Formulation of Two-Dimensional Elasticity Problems Professor M. H. Sadd

1. Formulation of Two-Dimensional Elasticity Problems Professor M. H. Sadd

2. 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.

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

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

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

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

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

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

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

10. Plane Strain Example

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

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

13. Correspondence Between Plane Problems Plane Strain Plane Stress

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

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

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

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

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

19. 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 Cartesian Coordinate Solutions Using Polynomial Stress Functions 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

20. Appears to Solve the Beam Problem: y d F x Stress Function Example