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  • ME5180/6900FiniteElementAnalysis

    Chapter15

    ThermalStressByAustinScheyer

    12/1/2016

  • Overview

    MotivationHeattransferreviewModeling Formulatethethermalstressproblem Derivetheforcematrix

    Onedimensional barelement Twodimensionplanestressandplanestrainelements

    Exampleproblem ANSYS

  • Motivation

    Thermalstressescanoccurinstructuresfortworeasons

    Restrictedmovement Differentcoefficientofthermalexpansion

  • HeatTransferReview

    FouriersLaw Conductiveheatflux

    " = )

    Newtonlawofcooling Conductiveheatflux

    " = + ,

  • ThermalStrain

    Thermalexpansion,.

    Coefficientofthermalexpansion(1/C) T Uniformchangeintemperature(C) L Originallength(m)

    Strain,.:

    T TL =L

    L .

    TT TL

    = =

    1

  • MechanicsofMaterialL .

    F Restorativeforce(N) E ModulusofElasticity A Crosssectionalarea

    Setthe

    Solvefortheforcegives:

    ThermalStress:

    RFLAE

    =

    R T = ETL FL

    A =

    F EAT=

    TF TA

    E = =

    L

  • OneElementBar

    IsotropicmaterialUniformtemperaturechange,T

    Force

    Where:

    Thus:

    { } { }TT TL

    = =

    { } [ ] [ ]{ }V

    TT Tf B D dV=

    [ ] 1 1BL L

    = [ ] [ ]D E=

    { }TE

    fETATA

    =

  • 1D ThermalStress

    Step1:Determinetheelementalcomponents Appliedthermalloading

    Stiffnessmatrix

    1

    B

    L

    2 3

    1 2

    { }(1) Ef ETATA

    =

    { }(2) Ef ETATA

    =

    { }(1) 1 11 12EAkL

    =

    { }(2) 1 11 12EAkL

    =

  • 1D ThermalStress

    Step2:ConstructtheglobalstiffnessmatrixWeknowthat

    Thus

    ApplyingtheboundaryconditionsUsingtheactivestiffnesstosolvefortheremainingdisplacements,thus

    1

    B

    L

    2 3

    1 2

    0{ } [ ]{ }F K d=

    1

    2

    3

    1 1 00 1 2 1

    20 1 1

    E uEA

    TAu

    LE uTA

    =

    1 30, 0u u= =

    2 0u =

  • 1D ThermalStress1

    B

    L

    2 3

    1 2

    1 1

    2 2

    3 3

    1 1 01 2 1 0 0

    20 1 1

    x

    x

    x

    F u E EEAF uL

    F

    TA TA

    TAu E TE A

    = =

    { } 0[ ]{ } { }F K d F=

    Step3:Solveforactualnodalforces Backsubstitutethedisplacementsintotheglobalstiffnessmatrix

    For: E=200GPA A=24cm2 L=1.2m =12.5x10-6 (mm/mm)/C

    1

    2

    3

    1800180

    x

    x

    x

    FF kNF

    =

  • ConstantStrainTriangle(CST)

    Theareaofthetriangleis: = 3+ 5 + A7

    Assumeatemperaturefield:3 + 5+ 7Where) isconstant

    Theshapefunctionsaredefinedas:3 , =

    3 5 , =

    5 7 , =

    7

    Shapefunctionmatrix = 3 5 7

    , = []357

    35

    7

    3

    3 5

    57

    7

  • ConstantStrainTriangle(CST)Temperaturegradient

    Heatfluxvector

    So

    11

    231

    32

    23

    NN NT Tx x x x TT NN N

    Ty y y y

    =

    [ ]B

    { }"

    "

    0"

    0x xx

    y yy

    TKq xq

    TKqy

    = =

    { }" [ ][ ]{ }q D B T=

    { }T

    [ ]D

    3

    3 5

    57

    7

  • ConstantStrainTriangle(CST)

    Applyingenergyprinciple

    Theequivalentforcevector

    [ ] [ ] [ ][ ] [ ] [ ]V

    TT

    S

    TK B D B dV h N N dS= +

    { } [ ] [ ] " [ ]V S S

    T T Tf N QdV N q dS N hT dS= + +

    { } Heat sourceQf =

    " 2{ } Heat flux on surface Sqf =3{ } Convection off surface Shf =

    { } { }[ ]TK T f=

  • Planestressandplanestrain

    PlaneStress

    PlaneStrain

    {0

    TT

    } =

    ( ){ 10

    TT

    } = +

  • ANSYS CircularPipe

    Given: Aluminum1100pipe E=69GPa r1=0.2m r2=1.0m =24x10^-6 K=177W/m*K Thickness=0.1m

    5

    3

    5

    3

  • ANSYS CircularPipe

    32Elements

    5

    3

    5

    3

    Mesh

    128Elements 512Elements

  • TemperatureDistribution

    Procedure DefineGeometry Definematerialproperties Defineelementtype(thermalsolidquad4node55)

    Specifyboundarycondition

  • ThermalStress

    Switchtheelementtypefromthermalsolidtostructuralsolid Redefineboundaryconditions Fromthermalanalysis

    32Elements 128Elements 512Elements

  • Refernces

    Logan, Daryl L. A first course in the finite element method. Cengage Learning, 2011.

    Chris Wilsons Notes Parsons,R.,etal"INVESTIGATIONOFTHEUSEOFTHEJAVAPROGRAMMINGLANGUAGEFORWEB-BASEDFINITEELEMENTMODELING"

  • Questions