机械工程中常用的数值分析方法_ 2
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0 0 0
0 0 0
0 0 0
K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36
K41 K42 K43 K44 K45 K46 K47 K48 K49 K51 K52 K53 K54 K55 K56 K57 K58 K59 K61 K62 K63 K64 K65 K66 K67 K68 K69 0 0 0 0 0 0 0 0 0 K74 K75 K76 K77 K78 K79 K84 K85 K86 K87 K88 K89 K94 K95 K96 K97 K98 K99
K41 K42 K43 K44 K45 K46 K47 K48 K49 K51 K52 K53 K54 K55 K56 K57 K58 K59 K61 K62 K63 K64 K65 K66 K67 K68 K69 0 0 0 0 0 0 0 0 0 K74 K75 K76 K77 K78 K79 K84 K85 K86 K87 K88 K89 K94 K95 K96 K97 K98 K99
Local and Global Coordinate
4,5,6
Local Coordinate
Global Coordinate
1,2,3
y*
x*
Local and Global Coordinate
y*
x*
Local and Global Coordinate
y* x*
Local and Global Coordinate
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ux1 u ? y1 ? θxy1 ? ux2 ? uy2 ? θxy2
Stiffness matrix of beam (6DOFs)
More Problems
7,8,9
4,5,6
?
1,2,3
4,5,6
1,2,3
Episode 5
Coordinate Transformation
Local and Global Coordinate
4,5,6
1,2,3
y* x*
Local Coordinate
Global Coordinate
Stiffness matrix of beam (9DOFs)
4,5,6,7,8,9 1,2,3,4,5,6
7,8,9
4,5,6
1,2,3
K11
K12 K13 K14 K15 K16
0 0 0
0 0 0
0 0 0
K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36
Stiffness matrix of beam (6DOFs)
Stiffness matrix of beam (6DOFs)
Stiffness matrix of beam (9DOFs)
4,5,6,7,8,9 1,2,3,4,5,6
7,8,9
4,5,6
1,2,3
Kห้องสมุดไป่ตู้1
K12 K13 K14 K15 K16
Local and Global Coordinate
Local and Global Coordinate
7,8,9
4,5,6
K11 1,2,3
K12 K13 K14 K15 K16
0 0 0
0 0 0
0 0 0
K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36
Items
Items
A unit displacement on 1st DOF generates the reaction forces.
Items
A unit displacement on 2nd DOF generates the reaction forces.
Items
A unit displacement on jth DOF generates the reaction force on ith DOF. An easy way to obtain stiffness matrix
Episode 4
FEM of Beams
Why beams
Spring
Bar
Beam
Stiffness matrix of beam (6DOFs)
f x1 ? f y1 ? mxy1 ? = f x2 ? f y2 ? mxy2 ?
1/ Stiffness Matrix: Singular, Symmetry, Items 2/ Stiffness Matrix of a Beam Element 3/ Coordinate Transformation of Stiffness Matrix
f x1 ? f y1 ? mxy1 ? = f x2 ? f y2 ? mxy2 ?
? f y1 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? 0 ? 1 ? 0 ? 0 ? 0 ? 0
Items
Symmetry
Symmetry
Symmetry
A unit displacement on jth DOF generates a reaction force on ith DOF
A unit displacement on ith DOF generates a reaction force on jth DOF
K41 K42 K43 K44 K45 K46 K47 K48 K49 K51 K52 K53 K54 K55 K56 K57 K58 K59 K61 K62 K63 K64 K65 K66 K67 K68 K69 0 0 0 0 0 0 0 0 0 K74 K75 K76 K77 K78 K79 K84 K85 K86 K87 K88 K89 K94 K95 K96 K97 K98 K99
Finite Element & Numerical Optimization in Mechanical Engineering
By Profs. Ji-Hong ZHU & Tong Gao
Episode 3
Properties of Stiffness Matrix
Singularity
Rigid Body Motion
Local and Global Coordinate
7,8,9
4,5,6
K11 1,2,3
K12 K13 K14 K15 K16
0 0 0
0 0 0
0 0 0
K21 K22 K23 K24 K25 K26 K31 K32 K33 K34 K35 K36
K41 K42 K43 K44 K45 K46 K47 K48 K49 K51 K52 K53 K54 K55 K56 K57 K58 K59 K61 K62 K63 K64 K65 K66 K67 K68 K69 0 0 0 0 0 0 0 0 0 K74 K75 K76 K77 K78 K79 K84 K85 K86 K87 K88 K89 K94 K95 K96 K97 K98 K99