{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
MCE 561 Computational Methods in Solid Mechanics One-Dimensional Problems
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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)
Background image of page 2
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 =
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}