Lecture Notes: Introduction to Finite Element Method |
Chapter 6. Solid Elements for 3-D Problems |
II. Finite Element Formulation
Displacement Field:
N
u = ∑N i ui
i =1 |
|
N |
|
v = ∑N i vi |
(8) |
i =1
N
w= ∑N i wi
i=1
Nodal values
In matrix form:
uv
w (3×1)
or
N1 = 00
0 |
0 |
N2 |
0 |
0 |
N1 |
0 |
0 |
N2 |
0 |
0 |
N1 |
0 |
0 |
N2 |
u =N d
L
L
L (3×3N )
u1v1w1u2v2w2M (3N
(9)
×1)
Using relations (5) and (8), we can derive the strain vector
ε =B d
(6×1) (6×3N)×(3N×1)
© 1997-2002 Yijun Liu, University of Cincinnati |
142 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 6. Solid Elements for 3-D Problems |
Stiffness Matrix: |
|
k = ∫BT EB dv |
(10) |
v
(3×N) (3N×6)×(6×6)×(6×3N)
Numerical quadratures are often needed to evaluate the above integration.
Rigid-body motions for 3-D bodies (6 components): 3 translations, 3 rotations.
These rigid-body motions (causes of singularity of the system of equations) must be removed from the FEA model to ensure the quality of the analysis.
© 1997-2002 Yijun Liu, University of Cincinnati |
143 |