Civil Engineering Reference
In-Depth Information
with what we will call the
geometric mapping matrix,
defined as
J
22
−
J
12
0
0
1
[
G
]
=
(9.79)
0
0
−
J
21
J
11
|
J
|
−
J
21
J
11
J
22
−
J
12
We must expand the column matrix on the extreme right-hand side of Equa-
tion 9.78 in terms of the discretized approximation to the displacements. Via
Equation 9.69, we have
∂
u
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
u
1
u
2
u
3
u
4
v
1
v
2
v
3
v
4
0
0
0
0
∂
r
r
r
r
r
∂
u
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
0
0
0
0
∂
s
s
s
s
s
=
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
∂
v
0
0
0
0
r
r
r
r
∂
r
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
∂
v
0
0
0
0
s
s
s
s
∂
s
(9.80)
where we reemphasize that the indicated partial derivatives are known functions
of the natural coordinates of the parent element. For shorthand notation, Equa-
tion 9.80 is rewritten as
∂
u
∂
r
∂
u
∂
s
=
[
P
]
{}
(9.81)
∂
v
∂
r
∂
v
∂
s
in which
[
P
]
is the matrix of partial derivatives and
{}
is the column matrix of
nodal displacement components.
Combining Equations 9.78 and 9.81, we obtain the sought-after relation for
the strain components in terms of nodal displacement components as
{
ε
}=
(9.82)
and, by analogy with previous developments, matrix
[
B
]
=
[
G
][
P
]
has been
determined such that
[
G
][
P
]
{}
{
ε
} =
[
B
]
{}
(9.83)
and the element stiffness matrix is defined by
k
(
e
)
=
t
[
B
]
T
[
D
][
B
]d
A
(9.84)
A
