Civil Engineering Reference
In-Depth Information
where [
k
] is the element stiffness matrix defined by
V
e
[
B
]
T
[
D
][
B
]
[
k
]
=
(9.24)
and we must keep in mind that we are dealing with only a constant strain trian-
gle at this point.
This theoretical development may not be obvious to the reader. To make the
process more clear, especially the application of Equation 9.21, we examine the
element stiffness matrix in more detail. First, we represent Equation 9.20 as
1
2
{}
T
[
k
]
T
=
{}−{}
{
f
}
(9.25)
and expand the relation formally to obtain the
quadratic
function
2
k
11
6
1
2
2
=
1
+
2
k
12
1
2
+
2
k
13
1
3
+
2
k
14
1
4
+···+
2
k
56
5
6
+
k
66
−
f
1
x
1
−
f
2
x
2
−
f
3
x
3
−
f
1
y
4
−
f
2
y
5
−
f
3
y
6
(9.26)
The quadratic function representation of total potential energy is characteristic of
linearly elastic systems. (Recall the energy expressions for the strain energy of
spring and bar elements of Chapter 2.)
The partial derivatives of Equation 9.21 are then in the form
∂
∂
1
=
k
11
1
+
k
12
2
+
k
13
3
+
k
14
4
+
k
15
5
+
k
16
6
−
f
1
x
=
0
(9.27)
∂
∂
2
=
k
21
1
+
k
22
2
+
k
23
3
+
k
24
4
+
k
25
5
+
k
26
6
−
f
2
x
=
0
for example. Equations 9.27 are the scalar equations representing equilibrium
of nodes 1 and 2 in the
x
-coordinate direction. The remaining four equations
similarly represent nodal equilibrium conditions in the respective coordinate
directions.
As we are dealing with an elastic element, the stiffness matrix should be
symmetric. Examining Equation 9.27, we should have
k
12
=
k
21
, for example.
Whether this is the case may not be obvious in consideration of Equation 9.24,
since
[
D
]
is a symmetric matrix but
[
B
]
is not symmetric. A fundamental prop-
erty of matrix multiplication (Appendix A) is as follows: If
[
G
]
is a real, sym-
metric
N
×
N
matrix and
[
F
]
is a real
N
×
M
matrix, the matrix triple product
[
F
]
T
[
G
][
F
]
is a real, symmetric
M
×
M
matrix. Thus, the stiffness matrix as
given by Equation 9.24 is a symmetric 6
×
6 matrix, since
[
D
]
is 3
×
3 and sym-
metric and
[
B
]
is a 6
×
3 real matrix.
9.2.2 Stiffness Matrix Evaluation
The stiffness matrix for the constant strain triangle element given by
Equation 9.24 is now evaluated in detail. The interpolation functions per
