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

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.

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

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

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

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

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

Plane Strain Formulation Strain-Displacement Hooke’s Law

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

Plane Strain Example

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

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

Correspondence Between Plane Problems Plane Strain Plane Stress

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

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

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

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

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

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

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