FEAElasticity

FEAElasticity - Procedures of Finite Element Analysis...

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Unformatted text preview: Procedures of Finite Element Analysis Two-Dimensional Elasticity Problems Professor M. H. Sadd Two Dimensional Elasticity Element Equation Orthotropic Plane Strain/Stress Derivation Using Weak Form – Ritz/Galerin Scheme Displacement Formulation Orthotropic Case 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 22 12 66 2 66 12 11 1 = + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ = + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ ∫ ∫ Ω Ω 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 Ω Ω Γ Ω Ω 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 Two Dimensional Elasticity Ritz-Galerkin Method ∫ ∫ ∫ ∫ ∫ ∫ ∫ Γ Ω Γ Ω Ω Ω Ω ψ + ψ = ψ + ψ = ∂ ψ ∂ ∂ ψ ∂ + ∂ ψ ∂ ∂ ψ ∂ = ∂ ψ ∂ ∂ ψ ∂ + ∂ ψ ∂ ∂ ψ ∂ = = ∂ ψ ∂...
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This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.

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FEAElasticity - Procedures of Finite Element Analysis...

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