10. lect10-2013-stud, Budownictwo Politechnika, metody obliczeniowe, wykłady
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//-->Linear ElastostaticsFEA in 3DMałgorzata StojekCracow University of TechnologyApril 2013MS(L-53 CUT)FE Models04/20131 / 32Linear Elastostatics in 3D IdomainΩ=Vboundaryc Felippastresstensorσ∼∂ΩD=Su∂ΩN=Ststressvectorσ·n=σn∼surface tractionbtFE Models04/20132 / 32MS(L-53 CUT)Linear Elastostatics in 3D IIA linear-elastic body instatic equilibrium,b=f:component (indicial) form:σij,j+bi=c FelippaMS(L-53 CUT)FE Models04/20133 / 32compacttensorform:∇·σ+f=divσ+f=∼∼Linear Elastostatics in 3D IIIcomponent (indicial) formcompacttensorformStrain-Displacement (Cauchy) Equations:εij=12(ui,j+uj,i)∼ε=12∇·u+∇T·uConstitutive Equations:σij=Cijklεklσ=C·ε∼∼ ∼whereCif afourth order elasticity tensorsatisfying general symmetries∼Cijkl=Cjikl=CijlkCijkl=Cklij→admits strain energy36 differentelastic moduli21 differentelastic moduli→MS(L-53 CUT)FE Models04/20134 / 32Governing EquationsStrong FormField Equations:Internal Equilibrium EquationsStrain-Displacement EquationsConstitutive EquationsBoundary Conditions:kinematic (essential)static (natural)NOTE:we have 15 unknownsu,ε,σ∼ ∼MS(L-53 CUT)FE Models−divσ=f∼∼inΩinΩinΩε=12∇·u+∇T·uσ=C·ε∼∼ ∼σ·n=bt∼u=ubon∂ΩDon∂ΩN& 15 field equations04/20135 / 32Principle of Virtual WorkDefinitionscalar (inner) product of second-order tensors33σ:ε=ε:σ=∼∼∼∼∑ ∑σijεij=σijεiji=1j=1DefinitionWeak Form (W)ZΩ∼ε(w):σ(u)dΩ=∼ZΩw·fdΩ+I∂ΩNw·bdS,t∀w∈VUWAGA - PL: wirtualne przemieszczenia;dopuszczalne przemieszczenia; przemieszczenia zgodne z wiezami˛MS(L-53 CUT)FE Models04/20136 / 32Displacement Assumed ElementNodal displacement vectorn−node element, order 3n×1de=ux1uy1uz1ux2uy2uz2. . .uxnuynuznShape function matrixof order 3×3nN1eNe=N1eN1eN2eN2eN2e. . .Nne......NneNneTDisplacement interpolation over elementux(x, y,z)u(x, y,z) =uy(x, y,z)=Nedeuz(x, y,z)MS(L-53 CUT)FE Models04/20137 / 323D Finite Elements Indof=4×3=12c FelippaMS(L-53 CUT)ndof=6×3=18FE Modelsndof=8×3=2404/20138 / 323D Finite Elements IIc FelippaMS(L-53 CUT)FE Models04/20139 / 323D Finite Elements IIIc FelippaMS(L-53 CUT)FE Models04/201310 / 32
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