机械工程中常用的数值分析方法-3

1、Finite Element & Numerical Optimization in Mechanical EngineeringByProfs. Ji-Hong ZHU & Tong Gao1Episode 6Balance of Solid Element2Solid structures3Solid structures4Finite element of solid structuresA triangle solid element and its three nodes5Problem of SolidSpringBarBeamSolid6Problem of SolidSolid

2、We need more equations from Elastic Mechanics(6*1)(6*1)7Balance of global elementEvery point is balanced.8Displacement interpolationDisplacement Vector on Each Point (x, y)InterpolationShape Function(2*1)(2*6)(6*1)9Displacement interpolation10Displacement interpolation(0,0)(8,-3)(7,6)11From displace

3、ment to strainThe Geometry EquationThe Geometry Matrix(3*1)(3*6)(6*1)12From strain to stressThe Physical EquationThe Elastic matrix (Hooke Matrix)(3*3)(3*6)(6*1)(3*1)13Balance of external work and internal energyBalance of 1) the work of external forces and 2) internal stored elastic energy.14Stiffn

4、ess matrix(6*6)(6*3)(3*3)(3*6)15All the Equations16Stiffness matrix1,23,45,67,86*66*6K11K12K15K16K17K18K21K22K25K26K27K28K51K52K55K56K57K58K61K62K65K66K67K68K71K72K75K76K77K78K81K82K85K86K87K88K11K12K13K14K15K16K21K22K23K24K25K26K31K32K33K34K35K36K41K42K43K44K45K46K51K52K53K54K55K56K61K62K63K64K65K6

5、61002/3-1-2/301/32/30-2/3-1/302/34/30-4/3-2/32/3004-2/3-4-1-2/3-4/3-2/37/34/3-2/3-1/3-2/3-44/313/31,23,45,67,8211002/3-1-2/301/32/30-2/3-1/302/34/30-4/3-2/32/3004-2/3-4-1-2/3-4/3-2/37/34/3-2/3-1/3-2/3-44/313/317Stiffness matrixK11K12K15K16K17K18K21K22K25K26K27K28K51K52K55K56K57K58K61K62K65K66K67K68K

6、71K72K75K76K77K78K81K82K85K86K87K88K11K12K13K14K15K16K21K22K23K24K25K26K31K32K33K34K35K36K41K42K43K44K45K46K51K52K53K54K55K56K61K62K63K64K65K66K11K12K13K14K15K16K17K18K21K22K23K24K25K26K27K28K31K32K33K34K35K3600K41K42K43K44K45K4600K51K52K53K54K55K56K57K58K61K62K63K64K65K66K67K68K71K7200K75K76K77K78K

7、81K8200K85K86K87K8818Stiffness matrixK11K12K13K14K15K16K17K18K21K22K23K24K25K26K27K28K31K32K33K34K35K3600K41K42K43K44K45K4600K51K52K53K54K55K56K57K58K61K62K63K64K65K66K67K68K71K7200K75K76K77K78K81K8200K85K86K87K881,23,45,67,86*66*6191/ From nodal displacement vector to displacement field: Interpolation2/ From displacement field to strain field: Geometry Equation3/ From strain field to stress fiel