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Elasticity2

# Elasticity2 - Formulation of Two-Dimensional Elasticity...

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

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Simplified Elasticity Formulations 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. 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.
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

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Two and Three Dimensional Problems x y z x y z Three-Dimensional Two-Dimensional x y z Spherical Cavity
Two-Dimensional Formulation x y z R x y z R 2 h Plane Strain Plane Stress 0 , ) , ( , ) , ( = = = w y x v v y x u u 0 ) , ( ) , ( ) , ( = τ = τ = σ τ = τ σ = σ σ = σ yz xz z xy xy y y x x y x y x y x << other dimensions

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

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Plane Strain Formulation 0 2 1 , , = = = + = = = yz xz z xy y x e e e x v y u e y v e x u e Strain-Displacement 0 , 2 ) ( ) ( 2 ) ( 2 ) ( = τ = τ μ = τ σ + σ ν = + λ = σ μ + + λ = σ μ + + λ = σ yz xz xy xy y x y x z y y x y x y x x e e e e e e e e e Hooke’s Law 0 , ) , ( , ) , ( = = = w y x v v y x u u
Plane Strain Formulation 0 ) ( 0 ) ( 2 2 = + + μ + λ + μ = + + μ + λ + μ y x F y v x u y v F y v x u x u Displacement Formulation 0 0 = + σ + τ = + τ + σ y y xy x xy x F y x F y x

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Elasticity2 - Formulation of Two-Dimensional Elasticity...

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