Civil Engineering Reference
In-Depth Information
Uniform axial pressure
q
z(v)
Internal
p
ressure
p
q
r(u)
(a)
(b)
Figure 2.6
(a) Cylinder under axial and radial pressure. (b) Cylindrical coordinate system
Figure 2.6(a) shows a thick tube subjected to radial pressure
p
and axial pressure
q
. Only
a typical radial cross-section need be analysed and is sub-divided into rectangular elements
in the figure. The cylindrical coordinate system, Figure 2.6(b), is the most convenient and
when it is used the element stiffness equation equivalent to (2.69) is
[
B
]
T
[
D
][
B
]
r
d
r
d
z
d
θ
[
k
m
]
=
(2.73)
which, when integrated over one radian, becomes
[
B
]
T
[
D
][
B
]
r
d
r
d
z
[
k
m
]
=
(2.74)
where the strain-displacement relations are now (Timoshenko and Goodier, 1982)
∂
∂r
0
r
z
γ
rz
θ
∂
∂z
u
v
0
=
(2.75)
∂
∂z
∂
∂r
1
r
0
or
{
} =
[
A
]
{
e
}
,where
u
and
v
now represent displacement components in the
r
and
z
directions.
