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MEM427_Lecture_5_Two-D_Elements

# MEM427_Lecture_5_Two-D_Elements - MEM427 Introduction to...

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Unformatted text preview: MEM427 Introduction to Finite Element Method Chapter 5 Two-Dimensional Elements; Numerical Integration 1 Lecture 5 Two-Dimensional Elements Global, Local, and Natural Coordinates Isoparametric Elements Numerical Integration Convergence of Solution MEM427 Introduction to Finite Element Method Chapter 5 Two-Dimensional Elements; Numerical Integration 2 { } [ ] {} { } [ ] {} { } [ ][ ] {} { } [ ] {} [ ] [ ] [ ][ ] ∫ = = = = = V T dV B D B k q k f q B D q B q S u ; ; ; ; σ ε General Procedure of Finite Element Formulation { } { } { } or or or D- 3 D- 2 D- 1 1 =- w v u v u v u u D {} D- 2 Beam Truss or or = j j i i j j i i j i v u v u v v u u q θ θ { } { } D- 3 D- 2 1 or or =- yz x xy y x D x γ ε γ ε ε ε ε { } { } D- 3 D- 2 1 or or =- zx x xy y x D x τ σ τ σ σ σ σ [ ] [ ] [ ] etc. , , , i.e., , properties Material : functions shape of s Derivative : functions Shape : G E D B S ν Displacement vector D.O.F. vector Strain vector Stress vector MEM427 Introduction to Finite Element Method Chapter 5 Two-Dimensional Elements; Numerical Integration 3 { } [ ] {} { } [ ] {} { } [ ][ ] {} { } [ ] {} [ ] [ ] [ ][ ] ∫ = = = = = V T dV B D B k q k f q B D q B q S u ; ; ; ; σ ε General Procedure of Finite Element Formulation l E, A x y i f j f i u j u l E, I x y 1 1 , M θ 2 2 , M θ 2 2 , V v 1 1 , V v &#129; &#130; ( 29 [ ] {} q S x c c x u = + = 2 1 ( 29 [ ] {} q S x c x c x c c x v = + + + = 3 4 2 3 2 1 {} { } j i T u u q = [ ] [ ] - = = l x l x S S S 1 2 1 [ ] [ ] - = ′ ′ = l l S S B 1 1 2 1 { } { } j i T f f f = [ ] [ ] [ ][ ] -- = = ∫ 1 1 1 1 l AE dV B D B k V T [ ] [ ] - +- +- +- = = 3 2 3 2 2 2 3 2 2 2 3 3 3 2 3 4 3 2 1 2 2 3 2 l l x x l xl l x x l xl l x x l l l x x S S S S S [ ] - +--- = 2 3 2 3 2 6 6 12 4 6 6 12 l l x l l x l l x l l x B [ ] [ ] [ ] ------ = = ∫ 2 2 2 2 3 4 6 2 6 6 12 6 12 2 6 4 6 6 12 6 12 l l l l l l l l l l l l l EI dx B B EI k l T {} { } 2 2 1 1 θ θ v v q T = { } { } 2 2 1 1 M V M V f T = Recall Truss Element Recall Beam Element MEM427 Introduction to Finite Element Method Chapter 5 Two-Dimensional Elements; Numerical Integration 4 σ σ Equilibrium for One-Dimensional State of Stress θ σ θ τ σ A Uniform State of Stress A Non-uniform State of Stress = + ⇒ = ∑ x x...
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MEM427_Lecture_5_Two-D_Elements - MEM427 Introduction to...

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