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

2. Simplified Elasticity Formulations
The General System of Elasticity Field Equations of 15 Equations for 15 Unknowns Is Very Difficult to Solve for Most Meaningful Problems, and So Modified Formulations Have Been Developed.
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.
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.

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

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

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

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

7. Examples of Plane Stress Problems
Thin Plate With Central Hole
Circular Plate Under Edge Loadings

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

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

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

13. Correspondence Between Plane Problems
Plane Strain
Plane Stress

14. Elastic Moduli Transformation Relations for Conversion Between Plane Stress and Plane Strain Problems
Plane Strain
Plane Stress E v
Plane Stress to Plane Strain
Plane Strain to 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. Polar Coordinate Formulationx1 x2  dr
rd d
Strain-Displacement
Hooke’s Law
Equilibrium Equations
Airy Representation

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

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

19. 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

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