机械工程中常用的数值分析方法_3

FiniteElement&NumericalOptimizationinMechanicalEngineeringByProfs.Ji-HongZHU&TongGaoEpisode6BalanceofSolidElementSolidstructures𝒇𝒇SolidstructuresFiniteelementofsolidstructuresAtrianglesolidelementanditsthreenodesProblemofSolid𝒇=𝒌𝒖𝒇=𝑬𝑨𝒍𝒖𝒇=𝟑𝑬𝑰𝒍𝟑𝒖SpringBarBeamSolid𝒇=?𝒖ProblemofSolidSolid𝒇=𝑲𝟔×𝟔𝒖WeneedmoreequationsfromElasticMechanics𝒖𝟏𝒙𝒖𝟏𝒚𝒖𝟐𝒚𝒖𝟐𝒙𝒖𝟑𝒚𝒖𝟑𝒙𝑲𝟔×𝟔?(6*1)(6*1)BalanceofglobalelementEverypointisbalanced.𝐹𝑥+𝑑𝑥+𝑇𝑦+𝑑𝑦−𝐹𝑥−𝑇𝑦+𝑋𝑑𝑥𝑑𝑦=0𝐹𝑦+𝑑𝑦+𝑇𝑥+𝑑𝑥−𝐹𝑦−𝑇𝑥+𝑌𝑑𝑥𝑑𝑦=0𝑇𝑥+𝑑𝑥⋅12𝑑𝑥+𝑇𝑥⋅12𝑑𝑥−𝑇𝑦+𝑑𝑦⋅12𝑑𝑦−𝑇𝑦⋅12𝑑𝑦=0𝐹𝑥+𝑑𝑥𝑇𝑦+𝑑𝑦𝐹𝑥𝑇𝑦𝐹𝑦+𝑑𝑦𝑇𝑥+𝑑𝑥𝐹𝑦𝑇𝑥𝑑𝑥𝑑𝑦Displacementinterpolation𝑼=𝑼𝑥,𝑦DisplacementVectoronEachPoint(x,y)𝑼𝑥,𝑦=𝑵𝑥,𝑦𝒖𝒖𝟏𝒙𝒖𝟏𝒚𝒖𝟐𝒚𝒖𝟐𝒙𝒖𝟑𝒚𝒖𝟑𝒙𝒖=𝑢1𝑥𝑢1𝑦𝑢2𝑥𝑢2𝑦𝑢3𝑥𝑢3𝑦InterpolationShapeFunction𝑵𝑥,𝑦(2*1)(2*6)(6*1)Displacementinterpolation𝒖𝟏𝒖𝟐𝒖𝒙𝒙=𝟎𝒙=𝟒𝒖𝒙=𝒂𝒙+𝒃𝒖𝟎=𝒂∙𝟎+𝒃=𝒖𝟏𝒖𝟒=𝒂∙𝟒+𝒃=𝒖𝟐𝒂=𝒖𝟐−𝒖𝟏𝟒𝒃=𝒖𝟏𝒖𝒙=𝒖𝟐−𝒖𝟏𝟒𝒙+𝒖𝟏=𝟏−𝒙𝟒𝒖𝟏+𝒙𝟒𝒖𝟐=𝟏−𝒙𝟒,𝒙𝟒𝒖𝟏𝒖𝟐=𝑵𝑥𝒖Displacementinterpolation𝒖𝟏𝒖𝟐𝒖𝒙(0,0)𝒖𝒙,𝒚=𝒂𝒙+𝒃𝒚+𝒄𝒖𝟏=𝒂∙𝟎+𝒃∙𝟎+𝒄𝒖𝟐=𝒂∙𝟖+𝒃∙(−𝟑)+𝒄𝒂=𝒖𝟑+𝟐𝒖𝟐−𝟑𝒖𝟏𝟐𝟑𝒃=𝟖𝒖𝟑+𝟕𝒖𝟐−𝒖𝟏𝟐𝟑𝒖𝒙,𝒚=𝑵𝒙,𝒚𝒖(8,-3)(7,6)𝒖𝟑𝒖𝟑=𝒂∙𝟕+𝒃∙𝟔+𝒄𝒄=𝒖𝟏Fromdisplacementtostrain𝐹𝑥+𝑑𝑥𝑇𝑦+𝑑𝑦𝐹𝑥𝑇𝑦𝐹𝑦+𝑑𝑦𝑇𝑥+𝑑𝑥𝐹𝑦𝑇𝑥𝑑𝑥𝑑𝑦TheGeometryEquation()()(),,,xxyyyxxyUxyxUxyyUUxyyxt¶=¶¶=¶¶¶=+抖ee𝜀𝑥𝑥,𝑦=𝜕𝑈𝑥𝜕𝑥𝜀𝑦𝑥,𝑦=𝜕𝑈𝑦𝜕𝑦𝜏𝑥𝑦𝑥,𝑦=𝜕𝑈𝑥𝜕𝑦+𝜕𝑈𝑦𝜕𝑥𝜺=𝜕𝑼𝜕𝒙=𝜕𝑵𝒙,𝒚𝒖𝜕𝒙=𝑩𝒖𝑩=𝜕𝑵𝒙,𝒚𝜕𝒙TheGeometryMatrix𝑩(3*1)(3*6)(6*1)Fromstraintostress𝐹𝑥+𝑑𝑥𝑇𝑦+𝑑𝑦𝐹𝑥𝑇𝑦𝐹𝑦+𝑑𝑦𝑇𝑥+𝑑𝑥𝐹𝑦𝑇𝑥𝑑𝑥𝑑𝑦ThePhysicalEquation𝜎=𝐸𝜀TheElasticmatrix(HookeMatrix)𝝈=𝑫𝜺=𝑫𝑩𝒖𝑫=𝑫𝑻(3*3)(3*6)(6*1)(3*1)211112EDnnnn轾犏犏犏=?犏犏-犏-犏犏臌𝑫=𝐸1−𝜈2⋅1𝜈0𝜈10001−𝜈2Balanceofexternalworkandinternalenergy𝒇,𝒖Balanceof1)theworkofexternalforcesand2)internalstoredelasticenergy.𝑊=12𝑓𝑖𝑢𝑖𝑖=12𝒇𝑇𝒖𝐶=12𝝈𝑇𝜺𝛺𝑑𝛺𝒇𝑇𝒖=𝝈𝑇𝜺𝛺𝑑𝛺=𝑫𝑩𝒖𝑇𝑩𝒖𝛺𝑑𝛺𝝈=𝑫𝜺=𝑫𝑩𝒖𝜺=𝑩𝒖=𝒖𝑇𝑩𝑇𝑫𝑩𝒖𝛺𝑑𝛺Stiffnessmatrix𝒇,𝒖𝒇𝑇𝒖=𝝈𝑇𝜺𝛺𝑑𝛺=𝑫𝑩𝒖𝑇𝑩𝒖𝛺𝑑𝛺=𝒖𝑇𝑩𝑇𝑫𝑩𝒖𝛺𝑑𝛺𝒇=𝑲𝟔×𝟔𝒖𝒇𝑇𝒖=𝒖𝑇𝑲𝒖=𝒖𝑇𝑩𝑇𝑫𝑩𝒖𝛺𝑑𝛺=𝒖𝑇𝑩𝑇𝑫𝑩𝛺𝑑𝛺𝒖𝑲𝟔×𝟔=𝑩𝑇𝑫𝑩𝛺𝑑𝛺(6*6)(6*3)(3*3)𝒇𝑇𝒖=𝒖𝑇𝑲𝒖=𝒖𝑇𝑩𝑇𝑫𝑩𝒖𝛺𝑑𝛺=𝒖𝑇𝑩𝑇𝑫𝑩𝛺𝑑𝛺𝒖(3*6)AlltheEquations𝑼𝑥,𝑦=𝑵𝑥,𝑦𝒖𝜺=𝜕𝑼𝜕𝒙=𝜕𝑵𝒙,𝒚𝒖𝜕𝒙=𝑩𝒖𝝈=𝑫𝜺=𝑫𝑩𝒖𝒇𝑇𝒖=𝒖𝑇𝑲𝒖=𝒖𝑇𝑩𝑇𝑫𝑩𝒖𝛺𝑑𝛺=𝒖𝑇𝑩𝑇𝑫𝑩𝛺𝑑𝛺𝒖𝑲𝟔×𝟔=𝑩𝑇𝑫𝑩𝛺𝑑𝛺𝒇=𝑲𝟔×𝟔𝒖𝒇,𝒖Stiffnessmatrix𝑲𝑲1,23,45,67,86*66*6K11K12K15K16K17K18K21K22K25K26K27K28K51K52K55K56K57K58K61K62K65K66K67K68K71K72K75K76K77K78K81K82K85K86K87K88K11K12K13K14K15K16K21K22K23K24K25K26K31K32K33K34K35K36K41K42K43K44K45K46K51K52K53K54K55K56K61K62K63K64K65K661002/3-1-2/301/32/30-2/3-1/302/34/30-4/3-2/32/3004-2/3-4-1-2/3-4/3-2/37/34/3-2/3-1/3-2/3-44/313/3𝑲𝑲1,23,45,67,8211002/3-1-2/301/32/30-2/3-1/302/34/30-4/3-2/32/3004-2/3-4-1-2/3-4/3-2/37/34/3-2/3-1/3-2/3-44/313/3StiffnessmatrixK11K12K15K16K17K18K21K22K25K26K27K28K51K52K55K56K57K58K61K62K65K66K67K68K71K72K75K76K77K78K81K82K85K86K87K88K11K12K13K14K15K16K21K22K23K24K25K26K31K32K33K34K35K36K41K42K43K44K45K46K51K52K53K54K55K56K61K62K63K64K65K66K11K12K13K14K15K16K17K18K21K22K23K24K25K26K27K28K31K32K33K34K35K3600K41K42K43K44K45K4600K51K52K53K54K55K56K57K58K61K62K63K64K65K66K67K68K71K7200K75K76K77K78K81K8200K85K86K87K88StiffnessmatrixK11K12K13K14K15K16K17K18K21K22K23K24K25K26K27K28K31K32K33K34K35K3600K41K42K43K44K45K4600K51K52K53K54K55K56K57K58K61K62K63K64K65K66K67K68K71K7200K75K76K77K78K81K8200K85K86K87K88𝑲𝑲1,23,45,67,86*66*61/Fromnodaldisplacementvectortodisplacementfield:Interpolation2/Fromdisplacementfieldtostrainfield:GeometryEquation3/Fromstrainfieldtostressfield:PhysicalEquation4/Fromstrainandstresstoelasticenergy5/Fromelasticenergytoelementstiffnessmatrix