin Plate WithCentral Hole Circular Plate UnderEdge Loadings 25

26. Plane Strain Formulation Strain-Displacement Hooke’s Law 26

27. Plane Strain Formulation Si R y S = Si + So So x Stress Formulation Displacement Formulation 27

28. Plane Strain Example 28

29. Plane Stress Formulation Hooke’s Law Strain-Displacement Note plane stress theory normally neglects some of the strain-displacement and compatibility equations. 29

30. Plane Stress Formulation Si R y S = Si + So So x Displacement Formulation Stress Formulation 30

31. Correspondence Between Plane Problems Plane Strain Plane Stress 31

32. Elastic Moduli Transformation Relations for ConversionBetween Plane Stress and Plane Strain Problems E v Plane Stress to Plane Strain Plane Strain to Plane Stress Plane Strain Plane Stress Therefore the solution to one plane problem also yields the solution to the other plane problem through this simple transformation 32

33. Airy Stress Function Method Plane Problems with No Body Forces Stress Formulation Airy Representation Biharmonic Governing Equation (Single Equation with Single Unknown) 33

34. Polar Coordinate Formulation Strain-Displacement x2 rd dr Hooke’s Law d  x1 Equilibrium Equations Airy Representation 34

35. Solutions to Plane ProblemsCartesian Coordinates y S R x Airy Representation Biharmonic Governing Equation Traction Boundary Conditions 35

36. Solutions to Plane ProblemsPolar Coordinates S R y  r  x Airy Representation Biharmonic Governing Equation Traction Boundary Conditions 36

37. Cartesian Coordinate Solutions Using Polynomial Stress Functions terms do not contribute to the stresses and are therefore dropped terms will automatically satisfy the biharmonic equation terms require constants Amn to be related in order to satisfy biharmonic equation Solution method limited to problems where boundary traction conditions can be represented by polynomials or where more complicated boundary conditions can be replaced by a statically equivalent loading 37

38. Plane Stress and Plane Strain Plane Stress - Thin Plate: 38 FDTP ON CE6602-STRUCTURAL ANALYSIS-II ORGANISED BY UNIVERSITY COLLEGE OF ENGG,THIRUKKUVALAI

39. Plane Stress and Plane Strain Plane Strain - Thick Plate: Plane Strain: Plane Stress: Replace E by and by 39 FDTP ON CE6602-STRUCTURAL ANALYSIS-II ORGANISED BY UNIVERSITY COLLEGE OF ENGG,THIRUKKUVALAI

40. Equations of Plane Elasticity Governing Equations (Static Equilibrium) Strain-Deformation (Small Deformation) Constitutive Relation (Linear Elasticity) 40

41. THANK YOU 41