MTU MEEM 4405 - Linear Strain Energy Density Review Notes

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ReviewLinear Strain Energy DensityMinimum Potential EnergyRayleigh-Ritz’s methodFinite Element MethodFunction and Derivative Continuity (Hermite Polynomials)Steps in FEM procedureNatural Coordinates1-d Coordinates2-D Triangular elements (Area Coordinates)Numerical Integration (Gauss Quadrature)Iso-Parametric ElementsStorage and Solution TechniquesErrors in FEMFEM ConvergencePatch TestMesh RefinementThermal AnalysisHeat ConductionThermal Stresses2-D Steady State Thermal AnalysisHeat Conduction Boundary Conditions:Convection Boundary Conditions:Radiation Boundary Condition:M. Vable Notes for finite element method: Review1ReviewLinear Strain Energy DensityUo12---σxxεxxσyyεyyσzzεzzτxyγxyτyzγyzτzxγzx+++++[]12---σ{}˜Tε{}˜12---ε{}˜TE˜[]ε{}˜=== σ˜{}σxxσyyτxy= ε˜{}εxxεyyγxyB[]d{}==Axial Torsion of circular shafts Symmetric bending of beamsUa12---EAxddu2= Ut12---GJxddφ2=Ub12---EIzzx22dd v2=• Any variable that can be used for describing deformation is called the general-ized displacement• Any variable that can be used for describing the cause that produces deformation is called the generalized force.• Work is the product of generalized force and the corresponding generalized dis-placement.• Work done by a force is conservative if it is path independent.• Non-linear systems and non-conservative systems are two independent descrip-tion of a system.• Conditions specified on deflections (and slopes) at the boundary are called Kine-matic or Essential boundary conditions.• Conditions specified on the internal forces and moments at the boundary are called Statical or Natural boundary conditions.• Functions that are continuous and satisfies all the kinematic boundary conditions are called kinematically admissible functions. • The virtual displacement is an infinitesimal imaginary kinematically admissible displacement field imposed on a body.• Virtual work is the work done by the forces in moving through a virtual displace-ment.• The total virtual work done on a body at equilibrium is zero.M. Vable Notes for finite element method: Review2Minimum Potential Energy• Of all the kinematically admissible displacement functions the actual displacement function is the one that minimizes the potential energy function at stable equilibrium.Potential energy function: Ω UW–=where, U is the strain energy and W is the work potential of a force. Rayleigh-Ritz’s method • Rayleigh-Ritz method is formal procedure to approximately minimize potential energy function. ux() Cifix()i1=n∑=where, Ci are undetermined constants (generalized displacement) fi(x) are kinematically admissible functions that are independent and form a complete set.The necessary condition at the minimum value of Ω is: Ci∂∂Ω0= i1n,=KijCjj1=n∑Ri= i1n,= K[]C{} R{}= KjiKij=UAWA2---------ΩA2--------–12---CjRjj1=n∑== = at equilibriumAxial Torsion of circular shafts Symmetric bending of beamsux() Cifix()i1=n∑= φ x() Cifix()i1=n∑= vx() Cifix()i1=n∑=KjkEAxddfjxddfkxd0L∫=KjkGJxddfjxddfkxd0L∫=KjkEIzz()x22dd fjx22dd fkxd0L∫=Rjpxx()fjx() xd0L∫=Fqfjxq()q1=m∑+Rjtx()fjx() xd0L∫=Tqfjxq()q1=m∑+Rjpyx()fjx() xd0L∫=Fqfjxq()q1=m∑Mqxddfjxq()q1=m∑++M. Vable Notes for finite element method: Review3Finite Element Method• The kinematically admissible displacement functions in finite element method is defined piecewise continuous over small (finite) domains called the ‘elements’. • The constants multiplying the piecewise kinematically admissible functions are the displacements of the nodes. • The kinematically admissible functions are called ‘interpolation functions’ as these functions can be used to interpolate the values of displacements between the nodes. ux() uie()fix()i1=n∑= δΩ δΩe()e1=n∑=KGe()[]T[]TKe()[]T[]= FG1e(){} T[]TF1e(){}=δΩe()δuGe(){}TKGe()[]uGe(){}FG1e(){}–()=Function Continuity (Lagrange Polynomials)Displacement Continuity: Axial, Torsion, 2-D and 3-D ElasticityTemperature Continuity: Heat Transfer 1-D, 2-D, and 3-DLangrange Polynomials: fixj()1ij=0ij≠=Function and Derivative Continuity (Hermite Polynomials)Displacement and Slope Continuity: Beams, Plates, Shells.Steps in FEM procedure1. Obtain element stiffness and element load vector.2. Transform from local orientation to global orientation.3. Assemble the global stiffness matrix and load vector.4. Incorporate the external loads5. Incorporate the boundary conditions.6. Solve the algebraic equations for nodal displacements.7. Obtain reaction force, stress, internal forces, strain energy.8. Interpret and check the results.9. Refine mesh if necessary, and repeat the above steps.Natural Coordinates• Coordinates which vary between 0 and 1 or -1 and 1. • Natural coordinates and non-dimensional coordinates.M. Vable Notes for finite element method: Review41-d CoordinatesξNode 1Node 2ξ=1ξNode 1Node 2ξ=1ξ=−1Possibility 1Possibility 2L1ξ() 1 ξ–()= L2ξ() ξ= L1ξ() 1 ξ–()2⁄= L2ξ() 1 ξ+()2⁄=2-D Triangular elements (Area Coordinates)• Use Pascal’s triangle to determine the nodes needed for complete polynomialsAIIJKAJAKLIAIA------=LI=0LI=1LJAJA------=LKAKA--------=LILJLK++ 1=LKLJ1234ξηξηζ12345678Bi-linear Tri-linearNumerical Integration (Gauss Quadrature)IFξ()ξd1–1∫wiF ξi()i1=n∑== IFξη,()ξd()ηd1–1∫1–1∫wiwjF ξiηj,()i1=n∑j1=n∑==IFξη,()ξd()ηd1–1∫1–1∫1–1∫wiwjwkF ξiηjζk,,()i1=n∑j1=n∑k1=n∑==• Stresses are found at Gauss points.M. Vable Notes for finite element method: Review5Iso-Parametric Elements• Displacements and coordinates are approximated by same interpolation functions.uNiξη,()uie()i1=n∑= vNiξη,()vie()i1=n∑= xNiξη,()xii1=n∑= yNiξη,()yii1=n∑=Jacobian Matrix• Matrix relating differentials of two coordinate system.ξ∂∂ux∂∂uξ∂∂xy∂∂uξ∂∂y+=η∂∂ux∂∂uη∂∂xy∂∂uη∂∂y+=J[]ξ∂∂Nixii1=m∑ξ∂∂Niyii1=m∑η∂∂Nixii1=m∑η∂∂Niyii1=m∑=x∂∂uy∂∂uJ[]1–ξ∂∂uη∂∂u=Ke()[] B[]TE[]B[]txd()yd()∫∫B˜[]TE[]B˜[]tJ ξd()ηd()1–1∫1–1∫==Storage and Solution Techniques1. Banded Matrix• The bandwidth is solely dictated


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MTU MEEM 4405 - Linear Strain Energy Density Review Notes

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