Civil Engineering Reference
In-Depth Information
z
(
r
u
)d
r
d
u
r
u
d
r
u
d
z
r
d
r
d
r
(c)
(a)
(b)
Figure 9.9
(a) Cross section of an axisymmetric body. (b) Differential element in
an
rz
plane. (c) Differential element in an
r
- plane illustrating tangential
deformation. Dashed lines represent deformed positions.
3.
All boundary (constraint) conditions are symmetric about the
z
axis.
4.
Materials properties are also symmetric (automatically satisfied by a linearly
elastic, homogeneous, isotropic material).
If these conditions are satisfied, the displacement field is independent of the
tangential coordinate
,
and hence the stress analysis is mathematically two-
dimensional, even though the physical problem is three-dimensional. To develop
the axisymmetric equations, we examine Figure 9.9a, representing a solid of rev-
olution that satisfies the preceding requirements. Figure 9.9b is a differential
element of the body in the
rz
plane; that is, any section through the body for
which
is constant. We cannot ignore the tangential coordinate completely,
however, since as depicted in Figure 9.9c, there is strain in the tangential direc-
tion (recall the basic definition of hoop stress in thin-walled pressure vessels
from mechanics of materials). Note that, in the radial direction, the element
undergoes displacement, which introduces increase in circumference and associ-
ated circumferential strain.
We denote the radial displacement as
u
, the tangential (circumferential) dis-
placement as
v
, and the axial displacement as
w
. From Figure 9.9c, the radial
strain is
u
u
1
d
r
+
∂
u
∂
u
ε
r
=
d
r
−
=
(9.88)
∂
r
∂
r
The axial strain is
w
w
1
d
z
+
∂
w
∂
∂
w
ε
z
=
d
z
−
=
(9.89)
z
∂
z
and these relations are as expected, since the
rz
plane is effectively the same as a
rectangular coordinate system. In the circumferential direction, the differential
element undergoes an expansion defined by considering the original arc length
