Civil Engineering Reference
In-Depth Information
9.5.1 Finite Element Formulation
Recall from the general discussion of interpolation functions in Chapter 6 that es-
sentially any two-dimensional element can be used to generate an axisymmetric
element. As there is, by definition, no dependence on the
coordinate and no cir-
cumferential displacement, the displacement field for the axisymmetric stress
problem can be expressed as
M
u
(
r
,
z
)
=
N
i
(
r
,
z
)
u
i
i
=
1
(9.95)
M
w
(
r
,
z
)
=
N
i
(
r
,
z
)
w
i
i
=
1
with
u
i
and
w
i
representing the nodal radial and axial displacements, respectively.
For illustrative purposes, we now assume the case of a three-node triangular
element.
The strain components become
3
∂
u
∂
N
i
∂
=
r
=
ε
r
u
i
∂
r
i
=
1
3
u
r
=
N
i
r
ε
=
u
i
i
=
1
(9.96)
3
∂
w
∂
N
i
∂
ε
z
=
z
=
w
i
∂
z
i
=
1
3
3
∂
u
z
+
∂
w
∂
∂
N
i
∂
∂
N
i
∂
rz
=
r
=
u
i
+
w
i
∂
z
r
i
=
1
i
=
1
and these are conveniently expressed in the matrix form
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
0
0
0
r
r
r
u
1
u
2
u
3
w
1
w
2
w
3
ε
r
ε
ε
z
rz
N
1
r
N
2
r
N
3
r
0
0
0
=
(9.97)
∂
∂
∂
N
1
∂
N
2
∂
N
3
∂
0
0
0
z
z
z
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
z
z
z
r
r
r
In keeping with previous developments, Equation 9.97 is denoted
{
ε
}=
[
B
]
{}
with
[
B
]
representing the 4
6 matrix involving the interpolation functions.
Thus total strain energy of the elements, as described by Equation 9.15 or 9.58
×
