Civil Engineering Reference
In-Depth Information
Simultaneous solution of the global equations (in this case, we inverted the global
stiffness matrix using a spreadsheet program) yields the nodal temperatures as
106
.
507
111
T
4
T
5
T
6
T
7
T
8
T
9
.
982
106
.
507
◦
F
=
89
.
041
90
.
966
89
.
041
If we now back substitute the computed nodal temperatures into the first three of the
global equations, specifically,
0
.
6906
T
1
−
0
.
1003
T
2
−
0
.
0713
T
4
−
0
.
2585
T
5
=
17
.
7084
+
F
1
−
0
.
1003
T
1
+
1
.
2654
T
2
−
0
.
1003
T
3
−
0
.
2585
T
4
−
0
.
2006
T
5
−
0
.
2585
T
6
=
35
.
4168
+
F
2
−
0
.
1003
T
2
+
0
.
6906
T
3
−
0
.
2585
T
5
−
0
.
0713
T
6
=
17
.
7084
+
F
3
we obtain the heat flow values at nodes 1, 2, and 3 as
=
F
1
F
2
F
3
52
.
008
78
.
720
52
.
008
Btu/hr
Note that, in terms of the matrix partitioning, we are now solving
[
K
cc
]
{
T
c
}+
[
K
ca
]
{
T
a
}={
F
c
}
to obtain the unknown values in
{
F
c
}
.
Since there is no convection from the edges defined by nodes 1-2 and 1-3 and the
temperature is specified on these edges, the reaction “forces” represent the heat input
(flux) across these edges and should be in balance with the convection loss across the lat-
eral surfaces of the body, and its edges, in a steady-state situation. This balance is a check
that can and should be made on the accuracy of a finite element solution of a heat trans-
fer problem and is analogous to checking equilibrium of a structural finite element
solution.
Example 7.5 is illustrated in great detail to point out the systematic proce-
dures for assembling the global matrices and force vectors. The astute reader
ascertains, in following the solution, that symmetry conditions can be used to
simplify the mathematics of the solution. As shown in Figure 7.11a, an axis
(plane) of symmetry exists through the horizontal center of the plate. Therefore,
the problem can be reduced to a two-element model, as shown in Figure 7.11b.
Along the edge of symmetry, the
y
-direction heat flux components are in balance,
and this edge can be treated as a perfectly insulated edge. One could then use
only two elements, with the appropriately adjusted boundary conditions to obtain
the same solution as in the example.
