FEAElasticity

# FEAElasticity - Procedures of Finite Element Analysis...

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Procedures of Finite Element Analysis Two-Dimensional Elasticity Problems Professor M. H. Sadd

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Two Dimensional Elasticity Element Equation Orthotropic Plane Strain/Stress Derivation Using Weak Form – Ritz/Galerin Scheme Displacement Formulation Orthotropic Case 0 0 22 12 66 66 12 11 = + + + + = + + + + y x F y v C x u C y x v y u C x F x v y u C y y v C x u C x strain plane and stress plane ) 1 ( 2 , strain plane ) 2 - )(1 (1 E stress plane 1 strain plane ) 2 - )(1 (1 ) - E(1 stress plane 1 66 2 12 2 22 11 ν + = μ = ν ν + ν ν - ν = ν ν + ν ν - = = E C E C E C C Material Isotropic xy xy y x y y x x e C e C e C e C e C 66 22 12 12 11 = τ + = σ + = σ Law s Hooke'
Two Dimensional Elasticity Weak Form 0 0 22 12 66 2 66 12 11 1 = + + + + = + + + + dxdy F y v C x u C y x v y u C x w h dxdy F x v y u C y y v C x u C x w h e e y e x e Mulitply Each Field Equation by Test Function & Integrate Over Element Use Divergence Theorem to Trade Differentiation On To Test Function y x y y x x y e y e e x e x e e n y v C x u C n x v y u C T n x v y u C n y v C x u C T ds T w h dxdy F w h dxdy y v C x u C y w x v y u C x w h ds T w h dxdy F w h dxdy x v y u C y w y v C x u C x w h e e e e e e + + + = + + + = + = + + + + = + + + Γ Γ 22 11 66 66 12 11 2 2 22 12 2 66 2 1 1 66 1 12 11 1 , (constant) ickness element th = e h

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Two Dimensional Elasticity Ritz-Galerkin Method Γ Γ ψ + ψ = ψ + ψ = ψ ψ + ψ ψ = ψ ψ + ψ ψ = = ψ ψ + ψ ψ = = = e e e e e e e ds T h dxdy F h F ds T h dxdy F h F dxdy y y C x x C h K dxdy
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