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New Lec2

# New Lec2 - MCE 561 Computational Methods in Solid Mechanics...

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MCE 561 Computational Methods in Solid Mechanics One-Dimensional Problems

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One-Dimensional Bar Element A = Cross-sectional Area E = Elastic Modulus f(x) = Distributed Loading dV u F dS u T dV e i V i S i n i ij V ij t δ + δ = δ σ Virtual Strain Energy = Virtual Work Done by Surface and Body Forces For One-Dimensional Case δ + + = δ σ udV f u P u P edV j j i i ( i ) ( j ) Axial Deformation of an Elastic Bar Typical Bar Element dx du AE P i i - = dx du AE P j j = i u j u L x (Two Degrees of Freedom)
One-Dimensional Bar Element δ + + = δ σ udV f u P u P edV j j i i } ]{ [ : Law Strain - Stress } ]{ [ } { ] [ ) ( : Strain } { ] [ ) ( : ion Approximat d B d B d N d N E Ee dx d u x dx d dx du e u x u k k k k k k = = σ = = ψ = = = ψ = + = L T T j i T L T T fdx A P P dx E A 0 0 ] [ } { } { } { ] [ ] [ } { N δd δd d B B δd + = L T L T fdx A dx E A 0 0 ] [ } { } { ] [ ] [ N P d B B Vector ent Displacem Nodal } { Vector Loading ] [ } { Matrix Stiffness ] [ ] [ ] [ 0 0 = = = + = = = j i L T j i L T u u fdx A P P dx E A K d N F B B } { } ]{ [ F d K =

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Linear Approximation Scheme [ ] Vector ent Displacem Nodal } { Matrix Function ion Approximat ] [ } ]{ [ 1 ) ( ) ( 1 2 1 2 1 2 1 2 2 1 1 2 1 1 2 1 2 1 2 1 1 2 1 = = = - = ψ ψ = ψ + ψ = + - = - + = + = = + = d N d N nt Displaceme Elastic e Approximat u u L x L x u u u u x u x u L x u L x x L u u u u L a a u a u x a a u x (local coordinate system) (1) (2) i u j u L x (1) (2) u ( x ) x (1) (2) ψ 1 ( x ) ψ 2 ( x ) 1 ψ k ( x ) – Lagrange Interpolation Functions
Element Equation Linear Approximation Scheme, Constant Properties Vector nt Displaceme Nodal } { 1 1 2 1 ] [ } { 1 1 1 1 1 1 1 1 ] [ ] [ ] [ ] [ ] [ 2 1 2 1 0 2 1 0 2 1 0 0 = = + = - + = + = - - = - - = = = u u L Af P P dx L x L x Af P P fdx A P P L AE L L L L L AE dx AE dx E A K o L o L T L T L T d N F B B B B + = - - = 1 1 2 1 1 1 1 } { } ]{ [ 2 1 2 1 L Af P P u u L AE o F d K

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Quadratic Approximation Scheme [ ] } ]{ [ ) ( ) ( ) ( 4 2 3 2 1 3 2 1 3 3 2 2 1 1 2 3 2 1 3 2 3 2 1 2 1 1 2 3 2 1 d N nt Displaceme Elastic e Approximat = ψ ψ ψ = ψ + ψ + ψ = + + = + + = = + + = u u u u u x u x u x u L a L a a u L a L a a u a u x a x a a u x (1) (3) 1 u 3 u (2) 2 u L u ( x ) x (1) (3) (2) x (1) (3) (2) 1 ψ 1 ( x ) ψ
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