Civil Engineering Reference
In-Depth Information
The global equations for the four-element model are then expressed as
0
.
6906
−
0
.
1003
−
0
.
0713
−
0
.
2585
0
0
0
0
0
180
180
180
T
4
T
5
T
6
T
7
T
8
T
9
−
.
.
−
.
−
.
−
.
−
.
0
1003
1
2654
0
1003
0
2585
0
2006
0
2585
0
0
0
0
−
0
.
1003
0
.
6906
0
−
0
.
2585
−
0
.
0713
0
0
0
−
0
.
0713
−
0
.
2585
0
1
.
3812
−
0
.
2006
0
−
0
.
0713
−
0
.
2585
0
−
0
.
2585
−
0
.
2006
−
0
.
2585
−
0
.
2006
2
.
5308
−
0
.
2006
−
0
.
2585
−
0
.
2006
−
0
.
2585
0
−
0
.
2585
−
0
.
0713
0
−
0
.
2006
1
.
3812
0
−
0
.
2585
−
0
.
0713
0
0
0
−
0
.
0713
−
0
.
2585
0
0
.
7484
−
0
.
2585
0
0
0
0
−
0
.
2585
−
0
.
2006
−
0
.
2585
−
0
.
2585
1
.
3812
−
0
.
0713
0
0
0
0
−
0
.
2585
−
0
.
0713
0
−
0
.
0713
0
.
7484
17
.
7084
+
F
1
35
.
+
F
2
4168
17
.
7084
+
F
3
35
.
4168
=
47
.
2224
35
4168
23
.
6112
35
.
4158
23
.
6112
.
Taking into account the specified temperatures on nodes 1, 2, and 3, the global equations
for the unknown temperatures become
1
.
3812
−
0
.
2006
0
−
0
.
0713
−
0
.
2585
0
T
4
T
5
T
6
T
7
T
8
T
9
94
.
7808
176
.
3904
94
.
7808
23
.
6112
35
.
4168
23
.
6112
−
0
.
2006
2
.
5308
−
0
.
2006
−
0
.
2585
−
0
.
2006
−
0
.
2585
0
−
0
.
2006
1
.
3812
0
−
0
.
2585
−
0
.
0713
=
−
0
.
0713
−
0
.
2585
0
0
.
7484
−
0
.
2585
0
−
0
.
2585
−
0
.
2006
−
0
.
2585
−
0
.
2585
1
.
3812
−
0
.
0713
0
−
0
.
2585
−
0
.
0713
0
−
0
.
0713
0
.
7484
The reader is urged to note that, in arriving at the last result, we partition the global matrix
as shown by the dashed lines and apply Equation 3.46a to obtain the equations governing
the “active” degrees of freedom. That is, the partitioned matrix is of the form
K
cc
T
c
T
a
F
c
F
a
K
ca
=
K
ac
K
aa
where the subscript
c
denotes terms associated with constrained (specified) temperatures
and the subscript
a
denotes terms associated with active (unknown) temperatures. Hence,
this
6
×
6
system represents
[
K
aa
]
{
T
a
}={
F
a
}−
[
K
ac
]
{
T
c
}
which now properly includes the effects of specified temperatures as forcing functions on
the right-hand side.
