Civil Engineering Reference
In-Depth Information
versus the deformed arc length. Prior to deformation, the arc length is
d
s
=
r
d
,
while after deformation, arc length is
d
s
=
(
r
+
u
)d
.
The tangential strain is
(
r
+
u
)(d
)
−
r
d
u
r
ε
=
=
(9.90)
r
d
and we observe that, even though the problem is independent of the tangential
coordinate, the tangential strain must be considered in the problem formulation.
Note that, if
r
=
0, the preceding expression for the tangential strain is trouble-
some mathematically, since division by zero is indicated. The situation occurs,
for example, if we examine stresses in a rotating solid body, in which case the
stresses are induced by centrifugal force (normal acceleration). Additional dis-
cussion of this problem is included later when we discuss element formulation.
Additionally, the shear strain components are
∂
u
z
+
∂
w
rz
=
∂
∂
r
(9.91)
r
=
0
z
=
0
If we substitute the strain components into the generalized stress-strain relations
of Appendix B (and, in this case, we utilize
=
y
), we obtain
E
r
=
)
[(1
−
)
ε
r
+
(
ε
+
ε
z
)]
(1
+
)(1
−
2
E
=
)
[(1
−
)
ε
+
(
ε
r
+
ε
z
)]
(1
+
)(1
−
2
(9.92)
E
z
=
)
[(1
−
)
ε
z
+
(
ε
r
+
ε
)]
(1
+
)(1
−
2
E
rz
=
)
rz
=
G
rz
2(1
+
For convenience in finite element development, Equation 9.92 is expressed in
matrix form as
−
1
0
r
z
rz
ε
r
ε
ε
z
ez
−
1
0
E
=
1
−
0
(9.93)
(1
+
)(1
−
2
)
1
−
2
0
0
0
2
in which we identify the material property matrix for axisymmetric elasticity as
1
−
0
1
−
0
E
[
D
]
=
1
−
0
(9.94)
(1
+
)(1
−
2
)
1
−
2
0
0
0
2
