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

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Unformatted text preview: MCE 561 Computational Methods in Solid Mechanics One-Dimensional Problems 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 + = Virtual Strain Energy = Virtual Work Done by Surface and Body Forces For One-Dimensional Case + + = 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 + + = 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 = = = = = = = = + = L T T j i T L T T fdx A P P dx E A ] [ } { } { } { ] [ ] [ } { N d d d B B d + = L T L T fdx A dx E A ] [ } { } { ] [ ] [ N P d B B Vector ent Displacem Nodal } { Vector Loading ] [ } { Matrix Stiffness ] [ ] [ ] [ = = = + = = = 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 = 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 = = = - = = + = + - =- + = + = = + = 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) 1 ( x ) 2 ( x ) 1 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 2 1 2 1 = = + = - + = + = -- = - - = = = 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...
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New Lec2 - MCE 561 Computational Methods in Solid Mechanics...

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