Civil Engineering Reference
In-Depth Information
so the body forces are allocated equally to each node. For the present example, the result is
f
r
1
=
f
r
2
=
f
r
3
=
0
.
88 lb
f
z
1
=
f
z
2
=
f
z
3
=−
1
.
03 lb
Note that, within the numerical accuracy used here, the total radial force and the total
axial force are the same for the two methods.
9.6 GENERAL THREE-DIMENSIONAL
STRESS ELEMENTS
While the conditions of plane stress, plane strain, and axisymmetry are frequently
encountered, more often than not the geometry of a structure and the applied loads
are such that a general three-dimensional state of stress exists. In the general case,
there are three displacement components
u
,
v
, and
w
in the directions of the
x
,
y
,
and
z
axes, respectively, and six strain components given by (Appendix B)
∂
u
∂
x
∂
v
∂
y
ε
x
ε
y
ε
z
xy
xz
yz
∂
w
∂
z
{
ε
} =
=
(9.103)
∂
u
y
+
∂
v
∂
∂
x
∂
u
z
+
∂
w
∂
∂
x
∂
v
z
+
∂
w
∂
∂
y
For convenience of presentation, the strain-displacement relations of Equa-
tion 9.103 can be expressed as
∂
∂
00
x
∂
∂
0
0
y
00
∂
∂
u
v
w
[
L
]
u
z
{
ε
}=
=
v
w
(9.104)
∂
∂
∂
∂
0
y
x
∂
∂
∂
∂
0
z
x
∂
∂
∂
∂
0
z
y
and matrix
[
L
]
is the
6
×
3
matrix of derivative operators.
