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Formulation of Two-Dimensional Elasticity Problems Professor M. H. Sadd PowerPoint Presentation
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Formulation of Two-Dimensional Elasticity Problems Professor M. H. Sadd. Simplified Elasticity Formulations.

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simplified elasticity formulations
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.

solution to elasticity problems

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

two and three dimensional problems
Two and Three Dimensional Problems

Two-Dimensional

Three-Dimensional

x

x

y

y

z

z

z

Spherical Cavity

y

x

two dimensional formulation

y

x

R

z

y

2h

R

z

x

Two-Dimensional Formulation

Plane Stress

Plane Strain

<< other dimensions

examples of plane strain problems

P

z

x

y

Examples of Plane Strain Problems

y

x

z

Long CylindersUnder Uniform Loading

Semi-Infinite Regions Under Uniform Loadings

examples of plane stress problems
Examples of Plane Stress Problems

Thin Plate WithCentral Hole

Circular Plate UnderEdge Loadings

plane strain formulation
Plane Strain Formulation

Strain-Displacement

Hooke’s Law

plane strain formulation1

Si

R

S = Si + So

So

y

x

Plane Strain Formulation

Stress Formulation

Displacement Formulation

plane stress formulation
Plane Stress Formulation

Hooke’s Law

Strain-Displacement

Note plane stress theory normally neglects some of the strain-displacement and compatibility equations.

plane stress formulation1

Si

R

S = Si + So

So

y

x

Plane Stress Formulation

Displacement Formulation

Stress Formulation

slide14

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

airy stress function method plane problems with no body forces
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

Strain-Displacement

Hooke’s Law

Equilibrium Equations

Airy Representation

x2

rd

dr

d

x1

Polar Coordinate Formulation
solutions to plane problems cartesian coordinates

y

x

S

R

Airy Representation

Solutions to Plane ProblemsCartesian Coordinates

Biharmonic Governing Equation

Traction Boundary Conditions

solutions to plane problems polar coordinates

S

R

y

r

x

Airy Representation

Solutions to Plane ProblemsPolar Coordinates

Biharmonic Governing Equation

Traction Boundary Conditions

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

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