Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1999 Yijun Liu, University of Cincinnati
144
III Typical 3-D Solid Elements
Tetrahedron:
Hexahedron (brick):
Penta:
Avoid using the linear (4-node) tetrahedron element in 3-D
stress analysis (Inaccurate! But it is OK for dynamic analysis).
linear (4 nodes) quadratic (10 nodes)
linear (8 nodes) quadratic (20 nodes)
linear (6 nodes) quadratic (15 nodes)
Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1999 Yijun Liu, University of Cincinnati
145
Element Formulation:
Linear Hexahedron Element
Displacement field in the element:
)
11
(
,
,
8
1
8
1
1
8
1
∑
∑
∑
=
=
=
=
=
=
i
i
i
i
i
i
i
i
i
w
N
w
v
N
v
u
N
u
6
5
y 8
7
2
1
4
3
mapping (x
↔ξ
)
x (-1
≤
ξ
,
η
,
ζ
≤
1)
z
η
(-1,1,-1) 4 3 (1,1,-1)
(-1,1,1) 8
7 (1,1,1)
o
ξ
(-1,-1,-1) 1
2 (1,-1,-1)
(-1,-1,1) 5
6 (1,-1,1)
ζ
Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1999 Yijun Liu, University of Cincinnati
146
Shape functions:
.
)
1
(
)
1
(
)
1
(
8
1
)
,
,
(
)
12
(
,
)
1
(
)
1
(
)
1
(
8
1
)
,
,
(
,
)
1
(
)
1
(
)
1
(
8
1
)
,
,
(
,
)
1
(
)
1
(
)
1
(
8
1
)
,
,
(
8
3
2
1
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
+
+
−
=
−
+
+
=
−
−
+
=
−
−
−
=
N
N
N
N
M
M
Note that we have the following relations for the shape
functions:
.
1
)
,
,
(
.
8
,
,
2
,
1
,
,
)
,
,
(
8
1
∑
=
=
=
=
i
i
ij
j
j
j
i
N
j
i
N
ζ
η
ξ
δ
ζ
η
ξ
L
Coordinate Transformation (Mapping):
)
13
(
.
,
,
8
1
8
1
8
1
∑
∑
∑
=
=
=
=
=
=
i
i
i
i
i
i
i
i
i
z
N
z
y
N
y
x
N
x
The same shape functions are used as for the displacement
field.
⇒
Isoparametric element.
Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1999 Yijun Liu, University of Cincinnati
147
Jacobian Matrix:
matrix
Jacobian
z
u
y
u
x
u
z
y
x
z
y
x
z
y
x
u
u
u
J
≡
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
)
14
(
ζ
ζ
ζ
η
η
η
ξ
ξ
ξ
ζ
η
ξ
⇒
∂
∂
=
∂
∂
∂
∂
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
∑
=
−
.
,
,
8
1
1
etc
u
N
u
u
u
u
z
u
y
u
x
u
i
i
i
ξ
ξ
ζ
η
ξ
J
and
)
15
(
,
1
∂
∂
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
−
ζ
η
ξ
v
v
v
z
v
y
v
x
v
J
also for w.
Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1999 Yijun Liu, University of Cincinnati
148
⇒
where d is the nodal displacement vector,
i.e.,
)
16
(
d
B
å
=
(6
×
1) (6
×
24)
×
(24
×
1)
d
B
å
=
=
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
∂
∂
∂
∂
=
=
)
15
(
use
zx
yz
xy
z
y
x
x
w
z
u
z
v
y
w
y
u
x
x
z
w
y
v
x
u
L
γ
γ
γ
ε
ε
ε
Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1999 Yijun Liu, University of Cincinnati
149
Strain energy,
)
17
(
2
1
2
1
)
(
2
1
2
1
d
B
E
B
d
å
E
å
å
E
å
å
ó
=
=
=
=
∫
∫
∫
∫
V
T
T
V
T
V
T
V
T
dV
dV
dV
dV
U
Element stiffness matrix,
)
18
(
∫
=
V
T
dV
B
E
B
k
(24
×
24) (24
×
6)
×
(6
×
6)
×
(6
×
24)
In
ξηζ
coordinates:
)
19
(
)
det
(
ζ
η
ξ
d
d
d
dV
J
=
⇒
)
20
(
)
(det
1
1
1
1
1
1
∫ ∫ ∫
− − −
=
ζ
η
ξ
d
d
d
T
J
B
E
B
k
( Numerical integration)
•
3-D elements usually do not use rotational DOFs.
Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1999 Yijun Liu, University of Cincinnati
150
Loads:
Distributed loads
⇒
Nodal forces
Area =A Nodal forces for 20-node
Hexahedron
Stresses:
d
B
E
å
E
ó
=
=
Principal stresses:
.
,
,
3
2
1
σ
σ
σ
von Mises stress:
2
1
3
2
3
2
2
2
1
)
(
)
(
)
(
2
1
σ
σ
σ
σ
σ
σ
σ
σ
−
+
−
+
−
=
=
VM
e
.
Stresses are evaluated at selected points (including nodes)
on each element. Averaging (around a node, for example) may
be employed to smooth the field.
Examples: …
pA/3 pA/12
p