机械工程中常用的数值分析方法_3

Stiffness matrix
(6*6)
(6*3)(3*3)(3*6)
All the Equations
Stiffness matrix 1,2 6*6 6*6 3,4 7,8 5,6
K11 K12 K13 K14 K15 K16 K11 K12 K15 K16 K17 K18
K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36
Displacement interpolation
Displacement interpolation (7,6)
(0,0)
(8,-3)
From displacement to strain
The Geometry Equation
(3*1) (3*6) (6*1)
The Geometry Matrix
K21 K22 K25 K26 K27 K28 K51 K52 K55 K56 K57 K58 K61 K62 K65 K66 K67 K68 K71 K72 K75 K76 K77 K78 K81 K82 K85 K86 K87 K88
K41 K42 K43 K44 K45 K46 K51 K52 K53 K54 K55 K56 K61 K62 K63 K64 K65 K66
Stiffness matrix 1,2 6*6 3,4 6*6
7,8
5,6
K11 K12 K13 K14 K15 K16 K17 K18 0 0 0 0 K21 K22 K23 K24 K25 K26 K27 K28 K31 K32 K33 K34 K35 K36 K41 K42 K43 K44 K45 K46
K21 K22 K23 K24 K25 K26 K27 K28 K31 K32 K33 K34 K35 K36 K41 K42 K43 K44 K45 K46
K51 K52 K53 K54 K55 K56 K57 K58 K61 K62 K63 K64 K65 K66 K67 K68 K71 K72 K81 K82 0 0 0 0 K75 K76 K77 K78 K85 K86 K87 K88
From strain to stress
轾 犏 1 n 犏 E 犏 D = ?犏 n 1 1 - n2 犏 犏 1- n 犏 犏 臌 2
The Physical Equation
(3*1) (3*3)(3*6)(6*1)
The Elastic matrix (Hooke Matrix)
Balance of external work and internal energy Balance of 1) the work of external forces and 2) internal stored elastic energy.
K51 K52 K53 K54 K55 K56 K57 K58 K61 K62 K63 K64 K65 K66 K67 K68 K71 K72 K81 K82 0 0 0 0 K75 K76 K77 K78 K85 K86 K87 K88
1/ From nodal displacement vector to displacement field: Interpolation
2/ From displacement field to strain field: Geometry Equation
3/ From strain field to stress field: Physical Equation 4/ From strain and stress to elastic energy 5/ From elastic energy to element stiffness matrix
Finite Element & Numerical Optimization in Mechanical Engineering
By Profs. Ji-Hong ZHU & Tong Gao
Episode 6
Balance of Solid Element
Solid structures
Solid structures
Balance of global element
Every point is balanced.
Displacement interpolation
Displacement Vector on Each Point (x, y)
(2*1)
(2*6) (6*1)
Interpolation
Shape Function
Finite element of solid structures
A triangle solid element and its three nodes
Problem of Solid Spring Bar
Beam
Solid
Problem of Solid
Solid
(6*1) (6*1)
We need more equations from Elastic Mechanics
Stiffness matrix
K11 K12 K13 K14 K15 K16
Kwenku.baidu.com1
K12 K15 K16 K17 K18
K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36 K41 K42 K43 K44 K45 K46 K51 K52 K53 K54 K55 K56 K61 K62 K63 K64 K65 K66
K21 K22 K25 K26 K27 K28 K51 K52 K55 K56 K57 K58 K61 K62 K65 K66 K67 K68 K71 K72 K75 K76 K77 K78 K81 K82 K85 K86 K87 K88
K11
K12 K13 K14 K15 K16 K17 K18 0 0 0 0