Civil Engineering Reference
In-Depth Information
The nonzero strain components in terms of displacements are
∂
u
ε
x
=
∂
x
∂
v
ε
y
=
(9.55)
∂
y
∂
u
∂
v
xy
=
y
+
∂
∂
x
For a four-node rectangular element (for example only), the column matrix
of strain components is expressed as
u
1
u
2
u
3
u
4
v
1
v
2
v
3
v
4
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
0
0
0
0
=
x
x
x
x
ε
x
ε
y
xy
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
0
0
0
0
{
ε
}=
y
y
y
y
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
∂
N
1
∂
∂
N
2
∂
∂
N
3
∂
∂
N
4
∂
y
x
y
y
x
x
x
x
(9.56)
in terms of the interpolation functions and the nodal displacements. As is cus-
tomary, Equation 9.56 is written as
{
ε
}=
(9.57)
with [
B
] representing the matrix of derivatives of interpolation functions and
{}
is the column matrix of nodal displacements. Hence, total strain energy of an
element is
[
B
]
{}
T
1
2
{}
1
2
{}
[
B
]
T
[
D
][
B
]d
V
T
[
k
]
U
e
=
{}=
{}
(9.58)
V
and the element stiffness matrix is again given by
k
(
e
)
=
[
B
T
][
D
][
B
]d
V
(
e
)
(9.59)
V
(
e
)
The interpolation functions for the four-node rectangular element per Equa-
tion 6.56 are
1
4
(1
N
1
(
r
,
s
)
=
−
r
)(1
−
s
)
1
4
(1
N
2
(
r
,
s
)
=
+
r
)(1
−
s
)
(9.60)
1
4
(1
N
3
(
r
,
s
)
=
+
r
)(1
+
s
)
1
4
(1
N
4
(
r
,
s
)
=
−
r
)(1
+
s
)
