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

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

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Presentation Transcript
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.

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-Dimensional

Three-Dimensional

x

x

y

y

z

z

z

Spherical Cavity

y

x

y

x

R

z

y

2h

R

z

x

Two-Dimensional Formulation

Plane Stress

Plane Strain

<< other dimensions

P

z

x

y

Examples of Plane Strain Problems

y

x

z

Examples of Plane Stress Problems

Thin Plate WithCentral Hole

Plane Strain Formulation

Strain-Displacement

Hooke’s Law

Si

R

S = Si + So

So

y

x

Plane Strain Formulation

Stress Formulation

Displacement Formulation

Plane Stress Formulation

Hooke’s Law

Strain-Displacement

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

Si

R

S = Si + So

So

y

x

Plane Stress Formulation

Displacement Formulation

Stress Formulation

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

Stress Formulation

Airy Representation

Biharmonic Governing Equation

(Single Equation with Single Unknown)

Strain-Displacement

Hooke’s Law

Equilibrium Equations

Airy Representation

x2

rd

dr

d

x1

Polar Coordinate Formulation

y

x

S

R

Airy Representation

Solutions to Plane ProblemsCartesian Coordinates

Biharmonic Governing Equation

Traction Boundary Conditions

S

R

y

r

x

Airy Representation

Solutions to Plane ProblemsPolar Coordinates

Biharmonic Governing Equation

Traction Boundary Conditions

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